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IEEE

Transactions on

Power Systems, Vol. 13, No. 4, November 1998

1211

A N INTERIOR-POINT

ETHODFOR NONLINEAR

OPTIMAL

OWERLOW

USINGVOLTAGE ECTANGULAROORDINATES

GERALD0 LEITE TORRES

gltorres@kingcong.uwaterloo.ca

VICTOR HUGO QUINTANA

’

quintana@kingcong.uwaterloo.ca

’

epartment of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON, CANADA.

Departamento de Engenharia Elbtrica e Sistemas de PotSncia, Universidade Federal de Pernambuco,

Recife,

PE,

BRAZIL.

ABSTRACT:The paper describes the solution of an optimal

power flow

(OPF) problem in rectangular form by an interior-

poin t me thod

(IPM) for nonlinear programming. Some OPF

variants when formulated in rectangular form have quadratic

objective and quadratic constraints. Such quadratic features

allow for ease of matrix setup, and inexpensive incorporation

of higher-order information in a predictor-corrector procedure

that generally improves IPM performance. The mathematical

development of the IPM in the paper is based on a general

nonlinear programming problem. Issues in implementation

to

solve the rectangular OPF are discussed. Computational tests

apply the IPM to both the rectangular and polar OPF versions.

Test results show that both algorithms perform extremely well.

KEYWORDS:Optimal Power Flow, Nonlinear Programming,

Interior-Point Method.

1.

INTRODUCTION

The

optimal power

p o w

(OPF) problem is

a

large scale

nonconvex

nonlinear programming

(NLP) problem, that

is complicated in realistic applications by the presence of

a large number of discrete variables. Given its importance

in power systems planning and operation activities, OPF

has been a subject of intensive study [1,2].

Quadratic

programming

(QP) and

linear programming

(LP) based

solution procedures benefit from efficient

interior-point

methods

(IPM’s) for QP and LP. Electric networks

are

nowadays operating heavily loaded, hence planning and

operation tools now must address strong nonlinearities,

in system behavior. Approximation-based optimization

techniques will be less attractive to cope with stressed

operation conditions. However, efficiently solving OPF’s

in

a

nonlinear manner is a very complex issue. Severa’l

conditions under which an O PF algorithm may fail to

converge are studied in [3].

In [4] the Newton’s method (for unconstrained opti-

mization) is combined with a Lagrange multiplier method

PE-OIO-PWRS-0-12-1997

A

paper recommended and approved by

the IEEE Power System Analysis, Computing and Economics

Committee

of

the IEEE Power Engineering Society for publication n the

IEEETransactions on Power Systems. Manuscript submitted January

16, 1997; made available

for

printing December 12, 1997.

(for optimization with equalities) and penalty functions

(for handling inequalities) t o solve large scale OPF’s in

a nonlinear manner. Well designed dat a struc ture and

efficient use of sparsity techniques made such an algorithm

very attractive and successful at the time. The major

difficulty turned out t o be th e efficient identification

of

binding inequalities, an issue later studied in [5]. In the

last few years, many applications of IPM’s t o solve Large

power system optimization problems star ted to appe w in

the power-engineering literature. Problems already solved

include state estimation [6], several OPF variants [7-91,

including power flow unsolvability [lo], to name a few.

A common feature of the works reported in [6-101 is

th at large scale NLP problems have been efficiently solved

by IPM’s for NLP derived from the

logarithmic bagprier

function approach. The logarithmic barrier approach was

introduced by Hrisch in 1955, and developed

as a

tool for

NLP in 1968 by Fiacco and McCormick

[ll].

Though

it was devised for solving general NLP problems, it was

in the LP field that its superb computational efficiency

(see [12]) was first demonstrated and broadly accepted

by the research community. Megiddo in [13] suggested

the application of the logarithmic barrier method to the

primal and dual LP problems simultaneously, known as

the

primal-dual logarithmic barrier

IPM in the category

of

path-following

methods. These variants incorporating

Mehrotra’s predictor-corrector steps [14] are,

at

present,

accepted

as

th e most computationally effective IPM’s.

In this paper an OP F problem is solved in a nonlinear

manner via a primal-dual IPM for NLP. The key feature

of this

OPF

formulation is tha t its objective function and

constraints, as the result of using voltages in rectangular

coordinates, are quadratic [15,16]. Desirable properties of

a quadratic function are: (a) its Hessian is constanl,, (b)

its Taylor expansion terminates at th e second-order term

without truncation error, and (c) the higher-order term

is easily evaluated. Such quadrati c features allow for ewe

of matrix setup and inexpensive incorporation of higher-

order information in

a

predictor-corrector procedure that

reduces the number of

IPM

iterations t o convergence.

