Sensitivity-Indices Based Risk Assessment of Large Scale Solar PV Investment Projects

Transcription

1 ACCEPTED TO IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, SEPTEMBER Senstvty-Indces Based Rsk Assessment of Large Scale Solar PV Investment Projects Indrajt Das, Student Member, IEEE, Kankar Bhattacharya, Senor Member, IEEE, Claudo Cañzares, Fellow, IEEE, and Wajd Muneer, Student Member, IEEE Abstract Large scale solar hotovoltac (PV) generaton s now a vable, economcally feasble and clean energy suly oton. Incentve schemes, such as the Feed-n-Tarff (FIT) n Ontaro, have attracted large-scale nvestments n solar PV generaton. In a revous work, the authors resented an nvestor-orentelannng model for otmum selecton of solar PV nvestment decsons. In ths aer, a method for determnng the senstvty ndces, based on the alcaton of dualty theory on the Karush-Kuhn-Tucker (KKT) otmalty condtons, ertanng to the solar PV nvestment model s resented. The senstvty of the nvestors roft to varous arameters, for a case study n Ontaro-Canada are resented and dscussed and these are found to be very close to those obtaned usng the Monte Carlo smulaton and fnte-dfference (ndvdual arameter erturbaton) based aroaches. Furthermore, a novel relatonsh s roosed between the senstvty ndces and the nvestor s roft for a gven confdence level to evaluate the rsk for an nvestor n solar PV rojects. Index Terms Solar hotovoltac, nvestor lannng, senstvty ndces, dualty theory, rsk assessment I. Nomenclature The man notatons used throughout the aer are stated below for quck reference. In the aer, matrces are n boldface, vectors are n bold and talcs, whle scalar quanttes are n talcs. Subscrts and suerscrts base Base case otmzaton outut DT Dualty Theory FD Fnte Dfference MC Monte Carlo MIN Mnmum value MAX Maxmum value new Re-otmzaton outut after arameter erturbaton o Proft at a gven Confdence Level % Quantty n ercentage Quantty n dollars Indces b Index for bnary varables, j Index for transmsson zones Ths work was suorted by Ontaro Centres of Excellence, Hydro One Inc., Frst Solar Inc. and London Hydro Research Grant for studes on Large- Scale Photovoltac Solar Power Integraton n Transmsson and Dstrbuton Networks. I. Das (das@uwaterloo.ca), K. Bhattacharya (kankar@uwaterloo.ca), C. Canzares (ccanzares@uwaterloo.ca) and W. Muneer (wmuneer@uwaterloo.ca) are wth the Deartment of Electrcal and Comuter Engneerng, Unversty of Waterloo, Waterloo, Canada. J k m o Set of ndexes of actve nequalty constrants Index of years of study horzon Index of all nequalty constrants Index of solar PV lan model arameters Set of arameters of the solar PV lan model Set of arameter base values of the solar PV lan model Functons f ( ) Objectve Functon F( ) Cumulatve Dstrbuton Functon g( ) Inequalty constrants h( ) Equalty constrants Parameters B, j Elements of zonal transmsson matrx,.u. Ca Conv Caacty of conventonal generaton, MW CF Conv Caacty Factor of conventonal generaton, % Caacty Factor of PV unt, % d k Dscount rate, % DB Intal number of years of zero nvestment FIT k Feed-n-Tarff, /kwh L Number of equalty constrants LbC PV Solar PV nstallaton labour cost, /kw LdC PV Solar PV nstallaton land cost, /kw M Number of nequalty constrants m J Number of non-zero actve nequalty constrants N Number of years of solar PV lan horzon n Number of varables OM PV Solar PV oeraton and mantenance cost, /kwh P, MAX j Maxmum ower transfer across zones, MW PD Power demand, MW TC PV Solar PV unt transortaton cost, /kw UC PV Cost of solar PV unt, /kw β k Annual budget of solar PV nvestor, β T Total budget of solar PV nvestor, δ MIN Mnmum bus angle, rad δ MAX Maxmum bus angle, rad µ Mean value of a data set σ Standard devaton of a data set Varables Caacty of solar PV generaton, MW Annual energy generaton from solar PV, MWh Annual energy generaton from conventonal, MWh New caacty of solar PV generaton, MW Power transfer across zones, MW E PV E Conv P, j

