Market Power and Price Volatility in Restructured Markets for Electricity

Transcription

1 PSERC akt Pow and Pi Volatility in Rstutud akts fo Eltiity Tim ount Copyigt 1999 IEEE. Publisd in t Podings of t Hawaii Intnational Confn On Systm Sins, Januay 5-8, 1999, aui, Hawaii. Psonal us of tis matial is pmittd. Howv, pmission to pint/publis tis matial fo advtising o pomotional puposs o fo ating nw olltiv woks fo sal o distibution to svs o lists, o to us any opyigtd omponnt of tis wok in ot woks, must b obtaind fom t IEEE. Contat: anag, Copyigts and Pmissions/IEEE Svi Cnt/445 Hos Lan/P.O. Box 1331/Pisataway, NJ , USA. Tlpon: + Intl

2 ARKET POER AND PRICE VOLATILITY IN RESTRUCTURED ARKETS FOR ELECTRICITY Tim ount Sool of Eltial Engining Univsity of Nw Sout als Sydny, Austalia and Dpatmnt of Agiultual, Rsou, and anagial Eonomis Conll Univsity 215 an Hall Itaa, NY Abstat T stutud makt fo ltiity in t UK as xpind a systmati pattn of pi spiks assoiatd wit t us of makt pow by t two dominant gnatos. Patly in spons to tis poblm, t sa of apaity ownd by any individual gnato aft stutuing was limitd in Vitoia, Austalia. As a sult, a mu mo omptitiv makt sultd wit pis substantially low tan ty w und gulation. Nvtlss, an ati pattn of pi spiks xists and t pi volatility is a potntial poblm fo ustoms. Tis pap agus tat t us of a unifom pi aution fo ltiity makts xabats pi volatility. A disiminatoy pi aution is poposd as a btt altnativ tat would du t sponsivnss of pi to os in foasting total load. 1. Intodution A numb of ountis av stutud ti makts fo ltiity fo a vaity of diffnt asons. Ty sa, owv, t objtiv of making t nw makt fo gnation mo omptitiv wit low avag pis. In t UK, ig pis ausd by t us of makt pow by t two lagst gnatos av bn a psistnt poblm (s von d F and Habod (1993), Nwbuy (1995), olak and Patik (1997), and Littlild (1998)). T ltiity makts in Sandinavia and Nw Zaland also av dominant gnatos, but ty a ownd by t stat and ty pati som fom of slf-imposd staint on t us of makt pow (s olak (1997) and Rad (1998)). Consquntly, pis a lativly stabl but a pobably ig tan omptitiv lvls. Givn t xpin in t UK makt, t stutuing of gnation in Vitoia, Austalia quid tat a majo pow plant sould b sold to a diffnt buy, fftivly limiting t sa of apaity ownd by any individual ompany (s Outd (1997) and olak (1997)). Hn, t foundation fo a lativly omptitiv makt wit six ompting gnatos was stablisd as an impovmnt ov t skwd pattn of ownsip in t UK. T subsqunt mging of t Vitoian makt wit t stat-ownd gnatos in Nw Sout als in ay 1997 did not ang t situation appiably. In fat, pis fll fut aft t mg. Altoug low pis fo ltiity in t Austalian makt a an nouaging sign, t a also ati pattns of pi spiks wi lad to ig pi volatility. Tis typ of pi bavio is illustatd in Figus 1 and 2. In Figu 1, avag daily pis a sown fo t

3 past ya (9/97 to 8/98), and t a many spiks and no obvious sasonal pattns Figu 1: Avag Daily Pis fo Eltiity in Nw Sout als (/k fom 9/97 to 8/98) Atual alf ouly spot pis fo a nt wk (23/8/98 to 29/8/98) a sown in Figu 2. On again, t pi spiks do not follow a gula daily pattn as ty do duing t wint monts in t UK. Pi volatility appas to b an intinsi poblm wit tis patiula omptitiv makt Figu 2: Spot akt Pis fo Eltiity in Nw Sout als (/kh fom 23/8/98 to 29/8/98) ost of t totial sa on autions as bn ditd to makts to sll itms and t bavio of buys. Assuming tat t logi fo stting offs in makts to buy itms is quivalnt to t logi fo stting bids disussd in t litatu, t sults of Ausubl and Camton (1997) suggst tat t off uvs submittd by slls in a multipl units aution will b ig and stp tan t tu maginal ost uvs if a unifom pi aution is usd (all sussful slls a paid t sam pi). Sin t diffn btwn t off and t maginal ost inass as t numb of units fo sal inass, tis bavio is an xampl of ow makt pow an b usd to inas t final pi. olak and Patik (1997) av sown tat t two lagst gnatos in t UK av usd ti makt pow sussfully to ais pis tis way. Bakman t.al. (1997) av usd xpimntal onomis to sow tat gnatos an aptu ongstion nts and mak xss pofits. In addition, Bnad t.al. (1998) av usd POEREB (a simulation modl of an ltiity makt usd to tst altnativ typs of aution at Conll Univsity) to sow tat patiipants an xploit oppotunitis fo makt pow in load pokts. Su bavio is not supising to most onomists. On of t impliations of aving stply slopd off uvs is tat t agggatd supply uv will b lativly pi inlasti. Consquntly, untainty in t load du to foasting os will b amplifid into ig pi volatility. Futmo, pi spiks a mo likly to ou wn t xptd load is ig and t lvl of makt pow is at its gatst. Pi spiks an also ou aft unxptd outags of gnatos o tansmission lins. In gnal, makt pow will mak pis mo volatil wn a unifom pi aution is usd, and all stutud makts fo ltiity av adoptd tis typ of aution. T main objtiv of tis pap is to dmonstat tat a disiminatoy pi aution, in wi gnatos a paid wat ty off, may b a btt fom of aution fo ltiity makts. T ason is tat t off uvs will b flatt and t agggatd supply uv mo pi lasti. Consquntly, t pi volatility assoiatd wit os in foasting t load will b small tan it is using a unifom pi aution vn if t is no appiabl makt pow. any onomists bliv tat disiminatoy pi autions a lss ffiint tan unifom pi autions baus Viky (1961) sowd tat buys would submit onst bids if ty paid t igst jtd bid, and not ti atual bids, in an aution to sll itms. Howv, ts sults do not gnaliz, as Viky ognizd, to situations in wi som individuals want to buy mo tan on itm. Swinkls (1997a and 1997b) as sown tat bot a unifom pi and a disiminatoy pi aution appoa t pftly omptitiv makt solution if t numb of patiipants is suffiintly lag. Futmo, ov 9% of t autions to sll tasuy bills in a sampl of 42 ountis us a disiminatoy pi aution (Batolini and Cottalli (1997)). Hn, t is no onvining ason to dismiss t onsidation of a disiminatoy pi aution fo ltiity makts on totial o mpiial gounds. 2