The paper is organized as follows. Section 2 formulates

an

O PF problem in redangular form. Section

3

presents

a

comprehensive derivat ion of the IPM from

a

general NLP

problem. Th e predictor-corrector method is described in

Section

4.

Some issues in implementation are addressed

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in Section

5.

Test results in Section

6

show that IPM

performance is equally good for the rectangular and polar

OPF versions. Conclusions in Section 8 close the paper.

2. OPTIMAL OWER

LOW ORMULATION

(Complex) voltages in rectangular form are used in [17]

to explore the idea of a n optimal multiplier t ha t improves

the load flow convergence, in

[18]

for power system state

estimation studies, and in

[19]

for calculation of power

system low-voltage solutions. In this section, a variant of

the

OPF

problem, the transmission losses minimization

problem, is formulated in rectangular form.

Let

Af

be the set of all buses, Af the set of all buses but

the swing, the set of generator buses, he set of buses

with fixed var sources, E the set of buses eligible for var

control, Nj the set of buses connected to bus j ,

B

the set

of branches in the system and 7 the set of transformers

with LTC. Let Afl e the size of set

N

nd so on. The

(complex) bus-voltages are defined in rectangular form

as

iri e i + j f i ,

V ~ E N

where ei and fi are, respectively, the real and imaginary

components of

Vi.

Without

loss

of generality, bus-1 is

selected as the swing bus, providing the angular reference

with el = VI and

fl

= 0. It is shown in [16] that the bus

active- and reactive-power injections can be expressed

as

V i E N* (1)

V

i

E

n/ (2 )

Pi = Gi: eie+ f i f ) +B;: f ie- i f ) ,

Qi = Gi: f i e-

eif)

-

Bi:

(eie+ f i f ) ,

and the activepower transmission losses

as

PLOSS= eTGe+

fTGf,

( 3 )

where

Gi,

is the ith row of the bus conductance matrix

G E

RINIXINI,

nd B;: is the ith row

of

bus susceptance

matrix

B E

RIKIXINI.otice that

(1)-(3)

are quadratic

functions of the voltage rectangular coordina,tes e and f .

A variety of O PF formulations can be derived from the

N L P problem model

min f ( x )

(4)

. t.

g(x)

=

0

h m i n

5 h(x) 5 hma"

-.

%mi

n

5

Ix

5

P a " ,

where, for the active-power losses minimization problem

in rectangular form,

(4)

an be

specified

as

follows:

x

E

R"

includes the bus-voltage components

e

and

f

but component f i , and the transformer tap settings t .

f

: R" + R can be one of PLOSSn (3) or the active-

power injection

at

the swing bus,

as

given by PI in I).

g :Rn m ncludes the bus active- and reactive-

power balance constraints

Gi: eie+ f i f ) +

Bi,

( f i e- i f ) - Pipe, V i E fl

(5)

G i : f i e eif) - Bi:(eie

+

i f ) - Qyp e V i

E

(6)

h

:R

+

RP ncludes the bus reactive-power and the

bus-voltage functional bound constraints

Gi:f i e- eif) -

Bi:

e ie+

f if) ,

Gi:

f i e i f ) Bi:(eie+ f if) ,

V i G

(7)

V i E E (8)

e ; + f . ,

V ~ E

V

(9 )

with t,he appropriate lower and upper bounds,

Branch flow limits can be handled in terms of the

branch active- and reactive-power flows, or the square

of the branch current magnitudes, which are quadratic

functions of the voltage components e and

f

(see

[E] .

Ix : lR

-+

IRq results in a sub-vector x of x with the

components of x tha t har e finite bounds. that

is, 2

=

t

..

with

2 m i n

=

t m i n

and

2max

tmax.

Except for the few terms involving the tap settings

t ,

implicit in the elements of G and B, all functions in the

activepower losses minimization problem are quadratic.

