IEEE Transactions on owr Systms, Vol. 13, No. 1, Fbruary 1998 159 Industrial owr Dmand Rspons Analysis for On-art Ral-Tim ricing J.G. Roos Univrsity of rtoria rtoria, Rpublic of South Africa I.E. Lan, Mmbr, IEEE Enrgy Efficincy Entrpriss rtoria, Rpublilc of South Africa Abstract- Dmand-sid managmnt (DSM) programs in th industrial sctor appar to b conomically fasibl du to th larg controllabl loads and rlativly low costs pr control point. Innovativ lctricity tariffs provid on of th most important DSM altrnativs. Bcaus ral-tim pricing (RT) is considrd as an xcllnt managmnt option which rflcts th ral cost of gnrating lctricity to th nd usr, th lctricity cost saving potntial of RT through dmand managmnt is prsntd in this papr. A uniqu analytical approach is followd to dscrib th potntial lctricity cost savings mathmatically in trms of variabls familiar to both th nd usr and utility. Ths variabls includ th installd powr consumption capacity of th plant, th plant s spar nrgy consumption capacity, and trms that dscrib th structur of th RT tariff. I. INTRODUCTION Sinc th introduction of DSM in th 197 s, load managmnt projcts mainly concntratd on rsidntial loads. Som of th projcts hav rsultd in a fair profitability, but many of th programs hav not succdd in achiving th stablishd objctivs, mainly du to th siz of load pr control point. Bjark [l] statd that it is likly that applications with low cost pr controlld load may b found in industry, whr th controllabl load pr control point is rlativly larg. Flory t a1 [2] rportd that at many utilitis 2-1% of th industrial customrs account for at last 8% of th lctricity usag, which mphasiss th conomic fasibility of DSM programs in th industrial sctor. In th South Afiican situation th industrial load dominats, which motivatd th local utility, ESKOM, to introduc a Ky Customr focus group to promot markting and customr srvics to its larg industrial customrs. In th viw of ths E-558-WRS--3-1997 A papr rcommndd and approvd by th IEEE owr Systm Enginring Committ of th IEEE owr Enginrin,g Socity for publication in th IEEE Transactions on owr Systms. Manuscript submittd August 5, 1996; mad availabl for printing March 26, 1997. obsrvations, this study focuss on dmand managmnt in th industrial sctor. Th formulation of utility DSM goals is largly influncd by th utility s charactristics and xtrnal oprating nvironmnt. Although utilitis can offr a wid rang of inducmnts and incntivs to ncourag customr participation in a particular DSM program, ultimatly it is th customr dcision to participat which influncs th succss of th activity. DSM approachs and tchniqus should involv a partnrship btwn th utility and its customrs, sking common ground to maximis mutual bnfit. This procss will vntually lad to a customisd pricing agrmnt btwn a supply utility and a customr who is willing to participat in th DSM program. artis involvd in a customisd pricing procss should b awar of th structurs of various tariff options, and thy should hav knowldg of th possibl impact of ths DSM tariffs on th prformanc critria of both th utility and th customr. Although tim of us (TOU) pricing rprsntd a significant stp towards fficint lctricity pricing, thr is a growing rcognition thal dynamic tariff forms can b mor fficint. Dynamic pricing broadly ncompasss tariff structurs that hav on or mor lmnts which can b calculatd and postd clos to th tim of applicability [4]. This dfinition mbracs svral concpts dvlopd in th pricing litratur, such as ral-tim (spot) pricing and othr forms of innovativ rats. Th thory bhind this pricing stratgy is wll documntd [5]. By rflcting th ral cost of lctricity to th consumr through variabl prics for spcific - gnrally on hour - tim priods, th utility provids th consumr with th information ncssary to mak conomically sound load managmnt dcisions. Bnfits of spot pricing for a customr ar shown to incras with [3]: th magnitud of pric changs ovr tim; th magnitud of th customr s storag capacity; th amount of his pak production capacity. Ths obsrvations wr mad in [3] by mans of a linar program (L) basd optiimisation algorithm. Th purpos of this papr is to add mor insight into th lctricity cost saving potntial of ral-tim pricing (RT) through intllignt dmand managmnt. Th analytical approach as illustratd, will nabl utilitis and industrial nd usrs of 885-895/98/$1. 1997 IEEE
