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dur i ng t hi s per i od. For each method, t he

avai l abl e paper s wi l l be r evi ewed.

The re ference j ournal s f rom whi ch the

publ i cat i ons are extr act ed are:

I EEE t r ansact i ons on Power Appar at us and

Syst ems

I EEE t r ansact i ons on Power Syst ems

I EEE t r ansact i ons on Automati c Cont r ol

I EEE Pr oceedi ngs

AI EE Transact i ons

I EE Proceedi ngs, Par t C

El ectr i c Power Syst ems Research J our nal

Oper at i ons Researc h

I nt er f aces

Amer i can Power Conf er ence Pr oceedi ngs

I FAC Symposi umon l ar ge scal e syst ems

PI CA Pr oceedi ngs

GENERAL

The background i nf ormat i on per t ai ni ng

to

t he

opt i m zati on t echni ques are di scussed i n

several r efer ences. Some of t he r ef erences

Cohen, A. I . and Sher kat , V. R. ,

Opti m zati on- Based Methods f or Oper ati on

Schedul i ng, Proceedi ngs of t he I EEE,

Vol . 75, pp. 1574- 1591, December 1987.

Wood, A. J . and Wol l enber g, B. F. , Power

Generat i on. Oper at i on

and

Cont r ol , J ohn

W l ey and Sons, New York, N. Y. , 1984.

Hi l l i er , F. S. , and L i eberman, G. J . ,

I ntr oducti on ODerat i ons Research,

Hol den-Day, I nc. , Oakl and, CA, 1990.

Baz ara a, M S. , J arv i s , J . J . , and Shera l i ,

H. D. , Li near Pr oar amm n and Net wor k

Fl ows, J ohn W l ey- and Sgons, New Yor k,

Nemhauser , G. L. , and Wol sey, L. A. ,

I nt eaer Combi nat or i al ODt i m zat i on,

J ohn W l ey and Sons, New York, N. Y. ,

1988.

Nemhauser , G. L., I ntr oduct i on

Q

Dvnam c

Pr oaramm ng, J ohn W l ey and Sons, New

Yor k, N. Y. , 1966.

Lasdon, L. S. , m zat f or Lar ue

Scal e Svst ems, M%%?% Yor k, N. Y. ,

1970.

Luenberger , D. G. , I nt r oduct i on t o Li near

and Nonl i near Pr o r amm ng,

Addi son- Wesl ey, Readi ng, MA, 1973.

Cooper , L. and St ei nber g, D. , Met hods and

ADDl i cat i ons of Li near Proar amm ng, WB.

Saunders, Phi l adel phi a, PA, 1974.

Gi l l , P. E. and Murr ay, W, Pract i cal

ODt i m zati on, Academ c Press, New York,

N. Y. , 1981.

J ensen, P. A. and Bar nes, J . W Network

Fl ow Pr oar amm nq, J ohn W l ey and Sons,

New York, N. Y. , 1980.

N. Y. , 1990.

Ref er ence 1 i s a gener al summary of t he

t echni ques used f or thi s pr obl em Ref erence 2

i s the now cl assi c t ext . Refer ences 3 t hrough

11

are var i ous Oper ati ons Research t ext s, i n

order, whi ch t he authors have f ound usef ul .

EXHAUSTI VE ENUMERATI ON

The

UC

probl em may be sol ved by enumer at i ng

al l possi bl e combi nati ons of t he generati ng

uni t s. Once t hi s pr ocess i s compl ete, the

combi nati on t hat yi el ds t he l east cost of

operati on i s chosen as t he opt i mal sol ut i on.

Thi s method f i nds t he opt i mal sol uti on once

al l t he system const r ai nt s and condi t i ons are

cons i der ed. The f i r s t t wo paper s ar e or i gi nal

att empt s t o r educe t he probl em t o mat hemat i cal

t erms.

Ker r , R. H. , Schei dt , J . L. . Font ana, A. J . ,

J r and W l ey J . K. Uni t Comm t ment ,

I EEE Transact i ons on PAS- 85, No. 5, pp.

417- 421, May 1966.

Hara, K. , Ki mura, M , and Honda, N. , A

Method f or Pl anni ng Econom c Uni t

Comm t ment and Mai nt enance of Thermal

Power Syst ems, I I EEE Tr ansacti ons on

PAS- 85, No. 5, pp. 427- 436, May 1966.

Happ, H H , J ohnson, R. C. , and W i ght ,

WJ . , Large Scal e Hydr o- Thermal Uni t

Comm i ment Method and Resul t s, I EEE

Transact i ons on PAS- 90, No.

3,

pp.

1373- 1384, May/ J une 1971.

PRI ORI TY

LIST

Thi s method arr anges t he generat i ng uni t s i n a

st art - up heur i st i c order i ng by operati ng cost

combi ned wi t h t r ansi t i on cost s. The

pr e-deter m ned or der i s t hen used t o comm t

t he uni t s such t hat t he syst em l oad i s

sat i s f i ed. Var i at i ons on t hi s t echni que

dynam cal l y r ank t he uni t s sequent i al l y. The

r anki ng process i s based

on

s pec i f i c

gui del i nes. The Comm t ment Ut i l i zati on Fact or

( CUF) and t he cl assi cal econom c i ndex Average

Ful l - Load Cost ( AFLC) can al so be combi ned t o

det erm ne t he pr i or i t y comm t ment order . The

CUF method can be appl i ed t o ei t her a

si ngl e-area UC or a mul t i - area uC.