The minimization of readive-power losses. or of cost of

power generations with quadratic cost curves.

are

examples

of

OPF variants with quadratic objective

and constraints. Sumerical advantages th at stem

from

this

form

are: (a) the Taylor expansion of the quadratic

function

f ( x )

=

?jx7Ax

terminates

at

the second-order

term without truncation error,

f(xk

+

A ~ ) f ( xk)+

( X ~ ) ~ A A X; AX?'AAX,

( IO)

(b) the Hessian of

f(x)

s constant

(Hf(x) =

A), and

(c) the higher-order term in (10) is easily evaluated as

f ( h x ) . The nonlinear OPF, whether in the rectangular

or polar

f or m,

is nonconvex because nonlinear equality

constraints and o r nonlinear functional bounds such

as

h?'" 5

h,(x)

5

,Fax cannot form a convex region.

3 .

LOGARITHMIC

ARRIER

P M FOR N L P

Though logarithmic barrier

IPM's

were originally derel-

oped to solve general NLP problems, research on I P M ' s

for large scale NLP has lately been motivated mainly by

the great performance of IPM's for

LP,

an area that has

received much attention and enjoyed incredible progress.

The I P M described here is of the same class of algorithms

as in

[7-91.

Though the IPM derivation is quite general,

implementation issues discussed in Section 5

are mostly

related to the solution

of

OP F in rectangular form.

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h(x);

31, S 2 , S3 and

S4

are diagonal matrices defined

by the components of

S I ,

s2 , s3 and s4, respectively; and

u stands

for

vectors

of

ones, of appropriate dimension.

Equations (18)-(22) with

(SI ,

s2 ,

s3

s 4 )

2

0 ensure pramal

feasibizity;

(13) along with

( z l ,

z1

zz) ,

z3, (z3 +z4)) 2 0

ensure dual feasibility; while (14)-(17) are perturbations

(p

0)

of

the complementar i ty condit ions

(pk

= 0).

3.1

Even though the KKT system (13)-(22) is nonlinear, its

solution is usually approximated by

a

single iteration of

Newton's method (the Newton direction is only a means

to follow the path

of

minimizers parameterized

by P I ' ) .

Applying Newton's method t o solve equations (13)-(22)'

th e following symmet ric indefinite system is obtained:

Solving for the Newton Direct ions

The first step in the I P M derivation is to transform all

inequality constraint s in the NLP problem model (4) into

equalities by adding non-negative slack vectors,

si

2

0, as

min

f(x)

s. t. g ( 4

=

0

-sl -

2- h min + h m a

0

-

0

-h(x)

- s 2 + h m a X

0

(11)

- sg

4- m i n +p a

-1x-s4+2m=

= 0

( s l , s Z , S 3 , s 4 ) 2 0.

The non-negativity conditions si 20 in (11) are handled

by incorporating them into logarithmic barrier terms, as

min j(x> ~ ' C ( I ~ S ~ ~ + I ~ S ~ ~ )p k C ( l n s 3 j + ~ n s 4 j )

s . t.

g(x> =

0

-sl - 2 -

+ p a x 0

-h(x) - s 2 + h m a x = 0

P

(1

j= 1 j=1

(12)

s3

-

s4

-

i n

+

p a x

-

0

-1 x - s 4 + j i ma x = 0,

where

p > 0

is

a barrier paramete r

that

is

monotonically

decreased to zero as iterat ions progress. The sequence of

parameters { / .A }E~enerates a sequence of subproblems

given by (12) and , under regularity assumptions (see [ll] ),

as pk

-1 o

the sequence { x ( p k ) } g o f solutions of (12)

approaches

x*,

a local minimizer of (11). The Lagrangian

function

L ,

of the equality constrained problem (12) is

L

=f x) - pk

( lns l j +1ns2,)- (lnssj +1nsl j )

P q

j = 1 j = 1

-

Tg(x)

- T

-

I

-

z

-

hmin hmax)

-

F

-

h(x)

-

2

+

h")

 

- - s3 - 4

-

i n + ax) -

T

(- Ix - 4 - a x )

where y

E Rm,z1 E R', z2

E

Rp ,

3

E Rg

and

z4

E

Etq

are vectors

of

Lagrange multipliers, called

dual variables.

A local minimizer of

(12)

is expressed in terms of a sta-

tionary point of

L,,

which must satisfy the Karush-Kuhn-.

