THERE is currently significant interest in the use of electrical

Transcription

1 Risk-Consrained Bidding and Offering Sraegy for a Merchan Compressed Air Energy Sorage Plan Soroush Shafiee, Suden Member, IEEE, Hamidreza Zareipour, Senior Member, IEEE, Andrew M. Knigh, Senior Member, IEEE, Nima Amjady, Senior Member, IEEE, Behnam Mohammadi-Ivaloo, Member, IEEE 1 Absrac Elecriciy price forecass are imperfec. Therefore, a merchan energy sorage faciliy requires a bidding and offering sraegy for purchasing and selling he elecriciy o manage he risk associaed wih price forecas errors. This paper proposes an informaion gap decision heory (IGDT)-based risk-consrained bidding/offering sraegy for a merchan compressed air energy sorage (CAES) plan ha paricipaes in he day-ahead energy markes considering price forecasing errors. Price uncerainy is modeled using IGDT. The IGDT-based self-scheduling formulaion is hen used o consruc separae hourly bidding and offering curves. The heoreical approach o develop he proposed sraegy is presened and validaed using numerical simulaions. Index Terms Compressed air energy sorage (CAES), bidding sraegy, informaion gap decision heory, IGDT, uncerainy. A. Index B. Parameers NOMENCLATURE Index for operaion inervals running from 1 o T. π E Forecased day-ahead elecriciy marke price for inerval. π NG Naural gas price. HR d Required hea rae of CAES for discharging mode. HR s Required hea rae of CAES for simple cycle mode. V OM exp Variable operaion and mainenance cos of expander. V OM c Variable operaion and mainenance cos of compressor. Pmax exp maximum generaion capaciy of expander. Pmax c maximum compression capaciy of compressor. E min minimum level of air sorage. E max maximum level of air sorage. E in Iniial level of air sorage. ER Energy raio. R r Robus profi level. Opporunisic profi level. R op This research projec has been parially funded by an NSERC Canada Research grans. S. Shafiee, H. Zareipour, and AM. Knigh are wih he Deparmen of Elecrical and Compuer Engineering, Schulich School of Engineering, Universiy of Calgary, Calgary, AB, Canada T2N 1N4 ( sshafiee@ucalgary.ca; h.zareipour@ucalgary.ca; aknigh@ucalgary.ca). N. Amjady is wih he Deparmen of Elecrical Engineering, Semnan Universiy, Semnan, Iran ( amjady@semnan.ac.ir) B. Mohammadi-Ivaloo is wih he Faculy of Elecrical and Compuer Engineering, Universiy of Tabriz, Tabriz, Iran ( mohammadi@ieee.org). C. Variables π E P i,d Acual day-ahead elecriciy marke clearing price a inerval. Power generaion in discharging mode in operaion inerval. P i,s Power generaion in simple cycle mode in operaion inerval. P c Power consumpion in charging mode in operaion inerval. OC Operaion cos of he plan a ime. u x Uni saus indicaor in eiher modes x, i.e., discharging (d), simple cycle (s), or charging modes (c) (1 is ON and 0 is OFF). α x Horizon of uncerain variable in robus (r) or opporunisic (op) cases. I. INTRODUCTION THERE is currenly significan ineres in he use of elecrical energy sorage sysems [1]. For example, he oal insalled elecriciy sorage capaciy in he US is forecased o grow from 22 GW in 2014 o GW by 2050 [2]. Bulk energy sorage sysems can serve load-shifing and peak capaciy services. Furhermore, high elecriciy price volailiy in some markes provides a business opporuniy for energy arbirage by hese sorage echnologies [3]. Compressed air energy sorage (CAES) is one of he maure bulk energy sorage echnologies wih he capabiliy of soring large amoun of energy. In addiion o he Hunrof and McInosh CAES Plans, which have been operaing for decades, here are more CAES projecs eiher announced, under consrucion or operaing in recen years [4]. For insance, a 317 MW CAES faciliy wih 96 hours of sorage is announced in Texas, he USA, which is scheduled o be commissioned by summer 2019 [5]. CAES echnology has he abiliy o operae as a gas urbine in case he air reservoir is depleed [6]. This mode makes CAES echnology differen from oher ypes of energy sorage, since unlike oher energy sorage echnologies, CAES is able o follow is scheduling plan and ake advanage of price spike in case of empy sorage reservoir. Several sudies have been presened in he lieraure ha focus on opimal self-scheduling CAES faciliies and esimaing he energy arbirage revenue in differen elecriciy markes [7] [9]. The feasibiliy of improving he economics of he CAES echnology by disribuing compressors near hea loads are analyzed in [10]. In hese sudies, i is assumed ha accurae elecriciy marke price forecass are available. Then, he self-scheduling problem of he sorage plan is formulaed as a profi maximizaion one. However, i is eviden from he lieraure ha price forecasing errors could vary from

