1

arXiv:1711.00317v2 [] 20 Dec 2017

Linear/Quadratic Programming-Based Optimal
Power Flow using Linear Power Flow and Absolute
Loss Approximations
Philipp Fortenbacher
Turhan Demiray
Research Center for Energy Networks (FEN)
ETH Zurich
Zurich, Switzerland
{fortenbacher, demirayt}@fen.ethz.ch

Abstract—This paper presents a novel method to approximate
the nonlinear AC optimal power flow (OPF) into tractable
linear/ quadratic programming (LP/QP) based OPF problems
that can be used for power system planning and operation.
We derive a linear power flow approximation and include a
convex reformulation of the power losses in the form of absolute
value functions. We show three ways how we can incorporate
this approximation into an LP/QP based OPF problem. The
usefulness of our OPF methods is analyzed for the IEEE test case
grids. As a result, the errors on voltage magnitudes and angles
are reasonable, while obtaining near-optimal results. We find that
our methods reduce significantly the computational complexity
compared to the nonlinear AC-OPF making them a good choice
for planning purposes.
Index Terms—Optimal Power Flow, Linear/ Quadratic Programming, Powerflow Approximation

I. I NTRODUCTION
A. Motivation
PTIMAL Power Flow (OPF) is indispensable for current
research in power system operation and planning. OPF
is widely used to find optimal expansion schemes [1], [2], [3]
for transmission networks in planning problems. Furthermore,
at operation level when OPF is incorporated as multi-period
problems, OPF enables to find optimal generator setpoint
schedules in unit commitment (UC) problems that minimize
operational costs [4], [5], [6].
Especially for planning problems it is crucial to have
tractable formulations of the multi-period OPF problem, since
the incorporation of the nonlinear original OPF problem would
impose a high computational burden. This is due to the
intertemporal coupling of long investment horizons. Moreover, transmission planning methods need to incorporate a
power flow approximation, since the nonlinear OPF cannot
deal with binary placement constraints. Therefore, current
planning methods include either the well-known lossless DC
power flow approximation [7], a lossless approximation of the
power flow in the full decision variable domain (active and
reactive power, voltage magnitudes and angles) [3] or a power
flow representation that does not operate in the full decision
variable domain [1], [2]. This can result in near-optimal or

O

infeasible solutions, when neither network losses nor the full
solution space are considered.
Also UC problems require often binary decisions e.g. consideration of startup costs in the full decision variable space.
This can only be achieved by mixed-integer programming
(MIP) frameworks that are either able to incorporate linear
or semidefinite programming (SDP) constraints. However,
relaxing planning or UC problems into a second order cone
(SOC) programming problem [8] or into an SDP problem [9]
is still a complex optimization problem. Consequently, linear
approximations are often the first choice to deal with the
complexity issue. This also explains why UC problems are
often divided into several stages [6] that reflect the binary
decisions in the first stage, using a linear DC approximation,
and then use the nonlinear OPF in the following stages at the
cost of optimality, and/ or computation time.
In sum, there is still a clear need of linear OPF approximations that work in the full decision variable space and capture
power losses. Hence, the objective of this paper is to find
a linear and tractable approximation of the OPF problem in
the full decision variable space of active/ reactive power and
voltage magnitudes/ angles for universal grid topologies.
B. Related Work
Finding reasonable linear power flow approximations for
OPF problems is not a new research field. The first approaches
included linearizing the power flow equations and passing this
information to an LP solver. However, since this approximation does not hold for the entire operating range, the LP
problem needs to be solved in an iterative way. Several papers
[10], [11], [12] have used this solution approach, where they
build the Jacobian of the power flow equations at a given
operating point. The work featured in [13], [14], [15] derives
a linear approximation of the AC power flow equations, but
does not show how these approximations can be incorporated
into an OPF problem. Mhanna et al. approximate the second
order cone constraints with linear constraints resulting in a
high number of constraints [16]. Castillo et al. [17] use also an
iterative approach to compute the optimal generator setpoints.
The linear OPF method of [18] does not capture losses and

2

A. Absolute Loss Approximation

p1 , q1 , v1 , θ1

The incurred active pl and reactive ql power losses can be
calculated by
y = g + jb

p2 , q2 , v2 , θ2

Fig. 1. Two-bus system to illustrate and derive the linear power flow and
absolute loss approximations.

only operates in the decision variable domain of voltage angles
and active power.