Tucker (KKT) first-order necessary conditions

V,

L,

= V ~ ( X ) , (x ) ~ Y

+

J ~ ( x ) ~ z ~P z ~0 (13)

V S L =

- / . A ' S ~ ~ U + Z ~0 (14)

VS,L,

=

-/.A'S,'U +

1

+ ~2 =

0

(15)

VS,L,

= -/.A'S,~U+

~3 0

(16:)

V S J

=

-/PSL1u

+

~3

+

~4 =

0

(17')

v, L,

=

-g(X)

=

0

(18)

VZ,L,

=

-SI

-

2

-

h"'" +hmax 0 (19)

V,,L, =

-h(x)

- 2 +hmax 0 (20)

V,J, =

= 0

(21)

V q L , = -1x -

4 +

= 0 (22)

-sa

-

4*- xmin +

where Vf(x )

E

R" is the gradie nt of f(x);J g (x ) E

RmX '

is the Jacobian of g(x);J ~ ( x ) Rpx s the Jacobian ctf

-ps ;2

0

0

0 0

0 I

0

O p S , 2 0 0 0

0

I I

0 0 p s ; Z o 1 0 0 0

0 0

O p S , 2 I I O 0

0 0

I

I 0 0 0 0

0

0 0

I 0 0 0 0

I

I O

0

0 0 0 0

0 I O 0

0 0 0 0

0 0 0 0

O P O J ;

- 0 0 0 0

0 0 0 0

0

0

0

0

0

J h

a

P

V 2 ,

-J,

X

0

0

0

0

0

0

0

0

0

-5;

(23)

where

m P

V i L , = H ~ ( x ' ) $Hgj (x ) + xZjHhi

(x').

The Newton direction can be obtained by solving

23)

directly or,

as

shown in [15], by solving the reduced system

j=1 j=1

for

A x

and

A y

first, and th en computing

As2 = - J ~ ( x ' ) A x

As1 = -As2

Az,

=

-pk(S )-2As1

AZZ= --~'(S~)-'AS: Az,

As4

=

-1Ax

AS^ = -As4

Az3 =

-p ' (Sk)-2A~3

A z 4 - / A ' ( S ~ ) - ~ A S ~

A z ~ ,

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where

J,

=V$L, +

pkJh(~k))'((St)-2

+ (S:)-')Jh(xk)

+ p k P ( ( s y+

( S y ) ?

(26)

r = v f x k ) , ( X ~ ) ~ Y ~J ~ ( x ~ ) ~ z ~

FZ:.

The evaluation of VBL, and J, involves a combination of

the objective function Hessian H ~ ( x )nd the constraint

Hessians

Hgj x)

and H h j (x ); hese Hessians, except for a

few entries, are constant in the OPF problem considered.

3.2

Updating the Variables

The new primal and dual variables are computed from

xk+ =xk + ctkA X

sf+ =

s f + akAsi ,

for i

=

1 ,2 ,3 ,4

y k + l

=

k

( 2 7 )

y

+cvkAy

z:+'

= zf + akAzi,

for

i = 1 ,2 ,3 ,4

where the scalar ab E

(0,

I] is the

step length

parameter.

3.3

Computing the Step Length

The maximum step length cvk in the Newton direction is

determined by

a

=

min

{yayaX,

aFax,

I.o}.

(28)

The scalar

y E

(0, l) is a

safety factor

to ensure that t he

next point will satisfy the strict positivity conditions;

a

typical value is y = 0.99995.

3.4 Reducing the Barrier Parameter

Although the duality properties of convex programming

cannot be fully extended

t o

general NLP, it is natural

to

ask

whether successful schemes used for reducing p k in

LP or convex QP could be extended to NLP. This has

been done in [7,8], and similar approach is used here.

The residual of the complementarity conditions, called the

complementarity

g a p , at

the current iterate

is

pk

= Z I ) T kI + (z,"+ z,")'.," + (z,")'s," +

(z,"

+Z,")'S,".

The sequence { p k } g o must converge to zero, and the

relationship between p k and pk, mplicit in the conditions

(14)-(17), suggests th at p k could be reduced based on a

predicted decrease of the complementarity gap,

as

where

8k

s

the

expected, but not necessarily realized,

cut in complementarity gap. The parameter

p E

(0 , l )

is called the

centering parameter

and is interpreted

as

follows. If Pk

=

1, the KKT system (13)-(22) define

a

centering direction, a

Newton step toward a point

at

the

barrier trajectory. A t the other extreme,Pk

0

gives the

pure Newton step, sometimes known

as

the

afine-scaling

direction.

To trade

off

between the twin goals

of

reducing

p k

and improving centrality,

pk

s dynamically chosen

as

Bk = max{0.95Pk-', O . l} ,

with 8

= 0.2.

3 5

Convergence Criteria

The I PM iterations ar e considered terminated whenever

.1

I

€1

Pk

I

cp

.;

5

€1 IlAXllaJ 5 EX

.

