Unified Unit Commitment Formulation and Fast Multi-Service LP Model for Flexibility Evaluation in Sustainable Power Systems

Transcription

1 IEEE Transacons on Susanable Energy Acceped for publcaon, November Unfed Un Commmen Formulaon and Fas Mul-Servce LP Model for Flexbly Evaluaon n Susanable Power Sysems Lngx Zhang, Suden Member, IEEE, Tomslav Capuder, Member, IEEE, and Perlug Mancarella, Senor Member, IEEE Absrac Classcal un commmen (UC) algorhms may be exremely me-consumng when appled o large sysems and for long erm smulaons (for nsance, a year) and may no consder all he feaures requred for flexbly assessmen, ncludng analyss of dfferen reserve ypes. In hs lgh, hs paper presens a novel flexbly-orened unfed formulaon of a large-scale schedulng model consderng mulple ypes of plans (ncludng sorage) and reserves, whch can seamlessly model bnary (BUC), mxed neger lnear programmng (MILP), and relaxed lnear programmng (LP) UC. Comparsons are carred ou on several case sudes for a reduced model of Grea Bran, assessng loss of accuracy (as measured accordng o varous mercs specfcally nroduced) agans compuaonal benefs n dfferen renewables scenaros wh more or less flexble sysems. I s demonsraed how he compuaonal me of he LP model s sgnfcanly less han he BUC and MILP approaches whle capurng wh relavely hgh precson all he relevan flexbly requremens and allocaon of mulple ypes of reserves o dfferen ypes of plans. The resuls ndcae ha he proposed fas LP model could be suable for varous compuaonally nensve flexbly sudes (e.g., Mone Carlo smulaons or plannng), wh sgnfcan reducon n smulaon me and only mnor errors relave o esablshed MILP models. Keywords: Flexbly, Lnear Programmng (LP), Mxed Ineger Lnear Programmng (MILP), Renewable energy sources, Energy sorage, Un commmen. BUC GM CCGT LP MILP MSG OCGT PFR ACRONYMS bnary un commmen generaon mx combne cycle gas urbnes lnear programmng mxed neger lnear programmng mnmum sable generaon open cycle gas urbnes prmary frequency response The wor s parly suppored by he projec FENISG - Flexble Energy Nodes n Low Carbon Smar Grd funded by Croaan Scence Foundaon under projec gran No Lngx Zhang and Perlug Mancarella are wh he Unversy of Mancheser, School of Elecrcal and Elecronc Engneerng, M13 9PL, Mancheser, UK (e-mal: lngx.zhang@posgrad.mancheser.ac.u, p.mancarella@mancheser.ac.u). Tomslav Capuder s wh he Unversy of Zagreb Faculy of Elecrcal Engneerng and Compung, Zagreb, Croaa (emal: omslav.capuder@fer.hr). PP PSPP RES SUR TUR power provson pumped-hydro sorage power plan renewable energy sources secondary up reserve erary up reserve NOMENCLATURE Indces,, T me perod (h) I, generaon un/cluser, se of generaon uns/clusers ss, oupu power segmen, se of segmens K, sorage un/cluser, se of sorage uns/clusers Varables oal C oal sysem operaonal cos ( ) flex, C cos of flexbly provder ( ) nflex, C cos of nsuffcen flexbly ( ) SU / SD, U generaor sar-up or shu-down ndex U ON, S U ON, shed over, generaor onlne/offlne commmen sae n BUC or he number of onlne uns n MILP/LP sorage generaon mode ndex e load sheddng volume (MW) e over-generaon volume (MW) FLEX INFLEX s / DOWN, sum of flexble componens (MW) sum of nflexble componens (MW) FLEX sorage dschargng or chargng power (MW) FLEX INFLEX D, D flexble, nflexble demand (MW), FLEX oal power oupu of generaor (MW) s, FLEX sorage ne power oupu (MW), s E, curaled wnd, solar generaon (MW) sorage energy conen (MWh) p s,, segmen of generaor oupu power (MW), f prmary frequency response conrbuon (MW)

2 IEEE Transacons on Susanable Energy Acceped for publcaon, November Sec _ spn / DOWN, r secondary up or down spnnng reserve conrbuon (MW) Sec _ s / r DOWN secondary up or down sorage reserve, Ter _ spn / DOWN, conrbuon (MW) r erary up and down spnnng reserve conrbuon (MW) Ter _ s / r DOWN erary up and down sorage reserve, conrbuon (MW) Ter _ sandng, cos M operaonal cos merc (%) GM r erary up sandng reserve conrbuon (MW) M generaon oupu merc (%) PP M power devaon merc (%) flexbly servces M flexbly servces devaon merc (%) Parameers r _ Sec r _ Ter smulaon me sep (h), secondary, erary reserve deploymen meframe (h) T, T maxmum perod for susanng secondary, r _ Sec r _ Ter W, PV D NL erary reserve (h) poenal wnd, solar producon (MW) oal demand (MW) C no-load cos of generaon un ( /h) SU SUE C, C generaon un sar-up fuel, emsson cos ( ) E s,, s, FU FU segmen generaon fuel, emsson cos ( /MWh) shed / over / cur C load sheddng, over-generaon or renewable generaon curalmen penales ( /MWh) MAX / MIN P maxmum or mnmum oupu of generaor (MW) S MAX / MIN F maxmum or mnmum sorage dschargng S MAX / MIN power (MW) P maxmum or mnmum sorage chargng s MAX / MIN, power (MW) E maxmum or mnmum sorage energy level (MWh) FC maxmum frequency response capably (MW/un) frequency response funcon slope rao F prmary frequency response requremen (MW) S PR maxmum percenage of sorage capacy for each reserve provson (%) G / DOWN maxmum avalable number of uns whn he cluser V generaon un maxmum rampng up or down rae(mw/h) / DOWN sup/ shdn T generaon un mnmum on, off me (h) T generaon un from off o on, on o off ranson perod(h) c/ dc Sec sorage chargng, dschargng effcency (%) / DOWM R secondary up or down reserve requremen Ter / DOWN (MW) R erary up or down reserve requremen (MW) T I. INTRODUCTION HE share of renewable energy sources (RES) n power sysems around he world s rapdly ncreasng and hs rend s expeced o connue n he fuure as polcy maers are seng more and more ambous goals for emsson reducons. The sochasc naure of RES has ncreased he uncerany and varably hsorcally presen n power sysems and, as he share of RES rses, so wll he flexbly requremens, hus challengng he prncples of oday s power sysems operaon and plannng. These arsng flexbly requremens queson he adequacy of exsng mare prncples and servces [1] and call for rehnng echncal lmaons of power sysem componen (e.g., as o wheher s possble o lower he mnmum generaon level of power plans or ncrease her ramp raes [2]). The common fnal goal of flexbly requremen sudes s o enhance sysems capably o cope wh he varably and uncerany of renewable producon, possbly also faclaed or suppored by he negraon of oher low carbon echnologes such as demand sde response and sorage [3], [4]. Assessmen of he operaonal capably of he enre power sysem s a compuaonally nensve as, amed a coordnang a large number of parcpans n order o manan he supply-demand balance over dfferen me horzons and servces. Tradonally, un commmen (UC) models have been developed o realscally represen dayahead and nra-day sysem (and mare) operaon and mulple reserves schedulng, ncludng reserves and n case frequency conrol requremens [5], [6]. The objecve of hese models s o schedule he resources avalable n he sysem n order o provde he desred servce (n case hrough a relevan mare) whle mnmsng he operaonal cos and complyng wh a number of sysem and uns echncal consrans such as mnmum up and down mes, ramp raes, mnmum sable generaon, and so forh. The complexy of hese models ncreases wh he negraon of large share of RES, parcularly f hey mean o capure mulple me scales and servces such as day-ahead and nra-day operaon, prmary and secondary frequency conrol, and erary reserves [7]. Based on he above, here s a connuous need for developng fas and relable models capable of assessng power sysems operaonal flexbly by capurng all relevan echncal consrans and, by dong ha, ndrecly provdng a bacbone for plannng fuure power sysems. On he oher hand, whle he concep of UC s nrnscally assocaed o he need for schedulng ndvdual power plans (and n case

