IEEE Transactions on Power Systems, VoI.6, No.1, February 1991 59

AN EXPERT SYSTEM WITH FUZZY SETS FOR OPTIMAL PLANNING

A.K. David, Member IEEE Hong Kong Polytechnic Kowloon, HONG KONG

Abstract

This paper describes a long range power system expansion planning program which incorporates three s ign i f icant features. I t i s an optimising program and uses dynamic programing f o r t racking an optimal expansion strategy. Secondly i t contains a r u l e based decision making mechanism t o incorporate engineering judgement based on design o f f i c e experience and expert opinions, and f i na l l y , since some of these considerations have t o be cast i n qua l i t a t i ve terms and need t o balance many con f l i c t i ng requirements, fuzzy set theory has been used t o formulate a por t ion o f the decision making procedure.

Key Uords: Power generation planning; Optimisation methods; Expert systems

INTRODUCTION

A previous paper by the authors 111 describes the in tegrat ion of dynamic programing opt imisat ion and expert system techniques i n long range generation expansion planning. The decis ion t o choose DP was based on i t s strength over other methods i n respect o f modelling the inherent and s ign i f i can t non- l inear i t ies of t h i s problem, i t s potent ia l f o r incorporating sophisticated production costing algorithms and the ease of including complex constraints such as r e l i a b i l i t y and qua l i t a t i ve considerations.

The reason f o r incorporating an expert system was two fold. F i r s t l y there are the sel f -evident advantages of guiding the decision making procedure a t each feasible stage o f the fu ture system by the knowledge and experience built up over many years by system planning engineers. The ES i s a means o f formal is ing and systematising t h i s experience and a computerised inference mechanism fo r using i t .

The second reason i s that an expert system can be exploi ted t o make the DP algorithm more viable. Conventional DP packages are v iab le only as f ine-tuning too ls since the i r broad based use soon runs up against the

90 SM 303-8 PWRS by the IEEE Power System Engineering Committee of the IEEE Power Engineering Society for presentation at the IEEE/PES 1990 Summer Meeting, Minneapolis, Minnesota, July 15-19, 1990. Manuscript submitted November 27, 1989; made available for printing April 24, 1990.

A paper recommended and approved

Zhao Rongda, Non-member IEEE T ian j i n Univers i ty

Tianj in, CHINA

'curse of dimensionality' i n any pract ica l problem of real world size. However, d iscr iminat ion in respect o f future s tate f e a s i b i l i t y decisions (windows i n state space) and constraints on stage t o stage t rans i t i on decisions (controls i n po l i cy space) can help carve out a heu r i s t i ca l l y constrained tunnel which prevents the explosion of the s tate space which occurs i n the l a t t e r stages of a conventional DP algorithm. The tunnel is, furthermore, a r e a l i s t i c engineering scenario since the ES i s based on planning experience and on rules dr iven by engineering considerations. These include considerations of f ue l source type, regulat ing t o peaking t o base-load capacity rat ios, transmission system and p lant locat ion re la ted effects, interconnection and exogenous considerations and other user defined ru les which may be formulated and readi ly incorporated i n the inference mechanism.

The f i n a l step i n t h i s development, which i s the subject o f t h i s paper, i s an extension and innovation carr ied out w i th in the expert system. The decision as t o whether a pa r t i cu la r hypothetical fu ture s tate of the system i s a feasible one or not depends on the balancing of several quant i ta t ive c r i t e r i a and qua l i t a t i ve judgments. The former are readi ly handled by the previous ES described i n 111 and addi t ional determinist ic or p robab i l i s t i c c r i t e r i a may be s im i la r l y incorporated i f they can be formulated as ru les and i f the relevant data can be provided. The novel feature t o be presented i n t h i s paper i s that fuzzy set theory has been introduced as a mechanism f o r incorporating the l a t t e r (qual i ta t ive judgments) in a l og i ca l l y consistent manner i n the program. The theoret ical approach developed by Baas and Kwakernaak 121 has been selected because o f i t s basic appropriateness, somewhat modified and extended t o s u i t the structure of t h i s problem, and incorporated i n the ES.