5

€1 llg(x"Ila3 L

€1

. 5

e1

u 5

€2

or

is satisfied, where

d k =f ( x ) -

yTg(x)-

z;(gmax ^min) - zqT(2jmax

2)

- zT(hmax-

hmin)

F(hmax- h(x))

where

ew =

IOv8,

eZ = l o p 4 , €1 =

and € 2

=

1 0 - 2 ~ 1

are typical values. If criteria

vf

5 €1 , ut 5 €1 and

u t

5 E I

are satisfied, then primal feasibility, scaled dual feasibil-

ity and complementarity conditions are satisfied, which

means that iterate k

is

a K K T point

of accuracy

€1. When

numerical problems prevent verifying this, the algorithm

stops a s soon

as

feasibility of the equality constraints is

achieved along with very small fractional change in the

objective value and negligible changes in the variables.

4.

PREDICTOR-CORRECTORROCEDURE

Mehrotra [14] developed

a

predictor-corrector procedure

which greatly improves the computational performance

of primal-dual IPM's. Rathe r than applying Newton's

method to (13)-(22) to obtain correction terms to the

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current estimate, the new point, say,

w k + l = w k

+

A w ,

is substituted into

(13)-(22)

directly, t o obtain

0 0 0 0 0 1 0

0

[ D o ' D z O

0

0

0

I I 0

O D 3 0 1 0 0 0

0

0

O D 4 1 I O 0

0

0 1 1 0 0 0 0 0

0 0 1 0 0 0 0 I

1 0 0 0 0 0 0

0

I O O O O O O J h

0 0 0 0 0

JE

V;L,

n

0 0 0

0

0 0 0

-Jg

0

0

0

0

0

0

0

0

0

- J i

X

X

A@

A s p

A s p

A z F

AziE

A$'

A Z ~ ~

Axsff

AJI"ff

and then an estirnate paff or

pk+'

is obtained as

This chooses

pa

t o be small when t he affine direction

produces a large decrease in complementarity and chooses

paff o be large otherwise. The actual new ste ps are chosen

as the solution to

(30)

with the right-hand side vector

where

D1

=

ST'Z1,

D2 =

S,'(Zl +

Z,), D3 =

ST1Z3

and D4 = SZ'(Z3 + 2 4 ) ; and Z1, Z Z ,

Z3, Zq,

A S , , A S S ,

AS3 and AS4 are diagonal matr ices defined by the vectors

z1, z2, z3, zq, As l, A s2, As3 and Asq, respectively.

The major difference between Newton systems (30) and

(23) is th at t he right-hand side vector of

(30)

cannot

be determined beforehand because of the nonlinear delta

t erms .

To solve

(30),

Mehro tra suggests first dropping the

p and de lta t e rms in the right-hand side vector and then

solving

for

the (pure Newton)

afine-scal ing direct ion

-z1

-8 1

-

2

-83

-23 -

4

-s3

- 4

-p Em""

-1x

- 4

+ -imax

-sl - 2- h""

+

hmax

-h(x) - z +h"""

-Vf(x) +Jg(x)2y - J ~ ( X ) ~ Z Z

F z ~

d x )

using the same system matrix in (30). These directions

are then used in two distinct ways: (a) to approximate

the

de lta t e rms

in the right-hand side of (30), and

(b)

t o

dynamically estimate the barrier parameter p.

To estimate p , the standard ratio test (28) is performed

to determine the step th at would actually be taken

if

the

affine direction given by

(31)

were used. An estimate

O F

the complementarity gap is computed from

paff

= (zf

+

aaffAz:ff)T(sf

+

aaffAsyff)

+ (z: + zi +

aaff(AzYff

+ A@))'(s," +

aaf fAsi f f )

+ (zt + a a f f A z g f f ) T ( ~ $aaffAsgff)

+ (z$+ zt + a a f f ( A z g f f A Z ; " ) ) ~ ( S ~aaffAsifffi

The addi tional effort in the predictor-corrector method

is in the ex tra linear system solution t o compute the affine

direction, and the extra ratio test used to compute

paff,

since the predictor and corrector steps are based on the

same matrix factorization. Wha t is usually gained is re-

duction in iterations and solution time. The higher-order

terms h ( A x ) and g ( A x ) , which differ this procedure from

that in [7] to solve the polar OPF, are computed only for

th e case when h(x) and

g(x)

are quadratic. Otherwise, it

would be too expensive to compute these terms.

