00004262_Youssef_1988_Dynamic Solution of Distribution Planning in Intermediate Time Range

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IEEE Transactions on Power Delivery, Vol. 3 , No . 1, January 1988DYNAM C SOLUTI ON OF D STRI BUTI ON PLANNI NG I N I NTERMED ATE TI ME RANGE

H K. Youssef , Student Member R. Hackam Seni or MemberDepart ment of El ect r i cal Engi neer i ng

Uni versi t y of Wndsor, W ndsor, Ont ari o, N9B 3P4, CanadaABSTRACT

Thi s paper appl i es t he t i me-phased mode of thel ong range di st r i but i on pl anni ng model i nt roduced i n arecent publ i cati on by t he authors. The modeldescr i bed i n t he previ ous work i ncorporates anaccyrate f ormul ati on of t he cont i nuous non- l i near costf uncti on. Once t he l oad-t i me vari at i on i n t he areaunder consi derati on i s known, t he model t akes i ntoaccount t he di f f erent t ypes of costs ( f i xed cost s,vari abl e cost s and t he cost of energy l osses) f or al lpl anned f aci l i t i es ( subst ati ons and feeders) and t henei ther t he opt humconst r uct i on or expansi on of thesystemi s obt ai ned t o sati sf y t he network operati onaland securi t y constr ai nt s. A real i sti c uti l i ty exampl ei s used t o campare t he dynamc and t he st ati csol ut i ons.

I NTRODUCTI ONThe pl anni ng of el ectr i cal - power networks i s acompl ex process i n whi ch t he appl i cat i on of comput i ngtechni ques has grown steadi l y. I n recent years ther e

has been a number of advances i n t he appl i cat i on ofmathemati cal programni ng t o t he sol ut i on ofdi stri buti on system pl anni ng [l-93. I n each of t hemodel s [l-31 approxi mati ons are necessari l y made i nvaryi ng degrees of det ai l , t o t he pr i mary f eedernetwork. M xed i nt eger l i near pr ogram ng has beenus ed i n these d e l s but was l i mted t o rel ati vel ysml l systems because of t he i neff i ci ency of t het echni ques used. Expl i ci t i ncl usi on of t h- dependantcosts and vol t age drop const r ai nts has been appl i ed[ 4] t o i mprove t he ef f i ci ency and t o ensure t hef easi bi l i ty of mxed i nteger l i near progr amng. Al i near t r ansshi pnt model has been used i n [5 ] whi chcontai ns al l t he detai l s of Adams and I aughton model[ l ] , f or a si ngl e t i me peri od, except f or t he f i xedcharges of the f eeder segments. However, due t o t heassumpti on of t he l i near i t y i nvol ved i n t he model , t hesol uti on may be unf easi bl e or non- opt i mal . D f f erentbranch and bound cr i t er i a have al so been used [6-81.

The di f f erent l i neari ty assuqti ons i nvol ved i n[l-81 l ead t o a near but not to t he opt i mal sol ut i on.I n t he model descr i bed i n [9] an accurate andconti nuous cost model i n the p e r f l ow of eachel ement of t he network and w t h t he t i me-dependentcapabi l i t y was presented. Thi s model si mul ates theactual r el ati on between t he pl anni ng var i abl es, t hei nstal l ati on ti me and t he p e r f l ow of each networkel emnt , and the f i xed and t he var i abl e costs of thatel emnt i n addi t i on t o the cost of i ts energy loss.The model al so has t he advant age of al l ow ng t hepossi bl e expansi on i n t he capaci t y of some or al l theexi st i ng f eeders and substati ons i nstead of

37 WY 174-6by the TEEE Transmiss ion and Dis t r i hu t io n Commit teeof the I E E E Power Englneering Society or presenta -t i on a t t h e IEEE/PES 1987 Wtnter Meetin g, !Jew Orleans ,Louis iana , February 1 - 6, 1987. Manusc ript submit-ted January 30, 1386; made ava i la b le or pr fn t ingDecember 1 5 , 1986.

A paper recommended and approved

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const ructi ng new ones. The number of var i abl esi nvol ved i n t he model i s rel ati vel y smal l er comparedto t he di scr ete model s previ ousl y empl oyed. I naddi t i on t he cont i nuous non- l i near nature of thef ormul at i on of t he model i s sui tabl e f or appl i cat i onsi n cont i ngency and rel i abi l i ty studi es.