WALITATIVE JWGIIEWTS

There are a nunber of judgments based on e i ther experience or expert opinion which in r e a l i t y are cruc ia l i n decision making i n a key sector such as the power supply industry. Usually i t i s not possible t o capture them w i th in the range of constraint formulations of conventional power sector optimising models C3-41 except as extremely r i g i d exogenous decisions or 'givens'. A n example could be the decision o f a country t o l i m i t re l iance on a cer ta in category of f ue l f o r s t ra teg ic or envirormental reasons. Frequently, t h i s would be presented as an ttimportantll, or It less importantt1, etc, c r i t e r i on . Hard, quant i tat ive, f igures cannot be eas i l y assigned t o i t . Nevertheless, ce r ta in key f i n a l decisions may, i n the end, turn on some such consideration. I f t h i s judgement could be incorporated wi th in themodel i t s e l f , a so lut ion which accomnodates t h i s s t ra teg ic opinion, and i s the optimal one among those that

0885-8950/91/0200-0059$01.00 0 1991 IEEE

I I "

60

do so adequately, could have been obtained and the arbitrarypost-optimisation ' cu t t ing and pasting' which n o w goes on can b;e avoided.

There are a Mmber of features that f a l l w i th in the category o f issues which can benef i t from qua l i t a t i ve representation w i th in a planning model. They vary in the i r degree o f engineering, economic, managerial or national strategic content. Fuel type sourcing i s an important consideration and, furthermore, the nature of t h i s problem may be en t i re l y d i f f e ren t f o r a b i g power or a small developing country. Consider fo r exanple the USA, Canada, Neu Zealand and Nepal t o appreciate the wide var iat ions i n strategic, domestic energy reserve conservation dr iven and technology modernisation dependent, considerations.

Technical considerations which, despite being technical, are more meaningful as qua l i t a t i ve judgments, rather thandeterminist ic or p robab i l i s t i c constraints, are matters such as the largest unit ra t i ng sui table on the system a t various fu tu re stages, l i m i t s on the r u b e r o f units, and generation mix preferences. The v i a b i l i t y o f re laxing cer ta in technical constraihts on the planning requirements, provided t h i s tenporary v io la t i on i s corrected s u f f i c i e n t l y quickly in subsequent years, i s another relevant rea l world consideration which i s more su i tab ly formulated i n the form o f qua l i t a t i ve judgments.

Finally, there are also judgments and preferences of a more managerial nature that play a ro le in u t i l i t y decision making. Some o f these are preferences and trade-offs, not hard constraints. These included such matters as preferences fo r supplier sourcing, standardization o f p lant and financing options. These judgments are o f ten qua l i t a t i ve (I8important8l, " f a i r l y inportantl l etc), and are also provis ional in the sense that cer ta in other considerations, perhaps also qua l i t a t i ve l y specified, should simultaneously be taken i n t o account i n a r r i v i ng a t a balanced decision.

USE OF FUZZY SET THEORY

Nature of Qua l i ta t i ve JudsRn ts

This sect ion describes how the method developed i n 121 has been adapted and incorporated in the ES which has been developed. In the DP algorithm each d i f f e ren t type o f avai lable generation opt ion (candidate plant option) const i tutes a separate dimension o f the problem s ta te space. A point i n t h i s multi-dimensional space speci f ies a hypothetical s ta te o f the system. W e t i s a t issue i s the s u i t a b i l i t y ( f e a s i b i l i t y ) and op t ima l i t y of such future states o f the system. The ES, wi th i t s rules and constraints, addresses the issue o f f e a s i b i l i t y and does so by extensive pruning out o f unsuitable states and impermissible stage t o stage transi t ions. The DP algorithm i s concerned u i t h the computationally e f f i c i e n t tracking o f the economically optimal evolut ion o f the system with respect t o time.

Every dimension, or candidate plant option, of t h i s state space i s defined as having a mmber of at t r ibutes. An a t t r i bu te may be any of the strategic, technical or managerial considerations discussed previously i n respect of which qua l i t a t i ve judgments have t o be incorporated i n decision making. Actually, macro level a t t r ibu tes are l i k e l y t o be deconposed i n t o smaller subsets of elemental a t t r ibu tes whose indiv idual 'ratings' and 'importances'

need t o be l og i ca l l y combined overa l l i n a r r i v i ng a t f i n a l decisions. Fig. 1 i l l u s t r a t e s a choice o f judgements one o f which may be assigned t o each par t i cu la r a t t r i bu te of each par t i cu la r option. A large va r ie t y o f such judgement 'templates' are bui l t i n t o the program and users may i n addition, def ine the i r om. Piecewise continuous functions have been chosen fo r computational convenience but more general forms may be used subject t o the three conditions stated i n the next section.