2 ACCEPTED TO IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, SEPTEMBER Power generated from conventonal sources, MW Power generated from solar PV, MW VaR Value at rsk, W b Bnary varables γ Lagrange multler for nequalty constrants δ Bus angle, rad λ Lagrange multler for equalty constrants ξ Senstvty ndex ρ Confdence level of a ortfolo of roft, % Net resent value (NPV) of roft, P Conv P PV II. Introducton The growng costs of fossl fuels couled wth concerns for envronmental emssons are strong motvatons for ncreaseenetraton of renewable sources of energy. Electrcty generated from renewable resources, artcularly solar hotovoltac (PV) has a large otental n the global energy market. Snce PV modules desgned for both commercal and resdental alcatons sutable for grd-connected and stand-alone systems have become readly avalable n the market, nvestments n solar PV rojects have been growng. Favourable government olces, ncentves and suort mechansms have fuelled ts growth to such an extent that t s now one of the fastest growng alternatve energy sources n the world [1]. Mnmal runnng costs, zero emssons and steadly declnng module and nverter costs of solar PV unts render these attractve to nvestors for large-scale solar PV generaton rojects. In [2], the economc lannng of solar PV ntegraton wth the ower grresents an assessment of the economc vablty of a grd-connected solar PV lant; a relatonsh for the breakeven catal cost for PV generaton as a functon of dfferent levels of PV enetraton s develoed, demonstratng that wth hgher enetraton ercentages, the breakeven cost decreases. In [3], the dstrbuted utlty concet s exlored and a market entry strategy s roosed for the ntroducton and growth of solar PV n utlty alcatons; a dffuson model strategy s develoed that brdges the ga between economc stand-alone secal alcatons and bulk ower roducton. Artcles [4], [5], [6], [7], [8] ublshed recently showcase the resent scenaro of grd ntegraton of solar PV generaton n the USA. An otmzaton framework, resented n [9], facltates a rosectve nvestor to arrve at an otmal nvestment lan n large-scale solar PV generaton rojects. Otmal decsons on locaton, szng and tme of nvestment that yelds the maxmum roft to the nvestor are determned. The set of nvestment decson varables s dscrete n nature and thus ncororates bnary varables n the otmzaton model. The objectve of the develoed mxed nteger lnear rogrammng (MILP) otmzaton model s to maxmze the Net Present Value (NPV) of nvestors roft. The data that are used to estmate the arameters of ths model are subject to errors, lack of recson, etc.; therefore, conclusons drawn from smulatons are senstve to the choce of nut data. Snce small changes n nut data can have sgnfcant effects on the results, t s essental to assess the senstvty of the results to varous model arameters and nut data, whch s the man objectve of the resent aer. In [1] and [11], a erturbaton aroach to comute senstvtes n otmzaton based models s dscussed. Ths method s used n [12] to derve general senstvty formulas for maxmum loadng condtons n ower systems. It rovdes generalzed senstvty exressons based on the soluton of a voltage stablty constraned otmal ower flow. These senstvty exressons use the dual varables (Lagrangan multlers) at the otmal soluton and the roertes of the Karush-Kuhn-Tucker (KKT) otmalty condtons. Reference [13] resents the alcaton of the method [11] to fnd locatonal margnal rce senstvtes wth resect to changes n nodal demands. In the current aer the concet of local senstvty analyss based on dualty theory (DT), [1] and [11], s aled to determne the lan senstvty ndces ertanng to the solar PV nvestment model [9]. The senstvty of the NPV of roft to changes n model nut arameters s determned. Ths aroach s comutatonally less exensve than the Monte Carlo smulaton based aroach [14] and s also more advantageous than the Fnte Dfference (FD) aroach, [15] and [16], as t determnes all the arameter senstvtes smultaneously. Addtonally, ths method allows determnng the senstvtes of the decson varables as well as the Lagrange multlers wth resect to all the model arameters, at once. However, snce ths analyss s based on a lnearzaton aroach, t may not yeld accurate results for large varatons n the nut data for a nonlnear system model; therefore a comarson of the senstvtes obtaned usng Monte Carlo smulaton and FD aroaches are resented. Furthermore, the solar PV nvestment rsk s assessed wth resect to varous arameters based on Value at Rsk (VaR) and Confdence Level (CL) ndces, whch are determned from the DT based senstvty ndces. The rest of ths aer s organzed as follows: Secton III rovdes a revew of the solar PV nvestment model develoed n [9]. Secton IV resents the methodology and formulaton of calculaton of senstvty ndces usng DT, FD and Monte Carlo aroaches. Secton V resents the develoment of the relatonsh of nvestment rsk ndces wth the roosed senstvty ndces. In Secton VI, the senstvty ndces are comuted and valdated based on MC smulatons and FD calculatons, and the derved relatonshs n Secton V are used to asses the rsk levels wth resect to varous arameters. Fnally, Secton VII hghlghts the conclusons and contrbutons from ths work. III. The Solar PV Investment Model A solar PV nvestment model, from the ersectve of an nvestor, reorted n [9], s brefly dscussed next for the sake of comleteness. The model s a MILP roblem comrsng contnuous and bnary varables. The objectve of the model s to maxmze and hence determne the otmal set of solar PV nvestment decsons, where: (Revenue Cost ) = (1) (1 + d k ) k k