2 2 5% o 36% depending on he forecasing mehod and marke srucure [11]. I could significanly affec he economics of energy sorage. Hence, price uncerainy should be considered when developing operaion scheduling and bidding/offering sraegies for he CAES uni. Previous sudies have invesigaed he bidding sraegies of generaion companies considering price uncerainy [12] [17]. A risk-neural approach for self-scheduling is used in [12], [13], assuming perfec forecas is available. Sochasic programming is also applied o model marke price uncerainies in self-scheduling of wind and hermal GenCos [14], [15]. Robus opimizaion is employed o hedge he risks associaed wih price uncerainy in [16], [17]. Furhermore, informaion gap decision heory (IGDT) is applied in [18] [20] o manage he risk of price forecas flucuaions in self-scheduling and bidding sraegies of generaion unis. Compared o a generaion company or a large load, he sory is somewha differen for a merchan CAES faciliy. The sorage plan should decide when o purchase elecriciy from he marke and also when o sell i o he marke o maximize is profi considering he operaional cos as well as he cos of purchasing elecriciy. In his decision making problem, price uncerainy should be incorporaed o manage he associaed risks. Accordingly, he CAES faciliy needs no only an appropriae offering sraegy for selling he elecriciy, bu also a proper bidding sraegy for purchasing energy from he marke. Self-scheduling and bidding sraegy of differen energy sorage faciliies has been repored in he previous lieraure [21] [27]. In [21] [23], sochasic programming is applied o opimize he operaion of an energy sorage faciliy co-locaed wih a wind farm considering he uncerainies associaed wih he marke price and wind generaion. Sochasic programming is also used in [24], [25] o invesigae he opimal bidding sraegy for an independen baery sorage in elecriciy marke. Sochasic programming requires large compuaional burden as well as he necessiy of knowing he probabiliy disribuion funcion (PDF) of he uncerain parameers. Robus opimizaion is also applied for he bidding sraegy of a wind farm combined wih energy sorage in [26], [27]. Robus opimizaion does no incorporae opporunisic acions in risk-consrained scheduling in order o ake advanage of favorable price variaions. Moreover, in sudies focusing on he bidding sraegy of an energy sorage faciliy in elecriciy marke [21] [27], only single block hourly bids and offers are consruced. In he day-ahead elecriciy marke, here is he possibiliy of submiing muli-sep bids and offers. Thus, he sorage operaor should be able o submi muli-sep bids and offers o he marke for purchasing and selling he elecriciy considering differen level of risk o manage he risk of price uncerainy more effecively. Thus, consrucing muli-sep bidding and offering curves for an energy sorage faciliy is imporan. In his paper, an IGDT-based risk-consrained bidding/offering sraegy is proposed for a merchan CAES, which paricipaes in he day-ahead energy markes, considering price uncerainy based on IGDT. The IGDT mehod applies o decision making problems in an uncerain environmen. The mehod enables he decision maker o formulae opimisic and pessimisic self-scheduling problems wihou any assumpion on he probabiliy disribuion funcion of he uncerain parameer and wih low compuaional load. Insead of maximizing plan s profi based on some assumpions on uncerain price flucuaions, he proposed robus formulaion maximizes he horizon of price uncerainy around forecased value and finds a scheduling soluion ha guaranees a cerain pre-deermined revenue. Furher, he proposed opporunisic mehod opimizes he operaion schedule o benefi from he favorable price flucuaions. In oher words, for he opporunisic case, he IGDT mehod invesigaes he minimum favorable price variaion so ha a higher profi han ha of he expeced one could be achieved. The proposed bi-level IGDT based mehod is convered o is equivalen single level opimizaion for boh robus and opporunisic formulaion. Then, he proposed IGDT-based robus and opporunisic