Shoul t s, R. R. , Chang, S . K . , Hel m ck, S. ,

and Gr ady, WM , A Pract i cal Appr oach to

Uni t Comm t ment , Econom c Di spatc h, and

Savi ngs Al l ocat i on f or Mul t i pl e- Ar ea Pool

Operat i on wi t h I mpor t / Expor t

Cons t r ai nt s ,

t

I EEE Tr ansact i ons on

PAS- 99, No.

2 ,

pp. 625- 633, Mar ch/ Apr i l

1980.

Lee, F. N. , Short - Ter mUni t Comm t ment

-

A New Met hod, I EEE Tr ansact i ons on

Lee, F. N. , The Appl i cati on of Comm t ment

Ut i l i zat i on Factor ( CUF) t o Ther mal Uni t

Comm t ment , I EEE Tr ansact i ons on PWRS- 6,

Lee, F. N. and Feng, Q., Mul t i - Ar ea Uni t

Comm t ment , I EEE Tr ansact i ons on PWRS,

Paper 91 WM 180- 0, New York, 1991.

heur i st i c of order i ng can be t r ansl at ed

r ul es and execut ed as an exper t svst emas

PWRS- 3, NO. 2, pp. 421- 428, May 1988.

NO. 2, pp. 691- 698, May 1991.

noted bel ow. Thus any of t hese Sechni hes can

be t r eat ed as exper t syst em approaches si mpl y

by usi ng an exper t syst emt ool .

DYNAM C PROGRAMM NG

Dynam c Pr ogr amm ng ( DP) sear ches t he sol uti on

space t hat cons i st s . of t he uni t s s t at us f or an

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opt i mal sol ut i on. The search can pr oceed i n a

f orward or backward di r ect i on. The t i me

peri ods of t he st udy hor i zon are known as t he

st ages of t he DP pr obl em Typi cal l y each

st age r epr esent s one hour

of

operat i on. The

combi nati ons of uni t s wi t hi n a t i me peri od ar e

known as t he st at es of t he DP pr obl em

For ward DP f i nds t he most econom cal schedul e

by st art i ng at t he i ni t i al st age accumul ati ng

total costs, t hen backtr acki ng f rom t he

combi nat i on of l east accumul ated cost st art i ng

at t he l ast st age and endi ng at t he i ni t i al

st age.

DP bui l ds and eval uat es t he compl et e deci si on

t ree t o opt i m ze t he probl em at hand. Thus,

DP suf f er s f rom t he cur se of di mensi onal i t y

because t he pr obl em gr ows r api dl y wi t h t he

number of gener at i ng uni t s t o be comm t t ed.

To r educe t he sear ch space and hence the

di mensi on of t he DP pr obl em several

approaches have been adopt ed. Most appr oaches

are based on t he above Pri ori t y Li st

t echni ques.

One such met hod uses t he pr i ori t y l i st

order i ng where t he l east cost l y uni t s t o

operat e ar e comm t t ed f i r st and t he most

cost l y uni t s ar e comm tt ed l ast . I n t hi s

case, t he pr obl em i s r educed by consi deri ng

combi nati ons of uni t s s equent i al l y t ur ned on

( of f ) i n pr i or i t y l i s t or der . Not e t hat

unavai l abl e, must - r un, f i xed, and peaki ng

uni t s are excl uded f r om t he present pri ori t y

l i st . Another appr oach i s t o adopt a var i abl e

pr i ori t y or deri ng scheme by organi zi ng t he

generati ng uni t s i nt o cl asses wi t hi n whi ch the

uni t s are pr i ori t i zed. A t hr eshol d and a

w ndow are def i ned i n each cl ass t o det erm ne

whi ch uni t s ar e aut omat i cal l y comm t t ed

( t hr eshol d) , whi ch uni t s ar e eval uat ed f or

comm t ment ( wi ndow) , and whi ch uni t s ar e not

consi dered at al l .

[ 19] Lowery, P. G. , Generat i ng Uni t Comm t ment

Transacti ons on PAS- 85, No. 5, pp.

[ 20] Guy, J . D. , Secur i t y Constr ai ned Uni t

Comm t ment , I EEE Transact i ons on PAS- 90,

No.

3, pp. 1385- 1389, May/ J une 1971.

[ 21] Le, K. D. , Day, J . T. , Cooper, B. L. , and

Gi bbons, E. W, A Gl obal Opt i m zati on

Method f or Schedul i ng Thermal Generat i on,

Hydr o Gener at i on, and Economy Pur chases,

I EEE Transacti ons on PAS- 102, No. 7, pp.

[ 22] Kusi c, G. L. and Put nam H. A. , Di spat ch

and Uni t Comm t ment I ncl udi ng Commonl y

Owned Uni t s, I I EEE Tr ansacti ons on

PAS- 104, No. 9, pp. 2408- 2412, September

1985.

by Dynam c Pr ogramm ng, I EEE

422- 426, May 1966.

1986- 1993, J ul y 1983.

[23] Snyder , W L. , Powel l , H. D. , J r . , and

Raybur n, J

.

C. Dynam c Pr ogramm ng

Appr oach t o Uni t Comm t ment , I EEE

Transact i ons on PWRS- 2,

No.

2,

pp.

339- 350, May 1987.

[ 24] Hobbs, W J . , Hermon, G. , Warner, S., and

Shebl B, G. B. An Enhanced Dynam c

Pr ogr amm ng Appr oach f or Uni t

Comm t ment , I EEE Tr ansact i ons on PWRS- 3,

NO. 3 pp. 1201- 1205, August 1988.

[ 25] Tong, S. K. and Shahi dehpour , S.M ,

Hydrot her mal Uni t Comm t ment wi t h

Probabi l i st i c Const rai nt s Usi ng

Segment at i on Met hod, I EEE Transact i ons

on PWRS- 5, No. 1. pp. 276- 282, Febr uar y

1990.

