1
Some Efficient Optimization Methods for Solving
the Security-Constrained Optimal Power Flow
Problem
Dzung Phan and Jayant Kalagnanam
Abstract—The security-constrained optimal power flow problem considers both the normal state and contingency constraints,
and it is formulated as a large-scale nonconvex optimization
problem. We propose a global optimization algorithm based on
Lagrangian duality to solve the nonconvex problem to optimality.
As usual, the global approach is often time-consuming, thus for
practical uses when dealing with a large number of contingencies,
we investigate two decomposition algorithms based on Benders
cut and the alternating direction method of multipliers. These
decomposition schemes often generate solutions with smaller objective function values than those generated by the conventional
approach and very close to the globally optimal points.
Index Terms—Security-constrained optimal power flow, Lagrangian duality, branch-and-bound, Benders decomposition,
alternating direction method of multipliers.
I. I NTRODUCTION
The optimal power flow (OPF) problem aims at determining
an optimal operating point for control variables which minimizes a given objective function subject to physical constraints
and control limits [1]. The most commonly considered objective function is total cost of generation. In current practices of
the real-time dispatch, the Regional Transmission Organizations (RTOs) and Independent System Operators (ISOs) often
perform contingency analysis to assure that the power system
remains balanced and no operational constraint is violated
during both the normal case and contingency scenarios when
the outage of a component incurs, the so-called N − 1
condition. The OPF problem with the contingency constraints
is often referred to as the security-constrained optimal power
flow (SCOPF) problem in the literature [2]. In this paper,
we are interested in the alternate current (AC) power flow
constraints for the problem. The SCOPF problem has been
widely classified into two classes: preventive [3] and corrective
[4] formulations.
The preventive formulation minimizes some cost function
by considering only the normal case control variables, that
are feasible for both the normal and contingency conditions.
For C contingency scenarios, the size of the problem is
approximately C + 1 times larger than that of the conventional
OPF. Consequently, a direct method for handling large-scale
power systems with numerous contingencies in a centralized
D. Phan and J. Kalagnanam are with the Department of Business Analytics
and Mathematical Sciences, IBM T.J. Watson Research Center, Yorktown
Heights, NY, 10598 USA, email: {phandu,jayant}@us.ibm.com. This
material is based upon work supported by the U.S. Department of Energy
under Award Number DE-OE0000190.
manner would result in a prohibitive memory usage and unacceptably large execution time. In practice, however, many postcontingency constraints are redundant [5]. As a result, a class
of algorithms based upon contingency filtering techniques has
been developed [5]–[7] that identify and only include those
potentially binding contingencies into the formulation. For
instance, the contingency ranking method in [6] is derived
by investigating a preventive SCOPF relaxation, where postcontingency conditions and the normal case are considered
separately. The ranking scheme makes use of the values of
dual variables and the decrease factor of regularized objective
function values, and then chooses contingencies with a severity
index exceeding some threshold for further consideration.
Other contingency filtering approaches [5] focus on determining a minimal subset of contingencies to be added based on the
post-contingency constraint violations. Additionally, a method
using the generalized Benders decomposition [8] is developed
in [9].
The corrective SCOPF formulation exploits the observation
that some contingency violations can resist up to some allowed
restoration time without damaging the devices [10]. It permits
post-contingency control variables such as power generation
outputs and transformer taps to be readjusted in order to
remove any violations caused by the contingency. The total
generation cost given by the corrective model is often smaller
than that of the preventive model. However, the model requires additional decision variables and constraints, and more
importantly it might need a large number of reschedules for
each contingency. The authors in [4] reformulate the problem
using only the base-case variables, where constraint reductions
are represented as implicit functions of the contingency-free
variables. The generalized Benders decomposition is used to
solve the reformulation. An extension of a contingency filtering technique from [5] is studied in [10], which includes an
additional OPF procedure to check the controllability of postcontingency states. A network compression technique [11]
shrinks the networks under contingency states based on the fact
that the impact of a failure is, in general, related to a localized
area of the power grid. The DC SCOPF approximation can
provide a good guess for binding contingencies once integrated in a contingency filtering framework [12]. The recent
paper [13] introduces a semidefinite program relaxation to the
problem and proves that a zero dual gap is obtained if strong
duality holds for every classical OPF problem associated with
each configuration; thus a globally optimal solution can be
recovered from the dual. The reader is referred to [14] for a
2
recent literature review on the topic.
In this work, we propose three approaches to solve the
corrective SCOPF problem. The first one is an exact method
based on a branch-and-bound algorithm for the rectangular
formulation, which guarantees to find a global solution. The
main idea comes from an extension of the technique for the
conventional OPF problem proposed in [15], which exploits
the quadratic structure of the problem together with the box
and spherical constraints. For practical purposes, we also
investigate two fast local algorithms based on decomposition
techniques for any coordinates, which treat each contingency
separately. The first local one is based on Benders decomposition, for which we derive new feasibility cuts by linearizing a
feasibility checking optimization problem. An adaptive cut is
proposed to handle the nonconvexity, which aims at improving
the quality of solution. The second heuristic is an application
of the alternating direction method of multipliers (ADMM)
[16]. The decomposition algorithms can be run on parallel
computers to reduce the running time. Note that these three
approaches can also be used to solve the preventive model.
The paper is organized as follows. In Section II, we review
the problem formulation in a general format as well as in
rectangular coordinates. Section III presents a Lagrangian duality and a rectangular branching procedure for a branch-andbound algorithm. In Section IV, we introduce a decomposition
scheme based on Benders cut, and in Section V we study
the application of ADMM for solving the SCOPF problem.
Section VI contains numerical experiments and Section VII
concludes the paper.
II. P ROBLEM F ORMULATION
In this paper, we are interested in the following corrective
SCOPF problem:
minuc ,xc :∀c∈{0}∪C f (u0 , x0 )
s.t. hc (uc , xc ) = 0, ∀c ∈ {0} ∪ C,
gc (uc , xc ) ≤ 0, ∀c ∈ {0} ∪ C,
|uc − u0 | ≤ uc , ∀c ∈ C,
(1)
where C = {1, 2, . . . , C} is the set of pre-specified C
contingencies, c = 0 represents the base case, the index
c ∈ C represents a contingency, uc is the control variables,
xc is the state variables, uc is the upper bound deviation
for control variables, hc and gc are operational constraints
such as power flow equations and transmission line limits. The
last ramping rate constraint is understood componentwise, i.e.,
|uic −ui0 | ≤ uic for every i. Note that some controls can operate
in a preventive mode when setting uic = 0, and they can model
for transformers, phase shifters and shunts but in a continuous
space. We assume the list of contingencies is given.
