Optimal Power Flow
over Radial Networks
S. Bose,, M. Chandy,
y,
M. Farivar, D. Gayme
S Low
S.
Caltech
March 2012
C. Clarke
Southern
California Edison
Outline
Motivation
Semidefinite relaxation
 Bus injection model
Conic relaxation
 Branch flow model
Challenge: uncertainty mgt
High Levels of Wind and Solar PV Will
Present an Operating Challenge!
Source: Rosa Yang, EPRI
Optimal power flow (OPF)
 OPF is solved routinely
y to determine
 How much power to generate where
 Market operation & pricing
 Parameter setting, e.g. taps, VARs
 Non-convex and hard to solve
 H
Huge literature
lit
t
since
i
1962
 In practice, operators often solve linearized
model and verify using AC power flow model
 Core
C
off many problems
bl
 OPF, LMP, Volt/VAR, DR, EV, planning …
Worldwide
energy demand:
16 TW
Wind power over land (exc. Antartica)
70 – 170 TW
electricity demand:
2.2 TW
wind capacity (2009):
159 GW
grid-tied PV capacity (2009):
21 GW
Solar power over land
340 TW
Source: Renewable Energy
Global Status Report, 2010
Source: M. Jacobson, 2011
Implications
Current control paradigm works well today
 Low uncertainty, few active assets to control
 Centralized, open-loop, human-in-loop, worst-case
preventive
ti
 Schedule supplies to match loads
Future needs
 Fast computation to cope with rapid, random,
large fluctuations in supply, demand, voltage, freq
 Simple algorithms to scale to large networks of
active DER
 Real-time
R l ti
d
data
t for
f adaptive
d ti
control
t l
Implications
Must close the loop
 Real-time feedback control, risk-limiting
 Driven by uncertainty of renewables
M t be
Must
b scalable
l bl
 Distributed & decentralized optimization
 Orders of magnitude more endpoints that can generate,
generate
compute, communicate, actuate
Control and optimization framework
 Theoretical foundation for a holistic framework that
integrates engineering + economics
 Systematic algorithm design, understandable global
behavior
 Clarify ideas
ideas, explore structures
structures, suggest direction
Application: Volt/VAR control
Motivation
 Static capacitor control cannot cope with rapid
random fluctuations of PVs on distr circuits
Inverter control
 Much faster & more frequent
 IEEE 1547 does
d
nott optimize
ti i
VAR currently (unity PF)
Load and Solar Variation
Load and Solar Variation
pic
g
i
p
Empirical distribution of (load, solar) for Calabash
of (load, solar) for Calabash
Improved reliability
p
y
g
c
p
,
p
for which problem is feasible
 i i  for which problem is feasible
Implication: reduced likelihood of violating voltage limits or VAR flow constraints
Energy savings gy
g
Summary
• More reliable operation
• Energy savings
Energy savings
Key message
Radial networks computationally simple
 Exploit tree graph & convex relaxation
 Real-time
Real time scalable control promising
Outline
Motivation
Semidefinite relaxation
 Bus injection model
Conic relaxation
 Branch flow model
Optimal power flow (OPF)
Problem formulation
 Carpentier 1962
Computational techniques:
 Dommel & Tinney 1968
 Surveys: Huneault et al 1991, Momoh et al 2001,
Pandya et al 2008
SDP formulation (bus injection model):
 Bai et al 2008,
2008 2009,
2009 Lavaei et al 2010
 Bose et al 2011, Sojoudi et al 2011, Zhang et al 2011
 Lesieutre et al 2011
Branch flow model
 Baran & Wu 1989,
1989 Chiang & Baran 1990,
1990 Farivar et al
2011
Models
s gj s cj
k
j
i
S j   S jk
branch
b
h
Sij
flow
S jk
k
bus
bus injection
i
Vi
zij   rij , xij 
I ij
j
k
Vj
I j   I jk
k
Models: Kirchoff’s law
Si   Sij  V I
*
i i
j
Vi
linear relation: I  YV
Vj
Sij  Vi Iij*
2
*
*
 V * V
V
V
VV
V
V
Sij  Vi  i * j   i*  i * j
zij
zij
 zij 
Vk
Bus injection model
Nodes i and j are linked with an
admittance

Kirchhoff's
Kirchhoff
s Law:
Classical OPF
min
g
f
P
 k k 
Generation cost
k G
kG
over
g
g
P
,Q
 k k , Vk 
subject to
P P  P
g
k
g
k
g
k
g
Qk  Q  Qk
g
g
k
V k |V
| k |  Vk
Generation power constraints
V l
Voltage
magnitude
i d constraints
i
KVL/KCL
power balance
nonconvexity
Classical OPF
In terms of V:
Pk  tr kVV
*
Qk  tr  kVV
min
kG
*
 Yk* Yk 
k : = 