4 T objtiv of tis pap is to ompa t ffts of using a unifom pi and a disiminatoy pi aution fo an ltiity makt. T following stion of t pap dfins t onditions fad by a singl gnato using a modifid quadati ost funtion. T optimum off uv is divd in Stion 3 fo a unifom pi aution, and t impliations fo t agggatd supply uv and pi volatility a dtmind. T sam stps a patd in Stion 4 fo a disiminatoy pi aution. In t final stion, t two supply uvs fo t two diffnt autions a ompad. T sults imply tat bot autions a advsly afftd by makt pow, but t pi volatility is mu low using a disiminatoy pi aution. 2. T Spifiation fo a Singl Gnato Consid a spot makt fo ltiity wit N gnatos patiipating. Ea gnato submits offs to an Indpndnt Systm Opato (ISO) and tis to maximiz xptd pofits (sot-un nt vnu) subjt to a known ost funtion. Using a unifom pi aution, t sam pi is paid to t gnatos wo submit t lowst offs to mt an xptd load E[Q tot ] (i.. dmand is pftly inlasti). T pi paid to gnatos is st at t intstion of t load and t ombind off uv fo all gnatos. Sin disontinuitis in t off uvs a uld out by t spifid fom of t ost funtions, t is no nd to distinguis btwn a Last Aptd Off and a Fist Rjtd Off aution. Ty a idntial. Using a disiminatoy pi aution, gnatos wo submit t lowst offs to mt E[Q tot ] a sltd, but t pis paid ospond to t atual offs. T fom of t sot-un ost uv fo gnation is spifid as a displad quadati, implying tat t maginal ost uv is a displad lina funtion. Tis fom is osn to appoximat t atual ost funtions divd by olak and Patik (&P) (1997) fo t UK. Using tis funtional fom maks it possibl to distinguis btwn off uvs tat alt t slop of t maginal ost uv and off uvs tat sift t loation of t maginal ost uv (.g. du t dg of displamnt fom t oigin). T latt bavio was found by &P to b a los appoximation to t offs submittd by t two dominant gnatos in t UK. Exampls of t two typs of off uv and t tu maginal ost uv a sown in Figu 3. P i / Figu 3: Altnativ Foms of Off Cuv Quantity Gnatd aginal Cost Exptd Dmand Siftd Intpt Hig Slop Fo t j t gnato, t sot-un ost uv is spifid as follows: Total ost fo gnato j C j (Q j ) = 1j + 2j Q j + 3j Q j 2 (1) w ij >, i = 1,2,3 and q j a known paamts (q j is t dg of displamnt fom t oigin), Q j is t lvl of gnation Q j = (Q j q j ) fo Q j > q j = otwis. Givn t fom of t total ost in (1), t maginal ost uv an b wittn: aginal ost fo gnato j C j (Q j ) = 2j + 2 3j Q j (2) = ( 2j 2 3j q j ) + 2 3j Q j if Q j > q j = 2j otwis. Fo simpliity, t off uvs a stitd to aving t sam lina fom as t maginal 3