We now review the model (1) in rectangular coordinates,
which will be used to develop a global optimization algorithm.
Because of the limitation of our approach, only the levels
of power generation are considered as control variables. Let
N denote the set of buses in the grid, G denote the set of
generators, and B denote the set of branches. Suppose that
every complex nodal voltage can be modeled in rectangular
coordinates vi = ei + jfi , for all i ∈ N , where j is the
imaginary unit. The net active power Pi and reactive power
Qi injections into bus i ∈ N are given by
∑(
)
Pi (e, f , G, B) =
ei (gij ej − bij fj ) + fi (gij fj + bij ej ) ,
j∈N
∑(
Qi (e, f , G, B) =
)
fi (gij ej − bij fj ) − ei (gij fj + bij ej ) ,
j∈N
where e ∈ R|N | and f ∈ R|N | are vectors with voltage
components, gij is the ij-th entry of the bus conductance
matrix G ∈ R|N |×|N | , and bij is the ij-th entry of the bus
susceptance matrix B ∈ R|N |×|N | . The active power flow Pij
from bus i to bus j is computed from
Pij = (e2i + fi2 )ḡii + (ei ej + fi fj )ḡij − (ei fj − ej fi )b̄ij ,
where ḡii is the self-conductance of branch admittance at
bus i, ḡij is the mutual conductance, and b̄ij is the mutual
susceptance of branch admittance from buses i to j.
Define Pig,c and Qg,c
as dispatchable active and reactive
i
powers, Pid and Qdi as active and reactive power demands at
bus i. To ease the notation, we assume that Pig,c = Qg,c
=0
i
and Pid = Qdi = 0 for any bus i that has zero generation
or zero demand, respectively. Without loss of generality, we
assign the index 1 as the reference bus f1c = 0 [2]. The SCOPF
problem of minimizing the generation cost becomes:
∑(
)
g,0 2
g,0
g,0
min
f
(P
)
=
c
(P
)
+
c
P
+
c
(2)
0i
1i
2i
i
i
g,c
g,c
c
c
ei ,fi ,Pi
s.t.
,Qi
i∈G
Pi (e , f , G , B ) = Pig,c − Pid , i ∈ N
d
Qi (ec , f c , Gc , Bc ) = Qg,c
i − Qi , i ∈ N
max,c
c c
c
c
Pij (e , f , Ḡ , B̄ ) ≤ Pij
, (i, j) ∈ B
c
f1 = 0
Pimin,c ≤ Pig,c ≤ Pimax,c , i ∈ G
Qmin,c
≤ Qg,c
≤ Qmax,c
,i ∈ G
i
i
i
(vimin,c )2 ≤ (eci )2 + (fic )2 ≤ (vimax,c )2 , i
c
|Pig,c − Pig,0 | ≤ P i , i ∈ G
c
g,0
|Qg,c
i − Qi | ≤ Qi , i ∈ G,
c
c
c
c
(3)
(4)
(5)
(6)
(7)
(8)
∈N
(9)
(10)
(11)
for all c ∈ {0}∪C, where Pmin,c and Pmax,c are active power
generation limits, Qmin,c and Qmax,c are reactive power
generation limits, vmin,c and vmax,c are voltage magnitude
c
c
limits, P and Q are pre-determined maximal allowed variation of power outputs, and c0i , c1i , and c2i are generation
c
cost parameters. The entries of Ḡc are ḡij
’s, and the entries
c
c
of B̄ are b̄ij ’s. Other forms of the objective function and
transmission line constraints can be considered by using the
same techniques described below if the quadratic feature still
holds, but for simplicity in the presentation, we present a
global algorithm only for the cost of generation and the active
power flow limits.
III. E XACT METHOD BASED ON BRANCH - AND - BOUND
In this section, we investigate an exact method based on
a spatial branch-and-bound approach to solve the SCOPF
problem in rectangular coordinates (2)-(11) to optimality.
The lower bound for the objective function is obtained by
Lagrangian duality, while the feasible set subdivision is based
3
on a rectangular bisection. Note that if control variables such
as transformer ratio and phase shift are considered, the power
flow balance equations are not quadratic, the exact method
cannot be applied.
A. Lagrange Dual Problem
We propose a Lagrange dual problem for the SCOPF
modeled by (2)-(11), which can provide a lower bound to
the global optimum in our branch-and-bound algorithm if
strong duality does not hold. In particular, we dualize the
complicating quadratic constraints, but retain the simple bound
constraints such as the boxes and spheres in the feasible set.
In many applications, the solution of the primal problem
can be achieved from the Lagrange dual problem, for example,
when strong duality holds. We will show later in the numerical
simulation that a zero gap is observed for almost our test
instances for the proposed dual problem. Let Ω be the feasible
set of the SCOPF problem and assume Ω is nonempty. We
form a new constraint for the summation of squared voltage
magnitudes
∑ min,c
∑
∑ max,c
Rc =
(vi
)2 ≤
((eci )2 + (fic )2 ) ≤
(vi
)2 .
i∈N
i∈N
i∈N
The SCOPF problem is equivalent to
min f (Pg,0 )
s.t. (3) − (11),
(ec , f c ) ∈ Rc , ∀c ∈ {0} ∪ C.
(12)
We define the Lagrangian L of (12) by incorporating
complicating quadratic constraints into the objective function
L(ec , f c , Pg,c , Qg,c , λc , β c , µc , γ c , θ c ) = f (Pg,0 )+
{
C
∑
∑ (
) ∑ c( d
λci Pid + Pi (ec , f c , Gc , Bc ) +
βi Qi +
c=0
i∈N
Qi (ec , f c , Gc , Bc )) +
)
Pijmax,c
+
∑
γic
(
and
∑
minec ,f c i∈N (λci Pi (ec , f c , Gc , Bc ) + βic Qi (ec , f c , Gc , Bc ))
∑
+(θic − γic )((eci )2 + (fic )2 ) + (i,j)∈B µci,j Pij (ec , f c , Ḡc , B̄c )
s.t. (6) and (ec , f c ) ∈ Rc .
