Branch Flow Model · jk, ` jk) and ( S kj, ` kj) are GH¿QHG for both directions, with the...
Transcript of Branch Flow Model · jk, ` jk) and ( S kj, ` kj) are GH¿QHG for both directions, with the...
Branch Flow Model
Masoud Farivar Steven Low
Computing + Math Sciences
Electrical Engineering
Arpil 2014
TPS paper
Farivar and Low
Branch flow model: relaxations and convexification
IEEE Trans. Power Systems, 28(3), Aug. 2013
This talk will only motivate why branch flow model
Outline
High-level summary
Branch flow model (BFM)
Advantages
BFM for radial networks
Equivalence
Recursive structure
Linearization and bounds
Application: OPF and SOCP relaxation
Branch flow model
i j k
s j
zij = yij-1
graph model G: directed
Vi -Vj = zijIij for all i® j
Branch flow model
power definition
power balance
s j
Ohm’s law
Sij : branch power
Iij : branch current
Vj : voltage
Sij =ViIij* for all i® j
Sij - zij Iij2
( )i® j
å + s j = S jkj®k
å for all j
loss
sending
end pwr
sending
end pwr
Bus injection model
I =YV
s j =VjI j* for all j
admittance matrix:
Yij :=
yikk~i
å if i = j
-yij if i ~ j
0 else
ì
í
ïï
î
ïï
I j : nodal current
Vj : voltage
s j
power balance
Ohm’s law
Vi -Vj = zijIij
Recap
Branch flow model Bus injection model
Sij =ViIij*
S jkj®k
å = Sij - zij Iij2
( )i® j
å + s j
s j = y jkH
k:k~ j
å Vj Vj -Vk( )H
X
(S, I,V, s) ÎC2(m+n+1)
V
(V, s) ÎC2(n+1)
solution
set
Advantages of BFM
It models directly branch power and current flows
Easier to use for certain applications
e.g. line limits, cascading failures, network of FACTS
Much more numerically stable for large-scale computation
Advantages of BFM
Recursive structure for radial networks [BaranWu1989]
Simplifies power flow computation
Forward-backward sweep is very fast and numerically stable
Linearized model for radial networks
Much more useful than DC approx. for distribution systems
Provide simple bounds on branch powers and voltages
Comparison of linearized models
Linear DistFlow
Includes reactive power and voltage magnitudes
useful for volt/var control and optimization
Explicit expression in terms of injections
Provides simple bounds to nonlinear BFM vars
Applicable only for radial networks
DC power flow
Ignores reactive power and fixes voltage magnitudes
Unclear relation with nonlinear BIM vars
Applicable for mesh networks as well
Outline
High-level summary
Branch flow model (BFM)
Advantages of BFM
BFM for radial networks
Equivalence
Recursive structure
Linearization and bounds
Application: OPF & SOCP relaxation
Vi -Vj = zijIij
Relaxed BFM
Relaxed model Branch flow model
S jkj®k
å = Sij - zij Iij2
( )i® j
å + s j
X
(S, I,V, s) ÎC2(m+n+1)
Pjkj®k
å = Pij - rij Iij2
( )i® j
å + p j
Qjk
j®k
å = Qij - xij Iij2
( )i® j
å + q j
ViIij* = Sij
Vi -Vj = zijIij
Relaxed BFM
Relaxed model Branch flow model
ViIij* = Sij
S jkj®k
å = Sij - zij Iij2
( )i® j
å + s j
X
(S, I,V, s) ÎC2(m+n+1)
S jkj®k
å = Sij - zij ij( )i® j
å + s j
vi - v j = 2 Re zij*Sij( ) - zij
2
ij
vi ij = Sij2
ij := Iij2
vi := Vi2
Xnc
(S, ,v, s) ÎR3(m+n+1)
these solution sets are
generally not equivalent
Branch flow model
Relaxed model Branch flow model
IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, 2014, TO APPEAR 9
h
h-1
C2(m+n+1)
R3(m+n+1)
Xnc
X X
X+
Fig. 1: Feasible sets X of OPF (9) in BFM, its equivalent set
X (defined by h) and its relaxations Xnc and X+ . If G is a tree
then X = Xnc.
that OPF (9) is equivalent to minimization over X and OPF-
socp is its SOCP relaxation. Moreover for radial networks
voltage and current angles can be ignored and OPF (9) is
equivalent to OPF-nc.
Theorem 9: X ⌘X ✓ Xnc✓ X+ . If G is a tree then X ⌘X =
Xnc ✓ X+ .
Let Copt be the optimal cost of OPF (9) in the branch
flow model. Let Copf , Cnc, Csocp be the optimal costs of OPF
(30), OPF-nc (31), OPF-socp (32) respectively defined above.
Theorem 9 implies
Corollary 10: Copt = Copf ≥ Cnc ≥ Csocp. If G is a tree then
Copt = Copf = Cnc ≥ Csocp.
Remark 8: SOCP relaxation. Suppose one solves OPF-socp
and obtains an optimal solution xsocp := (S,`,v,s) 2 X+ . For
radial networks if xsocp attains equality in (29) then xsocp 2Xnc and Theorem 9 implies that an optimal solution xopt :=
(S, I ,V,s) 2 X of OPF (9) can be recovered from xsocp. Indeed
xopt = h− 1(xsocp) where h− 1 is defined in (27). Alternatively
one can use the angle recovery algorithms in [57, Part I] to
recover xopt. For mesh networks xsocp needs to both attain
equality in (29) and satisfy the cycle condition (26) in order
for an optimal solution xopt to be recoverable. Our experience
with various practical test networks suggests that xsocp usually
attains equality in (29) but, for mesh networks, rarely satisfes
(26) [51], [57], [56], [28]. Hence OPF-socp is effective for
radial networks but not for mesh networks (in both BIM and
BFM).