5 ost uvs, and t off uv fo gnato j is dfind as follows: Off uv fo gnato j P j (Q j) = v 1j + v 2j Q j (3) w v 1j and v 2j > a onstants spifid by gnato j, and Q j is t lvl of gnation supplid at pi P j. T quantity supplid an b wittn as a funtion of t pi ivd to giv: Supply offd by gnato j S j (P j ) = (P j v 1j )/v 2j (4) Sin t ISO dispats gnatos using t off uvs submittd by t gnatos, t total paymnt fo mting load is minimizd by paying t sam pi P to all gnatos so tat Q tot = j S j (P). (Fo tis illustation, t osts of tansmission losss and onstaints a ignod.) T supply uv fo t ot (N 1) gnatos is t sum of ti supply uvs, and t osponding dmand fad by gnato j is t diffn btwn t total load and tis sum. Dmand fad by gnato j D j (P j ) = Q tot i j S j (P) (5) = [Q tot + i j (v 1i /v 2i )] [ i j (1/v 2i )]P Sin gnato j dos not know t paamt valus osn by ot gnatos fo ti off uvs (o t xat valu of Q tot tat will ou), it is assumd tat gnato j foms t following subjtiv xptation of t lina lationsip in (5): Subjtiv xptation of dmand by gnato j Q j = A 1j A 2j P (6) w A 1j > and A 2j > a onstants dtmind by gnato j. T xptd dmand fad by gnato j is sown in Figu 3, and it is spifid to go toug t point w t two off uvs oss. 3. T Optimum Off Cuv Using a Unifom Pi Aution T pofit funtion fad by gnato j ombins t xptd dmand lationsip (6) wit t tu ost funtion (1). Tis is t standad poblm fad by a podu wit makt pow, and t solution dtmins t optimum lvl of gnation Q j and t makt pi P. aximiz wit spt to Q j R j (Q j ) = PQ j C j (Q j ) (7) subjt to (6), w C j (Q j ) is t total ost dfind in (1). T fist od ondition fo maximizing (7) an b wittn: P - Q j /A 2j C j (Q j ) = (8) w C j (Q j ) is t maginal ost dfind in (2). Raanging (7) givs t following xpssion fo t optimum off uv: Optimum off uv fo gnato j P = C j (Q j ) + (1/A 2j )Q j (9) = ( 2j 2 3j q j ) + (2 3j + 1/A 2j ) Q j if Q j > q j = 2j + (1/A 2j )Q j otwis. In a omptitiv makt, 1/A 2j = and P = A 1j /A 2j = onstant. Consquntly, t optimum output fo gnato j is to st Q j so tat t osponding maginal ost quals t pi. In ou xampl, 1/A 2j > in (9), and as a sult, t optimum off uv is mo stply slopd tan t tu maginal ost uv. In addition, t optimum off uv as t sam intpt as t tu maginal ost uv, implying tat t dg of displamnt q j is unangd. T kink in t maginal ost uv and t optimum off uv ou at t sam valu Q j = q j. An xampl is sown in Figu 4. T fom of off uv in Figu 4 is onsistnt wit t fom of totial bid funtion divd by Ausubl and Camton fo an aution to buy mo tan on itm. T diffn btwn t off and t tu maginal ost inass as Q j gts lag. Using t sults in (9), t optimum paamt valus fo t off uv (3) an b wittn in tms of t paamts of t maginal ost uv (2) and t xptd dmand uv (6) as follows: Fo Q j > q j v 1j = ( 2j 2 3j q j ) and (1) v 2j = (2 3j + 1/A 2j ) Fo Q j q j 4

6 v 1j = 2j and v 2j = 1/A 2j (11) T optimum solution to t maximization in (7) givs a singl pi P and a singl quantity Q j, and t xpssions a: Optimum quantity and pi Q j = A 1j A 2j P fom (6) (12) P = ( 2j 2 3j q j + 2 3j A 1j + A 1j / A 2j ) / ( j A 2j ) fom (9). Ts two valus ould b dtmind on an off uv by inasing t slop of t maginal ost uv using (9) o by sifting t maginal ost uv by an appopiat amount to t lft. Fo t xampl in Figu 3, tis spial as osponds to t intstion of t two P i / Figu 4: Optimum Off Cuv fo On Gnato Quantity Gnatd aginal Cost Exptd Dmand Optimum Off off uvs wit xptd dmand at P = 3 and Q j = 12. (T paamt valus fo t maginal ost uv (2) a 2j = 1, 3j =.2 and q j = 1, and fo t xptd dmand uv (6), ty a A 1j = 42 and A 2j = 1. T siftd intpt osponds to stting q j = 7. ) T pimay ason fo submitting t off uv in (9) at tan sifting t maginal ost uv to t lft is tat it givs t lous of optimum pis and quantitis fo any valu of A 1j. Evn toug t xptd dmand fad by gnato j in (6) is onditional on t xptd bavio of gnatos, t is still untainty in t atual load Q tot, and onsquntly, in t valu of [Q tot + i j (v 1i /v 2i )] in (5) wi is psntd by A 1j in (6). T off uvs of t two dominant gnatos in t UK w sown by &P to ospond to witolding inxpnsiv apaity fom t makt (i.. making q j small and kping t slop 2 3j unangd). Tis is not onsistnt wit t optimum bavio implid in (9). T xplanation givn by &P is tat duing t apaity offd fom low ost gnatos is lss likly to inu govnmnt intvntion fom t ovsigt ommitt tan aising pis. Sin a typial ompany ontols a numb of diffnt pow plants of diffnt typs, t ost uv in (1) psnts all plants ontolld by gnato j. Hn, it is invitabl tat t quantitis of apaity availabl fo individual plants ang du to maintnan sduls and ot fatos. Fqunt angs in t pi offd fo gnation fom any spifi plant would b ad to justify. A poblm wit t obsvd bavio in t UK, owv, is tat it lads to makt inffiinis baus apaity fom t low ost plants is ld bak fom t makt, and t tu ost of gnation is ig tan it would b und bot pft omptition and t optimum off uv. Tis is a as w t tat of gulation may av a pvs fft on ffiiny but may still low t spot pi. In ontast, t optimum off uv in Figu 4 implis tat t anking of t tu maginal osts of gnating units is idntial to t anking of offs. 3.1 T Off Cuvs fo Idntial Gnatos In t numial xampl, t tu maginal ost at Q j = 12 is only C j = 18, wi osponds to 6 pnt of t optimum off 3. An obvious qustion is wt tis xampl is alisti. In t simplst as in wi all N gnatos av t sam ost uv and bav idntially, t N off uvs will also b idntial. T slop of t off uv is v 2j = (2 3j + 1/A 2j ) fo Q j > q j in (1), w A 2j, dfind in (6), is t subjtiv valu of t slop [ i j (1/v 2i )] in (5). n v 2i = v 2 and 3i = 3 fo all i, t following lationsip olds: 5