(16)
Note that the problem (15) is a separable convex quadratic
programming problem and it can be further decomposed into
|G| subproblems (associated with each i ∈ G). Problem (16)
is a nonconvex one, the additional spherical constraint Rc
and the quadratic structure of the objective function make it
tractable. Define the matrix Ac (λc , β c , µc , γ c , θ c ) by
−β c Gc −λc Bc
λc Gc −β c Bc
1
1:
1 1:
1
1:
1 1:
.
.
..
..
λc Gc −β c Bc
c
c
c
c
−β|N | G|N |: −λ|N | B|N |:
|N |
|N |:
|N | |N |:
β c Gc +λc Bc
λc Gc −β c Bc
1
1:
1 1:
1
1:
1 1:
.
..
...
(
+
c
c
c
c
β|N
| G|N |: +λ|N | B|N |:
µc ◦ Ḡc
−µc ◦ B̄c
µc ◦ B̄c
µc ◦ Ḡc
)
c
c
λc|N | Gc|N |: −β|N
| B|N |:
+ diag([θ c − γ c ; θ c − γ c ]),
i∈N
∑
(
µci,j Pij (ec , f c , Ḡc , B̄c )−
(i,j)∈B
(vimin,c )2
) ∑
− (eci )2 − (fic )2 +
θic
i∈N
(
The dual (14) is a concave maximization problem, the
optimal solution can be computed by a subgradient algorithm.
A subgradient of function q can be obtained by solving the
inner minimization problem (13). The task of solving (13) can
be done efficiently in our application for solving the SCOPF.
For a given (λ, β, µ, γ, θ), (Pg,c , Qg,c : c = 0, . . . , C)
and (ec , f c : c = 0, . . . , C) are decoupled in both the
objective function and the constraints of problem (13). Now,
the minimization problem (13) can be decomposed into C + 2
subproblems, which are relatively easy to be solved:
∑C ∑
minPg,c ,Qg,c f (Pg,0 ) − c=0 i∈G (λci Pig,c + βic Qg,c
i )
(15)
s.t. (7), (8), (10), (11)
i∈N
}
∑
)
max,c 2
c 2
c 2
c g,c
c g,c
) −
(λi Pi + βi Qi ) .
(ei ) + (fi ) − (vi
i∈G
where Gci: and Bci: are the i-th rows of matrices Gc and Bc ,
respectively, diag(x) is the diagonal matrix with x down the
diagonal, and “◦” denotes the componentwise product. Let
Āc (λc , β c , µc , γ c , θ c ) be the matrix obtained from Ac by
removing the (|N | + 1)-th row and (|N | + 1)-th column.
Likewise, f̄ c and R̄c are obtained from f c and Rc , respectively,
by deleting f1c . Then the latter (16) can be expressed as follows
The Lagrange dual function is given by
min (ec ; f̄ c )T Āc (ec ; f̄ c ) s.t. (ec , f̄ c ) ∈ R̄c .
q(λ, β, µ, γ, θ) = minec ,f c ,Pg,c ,Qg,c L(·)
The following theorem from [15] gives an optimal solution to
the problem (17).
s.t. (6) − (8), (10), (11)
(ec , f c ) ∈ Rc , ∀c ∈ {0} ∪ C,
(13)
where we abuse the multiplier notation by assuming λ =
(λ0 , . . . , λC ), β = (β 0 , . . . , β C ), etc. It is known that for
every {(λ, β, µ, γ, θ) : µ ≥ 0, γ ≥ 0, θ ≥ 0}, the value
of the dual function provides a lower bound on the optimal
objective function value of the primal problem. The best lower
bound is obtained by solving the dual problem:
max
λ,β,µ,γ,θ
q(λ, β, µ, γ, θ)
s.t. µ ≥ 0, γ ≥ 0, θ ≥ 0.
(14)
(17)
Theorem 1: Let uc be the eigenvector associated with the
smallest eigenvalue λmin of the matrix Āc + (Āc )T . The
solution of (17) is now determined as follows:
√∑
max,c 2
)
i∈N (vi
uc
, if λmin < 0
c∥
∥u
c c
√
(e ; f̄ ) =
∑
min,c 2
(v
)
i∈N
i
uc
, otherwise.
∥uc ∥
Thus we can solve the dual problem (14) efficiently, for
which a zero duality gap is numerically shown for many test
cases in the numerical experiments.
4
B. Global Optimization Algorithm
If strong duality does not hold for the dual problem (14),
we propose a branch-and-bound algorithm to obtain a globally
optimal solution of the SCOPF. The lower bound of a node is
achieved by solving the dual, while the branching procedure
is based on an iteratively rectangular bisection.
Rectangular bisection: The branching process in the branchand-bound algorithm is based on successive rectangular bisections of the bound constraints for the active and reactive
powers: Pg,c and Qg,c
)
}
{(
Pg,c
Pimin,c ≤ Pig,c ≤ Pimax,c , i ∈ G
c
B =
:
.
Qg,c
Qmin,c
≤ Qg,c
≤ Qmax,c
,i ∈ G
i
i
i
Consider a rectangle B in general form B = {x ∈ Rn : a ≤
x ≤ b}, a point v ∈ B, and an index j ∈ {1, . . . , n}. Suppose
that B is partitioned into two subrectangles determined by the
hyperplane {x : xj = vj }. We call a subdivision of B via
(v, j) is a bisection if the index j corresponds to a longest
side of B and v is a point of this side such that
In our application for solving the SCOPF, we apply the
scheme to B̂0 = B0 × B 1 × . . . × B C . Over a new generated
rectangle
min,c
B̂k = {(Pg,0 , Qg,0 , . . . , Pg,C , Qg,C ) : P̂k,i
≤ Pig,c ≤
max,c
P̂k,i
, Q̂min,c
≤ Qg,c
≤ Q̂max,c
, for all i ∈ G, c =
i
k,i
k,i
0, . . . , C},
the Lagrange dual function is given by
(ec , f c ) ∈ Rc , c = 0, . . . , C
IV. B ENDERS DECOMPOSITION ALGORITHM
We present an application of Benders decomposition to
solve the problem (1), where new feasibility cuts are introduced. They are different from the standard technique in the
literature by linearizing an optimization problem rather than a
contingency-related function. An adaptive cut is proposed to
handle nonconvexity to improve solution quality.