C. Equivalence
Theorem 9 establishes a bijection between X and the feasible
set X of OPF (9) in BFM. Theorem 4 establishes a bijection
between WG and the feasible set V of OPF (7) in BIM.
Theorem 1 hence implies that X ⌘ X ⌘ V ⌘ WG. Moreover
their SOCP relaxations are equivalent in these two models [58],
[28]. Define the set of partial matrices defined on G that are
2⇥ 2 psd rank-1 but do not satisfy the cycle condition (13):
Wnc := {Wnc | WG satisfies (12),WG( j ,k) ⌫0,
rank WG( j ,k) = 1 for all ( j ,k) 2 E}
Clearly WG✓Wnc✓W+G in general and WG = Wnc✓W+
G for
radial networks.
Theorem 11: X ⌘WG, Xnc ⌘Wnc and X+ ⌘W+G.
The bijection between X+ and W+G is defined as follows. Let
WG✓C2m+ n+ 1 denote the set of Hermitian partial matrices (in-
cluding [WG]00 = v0 which is given). Let x := (S,`,v,s) denote
vectors in R3(m+ n+ 1) . Define the linear mapping g : WG ! X+
by x = g(WG) where
Sjk := yHjk [WG] j j − [WG] jk , j ! k
` jk := |y jk|2 [WG] j j + [WG]kk− [WG] jk− [WG]k j , j ! k
v j := [WG] j j , j 2 N+
sj := Âk: j⇠k
yHjk [WG] j j − [WG] jk , j 2 N+
Its inverse g− 1 : X+ ! WG is WG = g− 1(x) where [WG] j j := v j
for j 2 N+ and [WG] jk := v j − zHjkSjk = : [WG]Hkj for j ! k. The
mapping g and its inverse g− 1 restricted to WG (Wnc) and X
(Xnc) define the bijection between them.
VI. BFM FOR RADIAL NETWORKS
Theorem 9 implies that for radial networks the model (24) is
exact. This is because the reduced incident matrix B in (26) is
n⇥ n and invertible, so the cycle condition is always satisfied
[57, Theorem 4]. Hence a solution in Xnc can be mapped to a
branch flow solution in X by the mapping h− 1 defined in (27).
For radial networks this model has two advantages: (i) it has
a recursive structure that simplifies computation, and (ii) it has
a linear approximation that provides simple bounds on branch
powers Sjk and voltage magnitudes v j , as we now show.
A. Recursive equations and graph orientation
The model (24) holds for any graph orientation of G. It has
a recursive structure when G is a tree. In that case different
orientations have different boundary conditions that initialize
the recursion and may be convenient for different applications.
Without loss of generality we take bus 0 as the root of the
tree. We discuss two different orientations: one where every
link points away from bus 0 and the other where every link
points towards bus 0. 6
Case I: Links point away from bus 0. Model (24) reduces to:
Âk: j ! k
Sjk = Si j − zi j ` i j + sj , j 2 N+ (33a)
v j − vk = 2Re zHjkSjk − |zjk|2` jk, j ! k 2 E (33b)
v j ` jk = |Sjk|2, j ! k 2 E (33c)
where bus i in (33a) denotes the unique parent of node j (on
the unique path from node 0 to node j), with the understanding
that if j = 0 then Si0 := 0 and ` i0 := 0. Similarly when j is a
leaf node7 all Sjk = 0 in (33a). The model (33) is called the
DistFlow equations and first proposed in [45], [46].
Its recursive structure is exploited in [47] to analyze the
power flow solutions given an (sj , j 2 N), as we now explain
using the special case of a linear network with n+ 1 buses
that represents a main feeder. To simplify notation denote
6An alternativemodel is to usean undirected graph and, for each link ( j ,k),the variables (Sjk, ` jk) and (Skj , `kj ) are defined for both directions, with theadditional equations Sjk + Skj = zjk` jk and `kj = ` jk.
7A node j is a leaf node if there exists no i such that i ! j 2 E.
The relaxed model is in general different from BFM
… but they are equivalent for radial networks !