16 lctricity to acquir a bttr knowldg of th bnfits that RT can offr. On of ths bnfits, i.. th lctricity cost saving potntial, will b addrssd in this papr. It will b mathmatically prsntd as a function of variabls that dscrib th structur of th ral-tim prics, as wll as th configuration of th industrial plant, which includs th spar nrgy consumption capacity of th nd usr and th installd powr consumption capacity. This approach is uniqu and contributs to knowldg in this fild of rsarch. A load schduling stratgy which may rsult in minimum lctricity costs to th nd usr, is prsntd in sction 11. Th fasibility of th stratgy dpnds on crtain assumptions, which will b givn. Th mathmatical modlling of th pric duration curv (hourly marginal rat duration curv) is introducd in sction 111. In sction IV mathmatical xprssions of th lctricity costs of an nd usr undr onpart RT tariff structurs ar drivd. Sction V prsnts th mathmatical xprssion of th lctricity cost saving potntial undr RT, togthr with som cas studis to graphically display th impacts of som important factors on th saving potntial. Conclusions follow in sction VI. 11. OTIMAL LOAD SCHEDULING STRATEGY An industrial nd usr of lctricity that is abl to curtail procsss on short notic in ordr to rspond to hourly varying nrgy tariffs, may b abl to bnfit from RT. By assuming: that th plant has adquat installd storag capacity or spar nrgy consumption capacity; that no losss du to load schduling occur; that th dmand lvls of th individual controllabl procsss in th plant can b controlld, without constraints, btwn a maximum lvl,, and a minimum lvl,,,; that ach individual controllabl procss has a crtain constant bas powr (or powr loss componnt) I, that dos not contribut to any production; that th sam production targt should b rachd undr controlld and uncontrolld conditions within th sam tim horizon of H hours; that an amount of E kwh of lctrical nrgy is rquird to produc th rquird production targt; that a on-part RT structur is considrd without a fixd cost componnt (thus only marginal rats apply). Th total lctricity costs EC (in cnts) ovr H hours of production can b givn as: EC = x,.hmr, + x2.hmr2 +... + xh.hmrh whr x, rprsnts th total hourly powr consumption (actually th avrag of hourly powr) of th procsss in hour i, whil hmr, is th hourly marginal rat (in ckwh) of th RT tariff structur in hour i. Th aim is to find th (1) valus of x, which will minimiz th objctiv function in (1) subjct to th following st of linar constraints: and (XI -b)+(~2-b)+... + ( x ~ -b)= E x~+x~+...+xh=e+~.h (2) mm 5 xi 5 mcuc, i = 1,2,3;.., H (3) whr b is th total hourly non-productiv powr (or bas powr) which is assumd to b constant ovr tim. By mans of an uppr-bounding dual linar programming algorithm [6] it has bn shown that th minimum lctricity costs will b obtaind if th procsss powr dmand lvls ar ithr at,,, whn hourly rats ar high, and at,, whn th hourly rats ar low - a rsult which can b xpctd as th optimum fasibl solution of a L problm will b on th boundary of th fasibl rgion, which is partly givn by (3). Thr xists a crtain hourly marginal rat cut-off valu, HMR,,,, which will provid th thrshold pric abov which th powr lvls should b shut down to,,,, and blow which th powr lvls should b st at,,,,. An hourly marginal rat duration curv (HMRDC) can b usd to graphically display this concpt and to form