A. Cut Generation
min(vj − aj , bj − vj ) = 0.5(bj − aj ).
q(λ, β, µ, γ, θ) = minec ,f c ,Pg,c ,Qg,c L(·)
s.t. (6), (10), (11)
Theorem 2: The rectangular branch-and-bound algorithm
generates an optimal solution of the SCOPF problem in a
finite number of iterations, or every accumulation point of the
sequence {xk } is a solution of the SCOPF.
Proof: Suppose the algorithm does not terminate in a
finite number of iterations. Since the branching procedure is
performed on the largest side of the rectangle at each iteration,
a nested sequence of bisections shrinks to a point. The proof
is competed by following the same reasoning used in Theorem
4.1 of [15].
(18)
(Pg,0 , Qg,0 , . . . , Pg,C , Qg,C ) ∈ B̂k .
Define the set F by
F = {(ec , f c , Pg,c , Qg,c : ∀c ∈ {0} ∪ C) : (3)-(6), (9)-(11)} .
For any box B̂, define ℓ(B̂) the lower bound of f (Pg,0 ) over
F ∩ B̂ given by the dual. Now, the branch-and-bound algorithm
can be stated as follows:
R ECTANGULAR BRANCH - AND - BOUND ALGORITHM
Initialize S = {B̂0 }, compute ℓ(B̂0 ) by the dual
and use a local solver to get an upper
bound u and a feasible point x.
Repeat until S = ∅
a) Select B̂ ∈ S such that ℓ(B̂) = min{ℓ(B̂) :
B̂ ∈ S}. Use the rectangular
bisection for B̂ to obtain B̂1 and B̂2 .
b) If B̂i ∩ Ω = ∅, set ℓ(B̂i ) = ∞;
Otherwise compute ℓ(B̂i ) by the dual.
c) Update the best upper bound u and the
associated feasible point xk by
applying a local solver over F ∩ B̂i .
d) Set S = {B ∈ S ∪ {B̂1 } ∪ {B̂2 } : ℓ(B) < u and
(u − ℓ(B))/u > ϵ, B ̸= B̂}
The following theorem shows that all accumulation points
of a branch-and-bound algorithm are the global optima.
We notice that the study of Benders decomposition for the
problem (1) amounts to considering the following problem
minx∈X f (x)
h(y) = 0, g(y) ≤ 0
Mx + Ny + b ≤ 0
(19)
yl ≤ y ≤ yu ,
where X ⊂ Rn , y ∈ Rm , M and N are matrices. The
connection between (1) and (19) can be explained as follows.
The variable x represents the base-case variable (u0 , x0 ) and
X is the feasible set of the base case. For each contingency,
the feasible set can be written in the form of the constraints
in (19), where y represents (uc , xc ), the power flow balance
constraint hc (uc , xc ) is h(y), the ramping rate constraints is
Mx + Ny + b ≤ 0, the flow limit and operational limit
constraints gc (uc , xc ) ≤ 0 are separated into g(y) ≤ 0 and
yl ≤ y ≤ yu . We only need to present the feasibility cut
generation for one contingency since every contingency can
be expressed in the same format.
Denote x̄ by the solution of the initial master problem
min f (x) s.t. x ∈ X.
(20)
Then we have to check if the following system is feasible
{h(y) = 0, g(y) ≤ 0, Mx̄ + Ny + b ≤ 0, yl ≤ y ≤ yu } (21)
The algorithm is terminated if (21) is feasible for the given
x̄. Otherwise, we generate a linear cut, which is added to the
master problem (20) to enforce the feasibility of the system
(21).
We now construct a linear cut if (21) is infeasible. For a
given x̄, consider the optimization problem
∑
miny,α
i αi
s.t.
h(y) = 0, g(y) ≤ 0
Mx̄ + Ny + b − α ≤ 0
yl ≤ y ≤ yu , α ≥ 0,
(22)
5
by introducing new variables αi ’s, whose optimal solution is
(ȳ, ᾱ). In general, in order to generate a linear cut based on the
dual solutions, we need the strong duality assumption. However, the problem arising from our application is nonconvex.
It motivates us to study the linearized version of (22).
The infeasibility of (21) implies that the optimal function
value of (22) is strictly positive. We make the substitution
z = y − yl and use the first-order approximations of h(y) and
g(y) at ȳ − yl , we yield the linear programming relaxation
∑
minz,α
i αi
h(ȳ) + ∇y h(ȳ)(z − (ȳ − y )) = 0
l
s.t.
g(ȳ) + ∇y g(ȳ)(z − (ȳ − yl )) ≤ 0
(23)
Nz − α + Mx̄ + b + Nyl ≤ 0
z ≤ yu − yl , 0 ≤ α, 0 ≤ z.
Because the optimal function value of (22) is strictly positive,
it follows from the first-order optimality conditions that the
optimal function value of (23) is also strictly positive. We
derive a linear cut based upon the dual variables of (23). The
Lagrange dual problem of (23) is defined as
max
min
π T (Nz − α + Mx̄ + Nyl + b) +
)
(
− (ȳ − yl )) + µT g(ȳ) +
) ∑
∇y g(ȳ)(z − (ȳ − yl )) +
αi
max
min
(
π≥0,µ≥0,η 0≤α
0≤z≤yu −yl
T
l
i
)T
T
NT π + ∇T
z
y h(ȳ)η + ∇y g(ȳ)µ
+π (Mx̄ + Ny + b) − η T ∇y h(ȳ)(ȳ − yl ) +
∑
µT (g(ȳ) − ∇y g(ȳ)(ȳ − yl )) +
αi (1 − πi )
=
π (Mx̄ + Ny + b) − η ∇y h(ȳ)(ȳ − y )
∑
ℓi (π, µ, η), where
+µT (g(ȳ) − ∇y g(ȳ)(ȳ − yl )) +
max
l
T
l
0≤πi ≤1,µ≥0,η
i
]
[
{
T
0 if NT π + ∇T
y h(ȳ)η + ∇y g(ȳ)µ i ≥ 0
ℓi (·)= [ T
] u
T
l
N π + ∇T
y h(ȳ)η + ∇y g(ȳ)µ i [yi − yi ] otherwise
Let (π̂, η̂, µ̂) be the optimal solution of the dual problem.