3 IEEE Transacons on Susanable Energy Acceped for publcaon, November energy sorage), hs level of deal may no be necessary for flexbly assessmen and generc uns could be modelled, provded ha hey are represenave of he flexbly ssue under nvesgaon. In hs lgh, hs paper brngs ogeher he followng conrbuons: - A novel flexbly-orened formulaon of UC modellng approach usng bnary, neger and connuous varables (he laer wo relaxng he problem and mprovng s soluon s compuaonal performance) s presened, showng how power sysem operaon can be seamlessly modelled n a unfed way for he hree classes of algorhms and specfcally ang no consderaon he echncal consrans relevan o flexbly provson. In parcular, he laer elemen s suppored by a formulaon ha hghlghs he neracon beween flexbly provders and nsuffcen flexbly ndcaors for ncreased flexbly requremens, parcularly due o RES. - Comparson of he schedulng and compuaonal me performance of dfferen classes of algorhms based on mercs specfcally nroduced and also consderng he provson of dfferen flexbly/reserve servces from dfferen plans (ncludng energy sorage). In hs respec, clear demonsraon s gven of he mer of he proposed fas UC algorhm based on lnear programmng (LP) and benchmared agans classcal mxed neger lnear programmng (MILP) approaches consderng dfferen generang un cluserng schemes as well as sandard bnary un commmen (BUC), for assessng low-carbon sysems flexbly. The res of hs wor s organzed as follows: In Secon II he curren research on sysem operaon modellng and flexbly assessmen s nvesgaed. Then, he flexblyorened unfed formulaon proposed for BUC, MILP and LP UC s descrbed n Secon III. Secon IV nroduces he mercs used for performance assessmen of he hree classes of algorhms. Secon V conans he descrpon of several case sudy applcaons for mulple RES and flexbly scenaros sarng from a reduced sysem model of Grea Bran. Secon VI analyses he resuls o demonsrae he feaures of he proposed model, hghlghng he pros and cons of an LP approach. Secon VII fnally concludes he paper. II. CURRENT RESEARCH A number of models have been proposed for fas assessmen of power sysem operaon, such as for nsance he one n [8]; however, ofen hese models do no suffcenly represen all he se of consrans and frequency conrol and reserve servces needed o explcly capure relevan sysem flexbly characerscs and mercs. Wh he adven of faser and faser opmzaon solvers commercally avalable, MILP mehods have recenly been developed n he drecon of reducng compuaonal me whle mananng reasonable accuracy n capurng all he relevan echncal and economc aspecs of sysem operaon, parcularly wh ncluson of mulple servces provded n dfferen mares. In hs respec, benefs of MILP modellng over more classcal Lagrangan Relaxaon echnques are well nown [9]. However, he complexy of MILP models s sll relavely hgh when dealng wh realsc large scale sysems wh hundreds of generaors. In addon, n many cases when performng flexbly analyss here s a need o carry ou sudes over long me frames (for nsance over a year n order o capure all he possble me varyng correlaon beween RES supply and demand and herefore ne load flexbly requremens), wh fne resoluon (30 mnues or below) and for mulple scenaros. Smlarly, MILP models have proven o be compuaonally oo expensve o be appled for plannng purposes [10] hus requrng planners o mae assumpons such as usng represenave wee smulaons nsead of enre year analyss and by dong so, mssng ou on capurng all he seasonal characerscs and recognzng poenal flexbly bolenecs and requremens. Several echnques o ncrease he compuaonal effcency of MILP models for large scale sysems have also been proposed, such as for nsance [11], where an effcen formulaon s proposed so as o requre fewer bnary varables and consrans. However, only one ype of reserve s dscussed here and he model s presened only for hermal uns. The auhors n [12], [13] use a cluserng algorhm o smulae mulple power sysems and neracons focusng on negraon of elecrc vehcles n he presence of large share of renewable sources. However, hey do no focus on assessng how accurae such an approach s relave o a full bnary UC model or on he comparson beween MILP and LP formulaons. An effecve approach has been proposed n [14], where a model based on neger varables raher han bnary varables s used o represen clusers of hermal generaors, hus ncreasng compuaonal effcency whle also consderng prmary frequency regulaon and erary reserve. On he same lne, and wh more focus on flexbly modellng, recen wor [15] has demonsraed how cluserng n MILP algorhms can sgnfcanly reduce he compuaonal demand of UC sudes wh mulple servces whle mananng very hgh level of accuracy. However, he auhors do no elaborae as o how energy oupu and dfferen servces, such as frequency response and reserves, are allocaed o specfc clusers, nor are he resuls compared o hose obaned when schedulng he same flexbly servces usng BUC. A smlar operaonal modellng approach has been appled o power sysem generaon plannng [16]; however, agan even wh reduced requremens n compuaonal me, analysng mulple servces and mulple scenaros requres focusng on represenave wees only. Ineresng wor nvesgang whch consrans can be relaxed for plannng fuure low-carbon sysems s presened n [17], bu agan no dscussng how hs relaxaon mgh affec he allocaon of specfc servces assocaed o flexbly provson (such as prmary frequency response (PFR) and secondary and erary reserves) from a sngle or a cluser of uns. In addon, energy sorage provdng mulple servces s no ncluded n he assessmen. Hence, generally speang no analyss could be found n he leraure wh respec o he use of LP approaches for UC, and especally for flexbly analyss purposes.