Al though t he model has been formul ated i n af l exi bl e f o mt whi ch makes i t sui tabl e f or l ong rangepl anni ng i n one st ep (e. g. 20 years) or t i me-phasedpl anni ng ( e.g. every 5 years) , i t was sol ved f or aspeci f i c case f or l ong range pl anni ng of 20 years [ 9 ] .I n t he present work t he model has been appl i ed t othe di str i but i on syst emdescri bed i n [ 9 ] but i ncl udi ngthe ref i nement of t he dynamc pl anni ng mode ( ever y 5years!. A compari son has been made between the st at i csol uti on obtai ned by t he model and the dynamcsol uti on ( over t he same peri od of t i me) obtai ned i nthi s work.

bDDEL DESCRIETI ONThe obj ecti ve of t he di st ri but i on systempl anni ngi s t o desi gn di st r i buti on systems t hat canef f i ci ent l y, econcani cal l y, and rel i abl y sati sf y t hel oad demands whi ch m ght grow i n ti me. The port i on ofthe di str i but i on system bei ng pl anned i ncl udes t hedi st r i buti on subst ati ons and t he pr i mary f eederci rcui ts. Expandi ng f rcman exi st i ng syst em( i f any),new subst ati on si t es and f eeder routes may be addedat cert ai n t h s t o meet t he changi ng l oadrequi rements.Know ng t he spat i al l oad vari at i ons i n t he areaunder consi derati on t he di st r i but i on engi neer has toconsi der di f f erent pl anni ng vari abl es. Thi s i ncl udesthe exi sti ng faci l i t i es (substat i ons and feeders) , al lthe possi bl e si t es f or new substat i ons, t he potenti alri ght - of - way and t he exi st i ng expandabl e f aci l i t i es.I n addi t i on t he i nst al l ati on and operati on t i me mustbe consi dered. The opt i mal desi gn must sati sf y t he

network operati onal and securi t y constr ai nt s w thm ni overal l cost f or al l el ements of t he systemThe f ol l ow ng constr ai nt s shoul d be sati sf i ed by t heopt i m sol ut i on:1. Vol taqe Const rai nts

The vol t age const rai nts ar e i ncl uded i n t he modeli n two f or m

T7 7 ESPi [pe b] 0,i 1 - vi 2 - zi ce=lU Lwhere Vn and Vn are t he per uni t upper and l ower

l i mt s on t he vol t age at node n, Vn i s t he per uni tvol tage at node n at t i me segment r , Vi 1 t he per uni t

rr

0885-8977/88/0100-0341$01 WO 1988 IEEE


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where NSEP i s the subset of al l potent i al andexpandabl e substat i ons, NFEP t he subset of al lpotent i al and expandabl e f eeders, NS, t he t otal numberof exi sti ng and potent i al subst ati ons, NF, t he t otalnumber of exi st i ng and potenti al f eeders, F, t hepresent wort h of capi t al i zed cost f or potent i al orexpansi on permt ted of subst ati on s whi ch i s anel ement of t he subset NSEP, Fi r t he present wort h ofthe capi t al i zed cost f or potent i al or expansi onperm t ted of f eeder i , f j , t he present wort h ofvari abl e costs f or el ement j ( f eeder or substati onexcl udi ng i t s energy l osses cost), and L., t he presentwort h of cost of energy l osses i n el emen? j .Fromequat i on (6) t i s cl ear that t he obj ecti vef uncti on i ncl udes t he var i abl e cost s of al l f eedersand substat i ons, t he capi t al i zed costs of t hepotent i al f eeders and t he subst ati ons, as wel l as t hecapi tal i zed cost of t he possi bl e expansi on perm t tedi n t he f eeders and t he subst ati ons. The mai ntenanceand t he operat i on cost s are i ncl uded i n t he var i abl ecosts. The cost of t he energy l osses are al soi ncl uded i n t he accept ed f orm of a non- l i nearrel at i onshi p w t h t he pl anni ng vari abl es. The costmodel i s a cont i nuous non- l i near f unct i on i n t heopti mzati on vari abl es whi ch i ncl udes the p e r f l owof t he el exent s, and t he const ruct i on ti mes of t hef uture f aci l i t i es. The network node vol tages are al soconsi dered as opt i m zat i on vari abl es. For mr e det ai l

about t he ccenponents of t he cost f unct i on C, acompl ete sumpary of t he mat hem i cal programni ngf ormul at i on i s gi ven i n Appendi x I .APPLI CATI ON

