Multiple Matrix Rank Constrained Optimization for Optimal Power

Flow over Large Scale Transmission Networks

Y. Shi

, H. D. Tuan

, S. W. Su

and A. V. Savkin

Faculty of Engineering and Information Technology, University of Technology, Sydney, NSW 2007, Australia

School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia

Keywords:

Optimal Power Flow (OPF), Transmission Networks, Rank-one Matrix Constraint, Nonsmooth Optimization,

Semi-Deﬁnite Programming (SDP).

Abstract:

The optimal power ﬂow (OPF) problem for power transmission networks is an NP-hard optimization problem

with numerous quadratic equality and indeﬁnite quadratic inequality constraints on bus voltages. The existing

nonlinear solvers often fail in yielding a feasible solution. In this paper, we follow our previously developed

nonsmooth optimization approach to address this difﬁcult large-scale OPF problem, which is an iterative

process to generate a sequence of improved solutions that converge to an optimal solution. Each iteration calls

an SDP of a moderate dimension. Intensive simulations for OPF over networks with a large number of buses

are provided to demonstrate the efﬁciency of our approach.

1 INTRODUCTION

The brain of a smart grid is advanced distribution

management system (DMS), which is responsible

for supervisory control and data acquisition in reac-

tive dispatch, voltage regulation, contingency analy-

sis, capability maximization and other smart opera-

tions. Optimal power ﬂow (OPF), which determines

a steady state operating point such that the cost of

electric power generation is minimized under oper-

ating constraints, lies at the heart of DMS ((Car-

pentier, 1962; Huneault and Galiana, 1991; Momoh

et al., 1999; Pandya and Joshi, 2008) and references

therein). The OPF problem is typically nonlinear

and nonconvex due to the multiple quadratic equal-

ity and indeﬁnite quadratic inequality constraints on

the voltages variables in expressing the bus intercon-

nections, hardware operating capacity and balance be-

tween power demand and supply. These nonlinear

constraints are difﬁcult so the state-of-the-art nonlin-

ear optimization solvers may converge to just station-

ary points (see (Bukhsh et al., 2013) and references

therein), which are even not necessarily feasible.

There has been a renewed attention on the ap-

plication of semi-deﬁnite programming (SDP) to the

OPF problem. As a nonconvex quadratic optimiza-

tion, OPF can be easily recast by convex quadratic

optimization with the additional nonconvex rank-one

constraint on the matrix W = VV

of the outer prod-

uct of voltage vector variable V (Bai et al., 2008;

Lavaei and Low, 2012). The matrix solution of the

semi-deﬁnite relaxation (SDR) by dropping the rank-

one matrix constraint, is of rank-one and therefore

provides the global OPF solution in the power dis-

tribution networks (Madani et al., 2015b) or in few

modiﬁed IEEE networks (Lavaei and Low, 2012).

Often, a low-rank matrix solution of SDR is of not

rank-one and cannot result in a feasible point of the

original nonconvex OPF problem (Lavaei and Low,

2012; Madani et al., 2015b; Madani et al., 2015a).

An another setback of using the such matrix variable

W ∈ C

n×n

for a network of n buses is that its dimen-

sion n(n +1)/2 increases dramatically with respect to

n. For instance, for moderate n = 150 and n = 300

such W is equivalent to 150 × 151/2 = 11.325 and

300 × 301/2 = 45.150 complex scalar variables. On

the other hand, all large scale networks are sparse in

the sense that the number of the ﬂow lines connecting

buses is relatively moderate. This means that only a

small portion of the crossed nonlinear terms V

∗

ap-

pearing in the constraints. References (Molzahn et al.,

2013; Andersen et al., 2014; Madani et al., 2015a)

suggest to formulate OPF as optimization in multiple

matrix variables of outer products of overlapped sub-

sets of voltages. Obviously, it is hardly expected that

SDR by dropping the rank-one constraints on all these

multiple outer products would have all rank-one solu-

tion. Consequently, the matrix solution of SDR does

384

Shi, Y., Tuan, H., Su, S. and Savkin, A.