C. Contribution
The contribution of this paper is the development of a novel
tractable Linear /Quadradic Programming (LP/QP) based OPF
methods that approximates the power flow over the entire
operating range and can be solved by off-the-shelf solvers.
This means that we do not require any iterations to solve the
OPF problem. Our problem links the full decision variable domain with linear power approximations and captures the power
losses by using absolute value loss approximations. Due to the
linear constraint formulations we can also account directly for
binary decisions that are present in UC commitment problems.
Our approach is universal to reflect any grid topology (meshed
and radial) and any voltage levels (low voltage, distribution,
and transmission grids). Unlike [16], we reduce the number
of constraints.
The remainder of this paper is organized as follows. Section II derives the power flow approximation. Section III
shows how this approximation can be included in an LP/QP
based OPF problem. Section IV analyzes the accuracy and
optimality of our suggested OPF method and Section V draws
the conclusion.

II. L INEAR A PPROXIMATION

(v12 + v22 )g − 2v1 v2 cos(θ1 − θ2 )g , (2)

ql = q1 + q2 =

−(v12 + v22 )b + 2v1 v2 cos(θ1 − θ2 )b . (3)

Let v1 , v2 = 1 then we can find an absolute power loss
approximation as a function of the voltage angle difference
for active and reactive power as follows
pl =

2(1 − cos(θ1 − θ2 ))g ≈

|θ1 − θ2 |2k1 g ,

(4)

ql =

−2(1 − cos(θ1 − θ2 ))b ≈

−|θ1 − θ2 |2k1 b ,

(5)

where k1 is a constant that represents the gradient of the
absolute function associated with the voltage angle difference.
Here, we approximate (1 − cos(θ1 − θ2 )) with k1 |θ1 − θ2 |. If
we let θ1 − θ2 = 0 then we obtain approximations in terms of
absolute values that are dependent on the voltage magnitude
difference:
pl =
ql =

(v1 − v2 )2 g ≈
2

−(v1 − v2 ) b ≈

|v1 − v2 |2k2 g ,

(6)

−|v1 − v2 |2k2 b ,

(7)

where k2 is a constant to approximate the losses associated
with the voltage magnitude difference. Here, we approximate
(v1 − v2 )2 with |v1 − v2 |2k2 .
By superposing (4), (6) and superposing (5), (7), we approximate the active papprox
and reactive power losses qlapprox as
l
follows:
papprox
= |θ1 − θ2 |2k1 g + |v1 − v2 |2k2 g ,
l
{z
} |
{z
}
|

(8)

qlapprox = −|θ1 − θ2 |2k1 b − |v1 − v2 |2k2 b ,
{z
} |
{z
}
|

(9)

2p∆θ
l

2ql∆θ

2p∆v
l

2ql∆v

that are convex reformulations of the exact power losses (2)
and (3).
B. Selection of k1 , k2

We first derive the linear power flow approximations based
on a 2-bus example and extend this result to capture tap
ratios, shunt elements, and line charging. Then, we introduce
nodal admittance matrices to reflect any grid topology and size
and incorporate this representation in an optimal power flow
problem that is compliant with an LP/QP framework.
Based on Fig. 1, the nodal active p1 , p2 and reactive q1 , q2
powers are given by the nonlinear AC power flow equations
that are for this case
p1 = v12 g −v1 v2 cos(θ1 − θ2 )g −v1 v2 sin(θ1 − θ2 )b
p2 = v22 g −v2 v1 cos(θ2 − θ1 )g −v2 v1 sin(θ2 − θ1 )b
q1 = −v12 b +v1 v2 cos(θ1 − θ2 )b −v1 v2 sin(θ1 − θ2 )g
q2 = −v22 b +v2 v1 cos(θ2 − θ1 )b −v2 v1 sin(θ2 − θ1 )g

pl = p1 + p2 =

,
,
(1)
,
,

where v1 , v2 are the per unit nodal voltage magnitudes, θ1 , θ2
are the voltage angles, g is the per unit line conductance and
b the per unit line susceptance.