Xnc
X
equivalent model
of BFM
BFM for radial networks
S jkj®k
å = Sij - zij ij + s j
vi - v j = 2 Re zij*Sij( ) - zij
2
ij
ijvi = Sij2
DiskFlow equations Baran and Wu 1989 for radial networks
power flow solutions: satisfy x := S, ,v, s( )
ij := Iij2
vi := Vi2
Advantages
• Recursive structure
• Linearized model & bounds
BFM for radial networks
Baran and Wu 1989 for radial networks
Recursive structure allows very
fast & stable computation
initialization
BFM for radial networks
Baran and Wu 1989 for radial networks
Recursive structure allows very
fast & stable computation
initialization
BFM for radial networks
S jkj®k
å = Sij - zij ij + s j
vi - v j = 2 Re zij*Sij( ) - zij
2
ij
ijvi = Sij2
Linear DiskFlow Baran and Wu 1989 for radial networks
Accurate & versatile linearized model
ij := Iij2
vi := Vi2
Linearization: ignores line loss • reasonable if line loss is much smaller
than branch power
BFM for radial networks
Pijlin = - pk
kÎTj
å , Qij
lin = - qkkÎTj
å
vilin - v j
lin = 2 rijPijlin + xijQij
lin( )
Accurate & versatile linearized model
- skkÎTj
å
s j
Sijlin
BFM for radial networks
Pijlin = - pk
kÎTj
å , Qij
lin = - qkkÎTj
å
vilin - v j
lin = 2 rijPijlin + xijQij
lin( )
Accurate & versatile linearized model
linear functions of injections s
Comparison of linearized models
Linear DistFlow
Includes reactive power and voltage magnitudes
useful for volt/var control and optimization
Explicit expression in terms of injections
Provides simple bounds to nonlinear BFM vars
Applicable only for radial networks
DC power flow
Ignores reactive power and fixes voltage magnitudes
Unclear relation with nonlinear BIM vars
Applicable for mesh networks as well
Vi -Vj = zijIij
Relaxed BFM
Relaxed model Branch flow model
ViIij* = Sij
S jkj®k
å = Sij - zij Iij2
( )i® j
å + s j S jkj®k
å = Sij - zij ij( )i® j
å + s j
vi - v j = 2 Re zij*Sij( ) - zij
2
ij
vi ij = Sij2
Sij ³ Sijlin
vi £ vilin
Outline
High-level summary
Branch flow model (BFM)
Advantages of BFM
BFM for radial networks
Equivalence
Recursive structure
Linearization and bounds
Application: OPF & SOCP relaxation
OPF & relaxation: examples
With PS
min riji~ j
å Iij2
+ ai
i
å Vi2+ ciiÎG
å pig
over (S, I,V, sg, sc )
OPF: branch flow model
real power loss CVR (conservation voltage reduction)
generation cost
OPF: branch flow model
min f x( )
over x := (S, I,V, sg, sc )
s. t. sig
£ sig £ si
g si £ sic £ si vi £ vi £ vi
OPF: branch flow model
min f x( )
over x := (S, I,V, sg, sc )
s. t. sig
£ sig £ si
g sic£ si
c £ sic vi £ vi £ vi
Sij =ViIij*
Vj =Vi - zijIij
branch flow model
Sij - zij Iij2
( )i® j
å - S jkj®k
å = s jc - s j
g
OPF: branch flow model
generation, volt/var control
Branch flow model is more convenient for applications
min f x( )
over x := (S, I,V, sg, sc )
s. t. sig
£ sig £ si
g sic£ si
c £ sic vi £ vi £ vi
Sij =ViIij*
Vj =Vi - zijIij
branch flow model
Sij - zij Iij2
( )i® j
å - S jkj®k
å = s jc - s j
g
OPF: branch flow model
demand response
min f x( )
over x := (S, I,V, sg, sc )
s. t. sig
£ sig £ si
g sic£ si
c £ sic vi £ vi £ vi
Challenge: nonconvexity !
Vi -Vj = zijIij
Branch flow model
Relaxed model Branch flow model
S jkj®k
å = Sij - zij Iij2
( )i® j
å + s j
X
(S, I,V, s) ÎC2(m+n+1)
Pjkj®k
å = Pij - rij Iij2
( )i® j
å + p j
Qjk
j®k
å = Qij - xij Iij2
( )i® j
å + q j
ViIij* = Sij
Vi -Vj = zijIij
Branch flow model
Relaxed model Branch flow model
ViIij* = Sij
S jkj®k
å = Sij - zij Iij2
( )i® j
å + s j
X
(S, I,V, s) ÎC2(m+n+1)
S jkj®k
å = Sij - zij ij( )i® j
å + s j
vi - v j = 2 Re zij*Sij( ) - zij
2
ij
vi ij = Sij2
ij := Iij2
vi := Vi2
Xnc
(S, ,v, s) ÎR3(m+n+1)
these solution sets are
generally not equivalent
Branch flow model
S jkj®k
å = Sij - zij ij + s j
vi - v j = 2 Re zij*Sij( ) - zij
2
ij
ijvi = Sij2
Baran and Wu 1989 for radial networks
power flow solutions: satisfy x := S, ,v, s( )
ij := Iij2
vi := Vi2
Advantages
• Recursive structure (radial networks)
• Variables represent physical quantities
Branch flow model
Relaxed model Branch flow model
IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, 2014, TO APPEAR 9
h
h-1
C2(m+n+1)
R3(m+n+1)
Xnc
X X
X+
Fig. 1: Feasible sets X of OPF (9) in BFM, its equivalent set
X (defined by h) and its relaxations Xnc and X+ . If G is a tree
then X = Xnc.
that OPF (9) is equivalent to minimization over X and OPF-
socp is its SOCP relaxation. Moreover for radial networks
voltage and current angles can be ignored and OPF (9) is
equivalent to OPF-nc.
Theorem 9: X ⌘X ✓ Xnc✓ X+ . If G is a tree then X ⌘X =
Xnc ✓ X+ .
Let Copt be the optimal cost of OPF (9) in the branch
flow model. Let Copf , Cnc, Csocp be the optimal costs of OPF
(30), OPF-nc (31), OPF-socp (32) respectively defined above.
Theorem 9 implies
Corollary 10: Copt = Copf ≥ Cnc ≥ Csocp. If G is a tree then
Copt = Copf = Cnc ≥ Csocp.
Remark 8: SOCP relaxation. Suppose one solves OPF-socp
and obtains an optimal solution xsocp := (S,`,v,s) 2 X+ . For
radial networks if xsocp attains equality in (29) then xsocp 2Xnc and Theorem 9 implies that an optimal solution xopt :=
(S, I ,V,s) 2 X of OPF (9) can be recovered from xsocp. Indeed
xopt = h− 1(xsocp) where h− 1 is defined in (27). Alternatively
one can use the angle recovery algorithms in [57, Part I] to
recover xopt. For mesh networks xsocp needs to both attain
equality in (29) and satisfy the cycle condition (26) in order
for an optimal solution xopt to be recoverable. Our experience
with various practical test networks suggests that xsocp usually
attains equality in (29) but, for mesh networks, rarely satisfes
(26) [51], [57], [56], [28]. Hence OPF-socp is effective for
radial networks but not for mesh networks (in both BIM and
BFM).