[ 26] Hsu, Y .Y . , Su, C. C. , L i ang, C. C. , L i n,

C. J . , and Huang, C. T. , Dynam c Secur i t y

Const r ai ned Mul t i - Ar ea Uni t Comm t ment ,

I EEE Tr ansact i ons on PWRS- 6,

No. 3 ,

pp.

Truncat ed DP i s anot her at t empt at r educi ng

t he s i ze of t he DP pr obl em I n t hi s case, a

smal l port i on of t he sol ut i on space i s

consi dered wi t hi n pr i or i ty l i st order i ng. The

pot ent i al uneconom cal comm t ment schedul es

are then tr uncated.

[ 27] Pang, C. K. and Chen, H. C. , Opti mal

Short - Ter mThermal Uni t Comm t ment , I EEE

Transacti ons on PAS- 95,

No.

4, pp.

1336- 1346, J ul y/ December 1976.

[ 28] Pang, C. K. , Shebl B, G. B. , and Al buyeh,

F. ,

Eval uat i on of Dynam c Pr ogr amm ng

Based Met hods and Mul t i pl e Ar ea

Repr esentat i on f or Thermal Uni t

Comm t ment s, I EEE Transact i ons on

PAS- 100, No.

3,

pp. 1212- 1218, Mar ch

1981.

Ref erence 24 pr esent ed a basi c obser vati on

whi ch has not been addr essed by ot her DP

appr oaches. Si mpl y st at ed, t he aut hor s

report ed that t he pr i nci pal of opt i mal i t y was

f ound t o be vi ol ated by t he syst em under

study. Speci f i cal l y, i t was more opt i mal t o

save sub- opt i mal sol uti ons duri ng t he f or ward

pr ocess. Such an observati on cl ear l y

quest i ons the use of DP f or t hi s pr obl em

The UC pr obl em may al so be decomposed i nt o

smal l er s ubprobl ems t hat ar e easi l y managed.

Each subpr obl em i s sol ved wi t h DP. The

subpr obl em coordi nati on i s achi eved ei t her

sequent i al l y wi t h Successi ve Appr oxi mati on

( SA) or i n paral l el wi t h a Hi erarchi cal

Approach

HA) .

I n SA, t he sol ut i on of eachsubprobl em

i s

subdi vi ded i nt o a smal l er gr i d

f or t he next subpr obl em and the i t er ati ve

procedure cont i nues unt i l no i mprovement i n

t he sol ut i on i s det ected. I n HA, t he

subpr obl ems ar e sol ved i ndependentl y of each

ot her . The i nt eract i on between t he

subprobl ems i s mani pul ated by a coordi nat or t o

converge t he sol uti on of t he subprobl ems to

t he overal l probl emsol ut i on.

~n al t ernat i ve t o t he decomposi t i on i s t o onl y

appl y SA to r estr i ct t he sol ut i on space of t he

DP approach. One of t he var i ant s of t hi s

appr oach uses Lagr angi an reduct i on of t he

search range by usi ng t he dual of t he UC

pr obl em Anot her vari ant uses a dual f unct i on

of t he rel axed ori gi nal UC probl em

[ 29] Van den Bosch, P. P. J . and Honder d, G. , A

Sol ut i on of t he Uni t Comm t ment Probl em

vi a Decomposi t i on and Dynam c

Pr ogr amm ng, f I EEE Tr ansacti ons on

[ 30] Ni eva, R. , I nda, A. , and Gui l l en, I . ,

Lagr angi an Reducti on

of

Search- Range f or

Lar ge Scal e Uni t Comm t ment , I EEE

Transact i ons on PWRS- 2, No. 2, pp.

1049- 1055, August 1991.

PAS- 104, NO. 7, pp. 1684- 1690, J ul y 1985.

465- 473, May 1987.

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131

Systems, IFAC Symposium on Large Scale

Systems, udine, Italy, June 1976.

Dillon, T.S., Edwin, K.W., Kochs, H.-D.,

and Taud, R.

J.

Integer Programming

Approach to the Problem of Optimal Unit

Commitment with Probabilistic Reserve

Determination, IEEE Transactions on

PAS-97, No. 6, pp. 2154-2166, Nov/Dec

1978.

Turgeon, A., Optimal Scheduling of

Thermal Generating Units, I IEEE

Transactions o n AC-23, No. 6, pp.

1000-1005, December 1978.

Pereira, M.V.F. and Pinto, L.M.V.G.,

ltApplication of Decomposition Techniques

to the Mid-Short-Term Scheduling of

Hydrothermal Systems, Proceedings of

the PICA, pp. 193-200, June 1983.

Shaw, J.J. and Bertsekas, D.P., lvOptimal

Scheduling of Large Hydrothermal Power

Systems, IEEE Transactions on PAS-104,

No. 2, pp. 286-294, February 1985.

Habibollahzadeh,

H.

and Bubenko, J.A.,

Application

of

Decomposition Techniques

to Short-Term Operation Planning of

Hydrothermal Power System, IEEE

Transactions on PWRS-1, No. 1, pp. 41-47,

February 1986.

Recently, expert systems have been applied to

the UC problem. UC

is

initially solved using

available optimization techniques such as DP

or variable window truncated DP and then the

solution

is

refined by satisfying heuristic

rules derived from knowledge of the system

operation and conditions.

Ouyang,

2

and Shahidehpour, S.M.,

Heuristic Multi-Area Unit Commitment

with Economic Dispatch, IEE Proceedings,

Part C, Vol. 138, No.