We have two degrees of freedom to approximate the power
losses with the constants k1 , k2 . To parametrize those parameters we define the design parameters ∆θd = θ1 − θ2 , ∆vd =
v1 − v2 . They specify a usual voltage magnitude and angle
difference between two nodes that are connected by a line. If
we solve the equations (4), (5) for k1 and (6), (7) for k2 , we
obtain the following parametrizations for k1 , k2
1 − cos ∆θd
|∆θd |
≈
,
(10)
|∆θd |
2
|∆vd |
k2 =
.
(11)
2
We choose reasonable values for ∆θd = 0.08 rad, ∆vd =
0.02 p.u. With this approach we normalize the losses with
respect to the susceptances and conductances. In this way it is
possible to consider universal grid topologies ranging from low
voltage grids usually having a high R/X ratio to transmission
grids possessing a high X/R ratio. Figures 2a and 2b show
k1 =

3

(a) Active power

(a) Active power losses

(b) Reactive power losses
Fig. 2. Power loss comparison for the parameters b = −10 and g = 1. The
blue surface indicates the exact losses, while the red surfaces are the absolute
power loss approximations.

the exact and the approximated power losses for the 2 bus
transmission system (Fig. 1) with the per unit susceptance
b = −10 and the per unit conductance g = 1. Note that
this formulation is an approximation and not a relaxation,
since there are loss regions that are underestimated above the
values ∆θd and ∆vd . This means that these errors translate to
an underestimation of voltage angles and magnitudes. Hence,
there is no guarantee that the approximated OPF solutions will
lie inside the feasible original solution space.
C. Linear Power Flow Approximation
We linearize the nonlinear power flow equations (1) by
using the following approximations cos(θi − θj ) ≈ 1,
vi2 − vi vj ≈ vi − vj , vi vj sin(θi − θj ) ≈ (θi − θj ) to obtain
p1
p2
q1
q2

≈ (v1 − v2 )g
≈ (v2 − v1 )g
≈ −(v1 − v2 )b
≈ −(v2 − v1 )b

−(θ1 − θ2 )b
−(θ2 − θ1 )b
−(θ1 − θ2 )g
−(θ2 − θ1 )g

+p∆θ
+ p∆v
l
l
+p∆θ
+ p∆v
l
l
+ql∆θ + ql∆v
+ql∆θ + ql∆v

,
,
,
,

(12)

in which we also add the convex reformulations of the
∆v ∆θ ∆θ
derived from the previous
power losses p∆v
l , ql , pl , ql
Section II-A. As a result, the power flow approximations
(12) are convex, which can also be graphically verified in
Figures 3a and 3b.

(b) Reactive power
Fig. 3. Power flow comparison. p1 , p2 represent the nonlinear power flow
equations (1) and papprox
, papprox
are the convex power flow approximations
1
2
(12).

D. Extension for Line Charging, Transformer Tap Ratios and
Shunts
The aforementioned two-bus example considers only a
series admittance y. In this section, we aim to extend our
approach to incorporate line charging, transformer tap ratios
and shunt elements. To capture these features, we use the
standard π branch model. The nodal admittance matrix Y b
for the two-bus system is then e.g.