C. Equivalence
Theorem 9 establishes a bijection between X and the feasible
set X of OPF (9) in BFM. Theorem 4 establishes a bijection
between WG and the feasible set V of OPF (7) in BIM.
Theorem 1 hence implies that X ⌘ X ⌘ V ⌘ WG. Moreover
their SOCP relaxations are equivalent in these two models [58],
[28]. Define the set of partial matrices defined on G that are
2⇥ 2 psd rank-1 but do not satisfy the cycle condition (13):
Wnc := {Wnc | WG satisfies (12),WG( j ,k) ⌫0,
rank WG( j ,k) = 1 for all ( j ,k) 2 E}
Clearly WG✓Wnc✓W+G in general and WG = Wnc✓W+
G for
radial networks.
Theorem 11: X ⌘WG, Xnc ⌘Wnc and X+ ⌘W+G.
The bijection between X+ and W+G is defined as follows. Let
WG✓C2m+ n+ 1 denote the set of Hermitian partial matrices (in-
cluding [WG]00 = v0 which is given). Let x := (S,`,v,s) denote
vectors in R3(m+ n+ 1) . Define the linear mapping g : WG ! X+
by x = g(WG) where
Sjk := yHjk [WG] j j − [WG] jk , j ! k
` jk := |y jk|2 [WG] j j + [WG]kk− [WG] jk− [WG]k j , j ! k
v j := [WG] j j , j 2 N+
sj := Âk: j⇠k
yHjk [WG] j j − [WG] jk , j 2 N+
Its inverse g− 1 : X+ ! WG is WG = g− 1(x) where [WG] j j := v j
for j 2 N+ and [WG] jk := v j − zHjkSjk = : [WG]Hkj for j ! k. The
mapping g and its inverse g− 1 restricted to WG (Wnc) and X
(Xnc) define the bijection between them.
VI. BFM FOR RADIAL NETWORKS
Theorem 9 implies that for radial networks the model (24) is
exact. This is because the reduced incident matrix B in (26) is
n⇥ n and invertible, so the cycle condition is always satisfied
[57, Theorem 4]. Hence a solution in Xnc can be mapped to a
branch flow solution in X by the mapping h− 1 defined in (27).
For radial networks this model has two advantages: (i) it has
a recursive structure that simplifies computation, and (ii) it has
a linear approximation that provides simple bounds on branch
powers Sjk and voltage magnitudes v j , as we now show.
A. Recursive equations and graph orientation
The model (24) holds for any graph orientation of G. It has
a recursive structure when G is a tree. In that case different
orientations have different boundary conditions that initialize
the recursion and may be convenient for different applications.
Without loss of generality we take bus 0 as the root of the
tree. We discuss two different orientations: one where every
link points away from bus 0 and the other where every link
points towards bus 0. 6
Case I: Links point away from bus 0. Model (24) reduces to:
Âk: j ! k
Sjk = Si j − zi j ` i j + sj , j 2 N+ (33a)
v j − vk = 2Re zHjkSjk − |zjk|2` jk, j ! k 2 E (33b)
v j ` jk = |Sjk|2, j ! k 2 E (33c)
where bus i in (33a) denotes the unique parent of node j (on
the unique path from node 0 to node j), with the understanding
that if j = 0 then Si0 := 0 and ` i0 := 0. Similarly when j is a
leaf node7 all Sjk = 0 in (33a). The model (33) is called the
DistFlow equations and first proposed in [45], [46].
Its recursive structure is exploited in [47] to analyze the
power flow solutions given an (sj , j 2 N), as we now explain
using the special case of a linear network with n+ 1 buses
that represents a main feeder. To simplify notation denote
6An alternativemodel is to usean undirected graph and, for each link ( j ,k),the variables (Sjk, ` jk) and (Skj , `kj ) are defined for both directions, with theadditional equations Sjk + Skj = zjk` jk and `kj = ` jk.
7A node j is a leaf node if there exists no i such that i ! j 2 E.
is nonconvex and effective superset of XXnc
restrict to get an
equivalent set
relax to get a
second-order
cone
Branch flow model
S jkk: j®k
å = Sij - zij ij( )i:i® j
å + s j
vi - v j = 2 Re zij*Sij( ) - zij
2
ij
ijvi = Sij2
Baran and Wu 1989 for radial networks
power flow solutions: satisfy x := S, ,v, s( )
ij := Iij2
vi := Vi2
nonconvexity
linear
Branch flow model
S jkk: j®k
å = Sij - zij ij( )i:i® j
å + s j
vi - v j = 2 Re zij*Sij( ) - zij
2
ij
ijvi ³ Sij2
Baran and Wu 1989 for radial networks
power flow solutions: satisfy x := S, ,v, s( )
ij := Iij2
vi := Vi2
second-order cone
linear
Cycle condition
A relaxed solution satisfies the cycle condition if
incidence matrix;
depends on topology
$q s.t. Bq = b(x) mod 2p
x
x := (S, ,v, s)
b jk (x) := Ð v j - z jkHS jk( )
Branch flow model
X :=x : satisfies linear
constraints
ìíî
üýþ
Ç jkv j = S2
cycle cond on x
ìíï
îï
üýï
þï
X+ :=
x : satisfies linear
constraints
ìíî
üýþ
Ç jkv j ³ S2
{ }
Theorem
second-order cone (convex) X ºXÍX+
relaxed solution: x := S, ,v, s( )
Branch flow model
Relaxed model Branch flow model
IEEE TRANS. ON CONTROL OF NETWORK SYSTEMS, 2014, TO APPEAR 9
h
h-1
C2(m+n+1)
R3(m+n+1)
Xnc
X X
X+
Fig. 1: Feasible sets X of OPF (9) in BFM, its equivalent set
X (defined by h) and its relaxations Xnc and X+ . If G is a tree
then X = Xnc.
that OPF (9) is equivalent to minimization over X and OPF-
socp is its SOCP relaxation. Moreover for radial networks
voltage and current angles can be ignored and OPF (9) is
equivalent to OPF-nc.