3 ACCEPTED TO IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, SEPTEMBER and Revenue = FIT k E PV (2) Cost = CC PV + OM PV E PV (3) Here = 5W b s an nteger varable, constructed usng b the auxlary bnary varables W b. Note that all varables, arameters and ndces n these and followng equatons are defned n Secton I. In (3), the Catal Cost (CC) s descrbed as: CC PV = UC PV + LbC PV + TC PV + LdC PV (4) The varous oeratonal, lannng and fnancal constrants of the MILP model are: Suly demand balance: ( ) P Conv + P PV PD = B, j δ δ k, j (5) Conventonal Energy Generaton Lmt: E Conv 876 Ca Conv Solar PV Energy Generaton Lmt: E PV 876 j CF Conv (6) (7) Transmsson Lne Flow Lmts: ( ) B, j δ δ k, j P MAX, j (8) Power angle lmts: Annual budget lmt: Total budget lmt: ( CC PV k δ MIN δ δ MAX (9) CC PV β k (1) + OM PV ) E PV βt (11) Dynamc Constrant on Solar PV Caacty Addton: k+1, = Intal Year Investment Constrant: + k = 1, 2,, (N 1) (12) k+1, = k = 1, 2,, DB (13) Termnal Year Investment Constrant: k+1, Solar PV Lfetme Constrant: k N (14) k+z+1, = CaPV k+z, NCPV k = 1, 2,, (N 1) (15) The nterested reader s referred to [9] for more detaled nformaton on the selecton of the model arameters. A. Dualty Theory IV. Calculaton of Senstvty Indces A descrton of the dualty theory based method for obtanng local senstvtes of all the arameters s dscussed n [1] and [11], and s brefly exlaned next, n the context of the aforementoned model. Thus, let consder a rmal Non-Lnear Programmng (NLP) roblem as follows: Mn. z = f (x, a) s.t. h(x, a) = b (16) g(x, a) c where h(x, a) = [h 1 (x, a),, h L (x, a)] T and g(x, a) = [g 1 (x, a),, g M (x, a)] T. In order to smlfy the mathematcal dervatons, the arameters a, b and c are assumed to be subsets of a set,.e., = [a b c] T. The senstvtes of the otmal soluton (x, λ, γ, z ) wth resect to local changes n the arameter can be obtaned by dfferentatng the objectve functon and the KKT condtons of otmalty of the NLP model (16). Thus, the matrx wth all dervatves,.e., the senstvty ndces, s gven by: where and [ dx dλ dγ F x 1 U = F xx H T x H x ] T dz = U 1 S (17) (18) S = [ F F x H ] T (19) F x(1 n) = [ x f (x, ) ] T F (1 ) = [ f (x, ) ] T H x(l n) = [ x h(x, ) ] T H (L ) = [ h(x, ) ] T F xx(n n) = xx f (x, ) + F x(n n) = x f (x, ) + L λ l xxh l (x, ) l=1 m J (2) (21) (22) (23) + γm xx g m (x, ) (24) m=1 L λ l xh l (x, ) l=1 m J + γm x g m (x, ) (25) m=1 The matrx U n (17) s generally nvertble, as the soluton to the otmzaton roblem (x, λ, γ, z ) s a regular nondegenerate ont. If any degenerate constrant,.e., a zerovalued Lagrange multler of an actve constrant, s resent, then the corresondng rows and columns n the matrces U and S are removed [12], makng U nvertble. Addtonally, the non-degenerate nequalty constrants are converted to equalty constrants [1]. Ths methorovdes, n one shot, the senstvty ndces for the objectve functon ξ DT = dz and dual varables, dλ and dγ dx, as well as the rmal, senstvtes wth resect to the

4 ACCEPTED TO IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, SEPTEMBER model arameters. The unts of ξ DT are /(unt of the arameter ), whch can be arorately normalzed to reresent n terms of /(1% change n ). Ths way, the senstvty ndces are scaled arorately n order to comare and rank them wth resect to ther severty. Addtonally, the dollar value of ξ DT can be reresented as a ercentage of base,.e., ξ % DT = 1 ξ DT base (26) where base s the NPV of the roft obtaned from solvng the solar PV nvestment model n Secton III. To aly the dualty theory based senstvty method to the solar PV nvestment model, the varables x anarameters are defned next. Thus, from the model dscussed n Secton III, the vector x of model varables are dentfed as follows: x = [ E PV E Conv δ ] T (27) and the vector of model arameters are defned as: = [ B, j Ca Conv LbC PV TC PV CF Conv LdC PV UC PV OM PV B. Monte Carlo