4 IEEE Transacons on Susanable Energy Acceped for publcaon, November III. UNIFIED UNIT COMMITMENT MODEL FOR FLEXIBILITY ANALYSIS IN SUSTAINABLE POWER SYSTEMS A. Objecve funcon Whle, as menoned above, here are a number of UC formulaons proposed n he leraure, he one presened here presens a flexbly-orened mahemacal descrpon dsngushng beween coss assocaed o flexble provders and coss assocaed o presence of nsuffcen flexbly or flexbly requremens (whch ogeher evenually represen drvers for addonal flexbly). The objecve funcon s expressed as n (1), where s emphassed how power sysem operaon s drven by he mnmsaon of oal operaonal cos (C oal ), expressed as he sum of cos of flexbly provders (C flex ) and penaly coss for nsuffcen flexbly (C nflex ): oal mnmse C C C flex nflex (1),, T, I In (1), C denoes operaonal cos, he me perods durng he consdered nerval T, and refers o a specfc un (n he case of BUC) or cluser of uns 1 (n he case of MILP or LP) n se I. The frs par of he objecve funcon (C flex ), s capurng all flexble uns ha can adjus her oupu accordng o mare or sysem requremens. Tradonally, hese are flexble convenonal generaors and more recenly sorage (whch are he focus of hs paper), bu n he conex of fuure power sysems new sources also arse, e.g., demand response and conrollable wnd power plans [17]. The operaonal cos of each generang un or cluser consumng fuel as npu s modelled as a sum of: sar-up cos (C SU ) 2 mes he number of sar-ups (U SU, ); operang cos, whch ncludes he no load cos (C NL ) and he segmen fuel cos (FU s, ) (a concep used for pece-wse lnearzaon of he generaor cos curve); and carbon emsson cos (C SUE and E FUs, ), where ndcaes he smulaon me sep. Ths s shown n Eq (2): C C U p FU FU flex NL ON E,, s,, s, s, ss C C U, SU SUE SU (2) On he oher hand, he second par of he objecve funcon (1) capures all he componens characerzng nsuffcen flexbly n he sysem, namely, penalsed by he cos of load sheddng (C shed ), over-generaon (C over ), and renewable curalmen (C cur ), as n Eq (3). The varables e shed, e over,, µ and Ω quanfy he energy of load shed, over-generaon, and wnd and phoovolac (PV) curalmen, respecvely. 1 As dscussed laer, he equaons below can be seamlessly appled o boh ndvdual uns and cluser of uns, and we wll refer o eher of hem dependng on he specfc case. 2 Ths erm ncludes he coss poenally assocaed o addonal fuel consumpons and emssons (n he case emssons are penalzed). nflex shed shed over over cur C, C e C e, C I B. Energy balances (3) As shown n Eq (4), he flexbly provders are he conrollable convenonal generaon uns (FLEX, ) and sorage (FLEX s, ), along wh, n general (alhough no dscussed n deal here), flexble demand (D FLEX ). Some research papers also nerpre curalmen of RES (µ and Ω ) as an addonal provder of flexbly for he purposes of economc sysem operaon [17], [18]. However, n hs paper we prefer o rea RES curalmen as an ndcaor of nsuffcen flexbly n he sysem, n lne wh curren operaonal pracces of sysem operaors. Therefore, he nonflexble energy balance erms n Eq (5) nclude nflexble demand and componens ndcang nsuffcen capably of he sysem o avod over-generaon (e over, ), load sheddng (e shed ), and harness all he renewable energy ha could be poenally produced (W and PV ). The over-generaon varables are used o ensure suffcen ably o provde reserve, specfcally durng perods of hgh renewable generaon and low demand. In hs sense he over-generaon varable (e over, ) ndcaes f he oupu level of he onlne uns needs o be ep hgher han demand n order o sasfy he downward reserve requremen. The defnon of he generaon-sde flexbly provders s shown n Eq (6), and s consraned by he maxmum oupu power (P MAX ) and mnmum sable generaon (MSG) level (P MIN ) of each generaor, as shown n (7). In (8), he sysem s oal load s modelled as he sum of flexble and nflexble shares. The operaonal consrans of sorage, whch s used n hs wor for energy arbrage purposes, are modelled n Eqs. (9) (13). More specfcally: Eq. (9) llusraes he flexbly ha sorage s can provde, wh FLEX DOWN, represenng downward s flexbly provson whle s chargng and FLEX, upward flexbly provson of dschargng operaon; he energy conen of sorage s consraned n (10) by energy nflows consderng chargng and dschargng effcences (η c, η dc ) as well as he energy conen n he prevous me s sep (E, 1 ); Eq (11) descrbes ha he energy conen of sorage s consraned by s mnmum and maxmum value s (E MIN s,, E MAX, ); boh chargng and dschargng operaons are consraned by mnmum and maxmum powers, as n (12) and (13); fnally, he wo flexble and nflexble componens (FLEX and INFLEX) need o be ep n equlbrum, as from (14). I s also worh hghlghng ha, whou loss of generaly, n he case sudes Pumped-Hydro Sorage Power Plan (PSPP) has been aen as ypcal example of sorage, so ha chargng and dschargng can be assocaed o urbne and pumpng operaon respecvely, whch s also refleced n he commmen varables and consrans n (12) and (13). (4) s FLEX,, I K FLEX FLEX FLEX D

5 IEEE Transacons on Susanable Energy Acceped for publcaon, November INFLEX over shed INFLEX D e, e ( W ) ( PV ) (5) FLEX, ps,, ss I (6) MIN ON MAX O N U, FLEX, P U, P (7) FLEX INFLEX D D D (8) s s sdown,,, FLEX FLEX FLEX (9) 1 DOWN E FLEX FLEX E s s dc s s,,, c, 1 smin s, smax (10) E E E (11) S S s SMAX S,,, MIN ON ON F U FLEX F U (12) 1 SMAX,, 1, S S s S MIN ON DOWN ON P U FLEX P U (13) FLEX C. Reserves INFLEX (14) All he flexble componens n (4) may n prncple conrbue o provson of reserves. In hs paper, he focus s pu on generaon uns as reserve provders whle more specfc modellng and analyss of reserves from sorage and flexble demand are lef o fuure exenson. The proposed model capures hree ypes of reserve, namely, prmary frequency response, secondary spnnng reserve and erary spnnng and sandng reserves [19]. 1) Prmary frequency response The oal prmary frequency response (F ) ha needs o be avalable n case of conngency s ypcally a consan value, for nsance defned n he Grea Bran based on he larges generang un. Ths needs o be provded by onlne generaors (f, ), as n (15), dependng on her characerscs. Typcally, hermal uns can provde around 10% of her nomnal capacy (FC ) whn he me frame of prmary reserve. f, F (15) I Provson of prmary frequency response s consraned by he spare capacy of he generaor excludng secondary and erary spnnng reserve provson, he upper lm of he response conrbuon of ndvdual generaors, and he slope funcon (γ ) when he oupu power s close o maxmum/mnmum value, as shown n Fg. 1 and formulaed n (16) and (17). Typcal echncal consrans are lsed n Table 1 of he case sudy usng smlar values as n [20]. ON,, f U FC (16), Sec _ spn Ter _ spn MAX ON,,, f, r r P U FLEX (17) Fg. 1. Lmaons of he generaor response conrbuons. 