A multi-level graded-precision model of large scale power systems for fast parallel computation

11,pp. 325-330,

Math1 Comput. Modeliing, Vol. Printed in Great Britain

.4 MIJLTI-LEVEL SYSTEMS

GRADED-PRECISION

FOR

G. Huang,

FAST

A. Abur,

Department

of Electrical USA

Ah&act. cality”

The

of the

algorithmic problem

power

is novel

system

arrangement, statement,

Elimination has

ings.

folds.

of larger

with

describing

less critical

level.

This

is disaggregated

algorithm

used

numerical

algorithm

of .lacobi these

at each

solut.ions

approximate

ods do not provide This

feature

part

of the

overall

system

Kevwords. tems;

methods

solution

the

Also,

fhis

modelling;

authors’

approach

one

computat.ional

tion

techniques

the

modelling

of obtaining

bark,

1948;

in the

between

modeling

one hand,

been

Zaborzsky

is novel

used

niques

are devised

modeling

node

gated which dure

t.o be used as the initial has a more starts

from

accurate

is that

of accuracy,

and

the

meth-

is completed.

a reasonable

scale

solution

convergence

systems;

in

of t.he

power

as a few folds has been

approach interface

solution

solution model.

sys-

i = 1 to i = M -

1.

of system

proce-

Speed-up

as

areas

such

wit.h network side,

325

the idea of aggregation/dis-

complexities

type

approaches that the

But. other

as optimal

load

constraints, numerical The

even though

sequential

study.

and distributed asymptotic

Jacobi

algorithms,

On the which

is

comput-

computational

among

type

that

ap-

and st,ate estimaalgorithms

are compared and

pot.entiaI

flow, economic

and are under

for parallel

time

among

the

algorithms.

to power flow

suitable

conclude

also dominates

issue in power

ing are re-evaluated.

lth

to

dominates

which is the most fundament,al

implement.ation mostly

not only

analysts,

will pro-

interesting

with application

tions are self-suggesting

at the

portion,

we discuss modelling

dispatch

tech-

for small size levels

It is also

day t,o day operation.

plication

more

of the numerical

on the

at the (i+l same

the interests

obtained

with

savings.

the critical

system

are disaggre-

The

larger

that

analysis

are computed

in parallel

systems

on the

and solution

the results

even

aggregation

(Kim-

the

larger

In this paper,

in

The

Solutions

to node

ith level of ~nodel and then

time

formulation

numerical

together.

from

type

for the choice

of t.his method

speed of convergence

aggrega-

systems

it dilutes

and problem

In this approach

a Jacobi

comput.ation

large

notice

t.o solve large

for a long

power

that

other.

one-pass),

of numerical

parallelly.

duce

utilizes

even though

scale

and algorithmic

independently

which

et al., 19851).

sense

for the (i + 1)th

degree

accelerat.es

graded-precision;

computational

here,

techniques

of large

model

while the traditional

the overall

is no antecedent

system

problems,

have

sav-

(; + l)th

only (i.e.

argument

advantage

speed

much

proposed

aggregaticn/disaggregation scale

estimate

implen~entation.

systems,

there

of t.he power

to the

t,o the

is independent,

solution;

approach

even larger

down

can be top-down

The

comput,ational up computation

and an aggregated

passed

procedure

and

to solve

processing.

knowledge,

in the literature

algorithms

produce

portion

unless

INTRODUCTION To the

t,he associated

to node of a set of the ith

with improving

and can be implement,ed

Multi-level

parallel

should

and

of “criti-

the models

to speed

from node

to the exact

solution

and

supply

The supporting

continually

when

is critical.

levels

is discussed.

converge

attractive

work,

the parallel

at each level.

a meaningful

is vrry

syst.rm

To simplify

degree

a truly engineering

is then

This

is introduced

design

and is used as the initial

are available

solut.ions

more

solution

mu16pass).

level.

creates

procedure

is employed

t.ype numerical

approximate

POWER

St at ion.

the

model

engineers

on the critical

This

aggrrgation/disaggregatjon (i.e.

M-level methods

parallelly

model

part.

(‘ollrge

It exp1oit.s

in our preliminary

sizes with

a detailed

or can be run iteratively

their

in numerical

are execut,ed

level criticality

level - the solution

experts

found,

algorithms

the

SCALE

power flow analysis

Conventionally,

of this interface

been

Systems

Solution

LARGE

University

ways.

by constructing

and then

approach,

A&M

in various

as one unit.

this numerically. several

Texas

to the large elecf.ric

approach

which

OF

COMPUTATION

Engineering

A novel approach

evaluated.