and Fnte Dfference d k FIT k P MAX, j PD (28) β k β k δ MIN δ MAX ] T These senstvtes, comuted based on a lnearzed aroach, are valdated here usng the well-known Monte Carlo smulaton aroach and a FD aroach. Ths allows to determne ther range of alcaton. The Monte Carlo smulaton s aled to the otmzaton model on an OAT (One-factor-at-A-Tme) bass. The arameters need to be erturbed symmetrcally around ther base values n order to obtan unskewed and unbased senstvty ndces. In a normally dstrbuterobablty densty functon (.d.f.) of a arameter, the exected value (mean) s the same as the medan of the dstrbuton; ths allows for a symmetrcal varaton of the arameter for varous standard devatons. Addtonally, for normally dstrbute.d.f., the mode of the varaton s also equal to ts medan, whch enables equal erturbaton of a arameter around ts determnstc value. Therefore, all the arameters n ths aer, are consdered to be normally dstrbuted for robablstc studes wth the standard devaton beng 1% of ts mean value,.e., ( ) σ % σ = 1 (29) where µ =. The solar PV nvestment model otmzaton rogram s run for 2 teratons wth the normally dstrbutearameter keeng other arameters unaltered. The standard devaton of the resultng s comuted as a ercentage of ts mean value, as follows: σ % = 1 µ σ µ (3) The rato of the standard devatons (n ercentage) of outut () and nut (a arameter) s ndcatve of the senstvty ndex comuted by the Monte-Carlo method: ξ % MC = σ% σ % (31).e., for 1% standard devaton n nut arameter ξ % MC = σ%. The senstvty ndex comuted usng Monte Carlo smulaton based aroach, can also be reresented as follows: Thus, from (31) and (32): ξ MC = σ σ % ξ % MC = 1 ξ MC µ (32) (33) For the FD aroach, each arameter s ncreased by 1% of ts base value ( ) and the s comuted agan from the solar PV nvestment model ( new ). The dfference between new and base denotes the change n for a 1% ncrease n a gven arameter whle other arameters reman unaltered. Ths change n s, n essence, the true senstvty ndex for the saarameter wth resect to,.e., new base = ξ FD (34) The senstvty ndex from (34) can also be reresented as a ercentage of base usng the followng exresson: ξ % FD = new base base (35) V. Rsk Analyss usng Senstvty Indces In rsk analyss, VaR s a measure that estmates how much a ortfolo could lose because of market movements for a gven robablty of occurrence of that ortfolo varable [17], and t s referred to as CL. VaR and CL are comuted from the cumulatve dstrbuton functon (c.d.f.) constructed from the.d.f. of the outut quantty. In the context of ths aer, the outut quantty s and the ortfolo s the normally dstrbute.d.f. of. Thus, from the c.d.f. of, a gven roft wth a corresondng cumulatve robablty F( ), ndcates confdence level ρ = 1 F( ), whch means that there s a ρ% lkelhood that and the VaR from the execteroft ( µ ) s gven by: VaR = µ (36) Monte Carlo smulatons are tycally used to comute the VaR and CL of the nvestment ortfolo. However, ths nvolves sgnfcant comutatonal burden arsng from the large number of smulaton cases requred and from establshng the c.d.f. of the nvestment ortfolo. It s demonstrated next, through mathematcal dervatons, that the roosed senstvty ndex ξ DT can be drectly utlzed to comute the rsk arameters VaR and CL, and hence reduce the comutatonal effort requred to determne these values. If a lnear relatonsh exsts between an nut arameter and the outut varable, the.d.f. of the outut s exected to reman smlar to the nut.d.f., and the mean of the nut dstrbuton s exected to generate the mean of the outut

5 ACCEPTED TO IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, SEPTEMBER dstrbuton. In ths aer, the mean of the normally dstrbuted.d.f. of a arameter s consdered to be the base value of the arameter ( ) beng erturbed. Hence, the mean of from every Monte-Carlo smulaton outut wll generally be equal to base,.e., µ = base (37) Then, from (26) and (33), the followng can be obtaned: ξ % MC ξ % DT = ξ MC ξ DT base µ (38) Thus, usng (37) n (38), the followng relatons are roosed: ξ MC ξ (39) and DT ξ % MC ξ % DT (4) From (26) and (31) n (4), the followng relaton holds: σ % = ξ % MC ξ % ξ DT = 1 DT (41) σ % base And from (3) n (41) and rearrangng terms, one has that: σ 1 = 1 ξ σ% DT (42) µ base From (37) n (42), a relatonsh between the standard devaton of the outut varable wth the senstvty ndex of the erturbearameter s obtaned, as follows: σ = σ% (43) ξ DT If more than one nut arameters are erturbed, the frst term