Multiple Matrix Rank Constrained Optimization for Optimal Power Flow over Large Scale Transmission Networks.

In Proceedings of the 5th International Conference on Smart Cities and Green ICT Systems (SMARTGREENS 2016), pages 384-389

ISBN: 978-989-758-184-7

not have any physical meaning.

To address the rank-one issue, in (Shi et al., 2015),

we have proposed a rank-reduced optimization to ad-

dress the rank-one for the matrix variable W of the

total voltages, which works very well and practically

for moderate networks up to 300 buses. In this paper,

we will further develop this rank-reduced optimiza-

tion technique for its large-scale OPFs up to a few

thousands buses.

The paper is structured as follows. Section 2 is

devoted to the problem formulation, its challenges

and its computational solution development. Section

3 provides simulation to show the efﬁciency of our

methods. The conclusions are drawn in Section 4.

The notation used in the paper is standard. More

speciﬁcally, j denotes the imaginary unit, M  0

means that the Hermitian symmetric matrix M is pos-

itive semi-deﬁnite, rank(M) is the rank of the ma-

trix M; ℜ(·) and ℑ(·) denote the real and imaginary

parts of a complex quantity. a ≤ b for two complex

numbers a and b is componentwise understood, i.e.

ℜ(a) ≤ ℜ(b) and ℑ(a) ≤ ℑ(b). h.,.i is the dot prod-

uct of matrices, while {A

}

i=1,...,n

denotes the matrix

with diagonal blocks A

and zero off-diagonal blocks.

2 LARGE-SCALE OPTIMAL

POWER FLOW PROBLEM AND

CHALLENGES

Consider an AC electricity transmission network with

a set of buses N := {1,2,··· ,n}. The buses are con-

nected through a set of ﬂow lines L ⊆ N × N , i.e.

bus m is connected to bus k if and only if (m,k) ∈ L.

Accordingly, N (k) := {m ∈ N : (m,k) ∈ L}. The

power demanded at bus k ∈ N is

= P

+ jQ

where P

and Q

are the real and reactive power.

A subset G ⊆ N of buses is supposed to be con-

nected to generators. Any bus k ∈ N \ G is thus

not connected to generators. Other notations are: (i)

Y = [y

]

(k,m)∈N ×N

∈ C

n×n

is the admittance matrix

(Zimmerman et al., 2011). Each y

is the mutual

admittance between bus k and bus m, so y

= y

∀ (k, m) ∈ L; (ii) V is the complex voltage vector, V =

,· · · ,V

]

∈ C

, where V

is the complex volt-

age injected to bus k ∈ N ; (iii) I is the complex cur-

rent vector, I = YV = [I

,· · · ,I

]

∈ C

, where I

is the complex current injected to bus k ∈ N ; (iv) I

is the complex current in the power line (k,m) ∈ L,

∑

m∈N (k)

= I

∑

m∈N (k)

; (v) S

= P

+ jQ

is the complex power transferred from bus k to bus

m, where P

and Q

represent the real and reactive

transferred power; (vi) S

= P

+ jQ

is the com-

plex power injected by bus k ∈ G, where P

and Q

represent the real and reactive generated power.