Theorem 9: X ⌘X ✓ Xnc✓ X+ . If G is a tree then X ⌘X =
Xnc ✓ X+ .
Let Copt be the optimal cost of OPF (9) in the branch
flow model. Let Copf , Cnc, Csocp be the optimal costs of OPF
(30), OPF-nc (31), OPF-socp (32) respectively defined above.
Theorem 9 implies
Corollary 10: Copt = Copf ≥ Cnc ≥ Csocp. If G is a tree then
Copt = Copf = Cnc ≥ Csocp.
Remark 8: SOCP relaxation. Suppose one solves OPF-socp
and obtains an optimal solution xsocp := (S,`,v,s) 2 X+ . For
radial networks if xsocp attains equality in (29) then xsocp 2Xnc and Theorem 9 implies that an optimal solution xopt :=
(S, I ,V,s) 2 X of OPF (9) can be recovered from xsocp. Indeed
xopt = h− 1(xsocp) where h− 1 is defined in (27). Alternatively
one can use the angle recovery algorithms in [57, Part I] to
recover xopt. For mesh networks xsocp needs to both attain
equality in (29) and satisfy the cycle condition (26) in order
for an optimal solution xopt to be recoverable. Our experience
with various practical test networks suggests that xsocp usually
attains equality in (29) but, for mesh networks, rarely satisfes
(26) [51], [57], [56], [28]. Hence OPF-socp is effective for
radial networks but not for mesh networks (in both BIM and
BFM).
C. Equivalence
Theorem 9 establishes a bijection between X and the feasible
set X of OPF (9) in BFM. Theorem 4 establishes a bijection
between WG and the feasible set V of OPF (7) in BIM.
Theorem 1 hence implies that X ⌘ X ⌘ V ⌘ WG. Moreover
their SOCP relaxations are equivalent in these two models [58],
[28]. Define the set of partial matrices defined on G that are
2⇥ 2 psd rank-1 but do not satisfy the cycle condition (13):
Wnc := {Wnc | WG satisfies (12),WG( j ,k) ⌫0,
rank WG( j ,k) = 1 for all ( j ,k) 2 E}
Clearly WG✓Wnc✓W+G in general and WG = Wnc✓W+
G for
radial networks.
Theorem 11: X ⌘WG, Xnc ⌘Wnc and X+ ⌘W+G.
The bijection between X+ and W+G is defined as follows. Let
WG✓C2m+ n+ 1 denote the set of Hermitian partial matrices (in-
cluding [WG]00 = v0 which is given). Let x := (S,`,v,s) denote
vectors in R3(m+ n+ 1) . Define the linear mapping g : WG ! X+
by x = g(WG) where
Sjk := yHjk [WG] j j − [WG] jk , j ! k
` jk := |y jk|2 [WG] j j + [WG]kk− [WG] jk− [WG]k j , j ! k
v j := [WG] j j , j 2 N+
sj := Âk: j⇠k
yHjk [WG] j j − [WG] jk , j 2 N+
Its inverse g− 1 : X+ ! WG is WG = g− 1(x) where [WG] j j := v j
for j 2 N+ and [WG] jk := v j − zHjkSjk = : [WG]Hkj for j ! k. The
mapping g and its inverse g− 1 restricted to WG (Wnc) and X
(Xnc) define the bijection between them.
VI. BFM FOR RADIAL NETWORKS
Theorem 9 implies that for radial networks the model (24) is
exact. This is because the reduced incident matrix B in (26) is
n⇥ n and invertible, so the cycle condition is always satisfied
[57, Theorem 4]. Hence a solution in Xnc can be mapped to a
branch flow solution in X by the mapping h− 1 defined in (27).
For radial networks this model has two advantages: (i) it has
a recursive structure that simplifies computation, and (ii) it has
a linear approximation that provides simple bounds on branch
powers Sjk and voltage magnitudes v j , as we now show.
A. Recursive equations and graph orientation
The model (24) holds for any graph orientation of G. It has
a recursive structure when G is a tree. In that case different
orientations have different boundary conditions that initialize
the recursion and may be convenient for different applications.
Without loss of generality we take bus 0 as the root of the
tree. We discuss two different orientations: one where every
link points away from bus 0 and the other where every link
points towards bus 0. 6
Case I: Links point away from bus 0. Model (24) reduces to:
Âk: j ! k
Sjk = Si j − zi j ` i j + sj , j 2 N+ (33a)
v j − vk = 2Re zHjkSjk − |zjk|2` jk, j ! k 2 E (33b)
v j ` jk = |Sjk|2, j ! k 2 E (33c)
where bus i in (33a) denotes the unique parent of node j (on
the unique path from node 0 to node j), with the understanding
that if j = 0 then Si0 := 0 and ` i0 := 0. Similarly when j is a
leaf node7 all Sjk = 0 in (33a). The model (33) is called the
DistFlow equations and first proposed in [45], [46].
