Optimal Power Flow
over Radial Networks
Steven Low
Computing + Math Sciences
Electrical Engineering
Caltech
Keynote, SmartGrid Workshop, Globecom
December 2011
Outline
Motivation
Renewable generation in distribution networks
OPF over radial networks
Caltech: S. Bose, M. Chandy, M. Farivar, D.
Gayme, J. Lavaei
SCE: C. Clarke, M. Montoya, R. Neal, R. Sherick
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
Challenge: uncertainty mgt
High Levels of Wind and Solar PV Will
Present an Operating Challenge!
Source: Rosa Yang, EPRI
Large-scale active network of
DER
DER: PVs, wind turbines, batteries, EVs, DR loads
Large-scale active network of
DER
Millions of active
endpoints introducing
rapid large
random fluctuations
in supply and demand
DER: PVs, wind turbines, batteries, EVs, DR loads
Volt/VAR control
Motivation: voltage support
Static capacitor control cannot cope with rapid
random fluctuations of PVs on distr circuits
Inverter control
Much faster & more frequent
IEEE 1547 does not optimize
VAR (unity PF)
Two timescale control
This talk: focuses on
inverter control
8
Outline
Motivation
Renewable generation in distribution networks
OPF over radial networks
OPF: nonconvex optimization
SDP relaxation
SOCP relaxation
Application: Volt/VAR control
Optimal power flow (OPF)
OPF is solved routinely to determine
How much power to generate where
Market operation & pricing
Parameter setting, e.g. taps, VARs
Non-convex and hard to solve
Huge literature since 1962
In practice, operators often use heuristics to
find a feasible operating point
Or solve DC power flow (LP)
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:
Bai et al 2008, 2009, Lavaei et al 2010
Bose et al 2011, Farivar et al 2011, Sojoudi
et al 2011, Zhang 2011
Lesieutre et al 2011
Network model
Nodes i and j are linked with an
admittance
Line: resistive and inductive
I: Vector of injection currents
Y:Admittance
matrix
Kirchhoff's
Law:
V: Vector of voltages
Classical OPF
min
g
f
P
∑ k( k )
Generation cost
k∈G
over
subject to
g
g
P
,Q
( k k , Vk )
g
k
g
k
g
k
g
k
g
P ≤P ≤ P
g
Generation power constraints
Qk ≤ Q ≤ Qk
V k ≤|Vk | ≤ V k
KVL/KCL
power balance
Voltage magnitude constraints
Classical OPF
In terms of V:
I k = (YV ) k = ekT YV
Pk = Pkg − Pkd
Sk :=Pk + iQk = Vk I k*
Qk = Qkg − Qkd
*
= e V ( e YV ) = ekT VV *Y *ek
T
k
T
k
= tr (Y *ek ekT ) VV *
1
424
3
Yk*
*
*




* *
* Yk +Yk
* Yk −Yk
= V Yk V = V 
V + iV 
V
2 
2i 


144244
3 144244
3
Pk
Qk
Classical OPF
In terms of V:
Pk = tr Φ kVV
*
Qk = tr Ψ kVV
min
k∈G
*
 Yk* +Yk 
Φk : = 