(a) Good (b) Fair

P I PI

(c) Rather Poor (d) Poor

Fig. 1 A Typical Set o f Ratings (r) and Membership ( j i ) corresponding t o D i f fe ren t Qua l i t a t i ve Judgements

The next issue o f importance i s the formation o f some overal l assessment o f the s u i t a b i l i t y or otherwise of a prospective s ta te o f the system. This means that the a t t r ibu tes o f each o f the d i f f e ren t types o f p lant i n a s ta te have t o be combined together wi th a weighting that re f l ec ts the amount o f each plant type in that s ta te and, furthermore, the d i f f e ren t a t t r ibu tes have t o be aggregated i n a manner that takes i n t o account the inportance o f each a t t r i bu te i n a r r i v i ng a t a f i n a l decision.

Uathematical M o d e l

The four examples in fig.1 describe the membership functions o f qua l i t a t i ve judgement such as llgoodll and llpoorll i n fuzzy set theory C21. The x-axis i s the ra t ing o f an a t t r i bu te h e r e rat ings are normalised in the range [0,11 and the y-axis i s the Inembership funct ion value which again l i e s in a membership range C0,ll. Thus i f in a par t i cu la r case some a t t r i bu te i s deemed t o be 11good81 i t has l i t t l e membership a t the l o w end of the ra t i ng scale and high membership a t the high end. I f llpoorll the converse holds, while intermediate judgements are compatibly represented. In t h i s usage the term ra t ing denotes the normalised value o f the appropriateness or goodness of the relevant a t t r i bu te or feature and can vary between ni 1 (zero) and uni ty. The membership value denotes the degree o f possi b i 1 i t y that the corresponding ra t ing may be deemed t o be a t rue representation o f the measure of t h i s a t t r ibu te .

Let dl, . . . , 4, be n avai lable plant options (dimensions) and al, . . . , % be m d i f f e ren t pert inent at t r ibutes; then rij represents the ra t i ng o f the j - t h a t t r i bu te o f the i - t h option and i t s membership funct ion i s denoted by jiij(rij). Wen a value judgement, such as 11good18, i s speci f ied fo r some rij the relevant pij w i l l be selected from the pool o f judgement 'templates'.

61

The fuzzy set Rij=C(rij,-pRij)> of dimension di wi th assuned t o have the respect t o a t t r i b u t e aj, V i , j , are

f o l 1 owing propert ies : i) Normalised and convex ii) Piecewise continuously d i f f e ren t i ab le iii) Monotonic non-decreasing i n the range [-*,PI

and monotonic non-increasing i n the range [p,*l where 0 ~ - l and fiRi3(p)=l.

There are tuo types o f weighting which w i l l be used i n the subsequent development of the theory. F i r s t l y there are the dimension weights wdi, V i= l t o n, which represent the re la t i ve contr ibut ion o f each o f the d i f f e ren t plant types t o the evaluation o f that pa r t i cu la r a t t r i b u t e in the system state. The contr ibut ion of each di t o the system state may reasonably be expected t o be proport ional t o the gross capacity Gi of that p lant type in the system and therefore the fol lowing weights are proposed f o r V i

Gi

n = G k

k= 1

wdi = -

This defines a non-fuzzy set o f dimension weights kid=cwdi>, v i , and i s independent o f j.

The second type of weights are a t t r i bu te weights U,., v j = l t o m, which represents the re la t i ve importance of t i e d i f f e ren t a t t r i bu tes t o f i n a l decis ion making. This represents a subject ive judgement and, as wi th the a t t r i bu te ra t ings themselves, are most appropriately represented by fuzzy sets and variables. Therefore these weights are denoted by concepts such as "very important1I, 88important88, ... , 88unimportant8a which have t o be supplied as expert options by the program user. The program w i l l then make an appropriate select ion from the same set o f judgement 'templates' referred t o previously. I t w i l l now be c lear that these 'templates' serve a general purpose funct ion and a set having a f i ne r o r a coarser graduation may be selected or set up as appropriate t o modelling requirements. The second type of weights, therefore, const i tu te a fuzzy set U,=CUaj) where Uaj=C(uaj, haj)), v j , and i s independent o f i.