3,

pp. 242-252, May

1991.

Ouyang, 2. and Shahidehpour, S.M.,

i i ~

Intelligent Dynamic Programming for Unit

Commitment Application, IEEE

Transactions on PWRS-6, No.

3,

pp.

1203-1209, August 1991.

Finally, fuzzy DP has been used to solve the

UC problem when the forecasted hourly loads

are not exactly known. For an optimal

solution, the DP model must express the

hourly loads, the cost, and the system

security in terms of fuzzy set notations.

[33 ] Su, C.C. and Hsu,

Y.Y.,

8tFuzzy Dynamic

Programming:

An

Application to Unit

Commitment, IEEE Transactions on PWRS-6,

All of the above techniques are use the same

vision to solve the problem. They see the

problem as a sequential decision process of

when to start the next unit and which unit to

start based on predicted (estimated) unit

operation. This theme

is

expanded to include

risk as shown below.

INTEGERWMIXED--PROGRAMMING

The solution

of

the UC problem based on the

Benders approach partitions the problem into a

nonlinear economic dispatch problem and a

pure-integer nonlinear UC problem. The

Mixed-Integer Programming (MI P) approach

solves the UC problem by reducing the solution

search space systematically through discarding

the infeasible subsets. Dual programming is

also suggested for the solution of the thermal

UC problem. The general solution concept is

based on solving

a

linear program and checking

for an integer solution. If the solution is

not integer, linear problems or subproblems

are continuously solved. The problems are not

similar because the number and type of integer

variables are changed while holding the

variables at a fixed integer value. Branching

is

the strategy adopted to determine which

variables to hold constant.

No.

3,

pp. 1231-1237, August 1991.

Garver, L. L. Power Generation

Scheduling by Integer Programming

-

Development of Theory, AIEE

Transactions No. 2, pp. 730-735, February

1963.

Muckstadt, J.A. and Wilson, R.C.,

An

Application

of

Mixed-Integer Programming

Duality to Scheduling Thermal Generating

Systems, IEEE Transactions on PAS-87,

Vol. 12, pp. 1968-1977, December 1968.

Dillon, T.S. and Egan, G.T., The

Application of Combinatorial Methods to

the Problems of Maintenance Scheduling

and Unit Commitment in Large Power

of these techniques may be viewed as a

means

of

discarding the paths (branches) which

are not expected t o yield a better solution.

BRANCH -ND BOUND

The Branch and Bound (B&B ) approach

essentially determines a lower bound to the

optimal solution and then finds a near-optimal

feasible commitment schedule. The

branch-and-bound tree is searched for the

best solution. The lower bound can be

determined from a dual optimization problem

that uses Lagrangian relaxation. Information

obtained from th e dual problem is instrumental

in producing dynamic priority lists even

though priority lists may not be necessary.

These lists are useful in the determination of

feasible solutions and help in the computation

of an.upper bound on the solution. Only few

nodes of the branch-and-bound tree are

examined to obtain near-optimal solutions if

an upper bound is found.

Lauer, G.S. Sandell, N.R., Jr.,

Bertsekas, D.P., and Posbergh, T.A.,

Solution of Large-scale Optimal unit

Commitment Problems,f' IEEE Transactions

on PAS-101, No.

1,

pp. 79-86, January

1982.

Cohen, A.I. and Yoshimura,

M.,

A Branch

and Bound Algorithm for Unit Commitment,

IEEE Transactions on PAS-102, No. 2, pp.

444-451, February 1983.

The concept of a tree is most appropriate if a

risk based approach'is to be used as discussed

below.

LINEAR PROGRAMMING

Several Linear Programming (L P) approaches

have been adopted to solve the large

UC

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probl em F i r s t , t he pr obl em i s decomposed

i nt o smal l er subprobl ems vi a t he Dant zi g-Wol f e

decomposi t i on pr i nci pl e. Each subprobl em i s

sol ved usi ng Li near Pr ogr amm ng. Second, t he

pr obl em i s sol ved wi t h t he revi sed si mpl ex

t echni que. A dual f or mul at i on wi t h a r educed

basi s

i s

adopt ed al ong wi t h rel axati on

t echni ques. A meri t - order l oadi ng easi l y

pr ovi des a st art i ng poi nt f or each schedul e.

Last , t he di scr ete deci si on l i near pr ogr amm ng

appr oach

i s

appl i ed t o t he UC pr obl em wi t h

heuri s t i cs such as pr i or i t y order i ng. Thi s

t echni que i s an LP pr obl em wi t h di screte

bounded vari abl es al l owed t o be ei t her t he

l ower or upper bound of t he i nt erval .

[44] S to t t , B. , Mar i nho, J . L . , and Al sac , O ,

Revi ew of Li near Pr ogr amm ng Appl i ed t o

Power Syst em Reschedul i ng, Proceedi ngs

of t he PI CA, pp. 142- 155, May 1979.

[ 45] Wai ght , J . G. , Bose,

A . ,

and Shebl e, G B. ,

Gener at i on Di spatc h wi t h Reserve Mar gi n

Const r ai nts Usi ng Li near Programm ng, n

I EEE Transact i ons on PAS- 100, No. 1, pp.

252- 258, J anuar y 1981.

[ 46] P i ekut owski ,

M

and Rose,

I . A. ,

A Li near

Pr ogramm ng Met hod f or Uni t Comm t ment

I ncor porati ng Generati on Conf i gur ati on,

Reserve, and Fl ow Const r ai nts,

I

I EEE

Tr ansact i ons on PAS7104,

NO

12, pp.