 2 
 Yk* −Yk 
Ψk : =

 2i 
d
f
P
+
P
∑ k( k k )
over
s.t.
V
g
k
g
k
P − P ≤ tr Φ kVV ≤ P − Pkd
g
d
k
*
d
k
g
*
d
k
Q k − Q ≤ tr Ψ kVV ≤ Q k − Q
2
k
*
V ≤ tr J kVV ≤ V
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 ΦkW ≤ P k
Q k ≤ tr Ψ kW ≤ Q k
2
k
2
k
V ≤ tr J kW ≤ V
W ≥ 0, rank W =1
convex relaxation: SDP
Semi-definite relaxation
Non-convex QCQP
Rank-constrained SDP
Relax the rank constraint and solve the SDP
Does the optimal solution satisfy the rank-constraint?
yes
We are done!
no
Solution may not
be meaningful
SDP relaxation of OPF
min
∑ tr M W
k
k∈G
over
W positive semidefinite matrix
s.t.
P k ≤ tr ΦkW ≤ P k
λ k , λk
Q k ≤ tr Ψ kW ≤ Q k
µ k , µk
2
k
V ≤ tr J kW ≤ V
2
k
Lagrange
multipliers
γ k,γk
W ≥0
A ( λk , µ k , γ k ) :=
∑ M + ∑( λ Φ
k
k∈G
k
k
k
+ µk Ψ k + γ k Jk )
Sufficient condition
Theorem
opt
A
If
has rank n-1 then
opt
W has rank 1
Duality gap is zero
A globally optimal V opt is found
All IEEE test systems (essentially) satisfy the
condition!
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
opt
always has rank n-1
W opt always has rank 1
OPF always has zero duality gap
Globally optimal V opt solvable efficiently
A
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
opt
always has rank n-1
W opt always has rank 1
OPF always has zero duality gap
Globally optimal V opt solvable efficiently
A
Also: B. Zhang and D. Tse, Allerton 2011
S. Sojoudi and J. Lavaei, submitted 2011
Outline
Motivation
Renewable generation in distribution networks
OPF over radial networks
OPF: nonconvex optimization
SDP relaxation
SOCP relaxation
Application: Volt/VAR control
Model
s gj s cj
k
j
i
power
injection Sij
i
Vi
S jk
zij = ( rij , xij )
I ij
j
Vj
radial (tree) network
k
Model
s gj s cj
k
j
i
power
injection Sij
Kirchoff’s Law: Sij =
S jk
∑S
k:j~k
2
jk
c
j
+ zij I ij + s − s
line
loss
g
j
load - gen
Model
s gj s cj
power
injection Sij
Kirchoff’s Law: Sij =
k
j
i
S jk
∑S
2
jk
c
j
+ zij I ij + s − s
g
j
k:j~k
Ohm’s Law: Vj = Vi − zij I ij
*
i ij
Sij = V I
Model
s gj s cj
k
j
i
S jk
Sij
∑P
Pij =
jk
2
c
j
+ rij I ij + p − p
given load
g
j
k:j~k
Qij =
c
g
Q
+
x
I
+
q
−
q
∑ jk ij ij j j
2
k:j~k
2
2
Vi = Vj + 2 ( rij Pij + xijQij ) − ( r + x
I ij
2
 P 2 + Q2 
ij
ij


=
2


 Vi
2
ij
2
ij
)I
2
ij
Baran and Wu 1989
for radial networks
Model
s gj s cj
k
j
i
Each node
described
by just Sij, |Vi|2
S jk
Sij
∑P
Pij =
jk
2
c
j
+ rij I ij + p − p
g
j
Given DG
k:j~k
Qij =
c
g
Q
+
x
I
+
q
−
q
∑ jk ij ij j j
2
VAR control
k:j~k
2
2
Vi = Vj + 2 ( rij Pij + xijQij ) − ( r + x
I ij
2
 P 2 + Q2 
ij
ij


=
2


 Vi
2
ij
2
ij
)I
2
ij
g
j
−qi ≤ q ≤ qi
Optimal
VAR control
min
∑ r l + ∑α v
ij ij
i~ j
i i
i
g
over (P,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)
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 + xij Qij ) − ( r + x ) lij
2
ij
 Pij2 + Qij2 
lij = 

 vi 
− qi ≤ q gj ≤ qi
2
ij
Optimal
VAR control
min
∑ r l + ∑α v
ij ij
i~ j
i i
i
g
over (P,Q, v, q , l)
lij := I ij
vi := Vi
2
s. t.
Pij =
∑P
jk
c
j
+ rij lij + p − p
g
j
k:j~k
2
Qij =
c
g
Q
+
x
l
+
q
−
q
∑ jk ij ij j j
k:j~k
• Linear objective
• Linear equality
• Quadratic equality
vi = v j + 2 ( rij Pij + xij Qij ) − ( r + x ) lij
2
ij
 Pij2 + Qij2 
lij = 