Evaluation of State Feas ib i l i t y

Let us now consider the assessment of each of the a t t r i bu tes in a state. The ra t i ng o f a s ta te wi th respect t o the a t t r i b u t e aj i s fuzzy, and i s represented by Rj=C(r j , pRj)), j=1, ... , m, where r j i s determined by funct ion

n f (x) = L wdirij

i = l

where x=(wdl, ... , bidn, rlj, ... , rn.). Hence the e f fec t o f the d i f f e ren t p lant types i n t i e s ta te are being combined together. Since the dimension weights wdi are constants, using Zadeh's extension p r inc ip le 151 the membership funct ion i s defined by

n IrRj(rj) SUP A

f (x)=r j i = l

I t i s not d i f f i c u l t t o obtain the fuzzy set

R .=C(r )I, j=1, ... , in, by the method suggested i n A. Ti; 3 w s o f r j and r.' f o r fiw(rj)=bo, coe[o,ll, are determined, respectively, i y

r j = wdlrlj+ ... + wdnrnj

r j '= wdlrlj'+ ... + wdnrnj'

as i l l u s t r a t e d i n Fig. 2.

Fig. 2 State-at t r ibutes as the Ueighted Fuzzy Aggregate o f Plant-at t r ibutes.

We can thus obtain the pR f o r j=1, ... , m. Next we undertake the combination of t i e d i f f e r e n t s ta te-at t r ibutes u i t h respect t o the importance of each at t r ibute. The f i n a l ra t i ng o r overa l l acceptabi l i ty o f a system state i s represented by fuzzy set R=<(r,fiR)). The r i s determined by the funct ion

m C w a j r j

j = l

m

j = l

g(y) =

"aj

where y=(wal, ... , U,, rl, ... , rm). The membership f a t i o n i s represented as

m m

g(y)=r j = l j=1 PR(r)= sup [ A haj ] A I A ' L R ~ 1.

Since f (x) has been set-up as a l inear formulation i n x, R j sa t i s f i es the conditions of 121 and pR can be s im i la r l y computed.

State-space Uindous

The f i n a l question now i s whether a par t icu lar s ta te whose overa l l r a t i ng membership funct ion p r o f i l e has been obtained as described previously i s an acceptable ( feasible) system state f o r the pa r t i cu la r planning stage being examined. Since the s tate has been rated as a fuzzy set what has been obtained i s an overa l l qua l i t a t i ve judgement o f i t s goodness i n a mathematical form. I t i s therefore proposed that i t be compared with a qua l i t a t i ve c r i t e r i o n of m i n i m acceptable goodness, which too has t o be cast in a mathematical form - that i s provided t o the program as a fuzzy cr i ter ion.

Let the overa l l r a t i ng o f the s tate be R=C(r, pR)), and the exogenously supplied fuzzy c r i t e r i o n be C=C(c, k)). The state i s declared feasible i f f the fuzzy p a r t i a l ordering re la t i on R>C-Cmax(C,R)=R and min(C,R)=Cl

. ..

62

exists. Fig. 3 gives some examples o f acceptable and unacceptable states. But sometimes the conclusions reached by t h i s d e f i n i t i o n i s not sa t is fac to ry as f o r example i n Fig. 4 . Here R+C but these can be l i t t l e doubt that R i s be t te r than C.

R C C C R > C

Fig. 3 Acceptable & Unacceptable States

E'

Fig. 4 A Case Which Sui ts the Weak Cr i te r ion Test

For t h i s reason we suggest the weak de f in i t ion . Define fuzzy sets C, and R I as

R t C i f f Rt>C'. K

Finally, the fo l lowing d e f i n i t i o n i s set up fo r decision making: The s ta te i s feasible i f f R t C . Here, the value o f

a i s supplied by the user and i s therefore a sui table candidate f o r expert opinion.

A n extension

a

Instead of having j us t one overa l l ra t ing f o r the system s ta te which combines together the goodness or otherwise o f a1 1 the a t t r ibu tes a more disaggregated approach may be used t o provide t i gh te r control. Separate fuzzy c r i t e r i a may be provided fo r each overal l state- a t t r i bu te rating. Then the s ta te w i l l be declared t o be acceptable i f a t t r i bu te by a t t r i bu te the s ta te passes each separate c r i t e r i o n test . The mathematical procedure i s s im i la r t o that described in the previous sub-section and the example i n the next section included a comparison o f the two methods. Hence we have two d i f f e ren t s t ra teg is ts t o determine which states are deemed t o be feasible - one based on an overa l l assessment o f s ta te goodness and the second based on an indiv idual a t t r i bu te by a t t r i bu te test .