3510- 3516, December 1985.

[ 47] Khodaver di an, E. , Br amel l er, A. , and

Dunnett ,

R

.M

Sem - Ri gor ous Thermal

Uni t Comm t ment f or Lar ge Scal e

El ect r i cal Power Syst ems, I EEE

Proceedi ngs, Vol .

1 33 ,

Par t C, No. 4, pp.

157- 164, may 1986.

The ext ensi on of such t echni ques to bi ddi ng

procedures

i s

st r ai ght f or war d [3,5,7]. The

essence of a bi ddi ng procedur e i s anal ogous t o

an open mar ket where each pl ant bi ds f or t he

next cont r act t o pr ovi de power and energy.

DYNAM C AND LI NEAR PROGRAMM NG

The UC pr obl em

i s

sol ved usi ng r egul ar DP or

DP wi t h successi ve appr oxi mati on of t he

sol uti on space. LP sol ves t he econom c

di spatch wi t hi n UC f or t he cal cul ati on of t he

pr oducti on cost or t he opt i mal al l ocati on of

f uel

.

Dant zi g- Wol f e decomposi t i on, when

us ed, par t i t i ons the l i near program i nto

smal l er , mor e manageabl e LP s ubpr obl ems. LP

w t h upper boundi ng i s al so an al t ernat i ve

sol ut i on t echni que t o t he econom c di spatch

probl em

Wai ght , J . G. , Al buyeh, F . , and Bose, A . ,

Schedul i ng of Gener at i on and Reserve

Mar gi n usi ng Dynam c and Li near

Programm ng, I EEE Transact i ons on

PAS- 100, No.

5

pp. 2226- 2230, May 1981.

Van Meet er en, H P Schedul i ng of

Generat i on and Al l ocat i on of Fuel Usi ng

Dynam c and Li near Pr ogr amm ng,

f

I EEE

Transact i ons on PAS- 103, No. 7, pp.

Shebl e, G. B. and Gr i gsby, L. L. , Deci si on

Anal ysi s Sol ut i on of t he Uni t Comm t ment

Probl em El ectr i c Power

.

Syst ems

Research, Vol . 10, No. 11, pp. 85- 93,

November 1986.

1562- 1568, J ul y 1984.

The essence of t hese techni ques i s t o pr ovi de

t he DP wi t h addi t i onal i nf ormati on t o gui de

t he sel ect i on of t he t r ee pat hs. Note t hat

r efer ence

5 0

vi ewed t he deci si on process as

how t o a l l ocat e t he f i nanci al r esour ces f or

uni t operat i on.

SEPARABLEPROGRAMMING

Separ abl e Pr ogr amm ng ( SP) assumes t hat t he

obj ecti ve f unct i on i s concave and t he

const r ai nts are convex wi t h onl y one non-z ero

vari abl e. Thi s speci al st r uct ur e can be

expl oi t ed by LP. The A- separ abl e progr amm ng

t echni que i s used wi t h general i zed upper

boundi ng LP t o sol ve t he UC pr obl em

[ 51] Rahman, S. , Power Syst em Operat i on

Schedul i ng usi ng Separ abl e Pr ogr amm ng,

El ectr i c Power Syst ems Research, Vol . 2,

No.

4 , pp. 292- 303, December 1979.

Thi s can al so be vi ewed as a t ype

of

LaGr angi an Rel axat i on appr oach.

NETWORK ELQH PROGRAMM NG

Net work F l ow (NF) Pr ogramm ng i s t he bas i s f or

schedul i ng most hydro syst ems. Thus i t woul d

be anot her anal ogy t o appl y t o t he uni t

comm t ment pr obl em The r esul t i s a nonl i near

obj ect i ve f unct i on and a l i near set of

constr ai nt s. Thi s pr obl emcan be sol ved wi t h

a r educed gr adi ent al gori t hm I t can al so be

sol ved wi t h a Frank- Wol f e t echni que. I n t hi s

case, net work f l ow r epl aces LP t o sol ve a par t

of t he pr obl em i f t he non- net wor k const r ai nt s

ar e not bi ndi ng.

[ 52] Br annl und, H. , Sj el vgr en, D. , and

Bubenko, J . A. , Shor t - Ter m Gener at i on

Schedul i ng wi t h Secur i t y Const r ai nt s,

I EEE Tr ansact i ons on PWRS- 3, No. 1, pp.

310- 316, Febr uar y 1988.

[ 53] Habi bol l ahzadeh,

H. ,

Fr ances, D. , and

Sui , U. , A New Gener at i on Schedul i ng

Pro gra m at Onta r i o Hydro , I EEE

Transact i ons on PWRS-5, No. 1, pp. 65- 73,

Febr uar y 1990.

The anal ogy of Net wor k Fl ows al so appl i es t o

t r ee based t echni ques as shown

i n

r efer ence

11.

LAGRANG AN-

The Lagr angi an Rel axat i on

LR)

opt i m z at i on

t echni que decomposes t he UC pr obl em i nt o a

mast er pr obl em and mor e manageabl e subprobl ems

t hat are sol ved i t era t i vel y unt i l a

The

ear - opt i mal sol ut i on

i s

obt ai ned.

subprobl ems ar e sol ved i ndependent l y. Each

subprobl em det erm nes t he comm t ment of a

si ngl e uni t . The probl ems are l i nked by

LaGr ange mul t i pl i ers t hat are added t o t he

mast er probl em t o yi el d a dual probl em The

dual probl em has l ower di mensi ons than t he

pr i mal pr obl em and

i s

eas i er t o sol ve. For

t he UC pr obl em t he pr i mal f unct i on i s al ways

gr eat er t han or equal t o t he f unct i on whi ch

i s

defi ned as weak dual ' i ty. The di f f erence

between t he t wo f unct i ons yi el ds t he dual i t y

gap f or whi ch the pr i mal f unct i on

i s

an upper

bound. The dual i t y gap provi des a measur e of

t he near - opt i mal i t y of t he sol ut i on.