MODEL

PARALLEL

W. K. Tsai

Texas

77843

0895-7177/88 $3.00 + 0.00 Pergamon Press plc

1988

the Newton

approaches.

We

it is well known that, Newton

Raphson

Proc. 6th Int.

326

type is superior ton

type

to Jacobi

algorithm

on the other

hand,

Jacobi

tic improvement tion.

Even

type

algorithm

Newton lelly.

worst,

algorithms

Furthermore, accelerated

eration

factors

analysis, complexity

Jacobi

than

if it. is implemented type

here,

the

paral-

algorithms

can

optimal

and it is much

of data

transfers

overheads,

dressing,

task

processor

approach

avoids

and thus

dimension

in

such

/w/

the Gaus-

roundoff

IS’

/

ad-

are consid-

as the computation

accumulate

prosimpler

memory

etc.

techniques

and as system

parallel

be

accel-

when factors

scheduling

the proposed

elimination

will not

the

and aggregation/disaggregation

introduced

Also,

have dras-

case

as communication

sian

for New-

complexity;

time

Jacobi

Modelling

by paralleliza-

by incorporating

the management

ered.

its time

complexity

has better

further

it is difficult

type algorithms

on time

in the

type

cedures

type,

1.0 improve

Conf. on Mathematical

error

progresses

increases.

‘:

Since

no parallel

all our

computing stead

machines

simulations time.

Jacobi

method.

of Gauss-Seidel made

at. this time,

sequentially.

According

Method

is used

in-

to our experi-

twice

ti

the iterations

to converge.

the convergence

To save

method

takesabout

met hod

accelerate

done

Gauss-Seidel

of Jacobi

ence,

are available

arr

No attempt

by accelerating

to

factors

is

in this paper.

Multi-level

aggregation/disaggregation

can be found in scientific as numerical

partial

(Hagemw,

1981).

grrgation

recently

(Huang

1985)

to be the fastest has been

et. al., 1986,

that

multi-

foundations

But

solvers

theoretical

available

par-

algorit.hm-texl.ure

just

1987;

proposed

by us

Zaborzsky

et al.,

tems,

As shown

in figure

is divided

into

(External

System

sents

the part

current.ly

which

analysis under

needs

in this pa-

for linear

equation For

ing judgement ple,

modified

and

it is

the

transmission tential

study.

problems

buses with cheap This

paper

second cepts

is divided

section, the

systems

are described.

derivation

analysis

In the

section.

of the

equations case.

of this algorithm The

put.er

simulation

cludes

the paper.

fifth

results

sect.ion and

con-

The third

for the two-level

implementation fourth

sections.

the aggregation/disaggregation

and modelling

explains

into six major

section

for power The

parallel

is discussed includes

the

final

in the

some

com-

section

con-

priority

since

reasons.

in the

PRECISION

General (Off-line

Ai-Level

Amreaation/Disagcrenation

Preparation

Stage

Thus

system

the

(z t

refinement (backbone

This

model

data

are stored

t,em data

them

drpends generation

and transmission connected

up 1.0

for the

but

the

identity

arc suppressed; terms

l)th

are kept of those

they

are ag-

in the backbone

level

system),+,

backbone

system.

system

2 (backbone

on the maintenance

is a

system.

system),.

off line and all the system

in the memory be updated

the third

Ihe whole system.

lines and buses

of the ith level backbone

schedule

first.

at. full capacity

similarly

cont.ains

is constructed can

po-

Big generation

run

buses

those

major

to the first level are ranked

syst.em,

as modified

that

and

have

or indirectly

of transmission

in the external Note

GRADED

tiers

backbone

further MODELLING

Thosr

until the Afth level which

gregated

MI,LTI-LEVEL

usually

We rank

identity

which

cost are also ranked

are

are dire&y

level.

For exam-

buses

generation

for economic

The

experience.

the first.

lines which second

The im-

to engineer-

are ranked they

the two or three

system.

lines:

of

and the rest.

according

and buses

and repre-

the importance

overall,

the external

transmission linrs

System),

where

1.0 the ith

and operating

major

System),

(Backbone

of dat.a is evaluated

sys-

to be

)t. The

is called

say i, the system

(Backbone

of the system

of the syst,em portance

1987).

and Huang,

1, at each level,

two parts:

is ranked

is the case for power

in (Tsai

equations,

the

are not discussed foundation

are available

nonlinear

A~reBation/Disaggreaatioll Modelling

knowledge

Theoretical

Al-level

concepts

1985a).

per.