 

1
(y + j b2c ) τ12 −y τ e−θ
ysh,1 0
s
Yb=
+
, (13)
0 ysh,2
−y τ e1θs
(y + j b2c )
where τ is the per unit tap ratio, θs is the transformer shift
angle, bc is the per unit capacitive reactance of the line, and
ysh,i are the per unit shunt admittances. We also define the
matrix Y 0b , in which we only consider the series admittance
with the complex tap ratios as follows


1
1
−y τ e−θ
y τ e−θ
0
s
s
.
(14)
Yb=
−y τ e1θs
y τ e1θs
This matrix is needed for the power flow approximation to
correctly represent the power flow contribution on the voltage
angles.
We next generalize our approach to account for any grid
topology. First, we define a grid that consists of nb buses, ng

4

generators, and nl lines. Any topology can be specified by constructing the node-branch incidence matrix C br ∈ Znl ×nb . We
introduce the generator active and reactive power injections
pg , q g ∈ Rng ×1 . The generator to bus mapping is specified
with the matrix C g ∈ Zng ×nb . We now can extend Y b ∈
∆v
∆θ
∆v
Cnb ×nb ,Y 0b ∈ Cnb ×nb , θ, v ∈ Rnb ×1 , p∆θ
∈
l , pl , q l , q l
nl ×1
R
to reflect any grid topology. The nodal active and
reactive power injections p, q ∈ Rnb ×1 are split into
p = C g pg − pd ,

(15)

q = C g qg − qd ,

(16)

where pd , q d ∈ Rnb ×1 are the active and reactive load vectors.
Under these definitions, we can find a more general matrix
representation for the active power flow approximation as


θ


 v 

0
T
T  p 
−={Y b } <{Y b } −C g |C br | |C br |  g  = −pd ,
 p∆θ

l
∆v
pl
(17)
and for the reactive power flow as


θ



 v 
0
T
T  q 
−<{Y b } −={Y b } −C g |C br | |C br |  g  = −q d .

 q ∆θ
l
∆v
ql
(18)

E. Branch Flow Approximation
To obtain tractable OPF problems, we linearly approximate
the active and reactive power line flows pf , q f ∈ Rnl ×1 as
pf = −={Y 0f }θ + <{Y f }v
q f = −<{Y 0f }θ − ={Y f }v

(19)
,

(20)

where Y f ∈ Cnl ×nb is the branch admittance matrix at the
from ends of all branches. The matrix Y 0f ∈ Cnl ×nb is the
adjusted version of the branch admittance matrix, in which
we neglect the shunt admittances. Note that the power line
losses are neglected by using this formulation.

III. A PPROXIMATED T RACTABLE O PTIMAL P OWER F LOW
P ROBLEMS
In this section we derive the formulations of the approximated OPF problem. First, we present a lossy LP/QP based
OPF problem that incorporates the power flow approximations
as linear constraints. Hence, we call this method LOLIN-OPF.
Secondly, we also define a lossless version of the OPF problem
that we call LIN-OPF. Thirdly, we also provide a mixed integer
formulation of the problem, where we include the power
flow approximations as linear constraints with binary decision
variables. This method we call MIP-OPF.

A. Lossy LP/QP based Optimal Power Flow (LOLIN-OPF)
Problem
With the introduced power flow approximations we
can now formulate the OPF problem within a standard
LP/QP framework. We specify the decision vector x =
∆θ
T
[θ v pg q g p∆θ
p∆v
q ∆v
l
l ql
l ] . The objective of the OPF
problem is to find the optimal active and reactive generator
powers that minimize either a linear or quadratic cost objective. The approximated lossy LP/QP based Optimal Power
Flow (LOLIN-OPF) problem is
LOLIN-OPF: min fp (pg ) + fq (q g )
x

s.t.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)

(17), (18)
k1 diag{g}C br θ − p∆θ
≤0
l
−k1 diag{g}C br θ − p∆θ
≤0
l
k2 diag{g}C br v − p∆v
≤
0
l
−k2 diag{g}C br v − p∆v
≤
0
l
k1 diag{b}C br θ − q ∆θ
≤
0
l
−k1 diag{b}C br θ − q ∆θ
≤0
l
k2 diag{b}C br v − q ∆v
≤
0
l
−k2 diag{b}C br v − q ∆v
≤
0
l
−s ≤ pf + Aq q f ≤ s
−s ≤ pf − Aq q f ≤ s
−s ≤ Aq pf + q f ≤ s
−s ≤ Aq pf − q f ≤ s
,