-LE of PROGRAll USE

An example i l l u s t r a t i n g i n some d e t a i l the use of fuzzy set theory, and also providing an overview o f the use o f the expert system and dynamic programing based optimisation, i s presented i n t h i s section. The size o f the system used, the nunber o f candidate generation options, and the nunber o f planning stages, are a l l kept down t o moderate values as t h i s helps t o b r ing out the sa l ien t features more clear ly.

The Problem

Table 1 i s the i n i t i a l conf igurat ion ex is t ing a t the comencement o f the planning exercise - stage zero, 1990. The planning horizon extends over 20 years div ided in to four stage, a t f i v e year ly intervals. The ant ic ipated peak demand is, f o r convenience, ass& t o be de termin is t i ca l l y predicted and i s show i n tab le 2. The candidate plant options which are avai lable f o r fu tu re addition and the corresponding economic data are presented in table 3. There are s i x types o f candidate plant options avai lable - four of them, items 1 t o 3 and i t em 6, are the same as current ly operating plant types, while items 4 and 5 are two new technologies. Each of the s i x items, or candidate plant options, const i tutes a dimension i n the s ta te space o f the optimisation problem. The las t c o l u m of table 3 places an upper l i m i t on the gross nunber o f un i t s o f each candidate plant type which may be i ns ta l l ed over the en t i re planning horizon. This places a very general global l i m i t on the widest coverage that the DP algorithm, in the absence of the ES, may attenpt t o range over.

Table 1 Current System Cunnitted & Ex is t ing Uni ts

Operating No. o f Type Size cost f.0.r. Uni ts (MU) (USt/Kuh) (XI

1 O i 1 350 0.02 2.0

2 Coa 1 650 0.01 4.0

2 Hydro 300 0.0013 1.0

2 G.T. 100 0.026 1 .o

Table 2 Planning Horizon Years

Peak(MW) Stage Year

I 1995 3000

2 2000 4200

3 2005 5500

4 2010 7000

Table 3 Future System Available Uni ts

Capacity Operating No. Type Size Cost Cost f.0.r. No. o f

(MU) (USWKU) (USS/KWh) (XI Units

1 O i l 350 500 0.02 2.0 2

2 Coal 650 650 0.01 4.0 3

3 Hydro 300 750 0.0013 1.0 2

4 Hvdro 600 900 0.002 1.5 2

5 Nu. 900 1500 0.004 5.0 2

6 C.T. 100 250 0.026 1.0 6 '

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Rules Added I n Following

Order

The Expert System

The ES, i n t h i s application, consists o f e ight sets o f rules. Rule 1 places an upper l i m i t on the m a x i m nunber of un i t s o f each types which may be ins ta l l ed a t each stage - t h i s i s a knowledge based input from the planning department. Rule 2 i s a very gross i n i t i a l del ineat ion o f the perimeter o f the Wmneltl. Rule 3 i s a s t ra teg ic decision, whi le 4 and 5 are ru les i l l u s t r a t i v e of expert system planning options i n log ica l (aaORoa, 18AND11) forms. Since the ES i s evaluated in PROLOG, log ica l depth in r u l e evaluation, and log ica l complexity in r u l e def in i t ion, can be eas i l y handled - ref . C11 discusses these issues more f u l l y . The p r i o r i t y rule, r u l e 6, i s an i l l u s t r a t i o n o f th is . IaHistor ical lb (or I%ackward Lookingoa) rules, which make possible the modeling of addi t ional s ign i f icant engineering o r management considerations i n respect o f constraint relaxation, have a lso been discussed in 111 and are not introduced i n t h i s example, up t o the l as t stage, because t h e i r conplexi ty i s not pert inent t o the matter o f using fuzzy l og i c t o represent qua l i t a t i ve considerations.

The modelling of qua l i t a t i ve considerations by fuzzy log ic i s located a t pos i t i on seven in the r u l e evaluation sequence. The most computationally heavy item, r e l i a b i l i t y assessment, i s the l as t r u l e t o be evaluated. A backward looking, one stage, review of t h i s c r i t e r i o n i s incorporated a t t h i s stage in the computation. A sunnary of the ru les follows.

Stages

1 2 3 4

R u l e s Used by the Expert System in This ExamLe

Uithout ES

nap . <p<p 11

IIORII ~

1.

2.

3.

4.

5.

6.

7.

8.