The model has been used t o produce opt i mal t i me-phased pl an (every 5 years) of t he subt ransmssi onsystemdescr i bed i n [9]. The cost s f or constr ucti ngsubt ransmssi on f eeders, whi ch are t ypi cal f or 556K&l a l u conductors, are esti mated at$66,OOO/Km. The cost of const ructi on of pr-di str i but i on feeders (336 K& al umhumconduct ors)i s esti mated at $40,OOO/Km. New di st r i but i onsubstat i on (5000 KVA) have an esti mated const ruct i oncost of $370,000. Tabl e 1 presents t he l oad- t i mevari at i ons where f or each l oad node f i ve par mt er sare l i sted: l oad node number, l oad at i t s saturat i onval ue, i ni ti al percentage, t he f i rst year of l oadgrowth, and the year t hat t he l oad saturates.TABLE 1 - Network l oad data.

198819832002200219851983198319922002200220021985

-OADNODENUPIREI;616263646566676869707172737475767770798081828384858687 r

LOADSATURATIONVALUE, kVA77708290737092009020750300040508840792010500644516650737011050205008600.20680110507440777074003931381023220415407670

INITIALPERCENTAGE

77.254.354.354.354.3100.0100.054.354.354.354.354.375.054.354.377.2100.048.454.394.0106.0100.050.952.325.855.091.3

FIRST YEAROF LOADCROWPH198319831983198319831983198319831983 ,198319831983198319831983198319831983198319831983198319831983198319831983

SATURATIONYEAR I20022002 1983198320022002200220022002199120022002 I

F i gwe 1 present s t he t ypi cal 44 KV di str i but i onsystem under consi derati on. The network has no

343

substati ons t o be bui l t w thi n the peri od of pl anni ng.114 f easi bl e f eeder s cont ai n 39 3- phase exi sti ngf eeders and 30 3-phase f utur e f eeders and 45 l oadbranches.The syst emsuppl i es the 27 l oad poi nts, l i sted i nTabl e 1, f romthree exi sti ng substati ons (SI and S2)have 140 MVA r ati ngs and $3 has 200 MVA. 1.0$/ m year vari abl e cost , i ncl udi ng t he energy l ossescost , at year 1983 i s used. The capi t al cost of t hef uture 3-phase f eeders are $66,OOO/Km i n t he year1983. The capaci t y l i mt of t he f eeders are 35 WA.The i nf l ati on rates consi dered are assumed i n thi sexampl e, as i n [9], t o be const ant at 5.0% and 6.7%,respecti vel y, f or t he f i xed and t he var i abl e costs.The i nter est rates are al so assumed t o be constant at17.0% and 10.0% f or f i xed and var i abl e costs,respecti vel y.The d e l has been t est ed on an I BM 3031 ccwputerat the Uni ver si t y of Wndsor. The programuses 10142words f or memry and i t t akes 39.8 mnutes (CPU t k )t o obtai n a compl ete sol ut i on.

RESULTS AND DI SCUSSI ONThe dynamc opt i mal sol uti on (every 5 years)of t hesyst em f or t he peri od 1983-2002, i s shown i n Fi gure2. Tabl e 2 i ncl udes power f l owpat terns t hr ough t he