(21)

where fp , fq are either linear or quadratic generator cost
functions. The constraints (17), (18) specify the active and
reactive power balance. The inequalities (21a-h) represent
the epigraphs of the approximated absolute power losses.
It is noteworthy that the power loss vectors only lie on
these hyperplanes if the Lagrange multipliers on the active
and reactive power balance constraints are positive. In other
words the constraints (21a-h) need to be binding to achieve a
physical meaningful solution. Negative multipliers (also called
locational marginal prices (LMPs)) would lead to fictitious
losses allowing to consume more active or reactive power
to lower the objective value. Negative LMPs could occur
in highly congested systems or for negative cost functions.
Since we often have the case that generator cost functions are
specified in the active power domain only, it is common to
have negative LMPs on the reactive power balance constraints.
However, to make use of efficient convex solvers, we remove
∆θ
the reactive power loss variables q ∆v
from the problem
l , ql
and discard the constraints (21e-h) at the cost of accuracy.
To generally include the power losses for negative LMPs, we
need to formulate a Mixed Integer problem (MIP) that will
be discussed in Section III-C. The constraints (21i-l) define
convex polygons to approximate the circular PQ capability
area. The vector s ∈ Rnl ×1 specifies the apparent line limits
and Aq = diag{aq ∈ Rnl ×1 } represents the derivatives of the
lines that form a convex polygon as shown in Fig. 4. Here,
we consider 8 convex segments to approximate the circular
PQ area.

5

pf
pf ≤

√


2 − 1 qf + s
| {z }
aq

Then, the Mixed Integer LP/QP based Optimal Power Flow
(MIP-OPF) problem is
fp (pg ) + fq (q g )
MIP-OPF: min
00
x

s
qf

Fig. 4. Illustrative example for the convex line flow approximation for one
power line. The gray line with the corresponding inequality specifies one
segment of the circular PQ area.

B. Lossless LP/QP based Optimal Power Flow (LIN-OPF)
Problem
If the LOLIN-OPF problem returns negative LMPs on the
active power balance and we would like to avoid solving an
MIP problem, we can define a lossless version of the OPF
problem at the cost of accuracy. We eliminate the power loss
vectors from (17) and (18), so that the decision vector is x0 =
[θ v pg q g ]T . Then, the lossless LP/QP based Optimal Power
Flow (LIN-OPF) problem is:
LIN-OPF: min
fp (pg ) + fq (q g )
0
x

s.t. (21i-l)
(22)
(a) −={Y 0b }θ + <{Y b }v − C g pg = −pd
0
(b) −<{Y b }θ − ={Y b }v − C g q g = −q d .

C. Mixed Integer LP/QP based Optimal Power Flow
(MIP-OPF) Problem
As discussed in Section III-A the optimal solution might
not be tight with respect to the power loss constraints.
An alternative but computationally more expensive formulation can be obtained by expressing the power loss constraints through a mixed integer programming (MIP) approach. For this, we restructure the decision vector to x00 =
[θ v m pg q g ∆θ ∆v bθ bv ]T . The power loss vectors are
now replaced by ∆θ, ∆v ∈ Rnl ×1 specifying the absolute
values of |θi − θj | and |vi − vj | in (8) and (9). We introduce
the binary variables bθ , bv ∈ Znl ×1 that are associated with
∆θ, ∆v. The approximated active and reactive power balance
can be adjusted to comply with the new introduced decision
vectors ∆θ, ∆v as



θ
 vm 
−={Y 0b } <{Y b } −C g 0
 


pg 
−<{Y 0b } −={Y b } 0 −C g  
−pd


 = −q d .
k1 diag{g}|C br |T k2 diag{g}|C br |T 
 qg 
T
T
 ∆θ 
k1 diag{b}|C br | k2 diag{b}|C br |
∆v
(23)



s.t.
(a)
(b)
(c)
(d)
(e)
(f)