For each bus k, it is obvious that S

− S

−P

)+ j(Q

−Q

) = V

∑

m∈N (k)

∗

. There-

fore, we can express the real generated power P

and reactive generated power Q

at bus k as

= P

+ ℜ(

∑

m∈N (k)

∗

) and Q

= Q

ℑ(

∑

m∈N (k)

∗

The objective of OPF is to minimize the follow-

ing cost function of real active generated power

: f (P

) =

∑

k∈G

+ c

) with c

0,c

given, which is a function of the bust volt-

ages V :

f (V ) =

∑

k∈G

+ ℜ(

∑

m∈N (k)

∗

))

+ ℜ(

∑

m∈N (k)

∗

)) + c

]. (1)

Accordingly, the following OPF problem is formu-

lated

min

V ∈C

f (V ) s.t. (2a)

−P

− jQ

∑

m∈N (k)

∗

,k ∈ N \ G, (2b)

min

≤ P

+ ℜ(

∑

m∈N (k)

∗

) ≤ P

max

,k ∈ G, (2c)

min

≤ Q

+ ℑ(

∑

m∈N (k)

∗

) ≤ Q

max

,k ∈ G, (2d)

min

≤ |V

| ≤ V

max

,k ∈ N , (2e)

| = |V

∗

| ≤ S

max

,∀(k,m) ∈ L (2f)

−V

| ≤ V

max

,(k,m) ∈ L, (2g)

|arg(V

) − arg(V

)| ≤ θ

max

,(k,m) ∈ L, (2h)

where (2b) is the equation of the balance between the

demand and supply power at bus k ∈ N \ G, (2c)-

(2d) are the power generation bounds, where P

min

and P

max

, Q

max

are the lower bound and upper

bound of the real power reactive power generations,

respectively, (2e) are the voltage amplitude bounds,

(2f)-(2h) are capacity limitations, where the line cur-

rents between the connected buses are constrained by

(2f), while (2g)-(2h) guarantee the voltage balance

in terms of their magnitude and phases (Zimmerman

et al., 2011).

Introducing W

= V

∗

,k = 1,..., n; m = 1, ..., n

and W = [W

]

k,m=1,...,n

, the problem (2) is recast to

Multiple Matrix Rank Constrained Optimization for Optimal Power Flow over Large Scale Transmission Networks

385

the following optimization problem

min

W 0

F(W ) s.t. (3a)

−P

− jQ

∑

m∈N (k)

∗

k ∈ N \ G, (3b)

min

≤ P

+ ℜ(

∑

m∈N (k)

∗

) ≤ P

max

,k ∈ G, (3c)

min

≤ Q

+ ℑ(

∑

m∈N (k)

∗

) ≤ Q

max

,k ∈ G, (3d)

min

)

≤ W

≤ (V

max

)

,k ∈ N , (3e)

∗

| ≤ S

max

,(k,m) ∈ L, (3f)

−W

≤ (V

max

)

,(k,m) ∈ L, (3g)

ℑ(W

) ≤ ℜ(W

)tan θ

max

,(k,m) ∈ L, (3h)

rank(W ) = 1, (3i)

where F(W ) =

∑

k∈G

+ ℜ(

∑

m∈N (k)

∗

))

+ ℜ(

∑

m∈N (k)

∗

)) + c

], which is convex

quadratic in W

SDR approach (see e.g. (Lavaei and Low, 2012)) is

to drop the difﬁcult rank-one constraint (3i) for SDR.

If the solution of such relaxed SDP is of rank-one,

i.e. it satisﬁes the nonconvex rank-one constraint (3i)

then it obviously leads to the global solution of the

nonconvex optimization problem (3). Otherwise even

a feasible solution of (3) is hardly obtained from a

solution of this SDR.

In (Shi et al., 2015), we suggest to address (3) by

the following spectral optimization

min

W 0

(W ) := F(W ) + µ(Trace(W )

−λ

max

(W )) s.t. (3b) − (3h), (4)

where λ

max

(W ) stands for the maximal eigenvalue

of W and µ is a penalty parameter. Note that

Trace(W ) − λ

max

(W ) = 0. The nonnegative quantity

Trace(W ) − λ

max

(W ) can therefore be used to mea-

sure the degree of satisfaction of the matrix rank-one

constraint (3i). Without square on the factor of µ,

the penalization Trace(W ) − λ

max

(W ) in (4) is exact,

meaning that the constraint Trace(W ) = λ

max

(W ) can

be satisﬁed by a minimizer of (4) with a ﬁnite value

of µ (see e.g. (Bonnans et al., 2006, Chapter 16)).