1.

FIG.

on aggregation/disag-

while a even faster

decomposition

such

solvers

concluded

Hackbush,

1977;

known

allel algorithm,

lit.erature

equation

which are based

(Brandt,

are, generally based

differential It has been

grid type solvers

arguments

computation

in advance.

from

time

The

sys-

to t.ime which

schedules,

and load forecasting

line outages, etc.

Proc. 6th Int. Co@ on Mathematical Model&g

C”mput.ati”nal

Procedures

of Multi-level

i?p.Rre~;ation/Disa~r~~atioll Online

Computation

The

computation

sion

model

either

Top-down

on multilevel

graded

be top-down

The

only

algorithms

preci-

or it.erate

are as follows:

phase bus.

1) Initialize

the estimate

2) Compute

the power

Step

3) If i < M,

from

2) to (i + 1)th

step

disaggregat,e

apk -

the e&mate

Sfep

2) Compute

the

at i = 1, set

Step

3) If i < hf:

from

ith level.

power Step

level

i=i-1,

a preselected

solution

go to 2).

Step

to (; -

For simplicity

level,

set

is discussed.

real

aggregation/disaggregation

all the

straight

Flow

where

flow solut.ion gorithm. it works

the

matrix areas

The

entries

the equation

about

out that

even

worst

reader

V to see the performance

t.er linear

refer

experience

syst,em

flow solution

approximation

for this can

magni-

_B’

ent.ries

0

are as defined

B’ according

to the

back-

gives:

equat,ion which

as:

area

bus numbers

2) =

I! 21J

In real-

of a system, at a standard

will result

and further

are defined

i : Ertcrnol area bus numbu j : Backbone bus number

case

to section

of the algorithm.

on the operating the

whose

BL(i, .i) = Bb,(j,

the system

It turns

power

C p 2

of the submat.rices

i, j are external

aggregation/

on our al-

we can linearize

and

done

its influence

it.y, based

angles

is

are

We linearize

no knowledge

the

of the

forward.

t.o study

well,

reciprocal bus k and j.

agGen-

and

very

or

and external

power flow equa-

step t,o derive

equations.

af @ = 0 to assume

negative

B:,(i, j) =

aggregation/disaggregation

t.ions as an indermediate

,qj)

C”S(& - 6j)

on the

B’ is the mat.rix

Equations

we will derive the linearized

disaggregation

cos(Ok-

above. Partitioning

only two-level

method

Power

B;,

AND

is relatively

Active

aP/ati

EQUATIONS

of the presentation,

power

The

as 0 = 0 and I7 = 1.0 yield:

ae

in this paper.

linearly,

loading

is

go to 3).

to M-level

Linearized Since

rate

aggregate

is the

i?P - = -B’

the

bone

gregation/disaggregation .41so, only

of the

in simula-

becomes:

v, BLj.

-vk

approximations

tude of voltages

5) Stop. AGGREGATION

presented

1) th

Otherwise,

DISAGGREGATION

eralization

value,

=

= 2

the solution

set i = i + 1. Compute

4) If i 2 2 and the convergence than

are included.

of the iine connecting

Further

flow solution

method.

disaggregat.e

reactance

flow solution.

smaller ith

power

numerical

B;,

where

Method

i = 1. for a designat.ed

But

the solution

80,

1) lnitialize

t.he resistance

i#k

set i = i + 1 and

Ag~regation/Disa.~regation

Step

resistances

of t,he Jacobian

method.

4) stop.

iterative

power injection

completely.

2 =$ vkti,

flow at the ith

go to st,ep 2). B.

at bus k with

at i = 1. set

numerical level,

net active

is omitted

the line

i = 1. Step

angle

of present,ation.

lines

submatrix

level for a designat,ed

to the swing

For simplirit,y system tion,

only:

Step

01. is the bus volt,age respect

Pkrpec is t,he specified at bus k.

St,age

based

can

over the M levels. A.