2) Secondary reserve Secondary up and down reserves (SUR and SDR) are o be provded by spnnng generaon (onlne) uns (r Sec_spn, ) and pumped-hydro sorage uns (r Sec_s, ), as shown n Eq. (18)- (19). The secondary reserve requremen s calculaed here based on he maxmum exsng generaon plan capacy n he sysem and he poenal volaly of he load and RES, whch are normally a resul of uncerany and varably [10], [21]. An neresng wor elaborang on varably and uncerany me frames of wnd generaon can be found n [22]. The spnnng reserve conrbuon of each generang un s lmed by s rampng ably assocaed wh he requred meframe of reserve deploymen (Δ r_sec ), as defned n (20) and (21). In addon, he generaor s spare capacy also needs o be consdered. A spare capacy consran for upward reserve s already consdered n he prmary up frequency response n Eq. (17). Eq. (22) adds spare capacy consran for downward spnnng reserves oo. The reserve conrbuon of sorage uns s consraned by he mnmum and maxmum energy conen and he maxmum perod requred for susanng he secondary and erary reserves (T r_sec, T r_ter ), as shown n (23) and (24). In addon, he maxmum percenage of sorage capacy used for each reserve provson S (PR ) s also lmed n Eqs. (25) and (26). Furher, he amoun of reserve provson from sorage s calculaed based on he operang mode (urbne or pumpng). More specfcally, as shown n Eq (27), he maxmum amoun of up reserve provson s calculaed eher wh he spare capacy n urbne mode or sorage npu power n pumpng mode. A smlar prncple s used o consran he downward reserve provson by usng spare capacy n pumpng mode or generaon oupu n urbne mode, as defned n (28). The sorage model used here s based on he wor carred ou n [23]. (18) Sec _ spn Sec _ s Sec r, r, R I K Sec _ spndown Sec _ sdown SecDOWN r, r, R I K Sec _ spn O N,, r_ Sec (19) r V U (20) Sec _ spndown DOWN ON,, r V U (21) r _ Sec r r FLEX P U (22) Sec _ spndown Ter _ spndow N MIN ON,,,, Sec _ s Ter _ s s smin, Tr _ Sec r, Tr _ Ter E, E r (23) Sec _ sdown Ter _ sdown smax s, Tr _ Sec, Tr _ Ter E, r r E (24)

6 IEEE Transacons on Susanable Energy Acceped for publcaon, November Sec _ s Ter _ s S SMAX,,, PR F r r (25) Sec _ sdown Ter _ sdown SDOWN SMAX,,, PR P r r (26) Sec _ s Ter _ s SMAX SON s sdown, r, F U, FLEX, FLEX, r (27) Sec _ sdown Ter _ sdown, r, r 3) Terary reserve S MAX SON sdown s 1,,, P U FLEX FLEX (28) Smlar logc as for secondary reserve can be appled for erary up and down reserves (TUR and TDR), wh he varably and uncerany of load and RES generaon dependng on he relevan deploymen me of erary reserve (Δ r_ter ). Agan, all parcpang uns, namely spnnng (r Ter_spn, ) and sandng (r Ter_sandng, ) and pumped-hydro sorage (r Ter_s, ) can be scheduled so as o leave headroom for provson of erary reserve, as n (29) and (30). r r r R (29) Ter _ spn Ter _ sandng Ter _ s Ter,,, I I K r r R (30) Ter _ spndown Ter _ sdown TerDOWN,, I K Ter _ spn O N,, r_ Ter r V U (31) Ter _ spndown DOWN ON,, r V U (32) D. Unfed UC formulaon r _ Ter Tradonal UC models [24] use bnary varables for modellng wheher a un s comng onlne, wheher s onlne and wheher s urnng offlne a a gven me; hs s modelled by U SU,, U ON, and SD U,. In [25], he auhor provdes a dealed analyss of UC and compares he approach usng bnary and neger varables n a clusered UC model; excellen correlaon of resuls of he models s shown. In he UC formulaon proposed here, U SU,, U ON, and SD U, can seamlessly be bnary, neger or connuous varables, and hs ndeed allows wrng a compac and unfed formulaon for he BUC, MILP and LP algorhms, respecvely. In parcular, and focusng our descrpon on he clusered cases, he number of he onlne uns U ON, s defned as n (33). Furher, he number of onlne uns a me should be less han he oal avalable number of uns whn he cluser (G ) mnus he offlne uns, as from (34). More specfcally, he consrans on he number of onlne uns n (34) can be explaned by consderng ha he onlne uns a me need o be fewer han he uns n he cluser mnus he number of uns whch are shu down whn T down + T sup + T shdn 1 before. Smlarly, he onlne uns need o be more han he uns ha sar up whn T up 1 before (Eq. (35)). ON ON SU SD,, 1,, U U U U (33) U U ON,, ON, SD G U (34) shdn DOWN sup 1, T T T 1 1, ( T 1) SU U, (35) DOWN In (34) and (35), T represens he mnmum downme, T he mnmum up-me, and T sup shdn and T he ranson mes from and o off-saus, respecvely. These ranson mes are calculaed from he un s rampng up (V ) or down (V DOWN ) raes, as shown n (36) and (37), respecvely: sup MIN T cel( P / V ) (36) shdn DOWN MIN T cel( P / V ) (37) The reason for explcly consderng hese ranson mes s ha when a un swches from onlne o offlne sae, he un aes me o ramp down o s fnal off-saus. Smlarly, when a un s swched on aes a ceran me before reachng s MSG level and herefore s onlne sae. Hence, s necessary o nclude rampng mes oo, and no mnmum up and down mes only, o fully accoun for he acual consrans relevan o ransons o and from he off-saus. The ably of a sngle un as well as un cluser o change operaonal pon beween wo me seps s bound by s rampng consran, whch s n urn relaed o he number of he exsng commed uns and he new uns comng onlne. In parcular, he ncrease n cluser s oupu s lmed by he rampng lms of he uns already onlne a he prevous me sep plus he maxmum oupu power of he new onlne uns. Ths s defned n (38): FLEX, FLEX, 1 ON SU sup, 1, U V U V T (38) In addon, he power ncrease ha can be acheved s also lmed by he maxmum power of he uns already onlne, as from (39): FLEX, FLEX, 1 U, 1 ON MAX, 1, SU sup P FLEX U V T (39) Smlarly, he decrease of he power oupu, formulaed n (33), s consraned by he down-rampng lms of he uns already onlne a he prevous me sep plus he oupu power of he uns ha have been shu down a he gven me sep. FLEX, 1 FLEX, ON DOWN SD DOWN shdn U, 1 V U, V T (40) The mahemacal model n equaons (33) (40) descrbes he operaon of he flexble generang uns, wh, as aforemenoned, U ON,, U SU, and SD U, poenally beng bnary, neger or connuous varables, dependng on he algorhm used (BUC, MILP or LP, respecvely). Ths s agan he core of he unfed formulaon ha we propose here, whch can seamlessly model he hree classes of algorhm. The followng secons presen and compare he resuls of all hree approaches, hghlghng how for LP relevan mercs (nroduced nex) are whn lmed margns of error bu whle runnng poenally several housand mes faser.