Its recursive structure is exploited in [47] to analyze the
power flow solutions given an (sj , j 2 N), as we now explain
using the special case of a linear network with n+ 1 buses
that represents a main feeder. To simplify notation denote
6An alternativemodel is to usean undirected graph and, for each link ( j ,k),the variables (Sjk, ` jk) and (Skj , `kj ) are defined for both directions, with theadditional equations Sjk + Skj = zjk` jk and `kj = ` jk.
7A node j is a leaf node if there exists no i such that i ! j 2 E.
Branch flow model
X :=x : satisfies linear
constraints
ìíî
üýþ
Ç jkv j = S2
cycle cond on x
ìíï
îï
üýï
þï
relaxed solution: x := S, ,v, s( )
X+ :=
x : satisfies linear
constraints
ìíî
üýþ
Ç jkv j ³ S2
{ }
Theorem
For radial network, X ºX ºXnc Í X+
Branch flow model
X :=x : satisfies linear
constraints
ìíî
üýþ
Ç jkv j = S2
cycle cond on x
ìíï
îï
üýï
þï
power flow solutions: x := S, ,v, s( )
X+ :=
x : satisfies linear
constraints
ìíî
üýþ
Ç jkv j ³ S2
{ }
OPF: minxÎX
f x( ) SOCP: minxÎX+
f x( )
OPF-socp
OPF solution
Recover V* cycle condition
Y
rank-1
OPF-ch OPF-sdp
Y
WG
* Wc(G )
*W *
Y, mesh
2x2 rank-1
Y radial
OPF-socp
cycle condition Y
x*
equality
Y radial
Y, mesh
OPF-socp
OPF solution
Recover V* cycle condition
Y
rank-1
OPF-ch OPF-sdp
Y
WG
* Wc(G )
*W *
Y, mesh
2x2 rank-1
Y radial
OPF-socp
cycle condition Y
x*
equality
Y radial
Y, mesh
OPF-socp
OPF solution
Recover V* cycle condition
Y
rank-1
OPF-ch OPF-sdp
Y
WG
* Wc(G )
*W *
Y, mesh
2x2 rank-1
Y radial
OPF-socp
cycle condition Y
x*
equality
Y radial
Y, mesh
Exact relaxation
Definition
A relaxation is exact if an optimal solution of the original OPF can be recovered from every optimal solution of the relaxation
OPF-socp
OPF solution
Recover V* cycle condition
Y
rank-1
OPF-ch OPF-sdp
Y
WG
* Wc(G )
*W *
Y, mesh
2x2 rank-1
Y radial
OPF-socp
cycle condition Y
x*
equality
Y radial
Y, mesh
Definition
Every optimal matrix
or partial matrix is
(2x2) rank-1
Definition
Every optimal relaxed
solution attains equality
1. QCQP over tree
graph of QCQP
G C,Ck( ) has edge (i, j) Û
Cij ¹ 0 or Ck[ ]ij¹ 0 for some k
QCQP
QCQP over tree
G C,Ck( ) is a tree
min x*Cx
over x ÎCn
s.t. x*Ckx £ bk k Î K
C,Ck( )
1. Linear separability
min x*Cx
over x ÎCn
s.t. x*Ckx £ bk k Î K
Key condition
i ~ j : Cij, Ck[ ]ij, "k( ) lie on half-plane through 0
QCQP C,Ck( )
Theorem
SOCP relaxation is exact for
QCQP over tree
Re
Im
1. Linear separability
“no lower bounds”
removes these Ck[ ]ij
sufficient cond
remove these Ck[ ]ij
1. Linear separability
sufficient cond
remove these Ck[ ]ij
1. Linear separability
Outline
Radial networks
3 sufficient conditions
Mesh networks
with phase shifters
Phase shifter
ideal phase shifter
11
3) For each link (j , k) ∈ E \ ET not in the spanning
tree, node j is an additional parent of k in addition
to k’s parent in the spanning tree from which ∠Vk
has already been computed in Step 2.
a) Compute current angle ∠I j k using (39).
b) Compute a new voltage angle θjk using the new
parent j and (40). If θjk = ∠Vk , then angle
recovery has failed and (S, , v, s0) is spurious.
If the angle recovery procedure succeeds in Step 3, then
(S, , v, s0) together with these angles ∠Vk ,∠I j k are
indeed abranch flow solution. Otherwise, theangles∠Vk
determined in Step 1 do not satisfy the Kirchhoff voltage
law i Vi = 0 around the loop that involves the link
(j , k) identified in Step 3(b). This violates the condition
BT ⊥ B − 1T βT = βT ⊥ in Theorem 2.
C. Radial networks
Recall that all relaxed solutions in Y \ h(X) are
spurious. Our next key result shows that, for radial
network, h(X) = Y and hence angle relaxation is always
exact in the sense that there is always a unique inverse
projection that maps any relaxed solution y to a branch
flow solution in X (even though X = Y).
Theorem 4: Suppose G = T is a tree. Then
1) h(X) = Y.
2) given any y, θ∗ := B − 1β always exists and is the
unique phase angle vector such that hθ∗ (y) ∈ X.
Proof: When G = T is a tree, m = n and hence
B = BT and β = βT . Moreover B is n × n and of
full rank. Therefore θ∗ = B − 1β always exists and, by
Theorem 2, hθ∗ (y) is the unique branch flow solution
in X whose projection is y. Since this holds for any
arbitrary y ∈ Y, Y = h(X).
A direct consequence of Theorem 1 and Theorem 4
is that, for a radial network, OPF is equivalent to the
convex problem OPF-cr in the sense that we can obtain
an optimal solution of oneproblem from that of theother.
Corollary 5: Suppose G is a tree. Given an optimal
solution (y∗, s∗) of OPF-cr, there exists a unique θ∗ such
that (hθ∗ (y∗), s∗) is an optimal solution of the original
OPF.