5 . COMPUTATIONALMPLEMENTATION

An outline of the

OPF

solution procedure is

as

follows:

STEP 0:

Run any load flow program and obtain the bus

voltage rectangula r coordinates.

STEP

1:

Set k I = 0, define bo and choose a starting point

that satisfy the strict positivity conditions.

STEP 2: Form the Newton system

at

the current point

and solve for th e Newton direction.

STEP 3:

Compute the step length a in the Newton di-

rection and update primal and dual variables.

STEP

4: If the new point satisfies convergence criteria,

stop. If not, set IC = k +

1

and go to STEP 5.

STEP 5: Compute the barrier parameter p k and go back

to

STlEP

2.

5.1

Star t ing Poin t

Although the sta rting point needs only to meet the strict

positivity conditions, the IPM performs better if some

initialization heuristic is used. Th e following heuristic has

0 Estimate xo as given by a load flow solution, or as a

flat start using the middle point between the upper

and lower limits for the bounded variables.

been

implemented:

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0

The primal slack variables are next initialized

as

s:; = min (max(O.l5h?, hi(xO)- h?’”), 0 . 8 5 h f ) ,

s3O = min

(max(0.155

s o

-

ha

- 3 0

2; i 1;1

?yin),

.85i?F),

‘ l j

- - A

j

-

s:j.

where h a = , Fax

-

hFin and

zf

I

- zyin,

0

The dual variable yp is set to -1 if associated with

0 The dual variables zy,

zg,

z: and 22 are given by

(5),

or set to 0 if associated with (6).

z;

=

p o ( s y u ,

z

=

p O ( S y u-

z:,

2; = po(s;)-’u,

z:

=

pO(S,o)-’u

-

5.g

Forming the Newton System

Each bus has three Hessians in the composition of BiL ,.

One Hessian associated with constraint

(5),

another with

constraint (6),

(7)

or

(8),

and another with constraint

(9) .

Given the structure of these Hessians, V:L, is computed

by

a

single search of the non-zeros of G and

B, as

0 for each j

E

{ 1 ,2 , . . ,IN/} nd all i E Nj > j , compute

V2Le j , e j 2 G j j X y - j j X g +A; ) ,

(32)

(33)

(34)

V:L,=i,ej

Gij AY +A;) - Bij(A4+ A;)

V:Lfi,ej =

Bij(X

- A

+Gij(Ap - A:)

0 for each tap

tij

and all 2 E

{ e i ,

e j , i

fj, t i j} ,

compute

e f t

where A is the negative of the

y(.)

associated with bus

j

in (5) A: = 1);A4 is the negative of the g . ) associated

with bus j in (6), or the x 2 ( associated with bus j in

(7) or (8);

and

A3

is the ~2~~ associated with bus

j

in

(9).

The mapping

(y,

z2 ++ (Ap,

X q

A’)

from “constraint

multipliers” to “bus multipliers” reduces significantly the

number of logical operators

for

efficient implementation.

The Hessian VZL, is symmetric and highly sparse, with

a block sparsity pattern similar to t ha t of the load flow

Jacobian. Marks

Q, Q

and in Fig. 1 show the locations

of the elements computed by (32)-(34); total In/(

+

2 / 8 1

non-zeros,

as

in matrix

B.

Mark

@

shows the locations

of

the elements computed

hy

(35) ; come

t o

517) non-zeros.

The polar version computes

2((N[ 231-

IN1

-

1) extra

elements; marks

0

0

nd

@

show their locations. T he

number of

floating point operations

(flops) to compute

V;L, is

5(NI

+ OlBl+

96 Iq in the rectangular form and

14 Nl+ 29181+ 61171 151N1I in the polar. In practice,

the variables ( Z Z ,e , ,

y)

or Z Z ,v,

0,

y) are arranged in a

special way, (A:,

Xf ,

A fi

e i ) or

A A vi,

d i ,

to form

5 x 5 or 4 x 4 blocks, similar t o the Newton’s OPF [4],

allowing for efficient ordering and block factorizat ion. The

number of blocks totals IN1 + IBI.

Fig.

1:

Location of non-zeros computed to form V:L.

5 . 3 Solvzng the Linear System

The computational effort

of

each iteration

of

the

I P M

is

dominated by the solution of (23) or (24). Therefore, it

is

vital t o consider efficient methods for their solution. Di-

rect methods usually consider the

normal-equations

form

or the

aogmented-system

form [20].