547 761 3876

275 546 2076

51

31

31 142 I /

I n s t a l l a t i o n a b i l i t y per stage

T v ~ e No. 1 2 3 4 5 6

Stage 1 2 1 2 1 0 2

Stage 2 2 2 2 1 0 2

Stage 3 2 2 2 1 1 2

Stage 4 2 2 2 1 1 3

Capacity l i m i t i ns ta l l ed 1.1PeakzPz1.4Peak l1ORDERn8 r u l e One nuclear must be ins ta l l ed a t stage 3. ~IOR" r u l e lldlll, ladZll cannot be i ns ta l l ed simultaneously a t the stages 1, 2. 8aAND8t r u l e I f X MU nuclear i s instal led, GT used must be greater than 0.1X MU. IIPRIOR I TY" ru 1 e 8gd311 must be ins ta l l ed before 18d411. Fuzzy 8tMIX88 r u l e The membership values o f loading capacity f o r Base, Peak and Emergency must be greater than given values. ltLOLPon r u l e LOLPz50 h r ./yr. (Normal) 50 hr./yr .(LOLPzI 00 h r ./yr. To review one stage backward.

(Relaxed)

Results U i t h w t Fuzzy ES

This problem was run by the computer program without including r u l e 7 - the qua l i t a t i ve c r i t e r i o n - and the resul ts are s m r i s e d in tab le 4. The ef fect o f adding the other seven ES ru les one a t a time in the sequence shown i s i l l us t ra ted . Instead o f having 51 feasible routes a t the end of the f i r s t planning stage the ES gradually whi t t les

"AND"

"LOLP" 243

23 54 105 S u b - o p t imal ROUTES (& STATES)

Qual i ta t ive FUZZY C r i t e r i a

The incorporat ion o f r u l e 7 i s now discussed. A simple r u l e has been selected f o r the example, although

strategic, technical or managerial judgements of a qua l i t a t i ve nature and with conplex interdependencies can be eas i l y included in the ES. What the W X b l r u l e says i s that i ns ta l l ed p lant i n the system must be capable of discharging the systems Base Load, Peaking and Emergency standby dut ies (a t t r ibutes) wi th not less than cer ta in acceptable levels of confidence. The s u i t a b i l i t y o f each of the s i x candidate p lant types f o r each of these dut ies (a t t r i bu te rating), the r e l a t i v e importance of each of these at t r ibutes (a t t r i bu te weighting), and the minimal acceptable system standard (cr i ter ion) , are qua l i t a t i ve l y speci f ied by the planning staff/management.

Using dl t o d, t o denote each o f the s i x dimensions of the s tate space ( s i x candidate p lant types) f i g . 5 (a) t o (c) i l l u s t r a t e the a t t r i b u t e ra t ings (rij) and " b e r s h i p functions ( B ~ ~ ) assigned by the experts t o each rat ing. This const i tutes the fuzzy set Rij={(rij, fiij)). The importance assigned by expert opinion t o each o f the at t r ibutes (a t t r i bu te weighting) i s once again a matter o f judgement, an expert input. These weighting are shown i n f i g . 6 and const i tu te the fuzzy set Uaj=C(waj, baj)) and are used i n a r r i v i ng a t a f i na l , or overal l , assessment of system s u i t a b i l i t y .

The dimension weightings wdi are computed, as already explained in a previous section, simply in proport ion t o the gross capacity o f each type of p lant i n the system, i n any pa r t i cu la r feasible state, a t any pa r t i cu la r stage.

Final ly, when the overa l l fuzzy ra t i ng o f the system state has been obtained i t has t o be compared with what the planners th ink i s acceptable. This once more i s a c r i t e r i o n

, , .. , I

Rules Added In Following

Order

Uithout ES

llpmir" "OR"

"AND"

"PRIOR I TY

Fuzzy W I X J l

"LOLP"

Sub- optimal ROUTES (& STATES)

64 The e f fec t o f including r u l e 7 i n the optimal planning

algorithm i s i l l u s t r a t e d i n tab le 5. The f i r s t s i x entr ies P o f the f i r s t c o l u m are the same as the corresponding

entr ies o f tab le 4. However, o f the 25 feasible routes 1.0 surviv ing a f t e r r u l e 6 a fu r ther s i x are eliminated by

r u l e 7, the fuzzy c r i t e r i m , and only 19 survive thereafter. The r e l i a b i l i t y c r i t e r i o n fu r ther reduces t h i s by 13 leaving only s i x feasible routes in stage 1. These s i x states project i n t o 294 routes i n stage 2 i f no ES i s enployed. This nunber i s reduced t o 81 before ru le 7 i s used, t o 73 thereafter, and f i n a l l y t o 21 when r e l i a b i l i t y constraints are included. The DP algorithm, selects 16 sub-optimal routes and states f o r fu r ther consideration. The progression through stages 3 and 4 i s s imi lar .