Model

327

in bet-

accelerate

the

convergence. The expression k is given

for the mismatch

active

i, j are backbone

power at bus

by:

®ated

,=1

External

Considering

the

and again

neglecting

active

power

Yk,

=

IYk, lL6~j

admit t ante

is the

kj th ent,ry of the bus

matrix.

vk is the bus voltage

magnitude

at bus k.

bus numbers

Iniections

II-model

for the transmission

the resistances

lines

of the lines,

the

flow from bus m to bus k, measured

at

bus k can be written

where

area

as:

Proc. 6th Int. Conf. on Mathematical

328

Hence,

the

linearized

power

injection

due IO the line flows coming buses

directly

is given

c.onnected

from

at bus

i,

the external

t,o bus i, i.e.

from

I’,

the

specified

and is a fixed

‘. ik,

backbone

gated

injection

buses,

equations

the linearized

can

be written

aggre-

in matrix

nal bus

angles

solution

values

F =

[Fb]i@b]

FT = [PI Pz .

where

linearized denotes

+

vect.or

is followed

volt.age

magnitudes:

of order

system

Nb,

and

T

6

where

bus

angles

step).

is solved

,

here

by a correction

the

equation

/ is the specified where ILT8s”pc

magnitude

for the volt-

the equations

in rectangular

the real and reactive

coordinates

and

parts:

for i # j,

neighbors

for the

of backbone

i is the backbone

where

unaggregated

for i = j,

m is the bus index

immediat,e

at the

age. Writing

=

E,

the exter-

backbone

P - 8 problem

only

equation

separating j)

i)

i;’ = {i;~li:~)~r;‘T

the transpose.

Fb(i

at their

the

iteration

above

[FeI[Qel

P,v,] is the backbone

bus injection

while

at each

the

bus

at bus

injection

computation,

are fixed

Since, for the

form:

at the value

net react,ive

bus i (in this

are updated

system

injection

value.

Q, is the computed

by

For t,he backbone

net

LAP, is t,he mismatched

where

area

j = ir,

Modelling

and j is the external

bus i.

bus number

area

bus number.

[Fb), [F,] are Nb x ,%rband N, x A’, matrices, and Nb, AV~are number nal area

buses

which

Fb(i,

is fixed

at the first

j) is the linearized

algorithm

and

injection

[6’s] is changing

from

iteration

Flow

except

Equations

are valid also for the aggregated

for the following:

G,r. +lB,k will now be the (i. I:)th entry entire system bus admittance mat.rix.

sys-

for the Jacobi

Power

The above equations (1)

of ext,ernal

level of comput.ation,

equation

Second-Level

solution

of the line ;j.

[Fe][Be] is the aggregated tem

and exter-

respectively.

: reachnce

r*)

of backbone

of the

P,, Qt will be the net act,ive and reactive power either specified (P, ) or comput.ed (Qi) based on the most recent values of e,, f,.

(2)

type

injections

to it-

eraCon. L&aggregation First

Level

Using

Power

the

gregate

Flow

Equations

The

iterative

obtained

Gauss-Seidel

equation

can be written

at the

first

algorit.hm,

level

the

ag-

P,-_iQ,

-

”

sir,,

: i #

slack

where

i

= t, + jf,

is the complex voltage of bus i. = G,k + jB,k is t.he i, k’th entry of the

bus admittance

matrix

of the isolated

backbone

Now

+F,(B;,)-I P,“PC is the modified

active

at

injections

i (linearized

& is the backbone

first

level.

that

can

those

eb

A P,

power

bus injection

are

subtracted

be computed angles

or up to several tion

of model,

the

data

obtained

is a piece locally

be

directly

tiers

depending

and

thus

both

However, fixed

only involves

number local

on the

are

fast decreasing of t.iers.

Thus

information.

node,

connected

it can be approximated rows with

from

the

of information

at each

either

8, are

to bus b construc-

locally

[B),]-’ is precomputed base. [BLe]-’ is in general

ing sparse along

solution

APJV,

Note

bus

information.

p, = p,sp=- (Fb - Fe(B:,)-‘B:,)

bus

level can easily

from

bus

where

system.

Angles

at the second

for one iteration

. i;

id%

j;,

of External

estin1at.e

as:

7

i;

initial

and

available stored

in

a full matrix.

by a matrix nonzero

haventries

the multiplication

Proc. 6th ht.

BASED

PARALLEL COMPUTING MULTI-LEVEL

CRADED

Conf: on Mathematical

procedures

ON

The

Choice

of Numerical

It is well

known

ture,

a parallelly

that

algorithm

PRECISION

TABLE

Raphson

power

analysis

several

putational son

not the best computers.

1982)

(Alvarado,

it will become munication steps.

at least

is required

When

if the

mension

size.

Thus

O( n log n).