Since matrix J, in

(24) has non-diagonal sparsity pattern, the augmented-

system form is the suitable form, in which the sparse

symmetric indefinite system is usually solved by a Bunch-

Parle tt factorization [20]. The ordering/symbolic factor-

ization phase attempts to choose

a

pivot ordering that

will lead to low fill-ins. When the actual factorization

is computed, interchanges that alter the predicted pivot

sequence may be required to retain numerical stability.

Once a stable pivot order has been determined, i t is reused

at subsequent iterations

as

long as

i t

continues to give

satisfactory factorizations and solutions. Effectiveness of

inexact

Kewton

directions,

as a

result

of

early termination

of an iterative method, is under study using WATSIT-B

(Waterloo Sparse, Ite rative Matrix Solver). WATSIT-B is

designed to take advantage of block structure in matrices.

5.4 Preventing Numerical Ill- Conditioning

Each binding constraint drives one of its slacks to zero,

which may cause numerical difficulties

as

p k -1

0. To

help preventing such difficulties, the bounds are perturbed

as h m i n

8/18/2019 007362Asda31 (1)

7/8

1217

Problem

Set

IEEE30

voltage limits are set as 3~5%ff nominal values for load

buses and k2 off specified values for generator buses.

The tap limits are set

as

&l o% off nominal values. To test

algorithm robustness, solution difficulty is increased by

choosing small sets of buses eligible for shunt var control

and setting tight reactivepower limits for generator buses.

Table 1 displays the test set and dimensions of the

NLP problems. Each problem has been solved by four

OP F codes, which combine the rectangular and polar

OP F versions with the s tanda rd IP M of Section 3 ( R P D

and P P D ) and t he predictor-corrector IPM of Section

4

(R PC and PPC ). Table 2 lists the number of iterations

to convergence, the number of binding constraints, and

th e percentages in power losses reductions. In all in-

stances the algorithms performed extremely well, with the

number of i terations insensitive to th e dimension of the

problems, even

as a

large number of binding constraints

is present. The polar and rectangular O PF versions have

converged with the same number

of

iterations. Table

3

details the convergence for the largest problem. Fig. 2

shows the objective functions provided by the four codes.

Table

4

isplays th e number of flops to compute V;L,,

the number of non-zeros in V:L,, and the number of non-

zeros in J,, for th e rectangular and polar versions. The

overhead incurred in MATLAB generally makes it slower

and hence CPU times are not compared. Though the

computation of

V? ,

in polar form profits from terms,

in J, and Jh, it still requires nearly double the flops the

rectangular form does. The pitfall of the rectangular

O F F

is the handling of voltage bounds

as

functional bounds

The implications can be easily examined in (26) , looking

at the extent

Jr x)

differs from

VZL,.

Observe that the

voltage bounds in polar form, as part of 2, simply affeci,

the diagonal of O:L,, while in rectangular form,

as

pari,

of

J ~ ( x ) ,

ontribute with new non-zeros to

J,

whenever

two connected buses have neighbour buses in common.

Size of Index Sets Dimensions in 4)

NI I1G1

I

I l

I

IF1

I

12311171

n

I m I

p

q

30 I 61 41 201 411 4 631 491 4 0 ) 4

-

IEEE 300-Bus System - Rectangular version

IC

~

I

2

4 6 8

10 12 14

16

420‘

IPM iterat ions

IEEE300-Bus

System

-

Polar version

fbk )

U1” U3”

P”

a

d(w”)

V2”

U4”

Pn

an

4.599E 00 3.699E 00 4.598E-01 8.620E 00 -

IPM iterat ions

”

Fig. 2: Objective function of IEEE 300-Bus System.

-2.548E OO 2.382E 00 1.554E 00 2.000E-03 -

4.497E 00 1.905E 00 2.546E-01 4.781E 00 3.9893-01

3.147E-01 1.2293 00 9.300E-01 9.984E-04 4.831E-01

Table 1: Problem set and dimension of NLP problem 4).

4.140E 00

4.27OE OO

4.202E 00

4.2663 00

1.821E-02 3.231E-02

1.767E-06 5.370E-01

9.080E-03 3.696E-03 7.042E-02 4.346E-01

7.673E-03 1.587E-02 7.815E-06

5.775E-01

3.2163-03

1.843E-03 3.Sl6E-02

3.921E-01

7

5 45

80

15

128

101 69

15

l ~ ~ ~ ~ k l;; 1.541

121

5211861 91 244 169i 1841 9

IEEE300

300 69 12 219 411 50 649 518 381 50

1o

Table

2:

Iterations, binding constraints,PLOSSeduction.