network al ong w t h the recei vi ng end vol t ages at eachnode for t he f our i nt er val peri ods of f i ve y e a r s each.Fromexamnati on of t hese resul t s, i t i s f ound t hat :1. Onl y 2 newf eeders have been added to t he networkSI-44 and S2-36 and they have been i n servi ce st art i ngat t he t hi r d t h eri od. Thi s i s campared t o t he onestep sol uti on of 20 years [9] where 15 new f eederswere used.2. Feeder S3-16 beccarr?s sl i ght l y overl oaded af t er t hef i rst t wo t i me peri od secpents (i . e. af t er 10 years).The per cent age over l oadi ng i n thi s f eeder i s (3.2% i nthe thi rd t i me segment and 6% i n the four th t hseqent ) . I f t he pcwer capaci t y of t hi s f eeder hasbeen i ncreased by 25% af ter the second t hperi od, assum ngt hat t he capi t al cost of t he f eeder i sl i nearl y proport i onal t o i ts capaci ty l i mt; thi sexpansi on w l l cost $44,980.3 . The var i abl e cost over t he pl anni ng peri od i s$7,047,688., w t h fi xed cost $181,053., produci ng$7,228,741. f or t he overal l cost. Thi s i s l ess t hanthe over al l cost produced f or t he one-st ep sol ut i on of20 years [9], al t hough t he systemi n t hi s sol ut i on i soperat i ng over t he whol e pl anni ng per i od. Even i f t heexpansi on suggest ed i n t he above conwnt i s made t oavoi d any overl oadi ng t he t otal cost w l l be$7,273,722., whi ch i s sti l l l ess.4. No substat i ons has been overl oaded i n any of t het i me per i ods.5. The t ransshi prent node 33 i s f ed f romtwo sourcesi n the fourt h t k ecpent and thi s non- radi al probl emcan be sol ved by t he l oad spl i t t i ng t echni que [6]. I naddi t i on l oad poi nt 86 i s al so fed t hrough t wo feedersdur i ng t he thi rd. and the f our t h t i me peri ods and thati s because t he demand at thi s l oad i s hi gher t han t hecapaci t y l i mt of a si ngl e f eeder dur i ng t hese twoperi ods. Agai n, thi s can be sol ved by the spl i t t i ngtechni que [6].

Fromt he above resul t s we can see that t he dynamcsol uti on obtained here i s bett er t han t he st at i csol uti on obt ai ned earl i er [9]. The ef f i ci ency of t hedynamc sol ut i on can be i nproved usi ng short er t i meseqent s (e.g. 5 secj mnts, of 4 years each). Thiswoul d make t he pl an easi er t o be modi f i ed i f


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344FIGURE 2 - Dynamic optimal solution of t he system (every 5 years).

( a ) After 5 y e a r sNo new feeders have beenadded. Al l feedersandsubstations powers arew i t h i n the capacityl im i t s . h d ia l i t y i ssatisf ied.

(b ) After 10 years

No &tion of new faci-lities and there is nooverloading. The networki s radial.

v321 7177 8 4 T 1 9 r

79f ?9

12 74J's i s

85

77

56

3 '

e37

38

6 8

e36

'1,e e46 45 174


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345

3 6 4 4

TWO new feeders havebeen added SI-44 andS2-36. Feeder S3-16 isoverloaded by 3.2% aboveits capacity limit.Load point 86 is fe d bytwo feeders.

id) After 20 years

No new facilities havebeen added. overloadingof feeder S3-16 has beenincreased to 6% aboveits capacity limit.Load point 86 is fedthrough two feeders.Transshipnt point 33is fed f r m two sources.

043

2 8 85 ?713 S2

2577 I'037

05365 s5

5 9 * y

73 7274 76 i26145

?8177

054 *37

157 L 2

A12 - 174r' s i s


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346TABLE 2. Opth- al power f l owpat terns t hrough t he network al ong w t h the recei vi ng end vol tages at each node f ort he f our t i me peri ods. PI t o P4 are t he powers i n per i ds 1 to 4. V1 t o vd. are the V O l h F S i n peri ods 1 to 4 .

F R O M TONODE NODEs2 4

s2 64 5

6 77 88 99 1010 118 127 1313 1414 15

1616 1717 1817 19

2020 21

2222 15

2626 27

2323 2424 28s1 2929 25

3030 3430 3131 3232 3333 17

353644

4 766 6236 629 6310 6411 651 3 6714 6812 665 6127 7926 78

s3

s 7

53

s3

s3

s1

s1s2s1

22 7418 70?9 7133 8416 6923 7524 8725 7729 8131 8232 8334 8528 8035 8644 8670 722 1 13

23981 46.2257416 45.833

25407 46.31620108 45.18117108 44.64116358 44.12911647 43.7325766 43.357750 44.617