of the roduct on the rght hand sde of the equalty n (43),.e., the standard devaton of the nut arameter erturbaton n ercentage, can be reresented by the equvalent standard devaton of the arameters erturbed based on a well-known exresson from multvarate normal dstrbutons [18]. The second term of the roduct can then be relaced by the weghted average of the senstvty ndces of the arameters that are erturbed. Hence, the standard devaton of the outut can be reresented n generc form as follows: σ σ = ( ) 2 σ % % ξ DT (44) σ % Note that (43) s a secfc case of (44) when = 1,.e., when only one nut arameter s erturbed. The relatonsh descrbed n (44) s very mortant as t allows comutng the standard devaton of a normally dstrbuted outut varable resultng from the nut of one or more normally dstrbuted arameter(s) wthout the need to carryout the comutatonally exensve Monte Carlo smulatons. A c.d.f. of a normally dstrbute.d.f. of s shown n Fg. 1. Although the c.d.f. extends to the left asymtotcally, t s assumed that F( MIN ) =, neglectng the asymtotc nature of the curve n that regon. A lnear aroxmaton of the c.d.f. Cumulatve Probablty F μ F μ σ Fg. 1. F MIN =.5 Lnear aroxmaton of c.d.f MIN μ σ Actual c.d.f. σ μ () Tycal c.d.f. lot of dectng the lnear aroxmaton segment. n the regon [ MIN, ( µ )] σ can be reresented by: ( µ = MIN + ) σ MIN ( F ( µ ) 1 ρ ) (45) σ 1 A standard normal.d.f. has µ = and σ = 1 and assumng a varable range [ τ, τ], where F( τ), a relatonsh between µ and MIN can then be obtaned, as follows: MIN = µ τ σ (46) From the Standard Normal Cumulatve Dstrbuton Functon Table [18], one has for τ = 4, F( 4). The value of µ σ corresonds to one standard devaton below the mean and thus, the value of F ( µ σ ) s as follows: F ( µ σ ) = F( 1) = α (47) Usng the same table gven n [18], t s obtaned that α = Thus, fnally, the relatonsh of for a CL of ρ% wth ξ DT roosed here, relacng, (37), (46) and (47) n (45), s as follows: ( ) τ 1 ( = base τ σ + σ 1 ρ ) (48) α 1 For τ = 4 and α =.1587: ( = base 4 σ σ 1 ρ ) (49) 1 The roosed equatons (36), (44) and (49) show that for a lnear otmzaton roblem, the VaR for a gven CL can be closely estmated usng the DT-based senstvty analyss wthout actually runnng the comutatonally exensve Monte Carlo smulatons. A. Senstvty Indces VI. Results and Dscussons The otmzaton model descrbed n [9] s solved usng the CPLEX solver [19] n the GAMS [2] envronment wth relatve otmalty tolerance of.1%. The nut arametrc data s taken for the case of Ontaro from [9] and extraolated for a study erod of 35 years (29-243). The model determnes that all solar PV generaton are to be nstalled n the zone, 35 MW of nstalled solar PV

6 ACCEPTED TO IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, SEPTEMBER TABLE I Senstvty Indces of Parameters n Dollars TABLE III Comarson of Calculated and Actual Standard Devatons Parameters ξ DT ξ MC ξ FD /1% /1% /1% FIT 9,733,465 9,76, 9,733,464 d -8,946,481 8,943,838-8,882,92 7,351,319 7,442,591 7,351,315 β T 6,29, 6,215,678 6,114,413 UC PV -5,543,933 5,546,823-5,543,933 OM PV -2,382,15 2,382,665-2,36,953 LbC PV -729,1 729,15-729,1 LdC PV -28,32 28,497-28,32 TC PV -22,421 22,115-22,421 TABLE II Senstvty Indces of Parameters n Percentage Parameters ξ % DT ξ % MC %o f base /1% ξ % FD FIT d β T UC PV OM PV LbC PV LdC PV TC PV caacty durng the fourth to sxth year, and 25 MW n the seventh year of the lan horzon. The resultng ( base ) s Mllon. The otmal soluton so obtaned, dentfes both zero and non-zero Lagrange multlers assocated wth both equalty and nequalty constrants. The constrants ertanng to non-zero Lagrange multlers are used for comutng the senstvtes based on the DT method, as exlaned n Secton IV-A. The varables E Conv and δ rove to be non-basc and have no sgnfcant mact on the PV nvestment, as t s evdent from the zero valued Lagrangan multlers assocated wth arorate constrants. A MATLAB code s develoed usng Symbolc Math Toolbox [21] and the functons ertanng to f (x, ) and h(x, ) are fed nto t. The matrces U and S, gven by (18) and (19), are comuted symbolcally n MATLAB, and then the numercal values of arametrc data and actve Lagrange multlers are substtuted to comute the senstvty ndces. The senstvty ndces are ranked as er ther severty n Table I. The table shows the dollar amount by whch base changes for a 1% ncrease n the base value ( ) of a arameter. Ths table also resents a comarson of the senstvty ndces, n dollars, comuted usng the Monte Carlo