th basis of th mathmatical xprssions which will follow. Fig. 1 illustrats actual discrt hourly marginal rat (hmr) valus for H hours, sortd from th highst to lowst valu to form th discrt HMRDC. Th corrsponding powr dmand lvls according to th proposd optimum schduling stratgy ar shown togthr with a non-chronological cut-off hour, Hour, whr transition occurs btwn th,,, and,, lvls. Whn this valu of H,,, is projctd upwards to th HMRDC, th valu of HMR,,, can b rad from th duration curv. It is vidnt from Fig. 1 that th shap of th HMRDC as wll as th valu of H,,, will hav an influnc on th valur of HMR,,,, which will hav a larg impact on th lctricity costs of th nd usr. It will b convnint to dvlop a mathmatical xprssion for th HMRDC which will dscrib th hourly marginal rats as a continuous fimction of nonchronological hours h. This xprssion will b usd latr to dtrmin th potntial lctricity cost savings undr RT. 111. MODELLING OF THE HMRDC Th load duration curv (LDC) offrs a tool by which DSM impacts can b quantifid into powr systm planning and opration. Modls of th LDC provid on of th most important tools in th analysis of lctric powr systms. Thr ar svral mthods attmpting to xprss th LDC mathmatically and a rcnt rport [7] prsntd an analytical approach which appars to giv crdibl rsults. Basd on this analytical mthod, a modl of th HMRDC is drivd [SI. With this modl th non-chronological hourly marginal rats hmr(h) ar dscribd in trms of four principal
161 Hourly Marginal Rats (clkwh) actual hmr valu for scond hour owr (kw) max avg 'min loss f cut Enrgy E (kwh) I I 4 ' ///,/v,/ ' "" ' Non-chronological Hours Fig. 1. HMRDC and corrsponding optimal powr dmand lvls / ( b H h paramtrs of th HMRDC, i.. th pak hour marginal rat, th bas hour marginal rat B, th tim horizon H, and th avrag valu of th hourly marginal rats ovr H hours, hmr.th last trm is dirctly proportional to th ara undrnath th HMRDC. Th rsulting mathmatical xprssion is givn as [8]: C.h - hmr(h)= B+(-B). ( I-- ;)ap [c/kwh]. (4) whr C is th curv shap factor. Th curv will hav a concav shap whn C < (lik th on shown in Fig. l), a convx shap whn C >, and a linar shap with a ngativ slop whn C =. Th valu of C is givn as [8]: whr th valus of R, ar th sam as that drivd in [7]. Th modlld curv will always intrsct with co-ordinats [hmr(h), h] = [, 1 and [B, HI. With rfrnc to Fig. 1, for th sam valu of H,,,, diffrnt valus of th shap factor C will rsult in diffrnt pric thrshold valus. Fig. 2 illustrats an xampl whr a st of actual hourly marginal rats is modlld in a duration curv. In this cas (5) TIME @lon-chronological HOURS] Fig. 2. Actual and modlld hourly marginal rat duration curvs = 33 ckwh, B = 3.2 cikwh, H = 48 hours, hmr= 1.2 ckwh, and C = -2.944. By inspction of Fig. 1, th total nrgy E rquird (in kwh) within H hours to rach th production targt is givn as: from which H,,, (in non-chronological hours) is drivd as: whr Q is th total spar nrgy consumption capacity (in kwh) of th controllabl procsss. If no load schduling is applid, it is assumd that th plant has to oprat on a constant powr dmand lvl of mg to produc th sam production targt in H hours. This valu will b btwn,,, and, with th sam ara E undrnath th powr curv. E = " -- H IV. ELECTRICITY COSTS TO THE END USER WITHOUT LOAD SCHEDULING Th basic structur of,a on-part RT consists of marginal nrgy rats applicabl to th hourly nrgy consumption of th nd usr. If on considrs no load schduling opration, and assums that th plant oprats at a constant powr dmand lvl of, to produc th production targt, an xprssion for th lctricity costs ECn[,,. (in cnt) is givn in (9) by using (4) and (8) (th footnot nls dnots no loud schduling). Th non-linar dpndncy of th lctricity costs to th paramtrs of th hourly marginal rats is vidnt from (9), whil it is linarly dpndnt on th spar nrgy consumption capacity Q of th plant.