(23), (21i-l)
−M (1 − bθ ) ≤ −C br θ ≤ M bθ
0 ≤ −C br θ + ∆θ ≤ 2M bθ
0 ≤ C br θ + ∆θ ≤ 2M (1 − bθ )
−M (1 − bv ) ≤ −C br v ≤ M bv
0 ≤ −C br v + ∆v ≤ 2M bv
0 ≤ C br v + ∆v ≤ 2M (1 − bv ) ,

(24)

where the constraints (24a-f) specify a big M formulation of
the absolute value functions in (8) and (9). The variable M
has a considerable influence on the feasibility of the problem.
It needs to be chosen sufficiently large to approximate the real
(practical) range of the absolute values. Too large values might
result in weak relaxations leading to branching the problem
and hence to an increased computation time.
IV. R ESULTS
In this section we aim to show the performance of our
suggested OPF methods compared to existing OPF methods.
A. Implementation
We implemented the suggested OPF methods within the
MATPOWER framework [19] and use the GUROBI [20]
solver for solving the LP/QP, MILP/MIQP problems. We test
our methods based on the IEEE reference grid cases that are
bundled within MATPOWER. Since all test cases comprise
quadratic generator cost functions, we solve for the LINOPF/LOLIN-OPF methods a QP problem and for the MIPOPF method an MIQP problem.
B. Voltage Angle and Magnitude Errors
First, we aim to analyze the accuracy of our methods.
Here, we compare the approximated OPF solution of voltage
magnitudes and angles of our methods with the exact power
flow solution that calculates the true voltage magnitudes and
angles. For the comparison, we set the PV buses in the power
flow according to the OPF solution and compute the root mean
square (RMS) errors on the magnitude and angle deviations.
Figures 5a and 5b show the angles and magnitudes for the
LOLIN-OPF method and for the power flow (PF) program for
the IEEE 118 test grid. Although there is a small deviation in
voltage angles, it can be observed that the curves match well.
The angle offset can be explained by the fact that our OPF
method overestimates the losses at the slack bus, so that this
error propagates through the system.
We summarize the angle and magnitude errors for other
IEEE test grids in the bar plots in Figures 6a and 6b. For
comparison, we calculate the root mean squared (RMS) errors
and also show the results for the MIP-OPF. It can be observed
that the errors of the MIP-OPF are on average lower. This is
due to the fact that the MIP-OPF also considers the reactive
power losses in the power flow approximation, while this is
not the case for the LOLIN-OPF and LIN-OPF. We did not

6

(a) Voltage angle comparison

(a) Angle error

(b) Voltage magnitude comparison
Fig. 5. LOLIN-OPF voltage magnitude and angle comparison with the power
flow solution (PF) for the IEEE118 test grid.

find a MIP-OPF solution for the IEEE 300 test case, since the
generators reactive power capability margin was not sufficient
to cover the overestimated reactive power losses.
We observe a higher angle error (≈ 20◦ ) for the LIN-OPF
for the IEEE 300 test case. This can be explained in that the
LIN-OPF version translates the underestimated active power
generation setpoints to higher errors in voltage angles. Note
that the DC-OPF solution would also generate such error.
However, for the LOLIN-OPF and MIP-OPF methods the
errors are reasonable and can be accepted.
The error patterns follow the same ordering from low
errors (MIP-OPF) to higher errors (LIN-OPF) except for the
cases IEEE 30 and IEEE 39. The different orderings of these
special cases can be explained by higher voltage magnitude
differences that lead to higher approximation errors for the
LOLIN-OPF and MIP-OPF.

(b) Voltage magnitude error
Fig. 6. Voltage angle and magnitude errors compared to the power flow
solution for different benchmark grids.

Fig. 7. Optimality comparison of our OPF methods (LOLIN-OPF, LIN-OPF,
MIP-OPF) and DC-OPF against the nonlinear AC-OPF.