smulaton aroach and the FD method. Smlar results are shown n Table II for the senstvty ndces comuted usng the three methods as a ercentage of base for a 1% ncrease n the base value of a arameter. From Tables I and II, t s found that the roft of an nvestor s most senstve to the Feed-n-Tarff rate. Observe that both the FD method and the DT based method yeld the roer Inut Perturbaton σ % FIT d β T Actual σ (Mllon ) Calculated σ (Mllon ) sgns dectng the ncrement or decrement of the base value wth resect to an ncrease n the arameter value; on the other hand, the Monte Carlo smulaton aroach faled to rovde ths nformaton due to the fact that the senstvty s comuted from Monte Carlo smulatons as a rato of the outut and nut standard devatons, and standard devatons always have ostve values. Note as well n Table I and II that the senstvty ndces comuted usng the DT based method are numercally very close to the true senstvty ndces comuted usng the FD method as well as those obtaned from the Monte Carlo smulaton aroach. Ths valdates the relatonshs of (39) and (4) for the gven model. B. Rsk Analyss The standard devaton of the outut varable when any number of nut arameter(s) are erturbed wth σ % can be comuted usng (44). A comarson between the calculated standard devaton of versus the actual value obtaned from Monte Carlo smulaton s resented n Table III. Note that these results valdate the roosed exresson (44) for varous combnatons of nut arameter erturbatons, thus justfyng the use of (49) to comute the VaR for varous CLs. Table IV resents a comarson of calculated usng (49) versus the actual obtaned from Monte Carlo smulatons MC for a selected set of arameters beng erturbed and for varous values of CL. Observe that the relaton roosed n (49) yelds a farly close value to the actual. It s to be noted that the relaton n (49) rovdes a more conservatve estmate of for each confdence level. These conservatve estmates of roft are artcularly benefcal to solar PV nvestors to evaluate ther rsk strateges,.e., ther VaR and the exected rate of return, and thus arrve at an arorate nvestment decson. The ercentage errors between comuted from (49) and the actual values MC obtaned from the c.d.f. lot resultng from the Monte Carlo smulatons wth resect to ρ s shown n Fg. 2 for four arameters, namely, FIT, d, and

7 Error (%) Error (%) Error (%) ACCEPTED TO IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, SEPTEMBER TABLE IV Comarson of NPV of Proft for varous CLs ρ (%) Parameter Perturbed by 1% FIT d ρ MC Mllon Mllon 99% % % % % % % % % % % % β T 95% % % % F+D F+D+C F+D+C+T Fg. 3. Percentage error versus confdence level when all arameters are erturbed smultaneously by 1% standard devaton (F: FIT; D: Dscount Rate; C: (); T: Total Budget) ρ (%) x (a) σ % = 1% (b) σ % = 2% (c) σ % CFPV = 3% x x 1 8 Fg. 4. Probablty dstrbutons of outut, wth erturbed by varous standard devaton ρ(%) F D T C -4 Fg. 2. Percentage error versus confdence level when arameters are erturbed by 1% standard devaton (F: FIT; D: Dscount Rate; C: (); T: Total Budget). -8 1% SD 2% SD 3% SD (, ndvdually erturbed by 1% from ther base values σ % = 1% ). Note that the error values are very low n each case and le n the range of -.9% to -1.75% for 95% or more CL values, and mroves as the CL reduces,.e., as the rsk averseness of an nvestor decreases. Fgure 3 lots the error for varous CLs for the same set of arameters, smultaneously erturbed n varous combnatons, by 1% from ther base values. Observe that the error s stll qute low and les n the range of -2% to -3% for 95% or more CL values, and mroves as the CL reduces. Ths demonstrates that the senstvty ndces, comuted usng the DT based method, very recsely calculates the rsk arameters of the nvestor, for the gven solar PV nvestment model, and rovdes sgnfcant comutatonal advantages over the Monte Carlo smulaton aroach. Furthermore, (48) and (49) can also be used to comute the VaR for varous CLs even when the outut s not normally dstrbuted. The latter s demonstrated based on comutng the ercentage errors when s erturbed wth 1%, 2% and 3% standard devatons, resultng n ouut.d.f.s that are not normally Fg. 5. Percentage error versus CL when erturbed by varous standard devatons. dstrbuted, as shown n Fg. 4. Observe that as that the hstograms n ths fgure move away from normal dstrbuton as the standard devaton ncreases, the ercentage errors n Fg. 5 reman wthn to -8% for these nut erturbatons. Furthermore, note also that the ercentage errors are always negatve, mlyng that a more conservatve estmate of VaR s obtaned whle usng (49). Thus, t can be argued, that (48) and (49), resultng from the lnear aroxmaton n Fg. 1, are also able to comute the VaR and CL for outut ortfolos not havng normal dstrbutons. C. Range of Valdty of DT Aroach In order to understand the range of erturbatons for whch the senstvty ndces comuted usng the DT based method are vald, arameters are ndvdually erturbed n the range of +9% to -9% of ther and s comuted for each erturbed value. Senstvtes are then comuted usng