Proof: Suppose (y∗, s∗) is optimal for OPF-cr (24)–
(25). Theorem 1 implies that it is also optimal for OPF-
ar. In particular y∗ ∈ Y(s∗). Since G is a tree, Y(s∗) =
h(X(s∗)) by Theorem 4 and hence there is a unique θ∗such that hθ∗ (y∗) is a branch flow solution in X(s∗).
This means (hθ∗ (y∗), s∗) is feasible for OPF (10)–(11).
Since OPF-ar is a relaxation of OPF, (hθ∗ (y∗), s∗) is also
optimal for OPF.
Remark 3: Theorem 1 implies that we can always
solve efficiently a conic relaxation OPF-cr to obtain a
solution of OPF-ar, provided there are no upper bounds
on the power consumptions pci , qc
i . From a solution of
OPF-ar, Theorem 4 and Corollary 5 prescribe a way to
recover an optimal solution of OPF for radial networks.
For mesh networks, however, thesolution of OPF-ar may
be spurious, i.e., there are no angles ∠Vi ,∠I i j that will
satisfy the Kirchhoff laws if the angle recovery condition
in Theorem 2 fails to hold. To deal with this, we next
propose a way to convexify the network.
VI. CONVEXIFICATION OF MESH NETWORK
In this section, we explain how to use phase shifters
to convexify a mesh network so that an extended angle
recovery condition can alwaysbesatisfied by any relaxed
solution and can be mapped to a valid branch flow
solution of the convexified network. As a consequence,
the OPF for the convexified network can always be
solved efficiently (in polynomial time).
A. Branch flow solutions
In this section we study power flow solutions
and hence we fix an s. All quantities, such as
x, y, X, Y, X , X T , are with respect to the given s, even
though that is not explicit in the notation. In the next
section, s is also an optimization variable and the sets
X, Y, X , X T are for any s; c.f. the more accurate nota-
tion in (4) and (5).
Phase shifters can be traditional transformers or
FACTS (Flexible AC Transmission Systems) devices.
They can increase transmission capacity and improve
stability and power quality [37], [38]. In this paper,
we consider an idealized phase shifter that only shifts
the phase angles of the sending-end voltage and current
across a line, and has no impedance nor limits on the
shifted angles. Specifically, consider an idealized phase
shifter parametrized by φi j across line (i , j ), as shown
in Figure 4. As before, let Vi denote the sending-end
kzij
i j! ij
Fig. 4: Model of a phase shifter in line (i , j ).
voltage. Define I i j to be the sending-end current leaving
node i towards node j . Let k be the point between
Phase shifter
3
which is equivalent to the requirement that the (implied)
voltage angle differences sum to zero around any cycle
c:X
( i ,j )2 c
βi j = 0 (mod 2⇡ )
where βi j = βi j if (i , j ) 2 E and βi j = − βj i if (j , i ) 2
E .
B. Model with phase shifters
Phase shifters can be traditional transformers or
FACTS (Flexible AC Transmission Systems) devices.
They can increase transmission capacity and improve
stability and power quality [3], [4]. In this paper, we
consider an idealized phase shifter that only shifts the
phase angles of the sending-end voltage and current
across a line, and has no impedance nor limits on the
shifted angles. Specifically, consider an idealized phase
shifter parametrized by φi j across line (i , j ), as shown
in Figure 2. As before, let Vi denote the sending-end
kzij
i j! ij
Fig. 2: Model of a phase shifter in line (i , j ).
voltage. Define I i j to be the sending-end current leaving
node i towards node j . Let k be the point between
the phase shifter φi j and line impedance zi j . Let Vk
and I k be the voltage at k and current from k to j
respectively. Then theeffect of the idealized phaseshifter
is summarized by the following modeling assumption:
Vk = Vi eiφi j and I k = I i j eiφi j
The power transferred from nodes i to j is still (defined
to be) Si j := Vi I⇤i j which, as expected, is equal to the
power Vk I ⇤k from nodes k to j since the phase shifter is
assumed to be lossless. Applying Ohm’s law across zi j ,
we define the branch flow model with phase shifters as
the following set of equations:
I i j = yi j
⇣Vi − Vj e− iφi j
⌘(9)
Si j = Vi I⇤i j (10)
sj =X
k:j ! k
Sj k −X
i :i ! j
Si j − zi j |I i j |2 + y⇤j |Vj |2 (11)
Without phase shifters (φi j = 0), (9)–(11) reduce to the
branch flow model (1)–(3).
The inclusion of phase shifters modifies the network
and enlargers the solution set of the (new) branch flow
equations. Formally, let
X := { x | x solves (9)–(11) for some φ} (12)
Unless otherwise specified, all angles should be inter-
preted as being modulo 2⇡ and in (− ⇡ , ⇡ ]. Hence we are
primarily interested in φ 2 (− ⇡ , ⇡ ]m . For any spanning
tree T of G, let “φ 2 T? ” stands for “φi j = 0 for
all (i , j ) 2 T ” , i.e., φ involves only phase shifters in
branches not in the spanning tree T . Define
XT :=n
x | x solves (9)–(11) for some φ 2 T?o
(13)
Since (9)–(11) reduce to the branch flow model when
φ = 0, X ✓XT ✓X.
II I . PHASE ANGLE SETTING
Given a relaxed solution y, there are in general many
ways to choose angles φ on the phase shifters to recover
a feasible branch flow solution x 2 X from y. They
depend on the number and location of the phase shifters.