The LaGr ange mul t i pl i er s ar e computed at t he

mast er probl em l evel . Once comput ed, t he

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133

Pumped- St or age Hydr o, I EEE Tr ansact i ons

1989.

On PWRS-4, NO. 3, pp. 1065-1073, August

[65] Vi r mani ,

S.,

I mhof , K. , and Mukhenj ee,

S.,

' I mpl ement ati on of a Lagrangi an

Rel axat i on Based Uni t Comm t ment

Probl em I EEE Transact i ons on

PWRS- 4,

No. 4, pp. 1373-1379, November 1989.

[66]

Tong, S. K. and Shahi dehpour , S. M. , An

I nnovati ve Appr oach t o Generat i on

Schedul i ng i n Large- scal e Hydr o Thermal

Power Syst ems wi t h Fuel Const r ai ned

Uni t s, I EEE Transact i ons on PWRS- 5, No.

[67] Ruz i c ,

S.

and Raj akovi c, N. , ''A New

Appr oach f or Sol vi ng Ext ended Uni t

comm t ment Probl em I EEE Transact i ons on

PWRS- 6,

N o .

1,

pp.

269-275,

Febr uar y

1991.

2, pp. 665-673, May 1990.

Lagr ange mul t i pl i ers ar e passed t o t he

subprobl ems. The sol ut i on of t he subprobl ems

by f or war d DP i s f ed back t o the mast er

pr obl em and updated mul t i pl i ers are obt ai ned

and used by t he subpr obl ems agai n. Thi s

process

i s

repeat ed unt i l t he sol ut i on

conver ges. For t he short - t ermUC pr obl em t he

mul t i pl i ers are updated thr ough a subgr adi ent

method wi t h a scal i ng f act or and t uni ng

const ant s t hat ar e det er m ned heur i s t i cal l y .

For t he l ong- t er mUC pr obl em t he mul t i pl i er s

are updated wi t h t he vari abl e met r i c method t o

prevent t he sol ut i on near t he dual maxi mum

f r om os ci l l at i ng.

[54]

F i sher , M L. , Opt i mal Sol ut i on o f

Schedul i ng Pr obl ems usi ng Lagr ange

Mul t i pl i er s

:

Par t I ,

t

Oper ati ons

Research, Vol . 21, pp. 1114-1127, 1973.

[55] Muckstadt , J . A. and Koeni g, S. A. , I I A n

Appl i cat i on of Lagr angi an Rel axati on t o

Schedul i ng i n Power Gener at i ng Syst ems,

Oper ati ons Resear ch, Vol .

25,

pp.

387-403, May/ J une 1977.

[56] Ber t sekas,

D.P.,

Lauer,

G S. ,

Sandel l ,

N. R. , J r . , and Posber gh, T. A. , Opt i mal

Shor t - Ter m Schedul i ng of Lar ge Scal e

Power Syst ems, I EEE Tr ansact i ons on

AC -28 ,

No.

1,

pp.

1-11,

J anuar y

1983.

[57]

Mer l i n,

A.

and Sandr i n, P. , A New Met hod

f or Uni t Comm t ment at El ect r i ci t 6 de

France, I I EEE Tr ansact i ons on PAS- 102,

NO. 5, pp. 1218-1225, May 1983.

[58] Fi s her , M. L . , A n Appl i cat i on Ori ent ed

Gui de t o Lagr angi an Rel axati on,

I

I nte r f aces , vol .

15,

No.

2,

pp.

10-21,

Mar ch/ Apr i l 1985.

[59] Cohen, A. I . and Wan, S. H. , A Method f or

Sol vi ng t he Fuel Const r ai ned Uni t

Comm t ment Pr obl em I EEE Transact i ons on

[60] Aok i , K. , Sat oh, T. , I t oh, M , I chi mor i ,

T. , and Masegi , K. , Uni t Comm t ment i n

a

Lar ge-scal e Power System I ncl udi ng Fuel

Const r ai ned Ther mal and Pumped- St or age

Hydr o, I EEE Transact i ons on

PWRS-2, N o .

4,

pp.

1077-1084,

November

1987.

[61] Zhuang, F and Gal i ana, F. D. , Towar ds a

more Ri gor ous and Pract i cal Uni t

Comm t ment by Lagrangi an Rel axat i on,

I EEE Tr ansact i ons on PWRS-3, No. 2 , pp.

763-773, May 1988.

[62] Bard , J . F. , Shor t - Term Schedul i ng of

Thermal El ectr i c Generat ors Usi ng

Lagr angi an Rel axati on, Oper ati ons

Research, Vol . 36, No.

5 ,

pp. 756-766,

Sept ember/ Oct ober

1988.

[63] Tong, S. K. and Shahi dehpour, S.M ,

Combi nat i on of Lagr angi an- Rel axat i on and

Li near Pr ogr amm ng Appr oaches f or Fuel

Const r ai ned Uni t Comm t ment Pr obl ems,

I EE Proceedi ngs, Vol .

136,

Par t C, No.

3,

[64]

Aok i , K. , I t oh, M. , Sa toh, T. , Nar a, K. ,

and Kanezashi , M , Opt i mal Long- Ter m

uni t Comm t ment i n Lar ge Scal e Syst ems

I ncl udi ng Fuel Const r ai ned Ther mal and

PWRS- 2,

No.