8 Error (%) Error (%) Error (%) ACCEPTED TO IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, SEPTEMBER E-5 2.E-5 1.E-5.E+ -1.E-5-2.E-5-3.E-5-4.E Parameter Perturbaton (%) F LbC LdC Fg. 6. Percentage error versus range of arameter erturbaton (F: FIT; LbC: Labour cost; LdC: Land cost) Parameter Perturbaton (%) TC O&M D T Fg. 7. Percentage error versus range of arameter erturbaton (TC: Transortaton cost; O&M: Oeraton & Mantenance cost; D: Dscount rate; T: Total Budget) Parameter Perturbaton (%) UC Fg. 8. Percentage error versus range of arameter erturbaton (UC: Unt cost; C: ). the FD aroach for the arameter. Fgures 6, 7 and 8 show the ercentage errors between the comuted DT and FD senstvtes. Observe that, for the arameters erturbed n Fg. 6, the error s close to zero; thus, one can conclude that the relaton between these arameters and the NPV s lnear n nature. In Fg. 7, the error les between -7% and 1%, thus demonstratng the nonlnearty of the solar PV nvestment model wth resect to these arameters. In Fg. 8, note that for the caacty factor of PV at, the error s C qute low for ostve erturbatons, but for negatve erturbatons below -3%, the error ncreases; ths s due to the fact that when the caacty factor of PV at s reduced below the PV caacty factor value at another zone, all the new PV nstallatons shft to that other zone; n ths case, the NPV s not affected by changes to the value of at. VII. Concluson Ths aer resented the alcaton of the dualty theory based method of comutng senstvty ndces for a solar PV lannng model from the ersectve of an nvestor. The senstvty ndces were drectly comuted from the solar PV otmzaton model and only nvolved the comutaton of a set of Jacoban matrces and some matrx oeratons. In the context of ths model, the senstvty ndces reresent the change n the net resent value (NPV) of nvestors roft when model arameters such as the Feed-n-Tarff, dscount rate, total budget, etc., are vared from ther resectve base values. The resented results demonstrate that the senstvty ndces obtaned usng the DT based method are very close to those obtaned usng the Monte Carlo aroach and the fnte dfference aroach, whch can be consdered the true values. Contrary to the Monte Carlo smulaton based aroach for determnng the senstvty of arameters, whch nvolves a large number of smulatons of the solar PV model n the order of thousands, ths aroach comutes the ndces drectly n one smulaton; thus, the comutatonal burden s sgnfcantly reduced. Moreover, whle the Monte Carlo smulaton aroach does not rovde any nformaton on the drecton of change of the NPV of roft when a arameter s erturbed, the DT based method yelds also the desred sgns of the senstvty values, whch s valuable nformaton for the nvestor. A novel nterretaton of the senstvty ndces s roosed thereafter, to evaluate the nvestors rsk arameters ertanng to the solar PV rojects selected by the model. Usng an aroxmaton of the cumulatve dstrbuton functon of nvestors roft, a lnear relaton s develoed between the senstvty ndces and nvestors roft for a certan confdence level. The roosed alcaton of the senstvty ndces to determne the rsk arameters rovdes valuable nformaton on nvestment rsks for an nvestor. References [1] Global Market Outlook for Photovoltacs untl 215, Euroean Photovoltac Industry Assocaton, EPIA Reort, May 211. [Onlne]. Avalable: [2] W. Ku, N. Nour, T. Pasck, A. Frester, A. Stranx, and M. Zons, Economc Evaluaton of Photovoltac Generaton Alcatons n a Large Electrc Utlty System, IEEE Trans. Power A. Syst., vol. PAS- 12, no. 8, , aug [3] J. Iannucc and D. Shugar, Structural Evoluton of Utlty Systems and ts Imlcatons for Photovoltac Alcatons, n Record of the 22nd IEEE Photovoltac Secalsts Conference, vol. 1, oct 1991, [4] G. Brnkman, P. Denholm, E. Drury, R. Margols, and M. Mowers, Toward a Solar-Powered Grd, IEEE Power Energy Mag., vol. 9, no. 3, , may-june 211.