7 v 2 = v 2 /(N 1) (13) = 2 3 (N 1)/(N 2) fo N > 2 Fo t valus v 2 =.5 and 2 3 =.4 in t xampl, t numb of gnatos is N = 6. Hn, t lativly lag diffn btwn t slops of t off uv and t maginal ost uv (t slop of t off uv is 25 pnt ig tan t ffiint valu) osponds to a lativly lag numb of ompting gnatos by t standads of t lti utility industy. T maximum makup of t slop fo N > 2 is 1 pnt wn N = 3. It is intsting to not tat t xpssion fo v 2 in (13) dos not inlud t as of two idntial duopolists. T ason is tat t load fad by t duopolists in tis xampl is ompltly inlasti, and t is no stabl Nas quilibium wn N = 2. In (13), t only situation tat is valid fo N = 2 is wn 3 = (i.. t maginal ost uv is flat), but vn in tis situation, t valu of v 2 is still indtminat. In gnal, t sam poblm xists wn t duopolists a not idntial. If a known maximum pi is st fo a makt by t ISO, on would xpt tat a duopolist would submit a flat off uv at t maximum pi (as long as t podu fo baking tis givs a fai sa of t load to a patiipant). Howv, tis solution would still not b a stabl quilibium. T tat of taliation in an aution tat is patd many tims, lik an ouly makt fo ltiity, is on possibl ason fo t duopolists to kp t pi at t maximum. 3.2 T Impliations fo Total Supply Givn t sults fo t optimum off uv, it is possibl to dtmin t impliations fo t agggat supply uv fo t N gnatos. Fom (4), t agggat supply uv an b wittn (assuming Q j > q j fo all j): N Q tot = j =1 S j(p) (14) = [ j 1/v 2j ] P [ j v 1j /v 2j ] w v 1j and v 2j a t optimum valus dfind in (13). Sin (14) is a lina funtion of P, it an b wittn as an xpliit funtion of P as follows: P = (Q tot + B 1 )/B 2 (15) w B 1 and B 2 a t intpt and t slop in (14), sptivly. T slops of t optimum off uvs v 2j in (1) a lag tan t ffiint valus (slops of t maginal ost uvs). Consquntly, t valu of t slop in (15) 1/B 2 is also lag tan t ffiint valu, and t supply uv basd on offs is mo pi inlasti tan t ffiint supply uv basd on maginal osts. In t spial as of N idntial gnatos, t slop of all N > 2 off uvs is v 2 = 2 3 (N 1)/(N 2). As a sult, t slop of t supply uv in (15) an b wittn: 1/B 2 = (2 3 /N) ((N 1)/(N 2)) (16) w (2 3 /N) is t slop of t ffiint agggat supply uv basd on maginal osts and ((N 1)/(N 2)) > 1 fo N > 2 psnts t fft of makt pow. Consid t total ost uv fo a singl gnating unit: (Q ) = Q + 3 (Q q ) 2 (17) If gnato j ontols k j units, tn t agggat ost uv (assuming all k j units opat at t sam lvl of output) an b wittn: j (Q j ) = 1 k j + 2 Q j + ( 3 /k j )(Q j k j q ) 2 (18) w Q j = k j Q. Und tis spifiation t avag osts fo (17) at Q and fo (18) at Q j a idntial, and so a t two osponding maginal osts. If t is a total of k tot = j k j units of gnating apaity, t last ost solution fo mting any lvl of load Q tot is not afftd by t pattn of ownsip of apaity among t N gnatos. Futmo, if t agggat ost uv in (18) is assumd to old fo any k j >, and not just fo intg valus, tn a gnato ontols (k tot /N) units of apaity in t as of N idntial gnatos. In tis spial as, t paamts dfining t ost fo a gnato a: 1 = 1 k tot /N, 2 = 2, (19) 3 = 3 N/k tot and q = q k tot /N 6