This is generally considered as a sufﬁciently nice

property to make such exact penalization attractive.

Function λ

max

(W ) is nonsmooth but is

lower bounded by λ

max

(W ) = max

||w||=1

W w ≥

(κ)

max

)

W w

(κ)

max

, where w

(κ)

max

is the normalized eigen-

vector corresponding to the eigenvalue λ

max

(κ)

i.e. λ

max

(κ)

) = (w

(κ)

max

)

(κ)

max

Therefore, for any W

(κ)

feasible to the convex con-

straints (3b)-(3h), the following convex optimization

problem provides an upper bound for the nonconvex

optimization problem (4)

min

W 0

(κ)

(W ) := F(W ) + µ[Trace(W )

−(w

(κ)

max

)

W w

(κ)

max

] s.t. (3b) − (3h)

(5)

because F

(κ)

(W ) ≥ F

(W ) ∀ W  0.

Suppose that W

(κ+1)

is the optimal solution of

SDP (5). Since W

(κ)

is also feasible to (5) with

(κ)

) = F

(κ)

), it is true that F

(κ)

) =

(κ)

) ≥ F

(κ)

(κ+1)

) ≥ F

(κ+1)

), so W

(κ+1)

is a better feasible point of (4) than W

(κ)

The common drawback of using the structure-

free matrix variable W ∈ C

n×n

is that its dimension

n(n + 1)/2 increases dramatically with respect to the

number n of the network buses. For instance, for

n = 150 and n = 300 such W is equivalent to 150 ×

151/2 = 11.325 and 300 × 301/2 = 45.150 complex

scalar variables. On the other hand, most large scale

networks are sparse in the sense that the number of

the ﬂow lines connecting buses is relatively moder-

ate. There is only a small portion of the crossed terms

∗

appearing in the nonlinear constraints (2b)-(2h)

so such matrix variable W contains many redundant

∗

. References (Molzahn et al., 2013; Andersen

et al., 2014; Madani et al., 2015a) decompose the set

N := {1,2,...,n} into I possibly overlapped subset

= {i

,..., i

} called by bags, such that for each

i = 1,2,..., I

∈ N (i

`+1

),` = 1, ..., N

− 1 and i

∈ N (i

The set of bags can be reset such that its bags are of

relatively same size.

Deﬁne the Hermitian symmetric matrix variables

= [W

]

k,m=1,..,N

∈ C

×N

,i = 1, 2, ...,I (6)

and W = diag{W

}

i=1,...,I

. By replacing W

= V

∗

in (1) and constraints (2b)-(2h), we have the following

equivalent optimization reformulation for (2)

min

W 0

F(W ) s.t. (3b)−(3h), rank(W

) = 1, i = 1,...,I .

(7)

Reference (Molzahn et al., 2013; Andersen et al.,

2014) consider (2) just by dropping all rank-one con-

straints in (7), while (Madani et al., 2015a) used a

penalized SDR for locating a low-rank semi-deﬁnite

solution W

. In the end, a rank-one solution could not

be found.

Now, we propose the following algorithm for

computational solution of (7).

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386

Large-scale Nonsmooth Optimization Algorithm

(NOA).