-

0 0.2 0.4 0.6 0.8

(a) Base Load

d 5 d2 dl d 3 d l d 6

Stages

1 2 3 4 51 294 459 1976 31 199 344 1208

31 97 / /

/ / 93 793 25 81 85 666

19 73 75 577

6 21 40 132

6 16 27 63

I 1.0 r

(b) Peaking

(c) Emergency

Fig. 5 Expert Opinions of Plant Types

Fig. 6 At t r i bu te Ueightings

Fig. 7 Overall C r i t e r i on and Overall State Rating

which i s best formulated i n f lex ib le , qua l i t a t i ve terms. This leads t o the fuzzy c r i t e r i o n funct ion i l l u s t r a t e d by f u l l l i nes i n f i g . 7 - an a s s 4 input based on the planners experience. The c r i t e r i o n funct ion could also be derived. by select ing some system conf igurat ioh as " jus t good enougha* and using the corresponding membership funct ion as the reference set (C of f i g . 7).

The appl icat ion o f the c r i t e r i o n i s i l l u s t r a t e d in f ig.7. A par t i cu la r s ta te (d,,d,,d,,d,,d5,d,)=(3,2,3,0,0,21 whose consideration i s posed a t a cer ta in stage in the problem i s used as an i l l us t ra t i on . The f i n a l membership funct ion pR of t h i s s ta te i s derived in accordance with the theory described i n the previous sections o f the paper. This i s p lo t ted on f i g . 7 and i s shown by dotted l ines. Using C f o r the fuzzy c r i t e r i o n ( f u l l l ines) and R f o r t h i s overal l fuzzy s ta te ra t i ng (dotted l ines) i t can be seen that the condi t ion R>C defined in the paper i s sa t i s f i ed and, therefore, t h i s i s an acceptable s ta te a t the relevant stage.

A single, overall, fuzzy system ra t ing has been derived and compared with an overa l l system c r i t e r i o n in the previous presentation. A s t r i c t e r approach would be, i f the system was required t o 81passa8 a c r i t e r i o n tes t a t t r i bu te by at t r ibute, rather than. an averaged out tes t over a l l at t r ibutes. In t h i s case the a t t r i bu te weights ( f i g . 6) are not used. Only the dimension weights wai and the fuzzy sets Rij are used t o der ive a set o f f i n a l (overal l ) s ta te a t t r i bu te ra t i ng membership functions pRj . The three fuzzy sets R j derived i n t h i s way fo r j=Base, Peaking and Emergency are shown by dotted l ines i n f i g . 8. Experts supply a set o f mininun requirements, c r i t e r i o n functions, i n respect of each a t t r ibu te , taking i n t o account the importance o f each a t t r i bu te i n the i r own judgement. These are shown by f u l l l ines on f i g . 8. It w i l l be seen that in t h i s exanple the s ta te f a i l s the c r i t e r i o n tes t i n respect o f the Base loading consideration and has t o be declared infeasible.

R E 'P 'PCE

Rules Added I n Following

Order

Fig. 8 A t t r i b u t e by A t t r i b u t e Analysis

Table 6 shows the problem so lu t ion s m r y f o r the case when separate a t t r i b u t e ra t ings are employed. In general, i t i s t o be anticipated, that a t t r i b u t e by a t t r i b u t e tes t ing u i l l be more rigorous than using a s ingle overal l index. This i s shoun up i n the table entr ies. Only f i v e sub-optimal routes and states survive up t o stage 1, 12 up t o stage 2, 21 t o stage 3 and 55 up t o stage 4 compared with corresponding values of 6, 14, 27 and 63 i n table 5.

Stages

1 2 3 1 .

Table 6 Number o f ROUTES Including the Fuzzy W I X t l Rule based on Attr ibute-bv-Attr ibute CaDabil i tv

Uithout ES

alp . 'P'P 11

~ ~~~

51 225 370 1632

31 123 281 1010 IIORII

"AND"

31 76 / /

/ / a2 658

"PRIOR I TY"

Fuzzv "MIXln

The use o f the ueak c r i t e r i o n tes t i s not i l l u s t r a t e d in t h i s example presentation since the methodology i s a f a i r l y simple extension of the approach discussed here.