 2 
 Yk* Yk 
k : =

 2i 
*
tr
M
VV
 k
over
V
st
s.t.
P  P  tr  kVV  P  Pkd
g
k
d
k
g
k
*
g
Q k  Q  tr  kVV  Q k  Qkd
g
d
k
*
V  tr J kVV  V
2
k
*
2
k
Key observation [Bai et al 2008]:
OPF = rank constrained SDP
Classical OPF
min
 tr M W
k
kG
over
W positive semidefinite matrix
s.t.
P k  tr kW  P k
Q k  tr  kW  Q k
2
k
V  tr J kW  V
W  0,
0 rank W 1
2
k
convex relaxation: SDP
Semi-definite relaxation
Non-convex
o co e QCQ
QCQP
Rank-constrained SDP
Relax the rank constraint and solve the SDP
Does the optimal solution satisfy the rank-constraint?
yes
W are done!
We
d
!
no
Solution
S
l ti may nott
be meaningful
SDP relaxation of OPF
min
 tr M W
k
kG
over
W positive semidefinite matrix
s.t.
P k  tr kW  P k
 k , k
Q k  tr  kW  Q k
 k , k
V  tr J kW  V
2
k
2
k
Lagrange
multipliers
 k, k
W 0
A  k ,  k ,  k  :=
 M    
k
kG
k
k
k
 k  k   k Jk 
Sufficient condition
Theorem
opt
A
If
has rank n-1 then
opt
 W has rank 1, SDP relaxation is exact
 Duality gap is zero
 A globally optimal V opt can be recovered
IEEE test systems (essentially) satisfy the
condition!
diti !
J. Lavaei and S. H. Low: Zero duality gap in optimal power flow problem.
Allerton 2010, TPS 2011
OPF over radial networks
Suppose
 tree (radial) network
 no lower bounds on power injections
Theorem
Aopt always has rank n-1
opt
always has rank 1 (exact
relaxation))
opt duality gap
 OPF always has zero
V
 Globally optimal
solvable efficiently

W
S. Bose, D. Gayme, S. H. Low and M. Chandy, OPF over tree networks.
Allerton 2011
OPF over radial networks
Suppose
 tree (radial) network
 no lower bounds on power injections
Theorem
Aopt always has rank n-1
opt
always has rank 1 (exact
relaxation))
opt duality gap
 OPF always has zero
V
 Globally optimal
solvable efficiently

W
Also: B. Zhang and D. Tse, Allerton 2011
S. Sojoudi and J. Lavaei, submitted 2011
QCQP over tree
QCQP C, Ck 
min
i
x *Cx
C
over
x  Cn
s.t.
x *Ck x  bk
kK
graph of QCQP
G C,Ck  has edge (i, j) 
Cij  0 or Ck ij  0 for some k
QCQP over tree
G C,Ck  is a tree
QCQP over tree
QCQP C, Ck 
min
i
x *Cx
C
over
x  Cn
s.t.
x *Ck x  bk
kK
Semidefinite relaxation
min
tr CW
over
W 0
s. t.
tr CkW  bk
kK
QCQP over tree
QCQP C, Ck 
min
i
x *Cx
C
over
x  Cn
s.t.
x *Ck x  bk
Key assumption
kK

(i, j)  G C,Ck  : 0  int conv hull Cij , Ck ij , k

Theorem
Semidefinite relaxation is exact for
S. Bose, D. Gayme, S. H. Low and
QCQP over tree
M. Chandy, submitted March 2012
OPF over radial networks
“no lower bounds”
removes these Ck ij
Theorem
opt
always has rank n-1
opt
 W always has rank 1 (exact
relaxation))
opt
 OPF always has zero
V duality gap
 Globally optimal
solvable efficiently
A
OPF over radial networks
bounds on constraints
remove these Ck ij
Theorem
opt
always has rank n-1
opt
 W always has rank 1 (exact
relaxation))
opt
 OPF always has zero
V duality gap
 Globally optimal
solvable efficiently
A
Outline
Motivation
Semidefinite relaxation
 Bus injection model
Conic relaxation
 Branch flow model
Model
s gj s cj
k
j
i
power
i j ti Sij
injection
i
Vi
S jkk
zij   rij , xij 
I ij
j
Vj
radial (tree) network
k
Model
s gj s cj
k
j
i
power
i j ti Sij
injection
Kirchoff’s Law: Sij 
S jkk
S
k:j~k
2
jk
 zij I ij  s  s
line
loss
c
j
g
j
load ‐ gen
Model
s gj s cj
power
i j ti Sij
injection
Kirchoff’s Law: Sij 
k
j
i
S jkk
S
2
jk
 zij I ij  s  s
c
j
g
j
k:j~k
Oh ’ L
Ohm’s Law:
Vj  Vi  zij Iij
Sij  V I
*
i ij
min
OPF
 r l   v
ij ij
i~ j
i i
i
g
over ((P,Q,
,Q, v,, q , l))
lij : I ij
vi : Vi
2
2

c
g
real power loss
CVR (conservation
s. t. Pij 
Pjk  rij lij  p j  p j
voltage reduction)
voltage reduction) k:j~k
Qij 
c
g
Q

x
l

q

q
 jk ij ij j j
k: j~k
vi  v j  2  rij Pij  xijQij    r  x  lij
2
ij
 Pij2  Qij2 
lij  