7 IEEE Transacons on Susanable Energy Acceped for publcaon, November TABLE I CONVENTIONAL GENERATOR CHARACTERISTIC IN CLUSTERING P MAX/P MIN C NL FU C SU T /T DOWN (MW/un) ( /(un.h)) ( /MWh) ( /un) (hour(s)) Technology No. generaors V /V DOWN (uns) (MW/h) γ Coal / / / Nuclear / / CCGT / / / FC (MW/un) IV. METRICS FOR LP, MILP AND BUC SCHEDULING ALGORITHM COMPARISON In order o assess he performance of he LP, MILP and BUC mehods, specfc mercs have been denfed wh respec o sysem operaonal cos and accuracy of generaon mx (GM) (as n [15]), as well as accuracy of flexbly servces (e.g., frequency response and secondary and erary up reserves) allocaon (as a novel conrbuon of hs wor). These mercs have been seleced owng o her ably o assess how wo algorhms, for example MILP and BUC, can schedule energy and dfferen ypes of reserves and allocae hem o dfferen plans. The dfferences beween he wo algorhms should hus emerge clearly when consderng he mercs proposed, namely, dfferences n oal cos, n energy generaon share per plan ype, n ype of generaon provdng energy and each reserve a every me sep, and n smulaon me. All hree mehods are compared wh each oher o oban nformaon n relave erms. A. Operaonal cos Ths merc (M cos ) s used o deermne he relave dfference of he oal sysem operaonal cos beween wo mehods, as n (41): M cos Mehod A/ B B. eraon mx oal oal Mehod A CMehod B oal CMehod B C (41) The followng merc (M GM ) s used o capure he average dfference of energy shares n beween algorhms. The energy share suppled by each ype of convenonal generaor s calculaed; hen he average absolue dfference of hese shares beween wo algorhms s deermned, as from (42). M GM Mehod A/ B Mean I p s,,, A p s,,, B T, ss T, ss p s,,, A p s,,, B T, ss, I T, ss, I (42) C. Power and flexbly servces provson In order o compare he performance of dfferen smulaon mehods n erms of generaon and flexbly servce provsons, he mean devaon s calculaed for power PP provson (PP) of all generaon (M Mehod A/B ) and flexbly flexbly servces (M servces Mehod A/B ) for all generaor ypes and a each me seps. For example, Eq. (43) s used o calculae he relave oal power oupu devaon: M p p PP Mehod A/ B Mean T, I s,, A s,, B ss ss ss, I s,, B p (43) For he flexbly servces, he calculaon mehod s he same, bu jus replacng he power oupu wh he conrbuon of each cluser on prmary frequency response (f, ), secondary reserve (r, Sec_spn ) or erary reserve (r, D. Smulaon me Ter_spn ). The smulaon me represens he me aen by he opmzaon solver, excludng npu daa loadng and oupu daa wrng processes. TABLE II CLUSTERS WITH DIFFERENT SEGMENT SIZES Segmen Number of clusers per each ype of plan sze Toal Coal plan Nuclear CCGT (MW) plan all A. Base sysem descrpon V. CASE STUDY DESCRIPTION In order o provde a comparson of he performance of LP, MILP and BUC algorhms, a reduced verson of he curren Grea Bran generaon porfolo s modelled consderng hsorcal demand daa [26] and wnd generaon daa [27] wh half-hour resoluon. Mos of he Grea Bran power plan capacy nformaon s aen from [28]. However, only 30GW ou of acual 35GW Combned Cycle Gas Turbne (CCGT) plan, 25 GW ou of acual 30GW coal plan, and 9.4GW nuclear plan capacy values are presened here, mang up 62 convenonal plan uns. Oher ypes of plans such as bomass, ol and desel plans are omed due o her relavely small capacy (2.6 GW n oal) and mnor generaon conrbuon o he sysem. The Open Cycle Gas Turbne (OCGT) plans are consdered here as a sandng reserve provder for he erary up reserve and are no ncluded n he smulaon of he generaon porfolo. The orgnal demand profle has a 56.6 GW pea, bu consderng he smaller generaon capacy n he model, has been scaled down o a profle wh a 49.7 GW pea. Ths reduced plan model has been smulaed accordng o a classcal BUC. The convenonal power plan characerscs for dfferen clusers by fuel ype are shown n Table II. The nne nuclear power plans are clusered no one equvalen un as s assumed ha hey wll operae as mus-run, herefore whou sar-up or shu down acons. The dfferen power plans characerscs such as sar-up cos, no load cos, rampng raes, ec. are scaled up/down based on her capaces. The frequency response requremen of he sysem

8 IEEE Transacons on Susanable Energy Acceped for publcaon, November TABLE III PROBLEM SIZE FOR ONE AND OBJECTIVE VALUE FOR THE DIFFERENT ALGORITHMS, ONE WEEK SIMULATION Model No. consrans No. varables No. non-zeros No. global enes Opmal objecve value ( M) BUC MILP, full sze, 3 clusers MILP, 2000MW, 5 clusers MILP, 1000MW,8 clusers MILP, 500MW, 12 clusers MILP, 250MW, 19 clusers MILP, 100MW,28 clusers LP, full sze,3 clusers s fxed o 1900 MW based on Naonal Grd regulaon [29] and he wnd and solar forecas error used for reserve requremen deermnaon s obaned from [21]. The sandng reserve provded by OCGT s fxed o 1524 GW based on Naonal Grd nformaon [30]. For he merc comparson, a wee smulaon (sarng from ) wh half-hour me seps s consdered. The demand paern of hs wee has been chosen as havng he leas devaon from he average demand paern of 13 wees n he sprng season of he year. B. Case sudes 1) Sze of clusers To assess he mpac of cluserng dfferen plans of each ype when runnng MILP and LP algorhms, dfferen numbers of clusers have been consdered based on fuel ype and segmen sze. The oal number of clusers hus range from 3 (uns clusered based on fuel ype only, as n Table I) o 28. In he laer case, he uns have been clusered based on fuel ype and 100MW segmens; hs means ha plans wh same fuel ype and sze n he range of MW have been clusered no one group, plans wh same fuel ype and capacy n he range MW no anoher group, and so forh. The deals of he dfferen clusers wh dfferen segmen sze are shown n Table II. 2) Renewable energy negraon scenaros Wh hgher RES peneraon level, flexbly requremens (for reserves, n parcular) wll change; hs, along wh he dfferen levels of energy produced by RES, wll n urn affec he operaons and ulsaon levels of dfferen generaors. Therefore, s mporan o nvesgae he mpac of dfferen RES scenaros on smulaon resul accuracy, parcularly n he comparson beween MILP and LP, whch s he core of hs wor. Four scenaros have hus been denfed (adaped from Naonal Grd s Gone Green scenaro [31]): - Tradonal sysem, wh no RES; - Curren sysem, wh 11.5GW wnd (8.86% energy peneraon level) and 5GW PV (1.26% energy level); sysem, wh 21.7GW wnd (16.32% energy level) and 6.7GW PV (1.69% energy level); sysem, wh 30.9GW wnd (23.24% energy level) and 8.2GW PV (2.07% energy level). 3) Scenaros wh dfferen levels of sysem flexbly To furher demonsrae comparsons of MILP and LP models, he four RES scenaros nroduced above have also been analysed consderng hree dfferen sysem characerscs n erms of flexbly, as from below: - The frs ( Normal ) flexbly scenaro s he base case wh characerscs gven n Table I. - In he second ( Flexble ) flexbly scenaro, he sysem s flexbly has been changed by alerng some ey echncal consrans, namely, sysem s MSG, ramp raes, and mnmum down mes. In hs case, followng [32], he sysem s MSG s reduced by reducng he MSG of coal and CCGT generaors from 40% o 30%; he ramp rae of coal and CCGT uns s ncreased by 66%; he mnmum down me of coal uns s reduced from 4 o 2 hours; - The hrd ( Flexble wh sorage ) flexbly scenaro s he same as he second one wh he addon of energy sorage for energy balancng and secondary and erary reserve provson, whch are he mos lely servces ha pumped-hydro plans would provde [32]. Ths also means, n pracce, ha he sysem s MSG s furher reduced and s ramp rae furher ncreased. As aforemenoned, a PSPP has been seleced as sorage, wh he curren Grea Bran characerscs [34], namely, 2.8 GW of nsalled capacy, and 27.6 GWh of energy volume [35], and assumng 87% sorage dschargng (urbne) effcency and 87% sorage chargng (pumpng) effcency. The maxmum percenage for reserve provson (PR S ) s se o 50%. C. Smulaon seup and problem sze The opmzaon solver appled here s FICO XPRESS 7.6 mxed-neger solver [36]. The npu, oupu and solver seng are confgured hrough MATLAB 2013a. The Mxed-Ineger Program (MIP) Gap s se o 0.1%. The me sep of smulaon profle s half hour and he daa s processed on a weely bass (so as o also capure ypcal arbrage operaons of sorage, n case). Ths resuls n 336 me seps n one opmzaon program. The duraon of one smulaon s lmed o 10 hours. The compuer used n he smulaon has an 8 core CPU wh 3.7 GHz cloc speed (Inel ) and 16 GB RAM. The number of consrans, varables, non-zeros and global enes (.e., bnary or neger varables) for dfferen UC models before pre-solvng, as well as her acheved objecve values, are shown n Table III for a one-wee smulaon. I