~ 162 H EC,, = 1 avg. hmr(h) dh V. ELECTRICITY COST SAVING OTENTIAL Whn th nd usr is applying optimum load schduling opration as proposd arlir, an xprssion for th lctricity costs undr load schduling opration, Els, (in cnt) is givn in (1) (whr th footnot Is dnots loud schduling). Hcur H EC, = 5 m,,. hmr(h) dh + 1 HCU, mm = [ B. C2. H + H( - B) xpc ] - m,n - C2 [ H( - B)(C + I)] +,. hmr(h) dh [pma."- 1 th diffrnc btwn,, and m,,; th lngth of th production priod H; th spar nrgy consumption capacity Q, which mainly dpnds on E,,, and H; th shap of th HMRDC, which dpnds on th curv shap factor C. Th shap factor C dpnds on: th pak hour marginal rat ; th bas hour marginal rat B; and th avrag of th hourly marginal rats hmr. Equation (13) is rathr complx and it is difficult to undrstand th impacts of th mntiond variabls on th valu of ECS. Som cas studis will b givn to graphically illustrat th potntial lctricity cost savings. Considr fiv possibl shaps of th HMRDC. Th avrag valu for ach curv is th sam, i.. hmr= 5.95 ckwh. Th bas marginal rat B of ach curv is 3 ckwh. H = 148 hours. Fig. 3 illustrats ths curvs with pak marginal rat valus as follows: C = -7.31, = 33 ckwh; C = -1.813, = 12.95 ckwh; C =, = 8.91 c/kwh; C =.498, = 7.97 ckwh, C = 1.56, = 7.3 ckwh. I\ [FOR EACH CURVE AVERAGE HOURLY MARGINAL RATE = 5 95 dkwhl L (1) Th potntial lctricity cost savings ECS (in cnt) to th nd usr ar th diffrnc in lctricity costs btwn schduld and unschduld opration. ECS = ECnls-ECIs (1 1) Th xprssion for th prcntag lctricity cost savings %ECS is givn as: %ECS = 1.- 2 4 6 8 IW 12 14 16 TIME [NONLCHRONOLOGICAL HOURS] Fig. 3. Modld HMRDCs for fiv valus of C By substituting (9) rsults: and (1) into (11), th following...--[ Q B.C2 +(-B)(xpc-C-l)] [cnt] C2 (13) Th variabl H,,, is a function of th spar nrgy consumption capacity Q and can b substitutd by (7) into (13). It is vidnt that th following factors will hav an influnc on th valu of ECS: * th production targt which will dtrmin E; th pak installd powr consumption capacity mux; A. Cas Study I Considr a fictitious plant with m, = 1 kw, m,, = 3 kw, and I,,, = 3 kw. Th maximum installd nrgy consumption capacity Em, availabl to produc products in H hours is: Emox = H. (mm - ioss 1 [kwhl (14) whil th maximum spar nrgy consumption capacity Q,, ovr that priod is: From (14) and (1 5): Qmm = H. ('ma - mzn ) [kwh]. (15) By substituting th givn powr ratings of this fictitious plant