8 Substituting (19) into (16), t slop of t supply uv in (15) an b wittn: 1/B 2 = (2 3 /k tot )((N 1)/(N 2)) (2) Consquntly, t slop of t ffiint supply uv basd on maginal osts (2 3 / k tot ), unlik t slop in (16), is t sam fo all valus of N. T numb of gnatos N affts only t diffn btwn t slop of t off uv and t slop of t maginal ost uv. Tis plas a la fous on t ol tat makt pow plays in inasing t spot pi P abov t ffiint lvl. Fo t numial xampl usd in Figu 4, t off uv osponds to N = 6 idntial gnatos wit a total load of Q tot = 72. Using k tot = 12 (quivalnt to a standad gnating unit of siz 6), t ost paamts fo t maginal ost of t standad gnating unit in (19) a 2 = 1, 3 =.4 and q = 5. T agggat supply uvs fo N = 3, 6 and 12 idntial gnatos a sown in Figu 5 P i / Figu 5: Total Supply Cuvs fo N Idntial Gnatos Total Quantity Gnatd ag. Cost N=12 N=6 N=3 togt wit t agggatd maginal ost uv (osponding to N = ). All t supply uvs and t positivly slopd pat of t maginal ost uv av t sam ngativ intpt ( 3). (T diffns in t slops among t supply uvs would b mo obvious if t was no displamnt of t ost uv (i.. q = ) baus in tat as t ommon intpt would b zo.) T supply uvs in Figu 5 a dfind fo valus of t total load abov k tot q = 6, and t xptd load in t xampl is Q tot = 72. Fo valus of Q tot < 6, t sult in (2) implis tat all off uvs would b flat baus t slop of t maginal ost uv is zo. Hn, all supply uvs would b t sam as t maginal ost uv at t onstant lvl 2 = 1. T supply uvs in Figu 5 a sown wit t sam fom as t optimum off uv in Figu 4 to giv on possibl oi of t fom, but t valus fo Q < 6 a not stitly optimum. 3.3 T Impliations fo Pi Volatility n offs a submittd to t ISO, t valus of B 1 and B 2 in (15) a fixd, and t makt solution fo t spot pi P is dtmind by t alizd valu of t total load Q tot. Consquntly, t untainty about Q tot is tanslatd into untainty about P by t slop 1/B 2 in (15). Using t sults fo N > 2 idntial gnatos in (2), t vaian of pi an b wittn: Va[P] = (1/B 2 ) 2 Va[Q tot ] (21) = ((N 1)/(N - 2)) 2 (2 3 /k tot ) 2 Va[Q tot ] T impotant onlusion fom (21) is tat t spot pi will b mo volatil tan t pi in an ffiint makt baus ((N 1)/(N 2)) > 1. T additional volatility du to makt pow gts small as t numb of gnatos inass baus ((N 1)/(N 2)) 1 as N. T situation in atual spot makts fo ltiity is mo ompliatd. In t UK, fo xampl, &P sow tat t kinkd off uvs in Figu 3 a asonabl appoximations to atual off uvs. Duing piods of low load, makt solutions gnally ou on t flat pat of t maginal ost uv. n load is ig, t spot pi is dtmind by t stp pat of t off uv. Hn, t is a mixtu of two gims stting t spot pi. In addition, t 7

9 is inasing vidn tat t two dominant gnatos xploit situations wn t total load is ig and t xptd dmand in (6) is most inlasti (A 2j is small), and ty submit offs tat dviat vn mo fom t maginal ost uv at ts tims (.g. duing t lat aftnoon on wkdays in t wint). Exploiting bad situations aft an unxptd outag of a gnato o a failu on t systm, fo xampl, is xatly t typ of bavio tat is likly to unlas t wat of govnmnt gulatos. In t UK, t blatant us of makt pow by t two dominant gnatos duing piods of ig load as sultd in punitiv ations fom t govnmnt in t fom of spial taxs on pofits and pssu to sll gnating apaity. 4. T Optimum Off Cuv Using a Disiminatoy Pi Aution In a disiminatoy aution, t pis ivd by a gnato ospond to t off uv submittd to t ISO. Evn toug t is no dit link btwn t makt laing pi P paid fo t last unit aptd fom gnato j and t pis paid fo t ot (Q j - 1) units aptd, t optimization poblm is vy simila to t situation using a unifom pi aution. If t subjtiv xptd dmand fad by gnato j is givn by (6), t objtiv is to maximiz pofits as bfo. A disiminatoy monopolist would b abl to xtat t full suplus btwn t dmand uv (6) and t maginal ost uv (2). Howv, t a limits on t ability of a gnato to ag disiminatoy pis. T most impotant on is tat t ISO will ank t offs fo individual units fom t last xpnsiv to t most xpnsiv. Hn, t makt laing pi P fo t maginal unit aptd fom gnato j must ospond to t igst off aptd fom gnato j. In ot wods, t off uv must b monotonially non-dasing (i.. v 2j in (3)). Und tis stition, vnu is maximizd fo any optimum Q j by spifying a flat off uv (i.. v 1j = P and v 2j = in (3)). T poblm wit tis statgy is tat it is not obust to untainty about t dmand uv in (6) du to untainty about total load, fo xampl. If t atual dmand was low tan xptd and t ISO apts Q j < Q j, t is no way to guaant tat t units aptd will b t ons wit t lowst osts. To t ISO, all units fom gnato j ost t sam if t off uv is flat. Hn, it is asonabl to spify a minimum positiv slop fo t off uv (v 2j = v 2m > in (3)) to nsu tat t ISO anks units otly. (Tis assumption also avoids t poblm of indtminany tat xists if all off uvs a flat fo N idntial gnatos.) it t slop of t off uv st at v 2m, t optimization poblm fo gnato j an b wittn as a modifiation to (7) as follows: aximiz wit spt to Q j. R j (Q j ) = PQ j - v 2m Q j 2 /2 - C j (Q j ) (22) subjt to (6), w C j (Q j ) is t total ost dfind in (1). T fist od onditions fo maximizing (22) an b wittn: P - Q j /A 2j - v 2m Q j - C j (Q j ) = (23) w C j (Q j ) is t maginal ost in (2). Raanging (23) givs t following xpssion fo t optimum pi and lvl of gnation: Optimum pi lous P = C j (Q j ) + (1/A 2j + v 2m )Q j (24) = ( 2j - 2 3j q j ) + (2 3j + 1/A 2j + v 2m )Q j if Q j > q j = 2j + (1/A 2j + v 2m )Q j otwis. Tis sult is almost idntial to t osponding xpssion fo t unifom pi aution in (9). T slop of t solution fo t optimum pi fo a givn valu of A 2j is stp in (24) du to t minimum slop v 2m tat is quid fo t off uv. T optimum pi lous and t optimum off uv a sown in Figu 6 using t sam maginal ost and xptd dmand uvs usd in Figu 4 fo a unifom pi aution. Assuming tat t minimum slop fo t off uv is v 2m = 6/11 = (to giv intg solutions fo P and Q j ), t sulting optimum pi is P = 31 (ompad to 3 in Figu 4) and t optimum quantity of Q j = 11 (ompad to 12 in Figu 4). Tis illustats t fft of t sligtly ig slop of t lous of optimum pis in Figu 6 ompad to t slop of t optimum off uv in Figu 4. Altoug ts diffns in t optimum pi and quantity a lativly small, t is also a majo diffn baus t slop of t 8