9 9 can be seen ha cluserng smlar uns and usng neger varables for he clusers of uns wh smlar echncal consrans (and hus smlar behavour) s an effecve way o reduce he problem sze, passng from bnary o neger varables. Ths should also sgnfcanly reduce he soluon me, as wll be shown laer. VI. ANALYSIS OF THE RESULTS A. Base case: cluserng by capacy sze The performance of he hree algorhms for he base case (aen here as curren sysem wh normal flexbly ) are compared for he dfferen cluser szes of Table II n Fg. 2 (showng he compuaonal speed, n logarhmc scale, of he dfferen algorhms vs dfferen cluser szes), whle Fg. 3 shows he performance comparson accordng o he dfferen mercs nroduced n Secon IV per cluser group. The values n he braces n he legend n Fg. 2 and Fg.3 represen he oal number of clusers. From Fg. 2, can be noced ha he BUC smulaon hs he predefned me consran whch s 10 hours. The smulaon me of MILP exponenally ncreases wh ncreasng number of clusers and reaches he smulaon me lm when he group number of clusers ncreases o 8. However, for he frs wo cluserng levels he MILP algorhm runs and 4300 mes faser han he BUC (ang as reference he 10h of me lm ha he BUC hs, whou convergng o he pre-se 0.1% gap see also Secon VI.D for furher dscusson). Ths confrms ha cluserng smlar uns and usng neger varables for he clusers of uns wh smlar echncal consrans (and hus smlar behavour), besdes beng an effecve way o reduce he problem sze, also sgnfcanly reduces he soluon me. Furhermore, he LP algorhm can be 3 mes (for he 3- clusers case) o mes (8-clusers case) faser han MILP, hus showng he compuaonal benefs of furher relaxng neger no connuous varables. I can be seen from Fg. 3 3 ha all he mercs consdered for dfferen cluser groups feaure errors smaller han 1.6% and he larges error occurs when comparng secondary up reserve beween LP and MILP algorhms. The secondary up requremen has he larges value among all hree flexbly servces n hs case. Also, he erary up reserve requremen s always hgher han he requremen of secondary up reserve, bu he former can be parally provded by sandng reserve; ha s why he spnnng reserve of erary up reserve s smaller han he secondary up reserve. For he MILP, s performance relave o BUC generally mproves wh more clusers. In addon, can be noced ha when he number of clusers ncreases from 19 o 28, he MILP performance doesn mprove noceably. Moreover, s neresng o noce ha, based on he resul of he cos merc, he MILP gves hgher oal coss (by 0.004% o 0.27%) relave o BUC 4. Ths s because he BUC can comm uns of smaller sze and wh wder choce relave o he uns n MILP, whch are characersed by her average capacy; hence, n spe of beng more consraned, BUC resuls no cheaper commmen, as no-load and sar-up coss can be lower. Wh regards o LP, snce he onlne number of uns s relaxed o a connuous value and he algorhms s less consraned, as could be expeced s characersed by lower commmen cos relave o boh MILP and BUC. In addon, he LP smulaon manans smlar performance across dfferen cluserng cases when usng he BUC performance as a benchmar (n he order of -0.15% for he cos merc, 0.5% for prmary frequency response, 0.8% for secondary up reserve, and 0.35% for erary up reserve). In fac, he LP algorhm allows generaors o be parally onlne, so he change n cluser sze wll no have a sgnfcan mpac on he smulaon resul. Furhermore, he dfference beween he performances of MILP and LP decreases wh ncreasng cluser number, as he maxmum value of any merc reduces from 1.5% o 0.8% n hs case. The above fndngs ndcae ha relaxng he MILP dscree varables o connuous ones (mang he problem an LP one), as proposed here, can resul no subsanal compuaonal effcency gan whou losng sgnfcan accuracy n he resuls relave o a full BUC. These benefs may be of hgh value n many cases, parcularly when runnng sraegc flexbly sudes for large scale sysems and long emporal horzons. Hence, n order o explore more he robusness of he MILP-o-LP relaxaon, more sudes are dscussed below. Fg. 2. Compuaon me of LP and MILP wh dfferen clusers and BUC: annual smulaon on a weely bass, curren sysem wh normal flexbly. B. Algorhm comparson for dfferen RES scenaros As aforemenoned, ncreasng he number of clusers reduces he dfferences beween MILP and LP. Therefore, n order o be conservave n he comparson beween a relaxed LP and a non-relaxed MILP, only he wors case wh hree clusers (by fuel ype) s consdered below 5. On he oher hand, he 3-cluser LP model also proves o be very close o a full BUC n erms of performance, whn 0.8% n any merc value, as from Fg I has o be noed ha, snce durng he smulaon was found ha he BUC could hardly converge o he desred 0.1% MIP gap due o he sze of he problem, n order o allow a le-for-le performance comparson and enable he resuls o converge o a unform MIP gap he hree algorhms have been esed on a daly smulaon nsead of a weely one. More dscussons on hs aspec are also provded n Secon VI.D. 4 However, has o be noced ha snce a 0.1% MIP gap has been used, hs s he hghes accuracy ha s meanngful n he comparsons carred ou by means of he cos merc. 5 However, alhough he deals are no shown here for brevy, our furher sudes ndcae ha oher cluser szes would brng smlar resuls and rends as n Fg. 3.

10 10 Fg. 3. Value of mercs n he smulaons of LP and MILP wh dfferen clusers and BUC: wee smulaon on a daly bass, curren sysem wh normal flexbly. Fg. 4. Value of mercs n he smulaons of LP and MILP wh hree generaon clusers and BUC: weely smulaon on a daly bass, a dfferen RES peneraon levels wh normal flexbly. The values of he consdered mercs for dfferen RES peneraon scenaros are shown n Fg. 4. The resuls generally ndcae ha he error ncreases wh he RES peneraon. Ths ncrease s parcularly evden n he secondary and erary reserve mercs, snce her requremens are also ncreasng wh he renewable peneraon level. Agan, he hghes error s ncurred for secondary up reserve a he 30.9 GW wnd capacy case, whch gves around 2.2% error beween LP and MILP. On he oher hand, ncreasng RES peneraon only causes mnor ncrease on he oupu power devaon and generaon oupu assessmen, wh hese mercs sayng whn 0.15%. As a ey fndng, can be noced ha he LP gves a beer or smlar performance relave o MILP across all he mercs a dfferen RES peneraon levels, f usng he BUC as benchmar. Therefore, agan hese sudes sugges ha here s only a mnor loss of accuracy n adopng he proposed relaxed LP as opposed o a commonly used clusered MILP for he purposes of assessng flexble sysem operaon, whle creang noceable benefs by reducng he requred compuaonal me. C. Analyss of dfferen scenaros of sysem s flexbly Furher sudes have been run o specfcally compare he performance of