163 35 3 2 25 9 p 2 z 2 5 25 2 p 2 z -C = -1 813 -c=o +C = 498 -C = 1 56 k 15 y1 p 1 3 L 5 1 2 3 4 5 6 7 8 9 1 SARE ENERGY CONSUMTION CAACITY [x OF MAXIMUM INSTALLED CAACITYI Fig 4 rcntag cost savings vs spar nrgy consumption capacity (paramtr C is varid and,,, = i, = 3% of mar) into (1 6), th maximum spar nrgy consumption capacity of th plant is 1% of th availabl installd nrgy consumption capacity. Fig. 4 displays th prcntag lctricity cost savings to an nd usr as a function of spar nrgy consumption capacity Q. Fiv curvs ar displayd which rprsnt th fiv possibl HMRDCs undr considration. It is clar that a larg concav HMRDC (C << ) will rsult in th highst cost savings, spcially whn th nd usr has spar nrgy consumption capacity of btwn 1 and 8%. A powr systm that is xprincing rlativly high loss of load probability (LOL) valus may rsult in HMRDCs with larg concav shaps (rlativly low marginal rats for most of th tim, with a fw xcptionally high marginal rats causd by high LOL valus). Rfr to [5] for mor dtail on LOL and th thory of spot-pricing. By logic rasoning no cost savings ar possibl whn th plant has zro spar nrgy consumption capacity, or at 1% spar nrgy consumption capacity whn no lctricity costs ar incurrd bcaus thr is no production. This is also vidnt from Fig. 4. B. Cas Study 2 Th sam as th prvious cas, but I,,,, = 1 kw. Equation (16) now indicats that th plant has a maximum spar nrgy consumption capacity of 78% of th maximum installd capacity. No cost savings ar possibl byond that lvl. Fig. 5 displays th prcntag lctricity cost savings for this cas. Th shaps and magnituds of th curvs look similar to th prvious cas, but th rang of spar nrgy consumption capacity whr savings can b incurrd is narrowr. It is again obvious that th HMRDC with a larg concav shap (C = -7.31) will rsult in th maximum possibl cost savings. C. Cas Study 3 Th sam as in cas 1, xcpt that mjh =,, = kw. This may rprsnt th cas whr th bas load and loss of th k 15 E Ln 1 5 $ k 5 8 1 2 3 4 5 6 7 8 9 loo SARE ENERGY CONSUMTION CAACITY R OF MAXIMUM INSTALLED CAAClTV Fig 5 rcntag cost savings vs spar nrgy consumption capacity (paramtr C is varid, idmln = 3% and L = 1% ~ ~ of ~ max) plant s procss(s) ar ngligibl and th procss(s) can b curtaild compltly (,,,, = kw). Th largr diffrnc btwn, and,,, rsults in highr prcntag cost savings, which is vidnt from Fig. 6 and th first trm in (13:). Again th larg concav shapd HMRDC (C = -7.31) will rsult in th largst cost savings ovr a wid rang of spar: nrgy consumption capacity. 1 2 3 4 5 EO 7 8 9 1 SARE ENERGY CONSUMTION CAACITY 1% OF MAXIMUM INSTALLED CAACI] Fig 6 rcntag cost savings vs spar nrgy consumption capacity (paramtr C is vand, and,,# = = % of mj D. Cas Study 4 As in cas 1,,= 1 kw,,,,= I,,, = 3 kw. Th aim of this cas study is to invstigat th impact of th avrag of th hourly marginal rats, hmr, on th potntial lctricity cost savings. Th shap factor of ach of th fiv HMRDCs is takn th sam, L. C = -7.3 1, whil B = 3 c/kwh for ach curv. Howvr, th pak rat,, varis for ach curv to rsult in a diffrnt avrag valu for ach curv. Highr avrag valus of th hourly marginal rats ovr a priod of H hours will rsult in highr prcntag lctricity cost savings, %ECS, as is displayd in Fig. 7.