10 Figu 6: Optimum Off Cuv fo On Gnato 6 subjt to a modifid (6) wit a lag intpt tan bfo (t simplst way to sift t dmand uv). T solution fo t optimum pi an b wittn as follows: P i / Quantity Gnatd aginal Cost Exptd Dmand Optimum Off Pi Lous P = C j (Q j ) + (1 / A 2j + v 2m ) Q (26) = ( 2j - 2 3j q j + 2 3j Q j ) + (2 3j + 1 / A 2j + v 2m ) Q w Q j = Q j + Q j is t nw optimum lvl of gnation. T solution in (26) as t sam slop as t solution in (24) but t intpt is siftd upwads and osponds to C j (Q j ). T solutions fo t optimum pis in (24) and (26) a sown in Figu 7. T maginal ost (ag. Cost), pi lous (P. Lous) optimum off (Opt. Off) and xptd dmand (Exp. Dmand) a idntial to t osponding lationsips in Figu 6 (t optimum solution is at P = 31 and Q j = 11). T nw optimum pi lous is P. Lous A&B, 6 Figu 7: Optimum Off Cuvs fo Additional Dmand optimum off uv fo t disiminatoy pi aution in Figu 6 is mu low ( vsus.5 in Figu 4). Tis as impotant impliations fo duing bot t inflation of t spot pi abov t tu maginal ost and t lvl of pi volatility. Fo Q j Q j, t optimum off uv is dfind by t slop v 2m and t solution to (24) fo P and Q j (v 1j = P - v 2m Q j and v 2j = v 2m ). T is still a qustion about t optimum offs fo Q j > Q j. Tis is no long dtmind automatially as it was using a unifom pi aution baus t optimum off uv is not t sam as t optimum lous fo P and Q j in (24). If Q j = Q j + Q j > Q j > q j, baus t dmand uv fad by gnato j as siftd up and to t igt, t optimization would b to maximiz t pofit fom t additional gnation Q j (baus t vnu fom Q j is alady dtmind). T objtiv funtion would b: aximiz wit spt to Q j R ( Q j ) = Q j P - v 2m Q 2 / 2 - [C j (Q j ) - C j (Q j )] (25) P i / Quantity Gnatd ag. Cost Opt. Off P. Lous A Opt. Off A Exp. Dmand P. Lous Opt. Off A Opt. Off B and two possibl optimum inmntal off uvs a sown in Figu 7. Ts ospond to two diffnt lvls of ig xptd dmand (Exp. Dmand A and Exp. Dmand B), and t optimum solutions fo Q j a 139 and 155, 9

11 sptivly (t intstions wit P. Lous A&B). Fo Exp. Dmand A, t optimum pi P = 3 is low tan t oiginal solution P = 31, and Opt. Off A is blow t oiginal Opt. Off. T ason is tat t oi of P affts t pis paid fo Q j but t oi of P only affts t pis paid fo Q j. Hn, it is not optimum to inflat t off P abov t tu maginal ost by su a lag amount. In ts situations, owv, t basi ul of aving a non-dasing off uv would b violatd. If t inas of xptd load is lag noug (Q j > 144), t optimum off uv is abov t oiginal Opt. Off. Opt. Off B is on xampl. Tating t poblm as a sis of inmntal stps fo Q j > Q j, a asonabl statgy is to xtnd t optimum off uv divd fo Q j < Q j until it as t maginal ost uv. Byond tat point, t offs would b qual to t tu maginal ost. T omplt optimum off uv fo a disiminatoy pi aution is sown in Figu 6. Howv, it would b pfabl, as a subjt fo fut sa, to onsid t ffts of untainty about t total load xpliitly in diving t optimum off uv. 4.1 T Impliations fo Total Supply Fo a disiminatoy pi aution, t slops of t optimum off uvs a st at t minimum valu v 2m fo all N gnatos. Consquntly, t sum of t N individual off uvs, osponding to (14), an b wittn: Q tot = [ j v 1j / v 2m ] + [N/v 2m ]P (27) w v 1j and v 2m a t paamts of t optimum off uv fo gnato j. T intpt in (27) is bigg tan it would b if t spot pi fll on t tu maginal ost uv. Howv, in most alisti situations t slop of t total supply uv (v 2m / N) will b substantially small tan t slops of bot t total supply uv using a unifom pi aution and t omptitiv supply uv basd on t tu maginal ost uvs. Futmo, total supply will b inasingly pi lasti as t numb of gnatos N gts lag. If t N gnatos a idntial and av ost uvs dfind by t standad paamts in (19), tn t omptitiv supply uv is indpndnt of N wit a slop of 2 3 /k tot fo Q tot > q k tot and a lvl of 2 fo Q tot q j k tot. T slop of t optimum lous of pis in (23) an b simplifid baus t slops of all optimum off uvs a always v 2m gadlss of t siz of N. Consquntly, t slop of t xptd dmand uv in (6) an b wittn: A 2j = (N - 1) / v 2m (28) T optimum pi lous in (23) is: P = C j (Q j ) + (v 2m N / (N - 1))Q j = ( q ) + (2 3 N / k tot + v 2m N/(N - 1))Q j (29) fo N idntial gnatos. It follows tat t xptd spot pi in t makt is dtmind by substituting Q j = E[Q tot ]/ N into (29). It is intsting to not tat, unlik t unifom pi aution, t onditions in (29) inlud t as of duopolists. It is only a monopolist wo would b abl to div t spot pi to t maximum allowd by t ISO. Using t sam paamt valus fo t ost funtion givn blow (2) and t minimum slop v 2m = 6/11, t optimum pi loi and t supply uvs fo N = 2,3,6 and 12 idntial gnatos a sown in Figus 8 and 9. T pi loi a simila in fom to t supply uvs fo a unifom pi aution in Figu 5, but t supply uvs in Figu 9 a mu mo pi lasti tan t supply uvs in Figu 5. In addition, t makt laing pis fo t sam numb of gnatos as Figu 5 a low fo t disiminatoy pi aution in Figu 9. T lativ flatnss of t supply uvs in Figu 9 will always old as long as t slop of t maginal ost uv fo a gnato is gat tan v 2m (t slop of t off uv using a unifom pi aution is gat tan o qual to t slop of t maginal ost uv). T anking of t pis btwn t two autions, owv, is dpndnt on t valus of t paamts. T lativly small slops of t supply uvs in Figu 9 imply tat makt pis will b lativly unafftd by untainly about Q tot. In ot wods, pi volatility will b mu low fo t supply uvs in Figu 9 ompad to t supply uvs in Figu 5. T quivalnt xpssion to (21) fo a disiminatoy pi aution is: 1