Initialization. Solve SDP

min

F(W ) s.t. (3b)− (3h) (8)

to ﬁnd its solution W

(0)

:= diag{W

i,(0)

}

i=1,...,I

. If

rank(W

i,(0)

) ≡ 1 stop: W

(0)

is the global solution of

the nonconvex optimization problem (7). Otherwise

deﬁne L

(0)

= {i ∈ {1,...,I } : rank(W

i,(0)

) = 1}

Step 1. For κ = 0, 1, .., and the normalized

eigenvector w

i,(κ)

max

corresponding to the eigenvalue

max

i,(κ)

) of W

i,(κ)

solve

min

(κ)

(W ) := F(W ) + µ

∑

i/∈L

(κ)

[Trace(W

)

−(w

i,(κ)

max

)

i,(κ)

max

] s.t. (3b) − (3h), (9a)

Trace(W

) − (w

i,(κ)

max

)

i,(κ)

max

≤ ε

tol

,i ∈ L

(κ)

(9b)

to ﬁnd its solution W

(κ+1)

:= diag{W

i,(κ+1)

}

i=1,2,...,I

Stop if rank(W

i,(κ+1)

) = 1 for all i = 1, 2, ....,I.

Otherwise deﬁne L

(κ+1)

= {i ∈ {1,...,I } :

rank(W

i,(κ+1)

) = 1} and reset κ = κ + 1.

Remark. The constraint (9b) implies that W

must be a rank-one solution, so it is introduced to

prevent good bags turning out to be bad.

Similarly to (Shi et al., 2015), the following

proposition holds.

Proposition 1. Initialized by any feasible point W

i,(0)

of SDP (8), {W

(κ)

} is a sequence of improved feasible

points of the nonconvex optimization problem

min

(κ)

(W ) := F(W ) + µ

∑

[Trace(W

)

−λ

max

)] s.t. (3b) − (3h), (10a)

which converges to a point satisfying ﬁrst-order nec-

essary optimality conditions.

3 SIMULATION RESULTS

The hardware and software facilities for our compu-

tational implementation are:

• Processor: Intel(R) Core(TM) i5-3470 CPU

@3.20GHz;

• Software: Matlab version R2013b;

• Matlab toolbox: Matpower version 5.1(Zimmer-

man et al., 2011) to compute the admittance ma-

trix Y from the power system data; CVX with

SDPT3 solver for SDP (5).

3.1 Polish-2383wp System

Polish-2383wp system is part of the European UCTE

system, with 2383 buses, 327 generators and 2896

transmission line, which lead to 2056 nonlinear con-

straints in (2b). Thus, there are 2383 × 2384/2 =

2840536 complex scalar variables in the W matrix.

However W is sparse matrix, most element are 0, thus

we apply the Large-Scale NOA as follow,

Initialization. Following (Madani et al., 2015a)

to decompose the 2383 buses into 2383 overlapped

bags. The largest bag size is 24, while the smallest

bag size is 1. After the bags decomposition, the to-

tal number of the complex decision variables in (8) is

23199 vs 2840536 complex decision variables in (3).

Solve (8) over the 2383 bags, we ﬁnd that there are

653 bad bags, which means the rank of the 653 bags

are more than 1. The largest bad bag size is 24, while

the smallest bad bag size is 2. Other 1720 good bags

are all rank-1. Then go to next step.

Step 1. Set ε

tol

= 10

−6

and put the 1720 good

bags in constraint (9b), solve (9), after 8 iterations,

the bad bags number turns to be 13, while the good

bags number turns to be 2370. Among the 13 bad

bags, six have size 3, four have size 4, one has size 5

and two have size 6.

Step 2. Put the 2370 good bags in constraint (9b),

solve (9), after 5 iterations, the bad bags number turns

to be 10, while the good bags number turns to be

2373. Among the 10 bad bags, six have size 3, three

have size 4, and one has size 6.

Step 3. Put the 2373 good bags in constraint (9b),

solve (9), after 5 iterations, the bad bags number turns

to be 5, while the good bags number turns to be 2378.

Among the 5 bad bags, two have size 3, two have size

4, and one has size 6.

Step 4. Put the 2378 good bags in constraint (9b),

solve (9), after 5 iterations, the bad bags number turns

to be 2, while the good bags number turns to be 2381.