A. Computing φ
For a network with phase shifters, we have from (9)
and (10)
Si j = Vi
V⇤i − V⇤
j eiφi j
z⇤i j
leading to Vi V⇤
j eiφi j = vi − z⇤i j Si j . Hence✓i −✓j = βi j −
φi j + 2⇡ki j for some integer ki j . This changes the angle
recovery condition in Theorem 2 of [2] from whether
there exists (✓, k) that solves (7) to whether there exists
(✓,φ, k) that solves
B✓ = β − φ+ 2⇡k (14)
for some integer vector k 2 (− 2⇡ , 2⇡ ]m . The case
without phase shifters corresponds to setting φ = 0.
We now describe two ways to compute φ: the first
minimizes the required number of phase shifters, and
the second minimizes the size of phase angles.
1) Minimize number of phase shifters: Our first key
result implies that, given a relaxed solution y :=
(S, ` , v, s0) 2 Y, we can always recover a branch flow
solution x := (S, I , V, s0) 2 X of the convexified
network. Moreover it suffices to use phase shifters in
branches only outside a spanning tree. This method
requires the smallest number (m − n) of phase shifters.
Given any d-dimensional vector ↵, let P (↵) denote
its projection onto (− ⇡ , ⇡ ]d by taking modulo 2⇡ com-
ponentwise.
Theorem 1: Let T be any spanning tree of G. Con-
sider a relaxed solution y 2 Y and the corresponding β
defined by (6) in terms of y.
2
On the other extreme, one can choose to minimize (the
Euclidean norm of) the phase shifter angles by deploying
phase shifters on every link in the network. We prove
that this minimization problem is NP-hard. Simulations
suggest, however, that asimple heuristic worksquitewell
in practice.
These results lead to an algorithm for solving OPF
when there are phase shifters in mesh networks, as
summarized in Figure 1.
Solve&OPF*cr&
Op. mize&phase&shi5ers&
N&
OPF&solu. on&
Recover&angles&
radial&
angle&recovery&condi. on&holds?& Y&mesh&
Fig. 1: Proposed algorithm for solving OPF with phase
shifters in mesh networks. The details are explained in
this two-part paper.
Since power networks in practice are very sparse, the
number of lines not in a spanning tree can be relatively
small compared to the number of buses squared, as
demonstrated in simulations in Section V using the IEEE
test systems with 14, 30, 57, 118 and 300 buses, as well
as a 39-bus model of a New England power system and
two models of a Polish power system with more than
2,000 buses. Moreover, the placement of these phase
shifters depends only on network topology, but not on
power flows, generations, loads, or operating constraints.
Therefore only one-time deployment cost is required
to achieve subsequent simplicity in network operation.
Even when phaseshifters arenot installed in thenetwork,
the optimal solution of a convex relaxation is useful in
providing a lower bound on the true optimal objective
value. This lower bound serves as a benchmark for other
heuristic solutions of OPF.
The paper is organized as follows. In Section II, we
extend the branch flow model of [2] to include phase
shifters. In Section III, we describe methods to compute
phase shifter angles to map any relaxed solution to an
branch flow solution. In Section IV, we explain how to
use phase shifters to simplify OPF. In Section V, we
present our simulation results.
I I . BRANCH FLOW MODEL WITH PHASE SHIFTERS
We adopt the same notations and assumptions A1–A4
of [2].
A. Review: model without phase shifters
For ease of reference, we reproduce the branch flow
model of [2] here:
I i j = yi j (Vi − Vj ) (1)
Si j = Vi I⇤i j (2)
sj =X
k:j ! k
Sj k −X
i :i ! j
Si j − zi j |I i j |2 + y⇤j |Vj |2 (3)
Recall the set X(s) of branch flow solutions given s
defined in [2]:
X(s) := { x := (S, I , V, s0) | x solves (1)–(3) given s}
(4)
and the set X of all branch flow solutions:
X :=[
s2 Cn
X(s) (5)
To simplify notation, we often use X to denote the set
defined either in (4) or in (5), depending on the context.
In this section we study power flow solutions and hence
we fix an s. All quantities, such as x, y, X, Y, X , X T ,
are with respect to the given s, even though that is not
explicit in the notation. In the next section, s is also an
optimization variable and the sets X, Y, X , X T are for
any s.
Given a relaxed solution y, define β := β(y) by:
βi j := \ vi − z⇤i j Si j , (i , j ) 2 E (6)
It is proved in Theorem 2 of [2] that a given y can be
mapped to a branch flow solution in X if and only if
there exists an (✓, k) that solves
B✓ = β + 2⇡k (7)
for some integer vector k 2 Nn . Moreover if (7) has a
solution, then it has a countably infinite set of solutions
(✓, k), but they are relatively unique, i.e., given k, the
solution ✓is unique, and given ✓, the solution k is
unique. Hence (7) has a unique solution (✓⇤, k⇤) with
✓⇤2 (− ⇡ , ⇡ ]n if and only if
B? B − 1T βT = β? (mod 2⇡ ) (8)
BFM without phase shifters:
BFM with phase shifters:
Convexification of mesh networks
OPF minx
f h(x)( ) s.t. x ÎX
Theorem
•
• Need phase shifters only
outside spanning tree
X =Y
OPF-ar minx
f h(x)( ) s.t. x ÎY
Y
X
OPF-ps minx,f
f h(x)( ) s.t. x ÎX
X
X
optimize over phase shifters as well
Convexification of mesh networks
OPF-ps minx,f
f h(x)( ) s.t. x ÎX
X
X
optimize over phase shifters as well
Optimization of f • Min # phase shifters (#lines - #buses + 1)
• Min : NP hard (good heuristics)
• Given existing network of PS, min # or
angles of additional PS
f2
Examples
With PS
Examples
With PS
Examples
With PS