 vi 
 qi  q gj  qi
2
ij
OPF using branch flow model min
 r l   v
ij ij
i~ j
i i
i
g
c
over (S,
(S I,V,
I V s ,s )
s. t.
s  sig  si g
g
i
si  sic
vi  vi  vi
Kirchoff’s Law: Sij 
S
2
jk
 zij I ij  s  s
c
j
g
j
k:j~k
Oh ’ L
Ohm’s Law:
Vj  Vi  zij Iij
Sij  V I
*
i ij
Solution strategy
OPF
branch flow model
nonconvex
exactt
for tree
Angle relaxation
eliminate phases
nonconvex
SOCP relaxation
convex program
always
exact
1. Angle relaxation
Angles of Ii , Vi eliminated !
Points relaxed to circles
l d
l
demands
c
g
P

r
I

p

p
 jk ij ij j j
2
Pij 
k: j~k
Qij 
c
g
Q

x
I

q

q
 jk ij ij j j
2
k:j~k
Vi  Vj  2  rij Pij  xijQij    r  x
2
I ij
2
2
 P 2  Q2 
ij
ij



2

 Vi

2
ij
2
ij
I
2
ij
Baran and Wu 1989
for radial networks
1. Angle relaxation
Angles of Ii , Vi eliminated !
Points relaxed to circles
l d
l
c
g
P

r
I

p

p
 jk ij ij j j
2
Pij 
generation
k: j~k
Qij 
c
g
Q

x
I

q

q
 jk ij ij j j
2
VAR control
k:j~k
Vi  Vj  2  rij Pij  xijQij    r  x
2
I ij
2
2
 P 2  Q2 
ij
ij



2

 Vi

2
ij
s  s  si
v i  vi  vi
g
i
g
i
g
2
ij
I
2
ij
si  s
c
i
1. Angle
relaxation
l ti
min
 r l   v
ijj ijj
i~ j
i i
i
over (S, l, v, s g , s c )
lij : I ij
vi : Vi
2
s. t. Pij 
c
g
P

r
l

p

p
 jk ij ij j j
k:j~k
2
Qij 
Q
jk
 xij lij  q  q
c
j
g
j
k:j~k
• Linear objective
• Linear constraints
• Quadratic equality
vi  v j  2  rij Pij  xijQij    rij2  xij2  lij
 Pij2  Qij2 
lij  
,
 vi 
s i  s  si
v i  vi  vi ,
si  si
g
i
c
2. SOCP
relaxation
l ti
min
 r l   v
ij ij
i~ j
i i
i
over ((S,, l,, v,, s g , s c )
s. t. Pij 
c
g
P

r
l

p

p
 jk ij ij j j
k:j~k
Qij 
Q
jk
 xij lij  q  q
c
j
g
j
k: j~k
vi  v j  2  rij Pij  xijQij    rij2  xij2  lij
Quadratic inequality
 Pij22  Qij22 
ij
lijij 
  ij
,
 vii 
si  s  si
v i  vi  vi ,
si  s
g
i
c
i
OPF over radial networks
OPF over radial
Theorem
Both relaxation steps are exact
Both relaxation steps are exact
– SOCP relaxation is (convex and) exact
– Phase angles can be uniquely determined
Original OPF problem has zero duality gap
M. Farivar, C. Clarke, S. H. Low and M. Chandy, Inverter VAR control for
distribution systems with renewables. SmartGridComm 2011
What about mesh networks ??
M. Farivar and S. H. Low, submitted March 2012
Solution strategy
Solution strategy
OPF
branch flow model
nonconvex
exactt
for tree
Angle relaxation
eliminate phases
nonconvex
SOCP relaxation
convex program
always
exact
??
OPF using branch flow model min
 r l   v
ij ij
i~ j
i i
i
g
c
over (S,
(S I,V,
I V s ,s )
s. t.
s  sig  si g
g
i
si  sic
vi  vi  vi
Kirchoff’s Law: Sij 
S
2
jk
 zij I ij  s  s
c
j
g
j
k:j~k
Oh ’ L
Ohm’s Law:
Vj  Vi  zij Iij
Sij  V I
*
i ij
Convexification of mesh networks
 
s.t.
x  X, s  S
 
st
s.t.
x  Y,
Y sS
 
s.t.
x  X, s  S
OPF
min f ĥ(x)
OPF ar
OPF-ar
min f h(x)
ĥ(x)
OPF-ps
min f ĥ(x)
x,s
x,s
x,s,
Theorem
Th
• XY
• Need
N d phase
h
shifters
hift
only
l
outside spanning tree
X
X
X
Y
Convexification of mesh networks
 
s.t.
xX
min f h(x)
ĥ(x)
 
st
s.t.
xY
 
s.t.
xX
OPF
min f ĥ(x)
OPF ar
OPF-ar
OPF-ps
min f ĥ(x)
x,s
x,s
x,s,
Theorem
Th
• XY
• Need
N d phase
h
shifters
hift
only
l
outside spanning tree
X
X
X
Y
Key message
Radial networks computationally simple
 Exploit tree graph & convex relaxation
 Real-time scalable control promising
Mesh networks can be convexified
 Design
g for simplicity
p
y
 Need few phase shifters (sparse topology)