Among the 2 bad bags, one is size 3, the other is size

Step 5. Put the 2381 good bags in constraint (9b),

solve (9), after 5 iterations, the bad bags number turns

to be 0. The optimal objective value is 1.9464 × 10

It should be noted that, in step 4, two multiplier µ

were added in the bags penalized term, to increase the

convergence speed of bag-2011 and bag-2254, where

µ = 4. In step 5, bag-711 was multiplied by 4 as well.

3.2 Polish-2736sp System

Polish-2736sp system represents the Polish networks

during summer 2004 peak conditions, with 2736

buses, 420 generators and 3504 transmission line,

Multiple Matrix Rank Constrained Optimization for Optimal Power Flow over Large Scale Transmission Networks

387

which lead to 2316 nonlinear constraints in (2b).

Thus, there are 2736 × 2737/2 = 3744216 complex

scalar variables in the W matrix. We apply the Large-

Scale NOA as follow,

Initialization. Following (Madani et al., 2015a)

to decompose the 2736 buses into 2736 overlapped

bags. The largest bag size is 24, while the smallest

bag size is 1. After the bags decomposition, the to-

tal number of the complex decision variables in (8) is

27298 vs 3744216 complex decision variables in (3).

Solve (8) over the 2736 bags, we ﬁnd that there is only

one bad bag with size 4, more speciﬁcally, the second

largest eigenvalue is only 3.4 ×10

−5

, which is closely

to rank-1 tolerance criterion 10

−5

Step 1. Set ε

tol

= 10

−5

, put the 2735 good bags in

constraint (9b), solve (9), after 1 iterations, the bad

bags number turns to be 0. The optimal objective

value is 1.3125 × 10

3.3 Polish-2737sop System

Polish-2737sop system is the Polish networks during

summer 2004 peak conditions, with 2737 buses, 399

generators and 3506 transmission line, which lead to

2338 nonlinear constraints in (2b). Thus, there are

2737 × 2738/2 = 3746953 complex scalar variables

in the W matrix. We apply the Large-Scale NOA as

follow,

Initialization. Following (Madani et al., 2015a) to

decompose the 2737 buses into 2737 overlapped bags.

The largest bag size is 24, while the smallest bag size

is 1. After the bags decomposition, the total number

of the complex decision variables in (8) is 27034 vs

3746953 complex decision variables in (3). Solve (8)

over the 2737 bags, we ﬁnd that there are three bad

bags, whose size are all 2.

Step 1. Set ε

tol

= 10

−5

, put the 2734 good bags in

constraint (9b), solve (9), after 1 iterations, the bad

bags number turns to be 0. The optimal objective

value is 7.8235 × 10

3.4 Polish-2746wop System

Polish-2746wop system is the Polish networks during

winter 2003-04 off-peak conditions, with 2746 buses,

514 generators and 3514 transmission line, which

lead to 2232 nonlinear constraints in (2b). Thus, there

are 2746 × 2747/2 = 3771631 complex scalar vari-

ables in the W matrix. We apply the Large-Scale NOA

as follow,

Initialization. Following (Madani et al., 2015a)

to decompose the 2746 buses into 2746 overlapped

bags.The largest bag size is 24, while the smallest bag

size is 1. After the bags decomposition, the total num-

ber of the complex decision variables in (8) is 29024

vs 3771631 complex decision variables in (3). Solve

(8) over the 2746 bags, we ﬁnd that there is only one

bad bag with size 2.

Step 1. Set ε

tol

= 10

−5

, put the 2746 good bags in

constraint (9b), solve (9), after 1 iterations, the bad

bags number turns to be 0. The optimal objective

value is 1.2084 × 10

3.5 Polish-2746wp System

Polish-2746wop system is the Polish networks dur-

ing winter 2003-04 evening peak conditions, with

2746 buses, 520 generators and 3514 transmission

line, which lead to 2226 nonlinear constraints in (2b).