Since strong duality holds for (23), we have
∑
l
T
T
l
i α̂i = π̂ (Mx̄ + Ny + b) − η̂ ∇y h(ȳ)(ȳ − y )+
∑
µ̂T (g(ȳ) − ∇y g(ȳ)(ȳ − yl )) + i ℓi (π̂, η̂, µ̂).
∑
From the definition
of ℓi (π, µ, η), we have i ℓi (π̂, η̂, µ̂) ≤
∑
0. Because i α̂i > 0, it implies that
π̂ T (Mx̄ + Nyl + b) − η̂ T ∇y h(ȳ)(ȳ − yl )+
µ̂T (g(ȳ) − ∇y g(ȳ)(ȳ − yl )) > 0.
Thus, we should impose the inequality (24) to the master
π̂ T (Mx + Nyl + b) − η̂ T ∇y h(ȳ)(ȳ − yl )+
µ̂ (g(ȳ) − ∇y g(ȳ)(ȳ − y )) ≤ 0,
T
l
(26)
Define U = {(y, α) : yl ≤ y ≤ yu , α ≥ 0}. Applying the
first-order optimality conditions for (22) yields
(
)
(
)
(
)
0
Nj:
∇y hj (ȳ)
∑
∑
+ j π̄j
+ j η̄j
1
−ej
0
(
) (
)
(27)
∇y gj (ȳ)
uy
∑
+ j µ̄j
+
= 0,
0
uα
(uy ; uα )T (ȳ − y; ᾱ − α) ≥ 0,
for all (y, α) ∈ U . We choose (y, α) = (yl , 0) and then
multiply (27) by (ȳ − yl , ᾱ), which yields
∑
T
l
T
T
l
i ᾱi + π̄ N(ȳ − y ) − π̄ ᾱ + η̄ ∇y h(ȳ)(ȳ − y )+
µ̄T ∇y g(ȳ)(ȳ − yl ) ≤ 0.
i
T
Proof: We prove by contradiction. Suppose that
(
π̄ T Mx̄ + π̄ T (Nyl + b) − η̄ T ∇y h(ȳ)(ȳ − yl )−
)
µ̄T ∇y g(ȳ)(ȳ − yl ) ≤ 0.
where Nj: is the j-th row of matrix N, 0 is the vector whose
entries are all 0, 1 is the vector whose entries are all 1, ej is
the j-th unit vector, (uy ; uα ) is a vector in the normal cone
of U at (ȳ, ᾱ), that is,
π≥0,µ≥0,η 0≤α
0≤z≤yu −yl
(
T
η h(ȳ) + ∇y h(ȳ)(z
=
multipliers corresponding to the constraints Mx̄ + Ny + b −
α ≤ 0, h(y) = 0 and g(y) ≤ 0, respectively, in problem (22).
Because of the complementary slackness condition, it follows
µ̄T g(ȳ) = 0. From (24), we derive the following result.
Theorem 3: For a given x̄, if the system (21) is infeasible,
then the following inequality is a cut for x̄
(
π̄ T Mx + π̄ T (Nyl + b) − η̄ T ∇y h(ȳ)(ȳ − yl )−
)
(25)
µ̄T ∇y g(ȳ)(ȳ − yl ) ≤ 0.
(24)
which acts as a linear cut.
Motivated by the above analysis, we propose an alternative
cut when (21) is infeasible. Let (π̄, η̄, µ̄) denote the Lagrange
Notice that π̄ T ᾱ = π̄ T (Mx̄ + Nȳ + b), then we have
∑
l
T
i ᾱi − π̄ (Ny + Mx̄ + b)+
(28)
η̄ T ∇y h(ȳ)(ȳ − yl ) + µ̄T ∇y g(ȳ)(ȳ − yl ) ≤ 0.
∑
Combining i ᾱi > 0, (26), and (28) gives a contradiction.
Note that our cut (25) is different from the Benders cut
used in [4], which is obtained by the linearization of an
implicit function with respect to the base-case variable for
each contingency. We explicitly use not only the Lagrange
multipliers of the linear constraints but also that of nonlinear
ones as well as the bounds on the decision variables.
B. Description of Benders Decomposition Algorithm
We now describe a Benders decomposition algorithm using
linear cuts (24) and (25). The master problem is expressed as
min f (u0 , x0 )
s.t. {h0 (x0 , u0 ) = 0, g0 (x0 , u0 ) ≤ 0, Pu0 + q ≤ 0}
(29)
where Pu0 + q ≤ 0 denotes the set of feasibility cuts at a
given step, P is a matrix, and q is a vector.
B ENDERS D ECOMPOSITION A LGORITHM
1. Solve the regular OPF to get (x10 , u10 ):
min f (u0 , x0 ) s.t. {h0 (x0 , u0 ) = 0, g0 (x0 , u0 ) ≤ 0}
6
2. For k = 1, 2, . . .
(a) For c = 1, . . . , C:
Solve a problem of∑form (22) to get
k
the optimal value
i αi , where u0 is in
place of x̄.
∑
∗ If
i αi > 0, then add the cut (24) or
(25)
∑ into the master problem (29).
∗ If
i αi = 0 for all c ∈ C, terminate
the algorithm.
(b) Solve the master problem (29) to get
(xk+1
, uk+1
).
0
0
message passing scheme based on the application of ADMM
for solving the optimal power scheduling problem in a distributed manner.
Let us briefly review the main idea of ADMM. Consider
the optimization problem with block separable structure
min {ϕ(x) + φ(y) : Mx + Ny = d, x ∈ X, y ∈ Y },
x,y
where X ⊂ Rn , Y ⊂ Rm , M ∈ Rp×n , N ∈ Rp×m and d ∈
Rp . The augmented Lagrangian function is given by
Remark 1 (Adaptive cuts): Because SCOPF is a nonconvex
optimization problem, if we use a linear cut without care, its
optimal solutions will likely be trimmed off. It might delete
subregions of the feasible set containing good feasible points.
Consequently, the generated solution is often far from the
global optimum. Even worse, the new cut can make the master
problem infeasible (see Figure 1). To alleviate this effect, we
propose to use a cut in an adaptive manner. Suppose that
c(x) ≤ 0 is a cut for x̄, it is easy to see that
Lβ (x, y, λ) =ϕ(x) + φ(y) + λT (Mx + Ny − d)+
β
∥Mx + Ny − d∥2 ,
2
where β > 0 is the penalty parameter.