3000 45.156

-2589 46.19326253 45.74120602 45.1775063 45.0216712 44.94418629 45.55014509 45.0327300 46.3072509 , 46.25819319 45.1357063 45.03822173 46.04415110 45.5037440 45.12916370 45.9378600 45.646

17304 46.5459618 46.4757686 46.511286 46.503

1 -8827 44.92426782 45.812

16565 46.2255299 46.316

I - -4711 44.1295881 43.7325766 43.3573000 45.1562589 46.193750 44.617

7416 45.8337063 45.038

12256 45.1354711 46.3075063 45.0216712 44.9548827 44.9245651 45.7417063 46.0447670 45.5038600 45.6467770 45.9377400 46.511286 46.503

9618 46.4751440 45.129

26782 45.812

4120 45.55814509 45.032

1 p2 V2 P3 v328270777026580201842018419434138376850750

-3000-6076311912447760157974

21545166501167360762345683922350215110744016370860021945141527793393

-1048831702

205006296-55976987685030003076750

777083921506455976015797410488671483927670860077707400393

14152744031702

489516650

46.15945.74846.30345.16444.52743.91843.44743.00144.503

45.83045.86645.58044.91044.72444.64545.39544.79246.13446.01944.82244.70746.01145.47045.09645.93745.64646.53246.42846.49046.486

44,60945.668

46.15946.303-43.91843.44743.00145.83045.86644.50345.74844.70744.82246.13444.72444.64544.60945.58046.01145.47045.64645.93746.49846.48646.42845.09645.668

45,39544.792

282707770

26163262632326322513160297935750

3000

-356336131283546968923722320166501004735632759397212483115110744016370860026479186867793I 393

-121493500072931621

20500- -729364848094793530003563750

777097211787264846968923712149777797217670860077707400393

186867440

350001621567016650

46.15945.7?846.30744.82444.09043.38542.83942.32344.06644.799

46.04145.42044.64344.42844.33645.35244.74946.19846.13144.50944.37645.97845.43745.06345.93745.64646.51846.38246.48446.97L

44.29445.57146.54346.55946.159-46.54343.38542.83942.32344.79946.04144.06645.74844.37644.50946.19844.42844.33644.29445.42045.97845.43745.64645.93746.48446.47246.38245.06345.57146.55945.35244.749

P 4 v428270 46.1597770 45.74R

26340 46.30626340 44.81926340 43.9885590 43.18618220 42.5669020 41.979750 43.964

-3000 45.579-7050 45.71537087 45.38828247 44.6147920 44.37010500 44.26523095 45.30916650 44.70614420 46.0257050 45.892

31730 44.19611050 44.04526160 45.94515110 45.4047440 45.030

16370 44.0388600 43.74734996 44.59423220 44.42411776 44.5414376 44.4093983 14.332-9827 44.33235000 43.6728290 46.5356540 44.54620500 46.159- -8290 46.5357370 43.1869200 42.5669020 41.9793000 45.6794050 45.715750 43.9647770 45.74811050 44.04520680 44.1967370 46.0257920 44.37010500 44.26513810 44.3328840 45.38811050 45.9457670 45.4048600 43.7477770 44.0387400 44.541393 44.409

23220 44.4247440 45.03035000 43.6726540 44.5466445 45.309

1

16650


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rf and rt are t he i nf l ati on and t he i nterest ratesrespecti vel y, K wei ght i ng f act or >>o, t o obtai nramp deci si on f unct i on, U t he power capaci t y l i mtof pwer segment e of suppl y S, Ue si ml ar t o Ue butf or f eeder i , t r real vari abl e representi ng i nstal l -ati on t i me of t he new suppl y f aci l i t y s at pl anni ngt i me segment r , t, si ml ar t o t, but f or new f eederfaci l i t y i , P posi t i ve real var i abl e representi ngpower f l ow i n power secpent e of suppl y s at pl anni ngit i me segment r , and Pre real var i abl e represent i ngpower f l ow i n power segment e of f eeder i at pl anni ngt i me segment r .