Besides

the system

dimension

Elimination cessor

scheduling.

plexity

of Jacobi

plemented

the

sequentially.

reduces

to O(n).

is used.

it will reduce

hand,

factor

may

be a realistic

are tried

factors

on a system.

need

any processor

rors

do not

Thus

Jacobi

since

Jacobi

type

scheduling

type

tains

it,

algorithm

system

with 2 levels

III:

algorithm

do not

algorithm

is preferred

er-

syst.em

57bus

that

Note mission

processor the

choice

since

can be easily cessors. number seven.

mapped

to the New-

syst.em

the

a clear

should

fift.een connections of connection

To adapt

power

system,

cessors

should

through

the

number

the

suffice.

since

interconnections power network

from

among

in this

there

processors

these

when heavy

four

to

of the the

pro-

purpose

will

will be enough to represent

topology.

the

transact.ions

SIMULATIONS

effect of aggregation/disaggregation

weak trans-

improves.

This

backbone

system

out. to be very

problem,

weak

it is important

t.ime to time,

of power

nat-

Since

especially

are involved.

CONCLIJSION A new duced

multilevel

graded

in this paper.

convergence

r&e.

model

it is also

Furthermore,

continually

available

with increasing

precision

This

mentable.

mode

easily

parallelly

approximate

for engineering

accuracy

is intro-

not. only improves imple-

solutions

are

use at each level

as the algorithm

progresses.

REFERENCES Alvarado,

F. L. (1976),

in Power

paratus

“Computational

Systems,”

IEEE

July/August.

A., (1977).

Harkbush,

W.,

(1985)

No.

Ap-

4, pp.

Adaptive

Solutions

Math.

of Camp.,

Problems,” April

1977.

“Multi-Crid

Springer

Complexity

OR Power

1976.

“Multi-Level

31. pp.333-380,

plications,”

Trans.

Vol.PAS-95,

and Systems.

1028-1037, Brandt. Vol.

The acceleration

con-

level contains

viewpoints.

from

t.o Boundary-Value SOME

the

turns

overflow lines

paral-

a hypercube

} processors

how

This

If the maximal

is k, then

of buses

network,

among

The

connections.

by general

connection number

buses.

t,his can be implemented

inter-processor

first

effect

engineering

of probus has

connection

such as the hypercube.

of bus

max{P,

ranges

interconnection

be irregular,

first level

syst.em cont,ains

indication

system

each

to the other

usually

It can be also implemented

with

to be a mat,power

network,

to the irregular

swit.chable

lel computers

seems

of the

power

-2 levels:

acceleration

be picked.

from

t,o monitor

t,o t,he interconnection

For a realistic

less than

array

topology

with 2 levels:

if t,he backbone

lines,

gives

progress.

Consideration

A reconfigurable

first level con-

9 buses.

ural

ural

:

the experience

method.

Architecture

with weak-

6 buses.

lines have potential Computer

test systems

6 buses.

tains

This

and the rounding

as the

IEEE

11: 30.bus

of n. If an accelerfurther.

system

I: 14-bus

will grow when more runs

accumu1at.e

ton Ra.phson

modified

when imparallelly

it will be reduced assumption

on the acceleration

Cases:

to O(log n) if the convergence

is used,

) 2 I 5.71 / 1.51 /

ened lines

aggregation/disaggregation

at each level is independent.

ation

pro-

the time com-

is O(n*)

If implemented

When

10e3

Ratio

as

complicated

algorit,hms

Speed-up

remains increase

due t.o t.he Gaussian

On the other

with tolerance=

Solution

even

wit,h the di-

errors

and it requires

at first level

com-

modest

complexity

increases

type

only

linearly

time

of iteration

91

How-

are observed

roundoff

616

is considered,

parallelly,

this,

steps,

dimension. global

are increased

57

Peters,

Elimination

20 times

1 III I

II

/ 30

System

first level Number

(

overhead

for the Gaussian

as 5 _

processors

com-

O( n log n), since

implemented

such

Number of buses in Backbone

it has

1976;

overhead

I

14

Raph-

if no communication

where n is the system

for

The

of quasi-Newton

if t.he communication

speed-ups

However,

machine.

Model

Bus No.

algorithm.

algorithm

by

EffeyxTwo-Level

Precision

litera-

sequential

parallel

in a parallel

will be O(n)

is considered,

rate

best

complexity

Wallach.

ever,

computing

for serial

time

method

1985;

parallel

is the well accepted

drawbacks

1. Acceleration Graded

implemented

is illustrated

below:

Algorithm

in the

is usually

Newton

for the two level model

the table

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