9 ACCEPTED TO IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, SEPTEMBER [5] A. Mlls et al., Dark Shadows, IEEE Power Energy Mag., vol. 9, no. 3, , may-june 211. [6] J. McCalley, W. Jewell, T. Mount, D. Osborn, and J. Fleeman, A Wder Horzon, IEEE Power Energy Mag., vol. 9, no. 3, , may-june 211. [7] A. Ells, M. Behnke, and J. Keller, Model Makers, IEEE Power Energy Mag., vol. 9, no. 3, , may-june 211. [8] K. Katrae and J. Aguero, Solar PV Integraton Challenges, IEEE Power Energy Mag., vol. 9, no. 3, , may-june 211. [9] W. Muneer, K. Bhattacharya, and C. A. Cañzares, Large-Scale Solar PV Investment Models, Tools, and Analyss: The Ontaro Case, IEEE Trans. Power Syst., vol. 26, no. 4, , nov [1] A. J. Conejo, E. Castllo, R. Mnguez, and R. Garca-Bertrand, Decomoston Technques n Mathematcal Programmng: Engneerng and Scence Alcatons. Berln: Srnger-Verlag, 26. [11] E. Castllo, A. Conejo, C. Castllo, R. Mnguez, and D. Ortgosa, Perturbaton Aroach to Senstvty Analyss n Mathematcal Programmng, Journal of Otmzaton Theory and Alcatons, vol. 128, , 26. [12] F. Mlano, A. Conejo, and R. Zarate-Mnano, General senstvty formulas for maxmum loadng condtons n ower systems, Generaton, Transmsson Dstrbuton, IET, vol. 1, no. 3, , may 27. [13] A. Conejo, E. Castllo, R. Mnguez, and F. Mlano, Locatonal Margnal Prce Senstvtes, IEEE Trans. Power Syst., vol. 2, no. 4, , nov. 25. [14] A. Saltell, Global Senstvty Analyss: The Prmer. Wley, 28. [15] A. Saltell, K. Chan, and E. Scott, Senstvty Analyss. Wley, 2. [16] J. Sobeszczansk-Sobesk, J. F. Barthelemyy, and K. M. Rley, Senstvty of Otmal Solutons of Problem Parameters, AIAA Journal, vol. 2, , [17] M. Shahdehour, H. Yamn, and Z. L, Market Oeratons n Electrc Power Systems: Forecastng, Schedulng, and Rsk Management. IEEE, Wley-Interscence, 22. [18] E. Suhr, Aled Probablty for Engneers and Scentsts. McGraw- Hll, [19] CPLEX 12, Solver Manual. [Onlne]. Avalable: [2] R. E. Rosenthal, GAMS - A User s Gude. [Onlne]. Avalable: [21] MATLAB Documentaton, The Mathworks Inc., 211. [Onlne]. Avalable: Indrajt Das (S 9) receved the B.E. degree n Power Plant Engneerng from Jadavur Unversty, Kolkata, Inda, n 1995 and the MSc degree n Electrcal Power Engneerng from Unversty of Strathclyde, Glasgow, UK, n Currently he s ursung hs PhD degree n Electrcal Engneerng at the Unversty of Waterloo, Canada. Hs research nterests are n renewable energy ntegraton nto ower systems. Kankar Bhattacharya (M 95 SM 1) receved the Ph.D. degree n electrcal engneerng from the Indan Insttute of Technology, New Delh, Inda, n He was n the faculty of Indra Gandh Insttute of Develoment Research, Mumba, Inda, durng , and then the Deartment of Electrc Power Engneerng, Chalmers Unversty of Technology, Gothenburg, Sweden, durng He joned the E&CE Deartment of the Unversty of Waterloo, Waterloo, ON, Canada, n 23 where he s currently a full Professor. Hs research nterests are n ower system economcs and oeratonal asects. Claudo Cañzares (S 85 M 91 SM F 7) receved the Electrcal Engneer Dloma from the Escuela Poltécnca Naconal (EPN), Quto, Ecuador, n Arl 1984, the M.S. and Ph.D. degrees n electrcal engneerng from the Unversty of Wsconsn-Madson, n 1988 and 1991, resectvely. He had held varous academc and admnstratve ostons at the E&CE Deartment of the Unversty of Waterloo, Waterloo, ON, Canada, snce 1993, where he s currently a full Professor, the Hydro One Endowed Char, and the Assocate Drector of the Waterloo Insttute for Sustanable Energy (WISE). Hs research nterests are n modelng, smulaton, control, stablty, comutatonal and dsatch ssues n sustanable ower and energy systems n the context of comettve markets and Smart Grds. Wajd Muneer (S 1) receved the B.E. degree n electrcal engneerng from N.E.D. Unversty of Engneerng and Technology, Karach, Pakstan, n 27 and the MASc degree from the Unversty of Waterloo, Waterloo, Ontaro, Canada n 211. Currently he s a Research Assstant n the same deartment at Waterloo. Hs research nterests are n ower systems oeraton, economcs anlannng, demand-sde management, sustanable energy systems, smart grds, and hybrd vehcles.