3,

pp. 608-614, August 1987.

pp.

162-174,

May

1989.

EXPERT SYSTEMS/AR= NEURAL NETWORKS

Expert syst ems combi ne t he i dent i f i cat i on of

exi st i ng probl ems wi t h t he UC al gor i t hms and

t he knowl edge of exper i enced power syst em

oper at or s and UC progr amm ng exper t s t o cr eat e

an expert system r ul e base ( pr ocedur al data

base). The exper t system i mpr oves t he

sol uti on by adj usti ng t he progr am s parameter s

t hr ough i nt eracti on wi t h the system s

operat or. Exper t Syst ems ar e mor e r ecentl y

r ef err ed t o as Knowl edge Base Syst ems ( KBS) .

Est i mates of Art i f i ci al Neur al Networks ( ANN)

paramet er s are based on a dat abase hol di ng

t ypi cal l oad curves and cor r espondi ng UC

schedul es. The patt ern of t he cur r ent l oad

cur ve

i s

compar ed t o t he i nf or mat i on i n t he

dat abase t o sel ect t he most econom cal

uc

schedul e. I n t he event t hat t he A sol ut i on

i s not f eas i bl e f or t he ent i r e UC per i od, i t

w l l be used as an i ni t i al s t ar t i ng poi nt f or

a near - opt i mal sol ut i on.

A s

r espond to

changes i n oper at i ng condi t i ons when present ed

w t h suf f i c i ent f act s , even t hough t hey are

t r ai ned of f - l i ne.

[68]

Mokht ari ,

S. ,

Si ngh,

J . ,

and Wol l enber g,

B. ,

A Uni t Comm t ment Exper t Syst em

Proceedi ngs of t he PI CA, pp. 400-405, May

1987,

[69] Shebl b, G. B. , Sol ut i on of t he Uni t

Comm t ment Pr obl em by t he Met hod of Uni t

Per i ods I I EEE Tr ansact i ons on

PWRS-5,

No. 1, pp. 257-260, Febr uar y 1990.

[70]

Wang, C. , Ouyang,

Z.,

and Shahi dehpour ,

S. M. , and Deeb, N . , uni t Comm t ment by

Neur al Net wor ks, Pr oceedi ngs of Amer i can

Power Conf erence, vol .

52,

pp.

245-250,

Apr i l 1990.

[71] Ouyang, 2 . and Shahi dehpour , S. M , Shor t

Ter m Uni t comm t ment Expert System

El ectr i c Power Syst ems Resear ch, Vol . 18,

No.

1,

pp.

1-13,

December

1990.

721

Tong,

S.K.,

Shahi dehpour ,

S.M. ,

and

Ouyang, A Heur i s t i c Shor t - Ter m Uni t

Comm t ment , I EEE Tr ansact i ons on

PWRS-6,

NO. 3, pp. 1210-1216, August

1991.

731 Ouyang, Z. and Shahi dehpour , S. M. ,

A

Mul t i - St age I nt el l i gent Sys tem f or uni t

8/11/2019 59.317549

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Commitment, IEEE Transactions on PWRS,

Paper SM 322-8, San Diego, 1991.

Sasaki,

H.,

Watanabe, M., kubokawa, J.,

Yorino, N., and Yokoyarna, R., A Solution

Method of Unit Commitment by Artificial

Neural Networks, IEEE Transactions on

PWRS, Paper 91 SM 437-4, San Diego, 1991.

Ouyang,

Z.

and Shahidehpour, S.M. ,

A

Hybrid Artificial Neural Network-Dynamic

Programming Approach to Unit Commitment,

IEEE Transactions on PWRS, Paper 91 SM

438-2, San Diego, 1991.

It is notable that considerable research is

underway to relate the trees of A s to those

of KBSs. It should be noted that strict

conversion, from one type of KBS to ANN, is

not possible. However, the addition of simple

decoupling neurons (n odes makes the

transition trivial for several practical

cases.

RI SKANALYSI S

Scheduling and rescheduling of generating

units are based on certain observable events

such as load changes and unit forced outages.

A

risk analysis is warranted to determine the

necessary additional generating capacity to

meet the system load and reserve requirements.

One such probabilistic analysis proposes that

the UC must satisfy two risk levels: one at

the isolated system level and another at the

interconnected level. Another approach adopts

a stochastic model that reflects the sequence

of events associated with scenario-based

sequential rescheduling decisions. The random

sequence model is expressed in terms of

available capacity to reduce the dimensions of

the problem.

[76] Chowdhury, N. and Billinton, R., Unit

Commitment in Interconnected Generating

Systems Using a Probabilistic Technique,

IEEE Transactions on PWRS-5, No. 4, pp.

1231-1237, November 1990.

[77] Lee, F.N. and Chen, Q., Unit Commitment

Risk with Sequential Rescheduling, IEEE

Transactions on PWRS-6, NO. 3, pp.

Risk Analysis is most recently extended to

include operational and planning events beyond

unit Commitment as discussed below.

a M U L A T m

ANNEAL1 G

The UC problem has been compared to the

annealing of a metal. When the metal is

cooled slowly (annealed ), its energy tends to

assume a globally minimal value. The states

of a metal correspond to the various feasible

solutions of the problem to minimize and the

energy of a state is analogous to the cost of

a feasible solution. Simulated Annealing

( s m ) generates near-optimal and fast

solutions. Feasible solutions are generated

randomly and are accepted as the next

generation to continue the solution process if

the cost of the current solution is less than

the previous one. Otherwise, the current

solution is accepted with a certain

probability.