12 P i / Figu 8: Optimum Pi Loi fo N Idntial Gnatos Total Quantity Gnatd ag. Cost N=12 N=6 N=3 N=2 Va[P] = (v 2m / N) 2 Va[Q tot ] (3) Consquntly, Va[P] as N in (3) but using a unifom pi aution in (21) Va[P] t vaian in a omptitiv makt ((2 3 /k tot ) 2 Va[Q tot ]). It sould b notd tat t low pi volatility is t spons to foasting os about Q tot (i.. t diffn btwn E[Q tot ] usd to dtmin t offs and t atual Q tot tat ous). akt pis will vay du to angs in t xptd load (.g. t daily load yl), but t pi lastiity of t supply uvs imply tat pi spiks a lss likly to ou using a disiminatoy pi aution tan a unifom pi aution. Ts issus a disussd fut in t onluding stion of t pap. 5. Summay and Conlusions T main objtiv of tis pap is to sow tat adopting a Disiminatoy Pi Aution (DPA) fo ltiity makts may b pfabl to t unt pati of using a Unifom Pi Aution (UPA) to dtmin spot pis. Evn toug it is diffiult to ank ts two autions onsistntly on t basis of onomi ffiiny o t lvl of t spot pi, t supply uvs will typially b mo pi lasti using a DPA. Consquntly, pi volatility ausd by os in foasting t total load on t systm will b low, and t pnomnon of unxptd pi spiks obsvd in t Austalian makt, fo xampl, ould b allviatd. Sin pi volatility is gnally not a dsiabl fatu of a makt fo ustoms o fo nw gnatos onsiding nty into t industy, lss pi volatility sould b bnfiial. Existing gnatos do bnfit fom t xistn of pi spiks, but t is no basis to judg wt avag pis will b ig o low using a DPA. Hn, pofits fo xisting gnatos ould b ig o low if t typ of aution is angd fom a UPA to a DPA. Using a UPA, t sults divd in Stion 3 sow tat offs submittd to t makt will flt t dg of makt pow ld by a gnato. T off uvs (and t agggatd supply uv) will b mo pi inlasti tan t tu maginal ost uvs (and t Figu 9: Total Supply Cuvs fo N Idntial Gnatos 12 1 P i 8 6 / Total Quantity Gnatd ag. Cost N=12 N=6 N=3 N=2 omptitiv supply uv). Ts aatistis 11