Using a Gauss-Seidel step in the augmented Lagrangian
multiplier framework, ADMM has the form
c(x) − τ c(x̄) ≤ 0, for any τ ∈ (0, 1),
λk+1 = λk + β(Mxk+1 + Nyk+1 − d).
yk+1 = argminy∈Y Lβ (xk+1 , y, λk )
(30)
is also a cut. By choosing τ adaptively based on a constraint
violation measure for x̄, we likely improve the quality of the
solution. For example, the reference factor can be the optimal
function value of the problem (22). The idea is that if the
constraint violation is large, we select τ close to 1, otherwise
τ is close to 0.
Fig. 1.
xk+1 = argminx∈X Lβ (x, yk , λk )
We notice that the SCOPF (1) can be reformulated by
introducing auxiliary variables u0c in a form suitable for
ADMM as
min f (u0 , x0 )
s.t. h0 (u0 , x0 ) = 0, g0 (u0 , x0 ) ≤ 0
hc (uc , xc ) = 0, gc (uc , xc ) ≤ 0, ∀c ∈ C
|uc − u0c | ≤ uc , ∀c ∈ C
u0 − u0c = 0, ∀c ∈ C.
Regular and adaptive Benders cuts
We define
x , (u0 , x0 ), y , (u1 , x1 , . . . , uC , xC ),
ϕ(x) , f (u0 , x0 ), φ(y) , 0,
X , {(u0 , x0 ) : h0 (u0 , x0 ) = 0, g0 (u0 , x0 ) ≤ 0},
Remark 2: To speedup the Benders algorithm, we can
switch off a contingency c if it was feasible for incumbent
uk0 because it is likely that no cuts are generated for later
. At the end, we need to check if the final u∗0 is feasible
uk+j
0
for it.
Y , (Y1 , . . . , YC ), where
{
Yc , (uc , xc , u0c ) : hc (uc , xc ) = 0, gc (uc , xc ) ≤ 0,
}
|uc − u0c | ≤ uc , ∀c ∈ C,
Mx + Ny = d , u0 − u0c = 0, ∀c ∈ C,
then the augmented Lagrangian is
Lβ (x, y, λ) = f (u0 , x0 )+
V. A LTERNATING DIRECTION METHOD OF MULTIPLIERS
This section introduces an alternative approach to solve
the large-scale SCOPF problem (1) by decomposing it into
a number of smaller subproblems related to each contingency.
By reformulating the original problem, we can apply the
alternating direction method of multipliers [16] to obtain
our algorithm. The first-order primal-dual algorithm ADMM
iteratively updates both primal and dual variables. In power
system analysis, to solve the classical OPF problem without
the security constraints, in [17], Kim and Baldick split the
power grid into a number of separate regions. By duplicating
the variables in overlap regions, they are able to decompose
the OPF problem by ADMM. Kraning et. al [18] develop a
C
∑
β
2
(λT
c (u0 −u0c )+ ∥u0 −u0c ∥ ).
2
c=1
The ADMM scheme for SCOPF is defined as follows
ADMM ALGORITHM
Initialize k = 0 and starting points u00c , λ0c .
Repeat
1. (xk+1
, uk+1 ) = argmin(u0 ,x0 )∈X f (u0 , x0 ) +
)
∑0 ( (0 k )T
λc (u0 − uk0c ) + β2 ∥u0 − uk0c ∥2 .
c∈C
2. For c = 1, . . . , C :
(xk+1
, uk+1 , uk+1 ) = argmin(uc ,xc ,u0c )∈Yc
( ck )T ck+1 0c
λc (u0 − u0c ) + β2 ∥uk+1
− u0c ∥2 .
0
k+1
k+1
k+1
k
λc = λc + β(u0 − u0c ).
3. Set k = k + 1.
7
TABLE II
CPU
TIMES AND NUMBER OF ITERATIONS FOR GLOBAL ALGORITHM
Until a stopping criterion is satisfied.
Note that if F, G, X, and Y are convex, ADMM is guaranteed to converge to the optimum. However, in our application,
X and Y are nonconvex, it can serve as a local algorithm.
At each iteration, we decompose the SCOPF into C + 1
subproblems with roughly the same size of the conventional
OPF problem, which is suitable for parallel implementation.
The feasible sets of subproblems in the ADMM algorithm have
not been modified over iterations, a significant computation
effort can be saved by using warm-start techniques for these
subproblems.
VI. C OMPUTATIONAL EXPERIMENTS
We use several standard power networks appearing in the
literature, whose structures and characteristics are summarized
in Table I. NE39 is the the New England system from [19],
while IEEE14, IEEE30, IEEE57, IEEE118 and IEEE300 are
available at http://www.ee.washington.edu/research/pstca/, and
PL3012 is the Polish network during winter 2007-08 evening
peak conditions from Matpower [20]. We artificially generated
TABLE I
T EST SYSTEMS CHARACTERISTICS
#
1.
2.
3.
4.
5.
6.
7.
Case
IEEE14
IEEE30
NE39
IEEE57
IEEE118
IEEE300
PL3012
|N |
14
30
39
57
118
300
3012
|G|
5
6
10
7
54
69
385
|B|
20
41
46
80
186
411
3572
|C|
10
15
20
30
100
150
300
#Var
418
1152
2058
3968
34744
111438
1992620
the list of contingencies; corresponding numbers of scenarios
are presented in the fifth column. The last column gives the
number of decision variables. In our experiments, a contingency relates to a transmission line outage; every generator
allows to readjust up to 5% of its maximal generation output
capacity.
All the experiments were performed on a personal computer
using a Matlab implementation with an Intel Xeon X5570 2.93
Ghz and 8Gb of memory. In the following tables, the basecase OPF cost “Base” represents the generation cost without
contingency constraints. Denote “Optimal” by the optimal
value generated by a global method. The total generation cost
is labeled by “Cost”, “Time” is the running time in seconds,
and “It” is the number of iterations.