1017-1023, August 1991.

[78] Zhuang, F. and Galiana, F.D., Unit

Commitment by Simulated Annealing, IEEE

Transactions on PWRS-5, NO. 1, pp.

311-318, February 1990.

AUGMENTE MG RA NG AN

This is an optimization technique that handles

static and dynamic constraints. The two types

of constraints are decomposed into subproblems

of reasonable size, homogeneous nature, and

well-known structure. The objective function

to optimize is unconstrained and continuously

differentiable. Ill-conditioning

is

avoided.

Artificial constraints and variables are added

to the original problem to decompose it.

Next, the artificial constraints are handled

via a dual approach and an augmented

Lagrangian (AL ) relaxation technique which

adds a quadratic penalty function to the

constraints. Finally, The Auxiliary Problem

Principle is applied to decompose the problem

by linearizing the nonseparable terms of the

cost function and by adding separable

quadratic terms (i f properly chosen) to the

cost function.

[ 791 Batut, J. and Renaud, A . Daily

Generation Scheduling Optimization with

Transmission Constraints: A New Class

of

Algorithms, IEEE Transactions on PWRS,

Paper 91 SM 429-1, San Diego, 1991.

DECISIONANALYSIS

Decision Analysis is the art of analyzing the

choice of decision options by subjectively

assessing the outcomes of each decision and

the probability of each outcome. The

reference which the authors have used IS:

[80] Raiffa, H. A., DecisiQn

An

alvsis

:

y n c e r t a w , Addison-Wesley Publishing

Company, Reading, Massachusetts, 1968.

Such an approach builds a decision tree to

show the outcomes possible from each decision.

Such a tree resembles the paths generated by a

full DP approach. If the outcomes are assumed

to be certain,then all of the above may be

viewed as tree based techniques. Many of the

above optimization techniques may be viewed as

different approaches to pruning the tree.

Some approaches prune by excluding paths above

a pre-determined cost value (truncated DP,

priority list modification, KBS). Other

methods prune by excluding branches (paths) by

estimating the minimum cost and/or feasibility

(MIP, B&B, LP,

SP,

LR, AL). Finally, the

latest approach is to select the best path by

evaluating the energy (quality of soluti on)

and randomly searching for alternative

decisions to reduce the energy

( S A n ) .

The

authors have started efforts on solving the uc

problem with Genetic Algorithms and will

present such work when definitive results are

available.

Since the tree approach is being used for bulk

power assessment, the key to future efforts

will be to explicityly link the above

techniques with decision trees. It has been

realized for some time, that the reason for

solving the unit commitment problem is not

just to determine a unit schedule. The

mtrodu ctorv Lect res Choices under

8/11/2019 59.317549

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135

Gerald B. Sheble'

( U

71, SM 85)

i s an

Associ at e Prof essor of E l ectr i cal Engi neer i ng,

I owa Stat e Uni ver si t y. Dr . Shebl 6 recei ved

hi s

B . S .

and M S. degrees i n El ect r i cal

Engi neeri ng fr om Pur due Uni ver si t y and hi s

Ph. D. i n El ectr i cal Engi neeri ng f rom Vi r gi ni a

Tech. Hi s i ndustr i al exper i ence extends over

f i f t een year s. Hi s academ c exper i ence

i ncl udes resear ch i n t he appl i cati on of power

systems t echni ques f or spacecr af t . Hi s

pr esent r esearch cent ers i n t he opt i mal

operat i on and schedul i ng

of

power s yst ems.

George Fahd (S 85)

r ecei ved hi s B. S. E. E. i n

December of 1985 and hi s M S. E. E. i n August of

1987 f r om t h e Uni ver s i t y of Al abama at

Bi r m ngham He recei ved t he Ph. D. maj ori ng i n

Power Syst ems f r om Auburn Uni ver si t y i n J une,

1991. He

hel d a post doctor ate posi t i on at

I owa St ate Uni ver si t y bef or e j oi ni ng Deci si on

Focus I ncor por at ed I n December of

1991.

Hi s

i nt erests i ncl ude uni t comm t ment model i ng,

economc di spat ch, opt i mal power f l ow,

opt i m zat i on t echni ques, power syst em

operat i on and cont r ol , and economy

t ransact i ons.

pr i mary r eason f or sol vi ng t he uni t comm t ment

probl em i s to provi de a cos t bas i s f or

t r ansact i on pr i ci ng. As such, resear ch f or

t he f utur e shoul d concent r ate on rel ati ng the

uni t schedul e t o t he avai l abl e t r ansact i ons

w th the i nt ent of sel ect i ng the l eas t cos t ,

yet re l i abl e, o pt i on.

SUMMARY

Thi s paper gi ves a l i st of t he ref er ences

avai l abl e for t he sol ut i on of t he thermal

UC

pr obl em A var i et y

of

t echni ques have been

appl i ed t o t hi s compl ex, nonl i near,

m xed- i nteger pr ogr amm ng pr obl em A cl ear

consensus i s pr esentl y tendi ng t oward t he

Lagr angi an Rel axat i on appr oach over ot her

met hodol ogi es. The Augment ed Lagr angi an i s a

r el ati vel y newcomer t hat i s not t horoughl y

t ested yet . Anot her area of r esearch i nvol ves

t he appl i cat i on of genet i c al gor i t hms t o t he

sol ut i on of t he UC pr obl em Thi s

s

a r andom

search met hod whi ch i s based on nat ur e' s

surv i val o f t he f i t t es t theory . Cl ear l y, the

met hods to be used f or t hi s probl em w l l

cont i nue t o evol ve.