LP and MILP approaches, agan wh hree clusers. As a resul, he cos, prmary frequency response and secondary up reserve mercs are shown n Fgs. 5, 6 and 7, respecvely, for all combnaons of RES and flexbly scenaros. These hree mercs have been seleced snce he cos mercs s he mos ndcave for sysem operaon, whle he prevous sudes have shown ha he oher wo mercs ypcally had he larges value. The smulaon meframe has been se o one wee, whch s also he ypcal maxmum cyclng perod for he PSPP sorage sze consdered here. The resuls llusrae how he values of all hree mercs generally decrease wh ncreasng flexbly of he sysem (meanng ha MILP and LP ge closer o each oher n erms of performance). In parcular, n Fg. 7 he wors case of he cos merc occurs for he 2024 scenaro, wh he merc s absolue value decreasng from 0.46% o 0.28% when he plans become more flexble and hen furher o 0.2% for he case wh sorage. For he prmary frequency response, as can be seen n Fg. 8, he merc value does no vary noceably when ncreasng he plans flexbly, and says n he order of 1.3% 1.4% across dfferen RES scenaros. However, when ncludng sorage, he value of he merc s slghly reduced o 1.07% 1.16%. A smlar rend can be seen n Fg. 9 n erms of secondary up reserve. For example, he merc n he 2024 sysem decreases from 2.25% o 1.89%. I s also of neres o noce ha, for each merc, he dfferences across RES scenaros for he Flexble plans wh sorage case ends o be lower han n he oher wo flexbly scenaros. In fac, sorage ncreases sysem flexbly, whch n urn suppors RES negraon, and as a resul he wo algorhms ge closer o each oher. Ths s also conssen wh he resuls of Fg. 3, whereby wh an ncrease n flexbly (whch n Fg. 3 can be assocaed o an ncrease

11 11 n number of MILP clusers) he LP and MILP algorhms perform closer o each oher. I s also neresng o hghlgh ha whle he sysem becomes more flexble, relevan savngs n operaonal coss may be assocaed o he value of flexbly. In hs respec, when usng MILP, he oal sysem coss beween normal plans and flexble plans wh sorage decrease by 1.9% (n he radonal sysem ) o 3% (n he 2024 sysem ). On he oher hand, he resuls n Fg. 5 ndcae ha MILP and LP perform exremely close o each oher (wh a dfference n assessng he value of flexbly n he order of 2 3%). Hence, consderng ha MILP algorhms are wdespread for flexbly sudes, can be sad once agan he proposed relaxed LP algorhm (whch s he focus of our wor) can also be very effecve o quanfy he economc value of flexbly, especally n scenaro and plannng sudes. Fg. 5. Value of cos merc n he smulaons of LP and MILP wh dfferen RES scenaros and sysem s flexbly characerscs, weely smulaon. Fg. 6. Value of prmary frequency response merc n he smulaons of LP and MILP wh dfferen RES scenaros and sysem s flexbly characerscs, weely smulaon. Fg. 7. Value of secondary up reserve merc n he smulaons of LP and MILP wh dfferen RES scenaros and sysem s flexbly characerscs, weely smulaon. D. Analyss and dscusson of he compuaonal me requremens The compuaonal me requremens of he LP model (wh hree clusers) for he smulaon of he Grea Bran power sysem s annual operaon eep n he order of 15 seconds across he mulple scenaros wh dfferen RES peneraon and flexbly cases. The smulaon duraon ncreases slghly (n he order of a few seconds) when sorage s also consdered, as hs adds more consrans o sysem operaon. In he case of he MILP model (wh hree clusers), he compuaonal me n he normal flexbly case ncreases from 45 seconds n he radonal sysem o 835 seconds n he 2024 sysem scenaro. In he scenaro when plans become more flexble he compuaonal me reduces, by n he order of 10% for radonal sysem o 70% for 2024 sysem. However addng sorage agan ncreases he compuaonal me, smlarly o LP, due o a larger number of consrans n he problem. In general, he rao of compuaonal me beween MILP and LP across mulple scenaros ranges beween abou 3 mes (for he case of radonal sysem wh normal plans ) and abou 50 mes (for he case of 2024 sysem wh normal plans ) n favour of LP. As a furher pon o hghlgh, was shown n Fg. 3 how he compuaonal me rao beween LP and MILP ncreases wh larger number of clusers. Ths resul s conssen wh he fac ha wh larger number of clusers MILP becomes more and more smlar o BUC, wh an ncreasng number of dscree varables. On he oher hand, he number of consrans and varables n he LP problem ncrease oo, bu he problem eeps beng fully lnear and herefore relavely easer o solve. Alhough he compuaonal comparson across scenaros s based on he 3-cluser model, as he one wh he larges error beween LP and MILP, our furher sudes confrm ha smulang more clusers would favour LP even more n erms of compuaonal me requred, n lne wh he general rend shown n Fg. 3 and Fg. 4. In order o furher nvesgae compuaonal me ssues and he ably of MILP and BUC o converge o he pre-se MIP gap, addonal smulaons for all hree algorhms and cluserng approaches have been conduced. In parcular, Fg. 8 llusraes he smulaon me for a weely smulaon n he base scenaro, for a number of LP and MILP clusers as well as for BUC, along wh he remanng MIP gap n hose cases when me lm was reached for he MILP and BUC algorhms. Furhermore, he fgure now also demonsraes he performance of he dfferen algorhms and cluserng approaches wh respec o a furher dmenson of problem complexy, namely, he number of reserves used n he smulaons. In fac, consderng mulple reserves s crcal o properly addressng flexbly ssues, bu may become compuaonally onerous for larger clusers. To confrm hs, dfferen cases named sngle reserve, wo reserves and full reserves have been run: sngle reserve means ha only one reserve ype s consdered, whch can be eher prmary, secondary or erary reserve; n he wo reserve cases, hree combnaons are analysed combnng he reserves n groups of wo,.e., prmary and secondary, prmary and erary, and secondary and erary 6 ; fnally, he full reserves cases mean 6 For smplcy, n order o show he resuls n a compac form and due o space reasons, he values repored n Fg. 8 represen he average compuaonal me of he hree reserves (n he sngle reserve cases) or of

12 12 ha all hree reserve ypes are ncluded n he algorhm, whch are also he general formulaon and case sudes consdered above n he paper 7. The resuls n Fg.8 ndcae ha ncreasng he number of reserves leads o longer average compuaonal me. For example, n he 8 clusers scenaro he MILP algorhm aes 16600, and seconds for he sngle, wo and full reserve cases, respecvely, wh seconds (=10h) hng he me lm wh a remanng MIP gap equal o 0.17%. Lewse, he BUC algorhm converges n abou seconds on average n he sngle reserve case, bu hs he pre-se smulaon me lm of 10h wh a remanng MIP gap of 0.19% and 0.49% n he wo reserves and full reserves cases, respecvely 8. Hence, he resuls from Fg. 8 clearly show how consderng mulple reserves can subsanally ncrease he problem complexy and capably and speed of he solver o converge o a soluon whn he pre-se gap, whle many of he UC algorhms ha are used for dfferen sudes and parcularly flexbly ones ofen only use one or a mos wo reserves. On he oher hand, as demonsraed above n he paper, he proposed relaxed LP algorhms manages o perform very close o and (much) faser han MILP and BUC when provdng full reserves, whch furher srenghens he conrbuon of hs wor. Snce he BUC algorhm and MILP algorhms wh hgh number of clusers may no converge whn he gven me cap when adopng wo or hree reserves, addonal analyses for a smaller problem (n erms of fewer smulaon me seps), namely, smulaon of one day, are shown n Fg. 9 for LP and MILP for dfferen number of clusers as well as BUC. The resuls once agan suppor he prevous fndngs of he paper. For a sngle day analyses all hree models converge o a unform MIP gap of 0.1% whn he compuaonal me lm. The performance of LP compared o MILP n erms of compuaonal me s n lne wh smlar resuls on a weely bass (Fg. 2 and Fg. 3). The smulaon mes of LP algorhms are beween 0.19 and 2.32 seconds across dfferen cluserng mehods, and hen ncrease o n a range of 0.25 o 362 seconds n he MILP cases and furher o 3090 seconds n BUC case. The compuaonal me dfference beween MILP and BUC (besdes LP) algorhms can hus now also be fully apprecaed. he hree reserve combnaons (n he wo reserve cases). erally speang, he dfferen algorhms are slower when usng prmary reserves han when usng secondary and n urn han when usng erary, and hs rend becomes more evden wh ncreasng number of clusers. 