 vi 
− qi ≤ q gj ≤ qi
vi ≤ vi ≤ vi
2
ij
SOCP
relaxation
min
∑ r l + ∑α v
ij ij
i~ j
i i
i
g
over (P,Q, v, q , l)
s. t.
Pij =
∑P
jk
c
j
+ rij lij + p − p
g
j
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 + xij Qij ) − ( r + x ) lij
2
ij
Quadratic inequality
 Pij2 + Qijij22 
≥ 
lij =

 vii 
− qi ≤ q gj ≤ qi
vi ≤ vi ≤ vi
2
ij
Exact convex relaxation
Theorem
• SOCP relaxation is (convex and) exact
• 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
Outline
Motivation
– Uncertainty and large scale
OPF over radial networks
– OPF: nonconvex optimization
– SDP relaxation
– SOCP relaxation
– Application: Volt/VAR control
SCE Calabash circuit
Ontario, CA
Load and Solar Variation
pic
g
i
p
Empirical distribution
of (load, solar) for Calabash
Improved reliability
(p , p )
g
i
c
i
for which problem is feasible
Implication: reduced likelihood of violating
voltage limits or VAR flow constraints
Energy savings
Summary
• More reliable operation
• Energy savings
M. Farivar, C. Clarke, S. H. Low and M. Chandy, Inverter VAR control for
distribution systems with renewables. SmartGridComm 2011
Backup Slides
Source: Steven Chu, GridWeek 2009
Features to capture
Wholesale markets
Day ahead, real-time balancing
Renewable generation
Non-dispatchable
Demand response
Real-time control (through pricing)
day ahead
balancing
renewable
utility
utility
users
users
Objective
Day-ahead decision
How much power
ahead market?
Pd should LSE buy from day-
Real-time decision (at t-)
How much x i should users consume, given
realization of wind power Pr and Pd ?
How to compute these decisions distributively?
How does closed-loop system behave ?
available info:
decision:
t – 24hrs
t-
ui (⋅), Fr (⋅)
Pr , Pd*
Pd*
x i*
Objective
Real-time (at
t-)
Given Pd and realizations of Pr , choose
optimal xi* = xi* ( Pd ; Pr ) to max social welfare
Day-ahead
*
P
Choose optimal d that maximizes expected
optimal social welfare
available info:
decision:
t – 24hrs
t-
ui (⋅), Fr (⋅)
Pr , Pd*
Pd*
x i*
Motivation: recap
Optimal power flow
[Bose, Gayme, L, Chandy]
Motivation: core of grid/market operation, but slow
& inefficient computation
Result: zero duality gap for radial networks
Impact: much faster and more efficient algorithm for
global optimality to cope with renewable fluctuations
Volt/VAR control
[Farivar, L, Clarke, Chandy]
Motivation: static capacitor-based control cannot
cope with rapid random fluctuations of renewables
Result: optimal real-time inverter-based feedback
control
Impact: more reliable and efficient distribution
network at high renewable penetration
Sample: optimal power flow
Motivation
Core of grid/market operation, but slow &
inefficient (NP-hard) computation
Result
Prove conditions that enable convex
relaxation
Fast relaxation algorithm for optimality
Surprising: all IEEE benchmark systems
satisfy our conditions
Impact
Can enable real-time control to cope with
renewable fluctuations
Sample: Volt/VAR control
Motivation
Static capacitor control cannot
cope with rapid random
fluctuations of renewables
Result
Radial network admits convex
relaxation
Optimal phase shifter placement
Optimal real-time inverter based
VAR control
Impact
More reliable and efficient
distribution network