25 60 76 538

14 46 60 485

COWCLUS ION

I'LOLP"

Sub- optimal ROUTES (& STATES)

The program described i n t h i s paper i l lus t ra tes hou an expert system f o r knowledge based inputs, qua l i ta t i ve judgements of a technical, managerial or strategic nature, and an optimisation method may be brought together. The appl icat ion around uhich the structure of the program has been developed i s long range power system planning. However, other appl icat ions uhere the l ink ing together of the three methodologies (expert system, q u a l i t a t i v e judgements and optimisation) i s s ign i f i can t u i l l also find the approach developed here t o be o f interest .

Qual i tat ive judgements ahd c r i t e r i a are integrated i n t o the program by the use of fuzzy set theory. Various q u a l i t i e s (or a t t r ibu tes) o f p lant are i d e n t i f i e d and given rat ings (or assessments o f goodness). The l a t t e r are q u a l i t a t i v e l y specified. These speci f icat ions are translated i n t o fuzzy sets by the program. The various rat ings o f the indiv idual parts of the system are combined together t o give one or more overal l fuzzy rat ings. F i n a l l y t h i s measure o f overa l l system goodness i s compared u i t h

5 14 30 112

12 21 55

65

q u a l i t a t i v e l y speci f ied c r i t e r i a ( m i n i m acceptable standards of overal l goodness) t o determine uhether each possible fu tu re s ta te of the system i s r e a l l y good enough t o the included in the fu r ther consideration of optimal plans.

REFERENCES

111 David, A. K. and Zhao, R. D., I I Integrating Expert Systems with Dynamic Programing i n Generation Expansion Planning1#, IEEE Transactions on Power System, Vol. PURS-4, NO. 3, pp. 1095-1101, August 1989.

121 Baas, Sjoerd H. and Kwakernaak, Huibert, "Rating and Ranking of Multi-Aspect Al ternat ives Using Fuzzy Sets", Automatica, Vol. 13, pp. 47-58, 1977.

131 EPRI, olElectric Generation Exwnsion Systemnn, 5 Vol., Report EL-2561, E l e c t r i c Power Research Ins t i tu te , August 1982.

141 Jenkins, R.T. and Joy, D. S., Wein Automatic Svstem Planning Package ORNL-4954, Oakridge National Laboratory, July 1974.

151 Kandel Abraham, llFuzzy Mathematical Techniques u i t h Amlicationsbl, pp. 15-16, Addison-Yesley Publishing Co., 1986.

BIOGRAPHIES

A. Kunar David ( M I 83) was born in Colombo, S r i Lanka on July 29, 1941. He received the B. Sc. Eng. Degree with f i r s t class honors from the Universi ty o f Ceylon, Colombo in 1963, and the D. I. C. and the Ph.D. degree from Imperial College, London i n 1969.

From 1963 t o 1966 and from 1970 t o 1980 he was attached t o the Universi ty of S r i Lanka, and during the per iod 1970 t o 1974 he uas also a part- t ime Director of the Ceylon E l e c t r i c i t y Board. He was a t the Universi ty of Zimbabue from 1980 t o 1983 before jo in ing the Hong Kong Polytechnic uhere he i s nou a Reader. He has held v i s i t i n g Professorships a t several ins t i tu t ions , including the Royal I n s t i t u t e o f Technology, Stockholm, Sweden, and Clarkson University, Potsdam, MY.

D r . David i s a member o f the Hong Kong Joint Chapter of the IEEE Pouer Engineering and Indus t r ia l Applications Societies.

Zhao Ronsda was born in Dalian, China on July 25, 1945. He graduated from Tsinghua University, Be i j ing and received h i s M. SC. degree from T i a n j i n University, T ian j in i n 1968 and 1982 respectively.

He uas a t the Fengning Pouer Bureau, Chengde, from 1969 t o 1979 as an e l e c t r i c a l Engineer and uorked i n the Operation and Planning Departments. In 1979 he joined T i a n j i n Universi ty as a Post-graduate student and joined the s t a f f i n 1982. He i s presently a Lecturer i n Power Engineering. He i s nou engaged i n power systems research i n Hong Kong Polytechnic on a col laborat ive program.