IEEE57

IEEE118

IEEE-300

4.232E 00 3.2643-03

7.924E-03

3.512E-06 5.739E-01

4.265E OO

1.584E-03 1.147E-03 2.19lE-02 2.867E-01

4.244E 00

1.8233-03 4.946E-03 1.9693-06 4.408E-01

4.2633 00 1.6033-03 6.1923-04 1.184E-02 3.478E-01

4.262E 00 8.6603-04 2.676E-03 9.582E-07 5.296E-01

Table

3:

Solution of

IEEE

300-Bus System by R P C code.

4.031E OO

3.947E-02 16.116E-02 13.724E-05

4.989E-01

-+- 4.278E OO 1.143E-02)7.536E-03)1.433E-01~.231E-01

Table 4:Number of non-zeros and flops.

Problem ~ ~ O D S

L,,

I Non-zeros in

VfL , , and

J,

set I Rect. I Polar I Rectannular I Polar

Some of the fill ins caused by functional voltage bounds

co-occur with

fill

ins caused by reactive-power constraints.

Hence, functional voltage bounds have little effect on the

factorization cost,

as

evidenced in Table 4. Branch flow

constraints in rectangular and polar form are alike.

8/18/2019 007362Asda31 (1)

8/8

1218

7 .

C O N C L U S I O N S

Tests performed under MATLAB confirm that I P M can

directly and efficiently solve nonlinear OPF problems, and

that computational performance is equally good in the

rectangular and polar variable spaces. Advantages of the

rectangular form explored in the paper are eaSe of matr ix

setup and incorporation of higher-order information in a

predictor-corrector procedure. A disadvantage is the need

to handle simple voltage bounds as functional bounds.

The research being carried out t o further improve the

I P M

performance considers different values of p for groups of

inequalities, and computation of cy and reduction of p

based on a suitable merit function to balance reduction

in infeasibility and in complementarity; this along with

more elaborate starting point choices can prevent non-

negative variables from becoming too close to zero

at

an

early stage,

a

condition that degrades

I P M

performance.

An approach t o handle discrete variables in the context

of

IPM's

for

NLP

is under consideration. The key idea is

to solve modified problems, yet continuous, derived from

(12)

by appending the conditions s1;sz; = 0, for all

i

E X ,

and s3;s4, = 0, for all

i

E X here 'H and X are index

sets of the discrete components hi(x) nd

Zi,

respectively;

none new variable is appended.

These conditions would

enforce a discrete solution, if one exists, by ensuring th at

one of sli

or

sZi, for each

i

E 31, nd one of s3; or sqi,

for each

i E A ,

become zero.

To

deal with multiple step

discrete variables a Phase I/Phase I1 solution scheme is

advocated. In Phase

I,

multiple step discrete variables are

treated

as

continuous. Let xi be the Phase

I

solution. In

Phase

11,

the limits are modified

so

that

,Fin

is the largest

discrete value smaller than

hi x*), hmax is

the smallest

discrete value larger th an

hi(x*),

nd so on. The approach

outlined

is

a sort of optimal rot .@ procedure, at best,

since discrete programs are inherently combinatorial.

ACKNOWLEDGMENTS

The authors would like to

thank

CAPES a n d UFPE

in Brazil

and NSERC in Canada for their f inancial support . Than ks

also

go t o

Dr.

Claudio C G i z a r e s for providing

the data

files.

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BIOGRAPHIES

VICTOR QUIl\iT.&?rA: Received the Dipl. Ing. (1959), M.Sc.

(1965)

and Ph.D. (1970) degrees

in

Electrical Engineering from the State

Technical University of Chile, the Universi ty of Wisconsin, Madis on,

and the University of Toronto,Onta rio, respectively. Since 1973 he

has been with th e University of Waterloo, Depart ment of Electrical

and Computer Eng., where he

is

a full professor. Dur ing 1990-1994

he served

as

a Dept. Associate Chairman for Graduate Studies.

GE RAL D0 T OR RE S: Received the B.Eng. (1987) and M.Sc.

(1991) degrees in Electrical Engineering from the Universidade de

Pernambuco and the Universidade Federal de Pernambuco (UFPE)

in Brazil, respectively. Since 1992 he

has

been with UFPE, where

he is

a

lecturer. He is currently

a

Ph.D. student in the Department

of Electrical and Com puter Eng., University

of

Waterloo, Canada.