C. Optimality
Another fact that needs to be discussed is the optimality
of our suggested OPF methods. We define optimality as the
objective value deviation between the introduced OPF methods
and the nonlinear AC-OPF. From Fig. 7 we find that the
objective values are higher for our OPF methods (LOLINOPF, MIP-OPF), since those methods overestimate the power
losses in the grid to some extent. However, the solutions
do not deviate more than 4% from the AC-OPF solution.
Furthermore, the DC-OPF and LIN-OPF solutions result in
the same values and have lower objective values compared to
the nonlinear OPF solution. This is due to the fact that these
methods do not incorporate power losses. In contrast to the
DC-OPF, the LIN-OPF also includes a voltage projection. The
errors introduced by MIP-OPF and LOLIN-OPF are almost
identical. We can see that on average, the absolute deviation

for the LOLIN-OPF (1.3%) and the MIP-OPF (1.6%) is lower
than for the DC-OPF and LIN-OPF (2.5%).
D. Complexity
To assess the computational complexity of our method, we
solve the same multi-period storage siting problem as proposed in [21]. Here, we replace the linear Forward-Backward
Sweep (FBS) OPF with our proposed LOLIN-OPF and the
nonlinear AC-OPF. Note that the LOLIN-OPF is a more
general OPF method than the FBS-OPF, since it can be used
for transmission and meshed networks. We extend the decision vector X = [x(1), . . . , x(N − 1)]T to reflect multiple
timesteps within the investment horizon N . The main objective
of the problem is to find the battery locations and sizes

7

V. C ONCLUSION

Fig. 8. Computation time comparison between LOLIN-OPF and nonlinear
AC-OPF and their corresponding lin-log regressions.

In this paper we presented novel tractable OPF methods
that work in the full decision domain of active/ reactive power
and voltage magnitudes/ angles. We linearly approximate the
power flow over the entire operating range avoiding the need
to iterate the OPF problem. Our OPF methods can be used by
efficient off-the-shelf LP/QP solvers. The obtained accuracy
in terms of voltage magnitudes and angles is reasonable and
we achieve near-optimal solutions compared to the nonlinear
AC-OPF. In a planning problem, we have shown that the LP/
QP based methods outperform the nonlinear AC-OPF in terms
of computational complexity.
R EFERENCES

that minimizes PV curtailment. The general problem can be
represented as

∗

J =

min T
X,z

(b)
(c)

!
cTp pg (k)
{z

costs

cTs z
|{z}

+

k=0

|
s.t.
(a)

N
−1
X

}

∀x : (21a)- (21l)
X
As
≤ bs
z
X min ≤ X ≤ X max

storage
investment

(25)

,

where z are the battery capacities, cs are the storage prices
and cp (k) are the linear generator costs. The constraint (25a)
incorporates N single-shot OPF problems, which are coupled
across different time instants via (25b). The matrix As and
vector bs reflect the storage coupling and are specified in [21].
The constraint (25c) determines the operating bounds of the
grid and the generation units. In this example, the cost function
represents the financial transactions associated with the import
and export of energy at the feeder node. PV curtailment is
mitigated and storage is installed when the storage prices are
low and the price spreads of importing electricity are rather
high. The grid has 18 buses and is configured with a high PV
penetration. More information on the full scenario definition
and parameters can be found in [21].
Problem (25) is a large-scale LP problem in which the
effective grid size grows as nb N with nb = 18 and the
decision variable size as N (7nb + 2nl ) with nl = 17. The
LP problem is solved with gurobi [20] on a high-performance
computing cluster using two 12-core Intel Xeon E5 processors.
For comparison, we solve the nonlinear AC-OPF version of
problem (25) with IPOPT and pardiso [22]. The computation
time results for both versions are shown in Fig. 8, where the
investment horizon corresponds to hours. The lin-log regression of the nonlinear AC-OPF is almost shifted in parallel
towards higher computation times. This means that on average
we achieve an improvement in computation time with the
LOLIN-OPF method of one order of magnitude with respect
to the nonlinear AC-OPF.

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