Thi s f ormul ati on makes i t possi bl e t o det erm nethe opti mal i nstal l ati on t h e of the new f acil i t i essi nce t hese t hes are consi dered expl i ci tl y. For mr edetai l see Reference [9].Vari abl e Costs

si s

S

i sS

Whi l e t he capi t al costs are onl y pai d once at t het i me of i nst al l ati on, t he var i abl e cost of an el ementj ( subst ati on or f eeder ) i s represented by the sumofequal payments pai d dur i ng al l t he years of servi ce.The equi val ent present wort h val ue of t hose paymentsi s f ormul ated as a f unct i on of t i me [lo]. As i n thecapi tal cost s, t he val ue of t he vari abl e cost at thepl anni ng t h e i s consi dered and t he corr espondi ng

H K. Youssef was born i nCai ro, Egypt , on August 26,1956. He recei ved t he B.Sc.(El ec. ) and M.Sc. ( Power andMachi nes) Engi neeri ng f r omCai ro Uni versi t y, Eg y p t , i n1979 and 1982, respecti vel y.He j oi ned t he Depart ment ofEl ectr i cal Power and Machi nesi n Cai ro Uni versi ty i n 1979 asDemonst rator then becameAssi stant Lecturer i n 1982.He has been on l eave ofabsence f rom Cai ro Uni versi tysi nce t hen and present l y work-i ng on hi s Ph. D. wee at Uni versi t y of Wndsor,Canada. H s f i el ds of i nterests are sol i d i nsul atorper f ormance under hi gh vol tage condi t i ons i n vaccumand tr ansmssi on and di st r i but i on systempl anni ng.

val ue at t he i nservi ce t h e i s cal cul ated based on thei nfl at i on rate. The term f j i n [6] can then bef ormul ated as f ol l ows:

where f i i s t he present wort h of t he vari abl e cost f orel ement i , El t he vari abl e cost of el emnt i at t het i me of pl anni ng, Q = (1 + rt), T the sumof t i mesegments t o the end of segment r , ESPi t he total powersegments perm t ted f or el ement i .The t erm f j does not i ncl ude the cost of theenergy l osses.Cost of Enerqy I Dsses

The cost of t he energy l osses i s rel ated t o t hepower rati ng of an e l a n t i n a non- l i nearrel ati onshi p and i t i s consi dered as a second orderrel ati onshi p i n our model . However, i t can berepresented by other di f f erent i ated non- l i nearrel ati onshi ps. As i n t he vari abl e costs, t he samprocedure has been f ol l owed and t he t ermL j i n [6] sf ormul ated as f ol l ows:

where L . i s t he present wort h of . t he cost of theenergy l osses i n el enent j , and C t he cost of theenergy l osses i n el ement j at t h e of pl anni ng.Reuben Hackam recei ved theB.Sc. degree f rat t heTechni on- I srael I nsti t ute ofTechnol ogy, Hai f a I srael , i n1960 and t he Ph. D. degree fromt he Uni versi t y of Li verpool ,Engl and, i n 1964. From 1964t o 1969, he was engaged i ni ndustr i al r esearch at t heNel son Research Laborat ori es,General El ectr i c-Engl i shEl ect r i c Ccanpani es, Staf f ord,Engl and. From 1969 t o 1978,he was on t he academc st af fof t he Uni versi ty of Shef f i el dEngl and, f i r st as l ecturer, t hen as Seni or Lecturer,and became a Reader i n El ectr i cal and El ectr oni cCngi neer i ng i n 1974. Si nce 1979, he has beenProfessor of El ectr i cal Engi neeri ng at t he Depart mentof El ectr i cal Engi neer i ng, Uni versi t y of Wndsor, y\ i i ndsor, Ontar i o, Canada. Duri ng 1980-1981, D.Ilackam was and si nce 1984 i s t he Head of thi sdepartment. From1964 t o 1969, he was al so Vi si ti ngStaf f Lecturer i n t he Depar t ment of Mathemati cs andComput i ng Sci ence at t he Staf f ordshi re Pol ytechni c,%af f ord, Engl and, and f r w 1970 t o 1978, Vi sit i ngLecturer i n t he Depar t ment of Mathemati cs andCoi nputi ng Sci ence, Sheff i el d Ci t y Pol ytechni c,Engl and. D r . Hackam i s a regi stered mof essi onal

Fngi neer i n the Provi nce of Ontar i o.

00004262_Youssef_1988_Dynamic Solution of Distribution Planning in Intermediate Time Range

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