A. Numerical Results for Global Algorithm
We present numerical results in Table II using the proposed
branch-and-bound algorithm in Section III. The projected subgradient method was used to solve for the dual (14). CPLEX
12.4 was used to solve for (15). The inverse-free preconditioned Krylov subspace projection algorithm [21] deals with
the eigenvector associated with the minimum eigenvalue in
Theorem 1. First of all, it is worth noting that the branch-andbound algorithm terminates at the first iteration, i.e., It = 1,
which implies the Lagrange dual problem gives a zero duality
Cases
IEEE14
IEEE30
NE39
IEEE57
IEEE118
IEEE300
PL3012
Base
80.81
89.06
361.54
417.37
1296.67
7197.23
25817.09
Optimal
81.04
89.74
391.61
419.55
1333.06
7197.48
Out of
It
Time
1
81.62
1
19.56
1
0.25
1
0.37
1
336.04
1
875.81
memory
gap. For these test cases, the branching procedure is not
needed. Thus solving the dual provides a global optimum
to the SCOPF problem. As usual, the main drawback of the
global scheme is the large computational time required, which
suffers from the scalability. However, its proved optimal results
are useful because they can be used to evaluate the solution
quality of any local approach. We can reduce the running
time and alleviate the memory requirement by utilizing the
decomposable structure of the dual to implement in parallel.
Note that instances NE39 and IEEE57 were solved quickly
because we used the Lagrangian multipliers of the primal as
the starting point for the dual, and these values are indeed the
optimal solution to the dual.
TABLE III
CPU
TIMES AND NUMBER OF ITERATIONS FOR GLOBAL ALGORITHM
C
Cases
Changes
IEEE30
Pd = 0.5Pd
{2 4 10} 36.3
Qd = 0.1Qd
Pd = 0.75Pd {2 6 15 22} 238.6
Qd = 0.2Qd
NE39
Base
Optimal
It
Time
36.4 103
218.4
291.2 156
305.7
For all above standard test cases, the Lagrange duality
relaxation solves the SCOPF problem, i.e., there is no need
to employ the branching step. We use some modified test
power systems suggested in [22], for which Lagrangian dual
relaxation has a non-zero duality gap for SCOPF, to illustrate
the benefit of the branching. The modifications are active and
reactive power demands. The power generation readjustments
are within 2% of maximal capacities. The stopping criterion
for the branch-and-bound algorithm was set at a relative gap
of ϵ = 0.01 between the lower bound and the upper bound.
The details of these tests and the algorithm performance are
shown in Table III.
B. Numerical Results for Local Algorithms
Version 3.10 of IPOPT [23] was used to solve the subproblems arising in these decomposition algorithms. The stopping
criterion for ADMM algorithm is
k+1
k
∥uk+1
− uk+1
0
0c ∥∞ ≤ 0.01 and ∥u0c − u0c ∥∞ ≤ 0.01.
We used a fixed parameter of β = 1 for the iterations. First,
we present the performance of Benders decomposition using
the cut (25) and the adaptive cut (30) with the following rule
for selecting τ in (30): if the optimal function value of (22) is
larger than 1, then we choose τ = 0.8; otherwise τ = 0.4. We
omit to report the results for the cut (24) because it has almost
8
CPU
TIMES OF
TABLE IV
B ENDERS [4], OUR PROPOSED B ENDERS DECOMPOSITION
ALGORITHMS , AND
ADMM
Benders [4]
Adaptive Benders
Prop. Benders
Adaptive Prop. Benders
ADMM
Cases
Optimal
Cost It
Time
Cost
It
Time
Cost It
Time
Cost
It
Time
Cost It
Time
IEEE14
81.04
83.25
5
0.61
81.08
8
2.09
84.18
3
0.53
81.12
4
1.41
81.07 35
2.20
IEEE30
89.74
failed
90.71 10
2.48
97.25
5
1.44
92.58 20
3.56
89.82 71
7.11
NE39
391.61
415.65
6
1.77
393.59
17
3.31
418.81
5
1.61
391.50 18
3.14
391.58 31
4.74
IEEE57
419.55
457.38
4
2.74
426.23
12
4.06
432.67
3
2.24
427.96
9
3.55
419.52 12
4.46
IEEE118 1333.06
failed
- 1333.55 23
16.42 1402.74
7
5.81 1332.92 18
12.63 1333.15 29
23.54
IEEE300 7197.48 7429.50
3
9.72 7197.39 14
21.94 7325.25
5
12.97 7197.36 18
26.15 7197.41 16
34.61
PL3012
26457.20
7 779.14 25923.47
13
1008.28 26137.92
8 831.16 25923.62 12
955.08 25921.58 23 3270.52
similar results with those of (25). If the optimal function value
of (22) is bigger than 0.01, we generate the feasibility cut for
this contingency. The performance of the standard Benders
decomposition algorithm in [4] and the application of our
adaptive cut scheme for this method are reported.
Table IV shows that both the standard Benders decomposition algorithm and our proposed algorithm have a fast
convergence rate (up to 8 iterations) if they are convergent;
however, they often generate solutions of poor quality. The
standard Benders cut from [4] failed to solve for the 30bus and 118-bus cases, which is due to the infeasibility
of the master problem when more linear cuts are added.
The difficulty with this sub-optimality and the infeasibility
is understandable since hyperplanes from the feasibility cuts
often prune the optimal solution from the nonconvex feasible
set as explained in Remark 1. The adaptive selection for both
Benders cuts gives much better solutions which are often close
to the globally optimal points. Moreover, it is also able to
cure the infeasibility issue. As expected, the deployment of the
technique leads to more iterations to converge in all cases. The
solution quality and running times for these adaptive Benders
cut algorithms are comparable.
ADMM is more robust in terms of solution quality. That
is, it often gives a slightly lower generation cost. In view
of a smaller number of iterations, the running times of the
adaptive Benders decompositions outperform ADMM. This is
also partially because of the fact that ADMM cannot exploit
the switch-off strategy presented in Remark 2, which requires
us to solve C + 1 subproblems per iteration. We notice that
all ADMM solutions are the essentially global optima. These
numerical results on serial codes show that these algorithms
are scalable, they are suitable for solving large-scale networks
with a large number of contingencies. The running times
will be reduced significantly if we exploit the decomposable
properties of these algorithms and implement in parallel in
order to meet the real-time operation requirement.
Table V gives the overall power generation (MW) and total
cost for OPF, SCOPF, and first four contingencies. The fourth
column, denoted by “Shifted”, shows the absolute value of
overall power (MW) shift between OPF and SCOPF solutions.