7 Regardng he modfcaon of relevan consrans n he problem formulaon when consderng dfferen number of reserves, f a specfc reserve ype s no consdered hen s relevan varable s o be se o zero for all consdered smulaon me seps. More specfcally, Eq. (16) s negleced when prmary frequency response s no consdered. Rampng consrans (20) and (21) are dscarded n he cases whou secondary reserve. Smlarly, Eqs (31) and (32) are excluded when erary reserve s no consdered. In addon, f up, s elmnaed from (17) f he prmary frequency response s no consdered; r Sec_spn, s aen ou from (17) and (22), for he cases whou secondary reserve; and r, Ter_spn s negleced n Eqs (17) and (22) for he cases whou erary reserve. 8 In hs respec, agan can be seen and somehow quanfed how BUC aes longer han MILP, as even when neher algorhm converges he gap for BUC s sll larger han for MILP. Fg. 8. Compuaonal me and opmaly gap of LP and MILP wh 3, 5 and 8 clusers and of BUC: weely smulaon, curren sysem wh normal flexbly and dfferen reserve numbers. Fg. 9. Compuaonal me of LP and MILP wh dfferen clusers and BUC: daly smulaon, curren sysem wh normal flexbly and full reserves. VII. CONCLUSIONS In hs paper, we have nroduced a flexbly-orened unfed formulaon ha can seamlessly model bnary (BUC), MILP and LP un commmen approaches by represenng he generang plans commmen varables as bnary, neger or connuous values, respecvely. In parcular, has been shown how an LP algorhm can be modelled whn he proposed formulaon by smply relaxng he varable represenng he onlne uns from neger (MILP) o connuous for dfferen generaors cluserng levels of he MILP represenaon. Frequency response and reserve servces and all relevan consrans have been explcly consdered, and he flexble and nflexble componens of he UC problem formulaon have been hghlghed. The performance of he hree mehods (BUC as well as MILP and LP wh dfferen cluserng levels) have been compared hrough dfferen mercs specfcally nroduced, namely, performance mercs such as relave dfferences n sysem s operaonal cos, power oupu composon per cluser, and allocaon of flexbly servces (prmary frequency response, secondary and erary up reserves) per cluser, as well as compuaonal effcency. To demonsrae he proposed approach, several case sudes have been run consderng a 62-un reduced verson of he Grea Bran power sysem. In he consdered base case, he MILP UC can perform 4300 o faser han BUC when usng he smaller number of clusers, bu reaches he pre-se me consran when he number of clusers ncreases o 8. On he oher hand, he LP UC can perform ens o housands mes faser han MILP across he 3 o 28 cluser cases. All performance mercs say whn around 1.5%, and ncreasng he number of clusers does no generally gve much

13 13 mprovemen n erms of resuls for LP cases (relave o BUC) bu mproves he MILP performance. Furhermore, for dfferen RES peneraon scenaros (run for a conservave 3- cluser case), he maxmum dfference n resuls beween he LP and MILP algorhms s confned o 2.2% for any performance merc. Ths dfference also decreases when he sysem becomes more flexble, mang he generang uns more flexble and hen furher addng sorage o he sysem. On he oher hand, he LP always brngs subsanal compuaonal gans, runnng abou 3 o 50 mes faser han a MILP, dependng on he scenaro. The complexy of he dfferen models when consderng dfferen ypes of reserve servces has also been nvesgaed n deal, also quanfyng he opmaly gap of MILP and BUC algorhms when he pre-se gap could no be reached whn he defned me lm was also nvesgaed n deal. In addon, he compuaonal performance of LP, MILP and BUC for a smaller case sudy (one-day smulaon) has also been analysed, confrmng he rends found for larger scale cases. The resuls of hs wor generally ndcae ha he proposed relaxed LP UC approach, n case wh low level of cluserng, can be used nsead of popular MILP UC approaches for sraegc flexbly analyss for large sysems wh long me horzons and smulaons, wh mnor (and n many cases neglgble) loss of accuracy n he dfferen mercs consdered, whle ganng noceable compuaonal speed. Hence, LP UC algorhms could be parcularly suable for me-consumng flexbly sudes ha mgh for nsance enal scenaro analyss or Mone Carlo/sochasc smulaons, as well as for plannng purposes. On he oher hand, gven he complexy of he ssues dscussed, he specfc resuls mgh also be affeced by he solver choce, solver sengs and deals of he problem formulaon, and more research s needed o comprehensvely address hese pons. Wor n progress ams a ncorporang demand sde resources n he analyss n a more comprehensve and sysemac way, n parcular for provson of reserves. 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14 14 [28] I. Macleay, K. Harrs, and A. Annu, DIGEST OF UNITED KINGDOM ENERGY STATISTICS 2014, [29] Naonal Grd, Wner Ouloo 2013/14, [30] [Naonal Grd, Shor Term Operang Reserve Fuel ype Analyss Season 8.5. [31] Naonal Grd, UK Fuure Energy Scenaros, [32] F. Teng, D. Pudjano, G. Srbac, F. Ferre, and R. Bove, Assessmen of he value of plan flexbly n low carbon energy sysem, n 3rd Renewable Power eraon Conference (RPG 2014), 2014, pp [33] D. Lumb and N. T. Hawns, Provson of power reserve from pumped sorage hydro plan, n IEE Colloquum on Economc Provson of a Frequency Responsve Power Reserve Servce, 1998, vol. 1998, pp [34] Parlamenary Offce of Scence and Technology, Elecrcy Sorage, no [35] Energy research Parnershp, The fuure role for energy sorage n he UK, [36] FICO, FICO XPRESS 7.6. FICO, Lngx Zhang (S 13) receved he B.Eng (Hons) degrees n Elecrcal and Elecronc Engnneerng from he Unversy of Mancheser, UK, and Norh Chna Elecrc Power Unversy, Chna, n He s currenly pursung a PhD degree n Elecrcal Energy and Power sysems group a he Unversy of Mancheser. Hs research neress nclude mulenergy sysem modellng, power sysem flexbly and energy sysem operaon and plannng. Tomslav Capuder (S 08 M 14) receved he Ph.D. degree n power sysems from he Faculy of Elecrcal Engneerng and Compung, Zagreb, Croaa, n He s currenly a Pos-Docoral Researcher wh he Faculy of Elecrcal Engneerng and Compung. Hs curren research neress nclude mulenergy generaon sysems, dsrc energy sysems, power and energy sysem flexbly, and operaon and plannng of energy sysems. Perlug Mancarella (M 08, SM 14) receved he Ph.D. degree n Elecrcal Energy Sysems from he Polecnco d Torno, Ialy, n Afer beng a Research Assocae a Imperal College London, UK, n 2011 Perlug joned he Unversy of Mancheser, UK, where he s currenly a Reader n Fuure Energy Newors. Perlug s research neress nclude mul-energy sysems modellng, power sysem negraon of low carbon echnologes, newor nvesmen under uncerany, and rs and reslence of smar grds. He s auhor of several boos and boo chapers and of over 200 research papers on hose opcs. Perlug s an Assocae Edor of he IEEE Sysems Journal, and he Char of he Energy worng group of he IEEE European Publc Polcy Inave.