164-7 1 2 3 4 5 6 7 8 1 SARE ENERGY CONSUMTION CAACITY r/o OF MAXIMUM INSTALLED CAACITV Fig 7 rcntag cost savings vs spar nrgy consumption capacity (paramtr is varid, and,,, = i, = 3% of,j VI. CONCLUSIONS Basd on a numbr of assumptions, an optimal load schduling stratgy was proposd to minimiz th lctricity costs of an industrial nd usr undr on-part RT. A mthod was prsntd which may b usd by an industrial nd usr to rspond adquatly to ral-tim lctricity prics. With th aid of an hourly marginal rat duration curv th thrshold valu of th hourly marginal rats can b dtrmind whr th nd usr should control his loads. An analytical approach was followd to dscrib th potntial lctricity cost savings to an industrial nd usr undr RT through intllignt dmand managmnt. Mathmatical xprssions ar givn to dscrib th cost savings in trms of a numbr of variabls familiar to th nd usr and utility. Ths variabls includ th plant s installd powr consumption capacity, th spar nrgy consumption capacity and trms which dscrib th structur of th RT tariff. Although idalisd to som xtnt bcaus of th numbr of assumptions that hav bn mad, ths mathmatical xprssions may provid valuabl insight into th dmand rspons potntial of an nd usr undr RT. Futur work may involv similar approachs to quantify th cost of unsrvd nrgy (CUE) which may rsult from dmand managmnt actions undr RT. Th CUE may consist of componnts (damag fictions) such as th cost of production losss du to inadquat storag capacitis, th cost du to switching losss (.g. procss rstart costs), tc. Evntually th conomic valu of RT may volv as th diffrnc btwn th potntial lctricity cost savings and th cost of unsrvd nrgy du to dmand managmnt. Th conomic valu of RT will srv to mdicat to th utility and nd usr whthr RT will b a fasibl DSM tariff altrnativ to implmnt at th nd usr s plant. VII. REFERENCE C.O. Bjbrk, Industrial Load Managmnt : Thory, ractic and Simulations, Elsvir Scinc ublishrs, Amstrdam, 1989. J. Flory, J. trs, L. Vogt, K. Kating, and B. Hopkins, Evaluating DSM: Can an nginr count on it? A short not papr summarizing a panl dcision, IEEE Transactions on owr Systms, vol. 9, no. 4, Nov. 1994, pp. 1752-1758. B. Daryanian, R.E. Bohn, and R.D. Tabors, Optimal dmand-sid rspons to lctricity spot prics for storag-typ customrs, IEEE Transactions on owr Systms, vol. 4, no. 3, Aug. 1989, pp. 897-93. A.. Sanghvi, Flxibl stratgis for loaddmand managmnt using dynamic pricing, IEEE Transactions on owr Systms, vol. 4, no. 1, Fb. 1989, pp. 83-93. F.C. Schwpp, M.C. Caramanis, R.D. Tabors, and R.E. Bohn, Spot ricing of Elctricity, Kluwr Acadmic ublishrs, Boston, USA, Edition I, 1988. A.J. Wood, and B.F. Wollnbrg, owr Gnration, Opration, and Control, John Wily & Sons, Nw York, 1984. S. Rahman, and Rinaldy, An fficint load modl for analyzing dmand-sid managmnt impacts, IEEE Transactions on owr Systms, vol. 8, no. 3, Aug. 1993, pp. 1219-1226. J.G. Roos, Incrmnting Industrial Cost Savings through Intllignt Dmand Managmnt, h.d. Thsis, Faculty of Enginring, Univrsity of rtoria, Rpublic of South Africa, April 1996. Johan G Roos was born in Bthlhm, Rpublic of South Africa, on Novmbr 7, 1962. H obtaind a Mastrs dgr in Elctrical Enginring (Cum Laud) from th Univrsity of rtoria in 199. H was prviously a Snior Lcturr in th Dpartmnt of Elctrical and Elctronic Enginring, Univrsity of rtoria. H is nrolld for a h.d. dgr in Elctrical Enginring at th sam Univrsity and will obtain th dgr in Sptmbr 1996. His rsarch intrsts includ nrgy control, dmand managmnt, lctricity tariffs and th application of artificial intllignc in lctrical dmand-sid managmnt. H is currntly a consultant in nrgy nginring to utilitis and industry. H is a mmbr of th South African Institut of Elctrical Enginrs. Ian E Lan (MIEEE) was born in Mordn, England, on March 8, 195. H obtaind a Mastrs dgr in Elctrical Enginring (Cum Laud) from th Univrsity of rtoria in 1974. H spnt 1 yars in various nginring and managmnt positions in th stl making industry, aftr which h obtaind a D.Eng. dgr from th Univrsity of rtoria in 1984. Dr. Lan was prviously rofssor of Enrgy Systms, Dpartmnt of Elctrical and Elctronic Enginring, Univrsity of rtoria, as wll as Dirctor of th Cntr for Nw Elctricity Studis in th sam Dpartmnt. Dr. Lan is currntly a consultant to nrgy utilitis, industry and th South African Govrnmnt. H is a fllow of th South African Institut of Elctrical Enginrs.