For cases with a large amount of power rescheduled such as
NE39, IEEE118 and PL3012, the generation costs of SCOPF
are noticeably bigger than those of OPF. In view of the
large distance between the solutions of OPF and SCOPF,
Benders decomposition algorithms tend to need more cuts,
thus more iterations, to converge. The total cost for each
contingency is always bigger than that of the normal case, i.e.,
the optimal function value of SCOPF. However, the overall
power generation for some contingencies can be smaller than
that of the SCOPF, for example, the third contingency of the
IEEE14 system.
TABLE V
C OMPARISON BETWEEN OPF AND SCOPF
SOLUTIONS
Case
OPF SCOPF Shifted Ctg. 1
Ctg. 2
Ctg. 3
Ctg. 4
1. MW
268.2
268.4
9.3
270.8
268.3
268.0
269.4
$
80.8
81.0
82.7
81.2
81.3
81.8
2. MW
295.1
292.8
66.9
292.6
294.8
292.9
292.5
$
89.1
89.7
90.4
91.8
90.6
89.9
3. MW 6194.2 6207.3 1038.0 6220.2 6215.2 6212.3 6214.8
$
361.5
391.6
390.0
400.3
389.2
389.5
4. MW 1267.2 1267.3 316.2 1267.1 1269.2 1267.4 1268.9
$
417.4
419.6
419.8
420.8
419.8
420.5
5. MW 4319.4 4309.6 738.8 4312.1 4311.1 4313.7 4312.3
$ 1296.7 1333.1
1333.7 1333.5 1334.5 1340.8
6. MW 23829.9 23830.4 103.1 23831.6 23827.1 23831.8 23828.9
$ 7197.2 7197.5
7202.6 7202.9 7203.6 7202.4
7. MW 27781.4 27794.3 579.5 27835.4 27814.7 27833.9 27813.5
$ 25817.1 25921.6
26075.8 25938.2 25931.4 25928.6
C. Stressed Networks
We tested the performance of these local algorithms on
stressed networks, where the upper limits on active power
flow in transmission lines were reduced. The modifications are
shown in Table VI, where the second column gives bus indices
connecting the modified lines, the third one presents the power
flows (MW) for the normal case of unmodified networks. We
select the lines among the highest values of power flows and
set the line limits (MW) significantly smaller than these values,
which are given in the fourth column. The new values of power
flows for the modified networks are given in the fifth column.
The last column shows the total MW adjustment between the
OPF and SCOPF.
TABLE VI
M ODIFIED
Cases
IEEE30
NE39
IEEE118
PL3012
Mod. Lines
1-2
6-28
2-3
29-38
26-30
68-116
34-40
98-99
191-193
TEST CASES FOR STRESSED NETWORKS
Org. Flow
120.1
16.8
522.6
930.0
226.0
184.1
678.9
874.7
425.2
Mod. Limit
100.0
10.0
450.0
800.0
100.0
100.0
580.0
750.0
370.0
Mod. Flow
79.5
6.1
425.8
800.0
100.0
100.0
580.0
746.9
369.5
Shifted
138.8
1410.5
951.1
872.4
9
TABLE VII
CPU
TIMES FOR STRESSED NETWORKS
Benders [4]
Adaptive Benders
Prop.
Cases
Optimal
Cost It
Time
Cost It
Time
Cost
IEEE30
94.6
failed
94.6 20
3.1
failed
NE39
397.5
failed
397.5 15
3.4
428.6
IEEE118 1346.3 1375.3
5
6.4 1347.8 13
10.2 1366.6
PL3012
26170.2 17 1280.5 26091.4 26 1445.7 26198.3
As shown in Table VII, both regular and our Benders
decomposition algorithms require more iterations for the networks with stress conditions, but the increases comparing with
the unstressed cases are moderate. Again, the adaptive cut
scheme not only alleviates the convergence failure of these
Benders cuts but also improves the solution quality. For many
cases, it helps to find an operating point very close to the
optimal point. The convergence rate of ADMM seems to be
insensitive to the stress condition of test systems, that is, the
number of iterations does not depend on the congestions. Its
found solutions are consistent, which are essentially global
optima.
VII. C ONCLUSIONS
We have developed three different approaches to solve the
SCOPF problem, namely a global optimization algorithm and
two decomposition schemes. Lagrangian duality is the main
tool for the global technique to compute a lower bound, and
the dual shows a zero duality gap for most of our test instances.
Decomposition algorithms are based on Benders cuts and the
alternating direction method of multipliers. We have shown
that if one uses the regular Benders cut without care, the
solution will be rather far from the optimal operating point.
By taking the nonconvexity into account properly, our adaptive
Benders decomposition often generates a solution of high
quality. The resulting ADMM algorithm is capable of yielding
a robust solution, which is numerically proved to be the global
optimum. These proposed decomposition schemes deal with
each contingency separately, therefore they are suitable for
large-scale problems and can be implemented in parallel to
handle many contingencies.
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Benders
Adaptive Prop. Benders
ADMM
It
Time
Cost It
Time
Cost It
Time
94.6 22
3.4
94.7 67
7.8
7
1.9
397.8 18
3.5
397.5 33
62
8
7.3 1347.0 21
14.1 1347.6 25
24.7
13 1165.1 26089.6 19
1295.2 26085.7 24 3582.5
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Dzung Phan is a Research Staff Member of the Business Analytics and Mathematical Sciences Department at IBM T.J. Watson Research Center, Yorktown
Heights, NY. His research interests lie in the field of the optimization theory
and algorithms including methods for problems arising from power systems.
He received a Ph.D. in Mathematics from the University of Florida in 2010.
Jayant Kalagnanam is a Program Manager in the Business Analytics
and Mathematical Sciences Department and has been at IBM T.J. Watson
Research Center as a Research Staff Member in 1996. Before joining IBM
Research he worked as a Research Faculty at the Department of Engineering
and Public Policy at Carnegie Mellon University where he developed large
cost minimization models for designing and retrofitting utilities with new
technologies. He graduated with a Ph.D. in 1991 from the same department.
After joining IBM research, he worked on developing optimization models
for production planning and scheduling in the context of manufacturing for
various industries including steel, semiconductor, mining and shipbuilding.