13 a onsistnt wit t typ of optimum bavio divd by Ausubl and Camton (1996) fo bids in a UPA to sll multipl units, and to t atual bavio dsibd by olak and Patik (1997) fo gnatos in t UK ltiity makt. Evidn sows tat t two dominant gnatos in t UK av bn willing to los som makt sa in od to gt ig pis and ig pofits. T sults fo a DPA disussd in Stion 4 dmonstat tat makt pow will inas makt pis. In gnal, t lativ siz of t inas ompad to a UPA is an mpiial issu. Evn toug Swinkls (1997a and 1997b) as sown tat bot typs of aution will appoa t ffiint omptitiv solution as t numb of patiipants inass (in an aution to sll multipl units), t is no totial basis fo anking t autions wn t numb of patiipants is small. T xampls in Stions 3 and 4 a onsistnt wit Swinkls sults, and t makt pi appoas t omptitiv pi as t numb of gnatos inass using it aution. T main distinguising fatu of t supply uv using a DPA is tat it is lativly pi lasti. As t numb of gnatos inass, t optimum pi lous in (29) gts los to t tu maginal ost uv but t slop of t off uv dos not ang. Using a UPA, t off uv and t optimum pi lous a idntial. Consquntly, t off uv fo a UPA stays t sam wn t xptd load angs, but t off uv sifts up and down in spons to ts angs using a DPA. Ts fatus a illustatd in Figu 1 fo a low load (L) and a ig load (H). T sults in Figu 1 psnt t agggatd supply fo N = 6 idntial gnatos using t sam paamt valus as Figu 5 fo a UPA and Figu 9 fo a DPA. In addition, t slop of t maginal ost uv fo Q tot > 9 is inasd fom 2 3 /k tot to 6 3 /k tot. Consquntly, t slop of t supply uv fo a UPA also inass fo Q tot > 9. In ontast, t slops of t supply uvs fo a DPA av t sam slops but diffnt intpts. T impliations of t diffnt supply uvs in Figu 1 a summaizd in Tabl 1. T makt pis sow ow makt pow inass t pis abov t omptitiv lvl fo bot autions. Dfining pofits as t diffn btwn total vnu and total osts (wit 1 = in (18)), pofits a substantially P i Figu 1: Total Supply Cuvs fo N = 6 Gnatos Total Quantity Gnatd ag. Cost UPA Supply DPA Lous DPA Supply L DPA Supply H 1 ig tan t omptitiv lvls fo bot autions. Fo xampl, pis a about 5% ig using a UPA and pofits a ougly doubl t omptitiv lvls. T low pofits fo a DPA flt bot t low pis and t fft of t slop of t supply uvs (vnu is t aa und t supply uv fo a DPA). Dfining t sala multipli of t vaian of load as t Volatility Fato ([((N - 1)/(N - 2)) 2 (2 3 /k tot ) 2 ] in (21) and [(v 2m /N) 2 ] in (3)), t impliations fo pi volatility a vy diffnt. T Volatility Fatos fo a UPA a ig tan t omptitiv lvls but ty a los to zo fo a DPA. T low Volatility Fatos a an attativ fatu of a DPA baus t ffts of untainty in t load will b dampnd ompad to t omptitiv supply uv. T opposit is t as using a UPA and fo tis typ of aution additional makt pow would inas t Volatility Fatos vn mo. Consquntly, pi volatility and pi spiks a likly to b mu mo of a poblm fo a UPA tan ty a fo a DPA. It is not la wy t UPA as bn osn fo ltiity makts (pis may diff among gnatos baus of tansmission losss, but a unifom pi is paid fo all gnation fom a givn loation). In 12

14 ontast, most makts fo slling Tasuy Bonds in diffnt ountis us a DPA. Tabl 1: Caatistis of Supply fo N = 6 Idntial Gnatos Load () Comptitiv UPA DPA akt Pi (/) Pofits ( ) Volatility Fato x Givn t impotan of pi volatility in omptitiv makts fo ltiity, lik t Austalian as, t is a la nd fo fut sa on disiminatoy pi autions. Tis would povid an xllnt oppotunity to us an xpimntal stting, su as Powb, to idntify t lativ mits of a disiminatoy pi aution and t onvntional unifom pi autions fo ltiity makts. Finding out mo about ow ts autions pfom und untainty is an ssntial stp fo undstanding t auss of pi volatility in ltiity makts and fo idntifying ways to du it. Rfns [1] Ausubl, Lawn. and Pt C. Camton. July Dmand Rdution and Inffiiny in ulti- Unit Autions. Univsity of ayland, Dpatmnt of Eonomis oking Pap No Btsda, D. [2] Bakman, Stvn R., Stpn J. Rassnti, and Vnon L. Smit. Januay 21, Effiiny and Inom Sas in Hig Dmand Engy Ntwoks: o Rivs t Congstion Rnts n a Lin Is Constaind? Univsity of Aizona Eonomi Sin Laboatoy oking Pap. Tuson, AZ. Puasing Elti Pow. ttp:// du/psbin/tst/gt/publiatio/ [5] Littlild, S.C. (1998). Rpot on Pool Pi Inass in int 1997/98. ttp://opn.gov.uk.off /doumnts/wintpool.x. [6] Nwby, David Pow akts and akt Pow. Engy Jounal, vol. 16, no. 3: [7] Outd, H A Rviw of Eltiity Industy Rstutuing in Austalia. Fotoming in Elti Pow Systms Rsa, Spial Issu on Elti Pow in Asia. [8] Rad, E. Gant Eltiity Sto Rfom in Nw Zaland: Lssons fom t Last Dad. oking Pap, Dpatmnt of anagmnt, Univsity of Cantbuy, Nw Zaland. [9] Swinkls, Jon. 1997a. Asymptoti Effiiny fo Unifom Pi Autions. anusipt, asington Univsity in St. Louis. [1] Swinkls, Jon. 1997b. Asymptoti Effiiny fo Disiminatoy Pivat Valu Autions wit Agggat Untainty. anusipt, asington Univsity in St. Louis. [11] von d F, N.. and D. Habod Spot akt Comptition in t UK Eltiity Industy. Eonomi Jounal, vol 13: [12] Viky, Countspulation, Autions, and Comptitiv Sald Tnds. Jounal of Finan, vol. 16: [13] olak, Fank A akt Dsign and Pi Bavio in Rstutud Eltiity akts: An Intnational Compaison. oking Pap, Dpatmnt of Eonomis, Stanfod Univsty [14] olak, Fank A. and R. H. Patik. Fbuay T Impat of akt Ruls and akt Stutu on t Pi Dtmination Poss in t England and als Eltiity akt. Sltd Pap psntd at t POER Confn, a 1997, Univsity of Califonia, Bkly. Bkly, Califonia. [3] Batolini, Lonado and Cottalli, Calo (1997). Tasuy Bill Autions: Issus and Uss in aoonomi Dimnsions of Publi Finan, ditd by.i. Blj and T. T-inassian, Routldg. [4] Bnad, Jon, Timoty ount, illiam Sulz, Ray D. Zimmman, Robt J. Tomas, Riad Sul (1998). Altnativ Aution Institutions fo 13