Thus, there are 2746 × 2747/2 = 3771631 complex

scalar variables in the W matrix. We apply the Large-

Scale NOA as follow,

Initialization. Following (Madani et al., 2015a)

to decompose the 2746 buses into 2746 overlapped

bags.The largest bag size is 24, while the smallest bag

size is 1. After the bags decomposition, the total num-

ber of the complex decision variables in (8) is 28257

vs 3771631 complex decision variables in (3). Solve

(8) over the 2746 bags, we ﬁnd that there is only one

bad bag with size 2.

Step 1. Set ε

tol

= 10

−5

, put the 2746 good bags in

constraint (9b), solve (9), after 1 iterations, the bad

bags number turns to be 0. The optimal objective

value is 1.6207 × 10

3.6 Polish-3012wp System

Polish-3012wp system represents the Polish networks

during winter 2007-08 evening peak, with 3012 buses,

502 generators and 3572 transmission line, which

lead to 2510 nonlinear constraints in (2b). Thus, there

are 3012 × 3013/2 = 4537578 complex scalar vari-

ables in the W matrix. We apply the Large-Scale NOA

as follow,

Initialization. Following (Madani et al., 2015a)

to decompose the 3012 buses into 3012 overlapped

bags.The largest bag size is 25, while the smallest bag

size is 1. After the bags decomposition, the total num-

ber of the complex decision variables in (8) is 30996

vs 4537578 complex decision variables in (3). Solve

(8) over the 3012 bags, we ﬁnd that there are 381 bad

bags, which means the rank of the 381 bags are more

than 1.The largest bad bag size is 25, while the small-

est bad bag size is 2. Other 2631 good bags are all

rank-1. Then go to next step.

Step 1. Set ε

tol

= 10

−5

and put the 2631 good

bags in constraint (9b), solve (9), after 5 iterations, the

bad bags number turns to be 5, while the good bags

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388

number turns to be 3007. Among the 5 bad bags, two

have size 3, two have size 4, and one has size 5.

Step 2. Put the 3007 good bags in constraint (9b),

solve (9), after 5 iterations, the bad bags number turns

to be 0. The optimal objective value is 2.6406 × 10

3.7 Polish-3120sp System

Polish-3120sp system represents the Polish networks

during summer 2008 morning peak, with 3120 buses,

505 generators and 3693 transmission line, which

lead to 2615 nonlinear constraints in (2b). Thus, there

are 3120 × 3121/2 = 4868760 complex scalar vari-

ables in the W matrix. We apply the Large-Scale NOA

as follow,

Initialization. Following (Madani et al., 2015a)

to decompose the 3120 buses into 3120 overlapped

bags.The largest bag size is 25, while the smallest bag

size is 1. After the bags decomposition, the total num-

ber of the complex decision variables in (8) is 32637

vs 4868760 complex decision variables in (3). Solve

(8) over the 3120 bags, we ﬁnd that there are 21 bad

bags,The largest bad bag size is 7, while the smallest

bad bag size is 2. which means the rank of the 21 bags

are more than 1. Other 3099 good bags are all rank-1.

Then go to next step.

Step 1. Set ε

tol

= 10

−5

and put the 3099 good bags

in constraint (9b), solve (9), after 5 iterations, only 2

bad bags with size 2 are still bad, while other bags

turn good.

Step 2. Put the 3118 good bags in constraint (9b),

solve (9), after 4 iterations, the bad bags number turns

to be 0. The optimal objective value is 2.1778 × 10

It should be noted that, in step 2, the penalized

term of bag-81 was multiplied by 2 to increase the

convergence speed.

4 CONCLUSIONS

OPF over power transmission networks is a difﬁcult

nonconvex optimization problem with numerous non-

linear equality and inequality constraints. We have

developed a large-scale nonsmooth optimization al-

gorithm to compute its optimal solution, which is

efﬁcient and practical for networks with reasonably

large numbers of buses. Applications of NOA to

OPF over three-phase power transmission networks

are currently under consideration.

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