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


Advantages of BFM


Equivalence

Recursive structure


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


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


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


S jkj®k

å = Sij - zij ij + s j


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


Baran and Wu 1989 for radial networks

Recursive structure allows very

fast & stable computation

initialization


S jkj®k



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


Pijlin = - pk

kÎTj

å , Qij

lin = - qkkÎTj

å

vilin - v j

lin = 2 rijPijlin + xijQij

lin( )


- skkÎTj

å

s j

Sijlin


Pijlin = - pk

kÎTj

å , Qij

lin = - qkkÎTj

å

vilin - v j

lin = 2 rijPijlin + xijQij

lin( )


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


ViIij* = Sij

S jkj®k

å = Sij - zij Iij2

( )i® j

å + s j S jkj®k


å + s j


2

ij

vi ij = Sij2

Sij ³ Sijlin

vi £ vilin

Outline

High-level summary


Advantages of BFM


Equivalence

Recursive structure


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


min f x( )

over x := (S, I,V, sg, sc )

s. t. sig

£ sig £ si

g si £ sic £ si vi £ vi £ vi


min f x( )


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


generation, volt/var control

Branch flow model is more convenient for applications

min f x( )


s. t. sig

£ sig £ si

g sic£ si


Sij =ViIij*

Vj =Vi - zijIij

branch flow model

Sij - zij Iij2

( )i® j

å - S jkj®k

å = s jc - s j

g


demand response

min f x( )


s. t. sig

£ sig £ si

g sic£ si


Challenge: nonconvexity !

Vi -Vj = zijIij

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


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 + s j


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



2

ij

ijvi = Sij2



ij := Iij2

vi := Vi2

Advantages

• Recursive structure (radial networks)

• Variables represent physical quantities

Branch flow model



h

h-1

C2(m+n+1)

R3(m+n+1)

Xnc

X X

X+



then X = Xnc.






Xnc ✓ X+ .




Theorem 9 implies
















BFM).

C. Equivalence











G for

radial networks.






by x = g(WG) where



v j := [WG] j j , j 2 N+

sj := Âk: j⇠k


























Âk: j ! k



v j ` jk = |Sjk|2, j ! k 2 E (33c)












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


2

ij

ijvi = Sij2



ij := Iij2

vi := Vi2

nonconvexity

linear

Branch flow model

S jkk: j®k

å = Sij - zij ij( )i:i® j

å + s j


2

ij

ijvi ³ Sij2



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



h

h-1

C2(m+n+1)

R3(m+n+1)

Xnc

X X

X+



then X = Xnc.






Xnc ✓ X+ .




Theorem 9 implies
















BFM).

C. Equivalence











G for

radial networks.






by x = g(WG) where



v j := [WG] j j , j 2 N+

sj := Âk: j⇠k


























Âk: j ! k



v j ` jk = |Sjk|2, j ! k 2 E (33c)












Branch flow model


constraints

ìíî

üýþ

Ç jkv j = S2

cycle cond on x

ìíï

îï

üýï

þï

relaxed solution: x := S, ,v, s( )

X+ :=


constraints

ìíî

üýþ

Ç jkv j ³ S2

{ }

Theorem

For radial network, X ºX ºXnc Í X+

Branch flow model


constraints

ìíî

üýþ

Ç jkv j = S2

cycle cond on x

ìíï

îï

üýï

þï

power flow solutions: x := S, ,v, s( )

X+ :=


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

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


“no lower bounds”

removes these Ck[ ]ij

sufficient cond

remove these Ck[ ]ij


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


X

X

optimize over phase shifters as well

Convexification of mesh networks

OPF-ps minx,f


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

branch flow model · jk, ` jk) and ( s kj, ` kj) are gh¿qhg for both directions, with the...

Documents