scheduling problems are applied o consruc hourly offering and bidding curves o submi o he marke for each hour, in order o ake differen levels of risk of price predicion ino accoun. This approach enables he faciliy operaor o no only ac conservaively in he marke by including pessimisic bids and offers, bu also ake advanages of favorable price spikes by considering opimisic bids and offers in he consruced bid and offer curves. The simple cycle mode of operaion for he CAES faciliy is also inegraed ino he proposed approach o illusrae is imporance when providing energy arbirage. I should be noed ha he proposed sraegy is for bidding and offering ino he day-ahead marke. Paricipaion in he real-ime marke is no considered in his sudy. As a conclusion, he conribuions of his paper can be saed as follows: Proposing a non-probabilisic risk-consrained operaion scheduling for a merchan CAES plan based on IGDT mehod. Proposing a process for consrucing hourly bidding and offering curves o hedge he risk associaed wih he price uncerainy in a day-ahead marke considering a combinaion of risk-averse and risk seeking sraegies. Convering he proposed bi-level IGDT-based opimizaion problem for he robus and opporunisic funcions separaely o heir equivalen single level formulaions. The background on he CAES echnology, a generic formulaion for he self-scheduling problem of a CAES and he IGDT mehod are presened in Secion II. In Secion III, he proposed IGDT-based robus and opporunisic selfscheduling opimizaion problems and heir equivalen single level formulaion are proposed. Secion IV explains he process o consruc separae hourly offering and bidding curves. Simulaion resuls are presened and discussed in Secion V. The paper is concluded in Secion VI. II. BACKGROUND A. CAES self-scheduling Formulaion A merchan CAES plan designed for energy arbirage purchases elecriciy during low price periods o power large compressors o compress air ino underground sal caverns. The sored air is laer used o power modified gas urbines (air expanders) during peak price hours. Unlike oher energy sorage echnology, he CAES echnology considered in his paper requires naural gas as he inpu fuel. The naural gas supply provides a fracion of he oupu power during discharge mode and also enables he faciliy o operae as a simple cycle gas generaor when he sored air is depleed.

3 3 Oher ypes of CAES ha do no require NG have been proposed [28] bu hese are no considered in his formulaion. The efficiency of a CAES faciliy is expressed based on is hea rae and energy raio. Hea rae expresses he amoun of fuel burned per uni of peak elecriciy generaed by he expander. Energy raio indicaes he amoun of energy ha he compressor of he plan consumes per uni of energy ha he expander generaes during he peak hours [9]. In his secion, he objecive funcion for he self-scheduling of a merchan CAES and he associaed consrains are described. The goal of he CAES plan is o maximize profi hrough energy arbirage as a paricipan in he elecriciy marke. The objecive funcion and consrains for he selfscheduling opimizaion are as follows. max T =1 Subjec o: OC = [P i,d [(P i,d P c ) π E OC ] (1) (HR d π NG + V OM exp )] (2) + [P i,s (HR s π NG + V OM exp + V OM c )] + [P c V OM c ] T u c + u d + u s 1 T (3) 0 P c Pmax.u c c T (4) 0 P i,d Pmax.u exp T (5) 0 P i,s Pmax.u exp T (6) E min E E max T (7) E +1 = E + P c P i,d ER T (8) E (0) = E in (9) The firs erm of objecive funcion (1) is he revenue from elecriciy sales o he marke from discharging he sored air or purely using gas, i.e., he simple-cycle gas mode and also he cos of purchasing he elecriciy from he marke. The second erm of objecive funcion represens he operaing cos of he plan. The operaing cos is expressed in hree erms in (2). These erms are respecively operaing cos of generaion in discharging mode, operaing cos of generaion in simple cycle mode, and he variable cos of compressor in charging mode. Noe ha in simple-cycle mode, fuel consumpion of CAES would increase from he opimal design poin (almos wice) [29]. A CAES plan is no likely o operae a hese high hea raes unless forced by he circumsances (e.g., here are high prices in he marke o ake advanage of). The operaional consrain is expressed in (3) i.e., he CAES can operae in only one specific mode a a ime. The charging and discharging power and energy limis of he CAES are specified by (4)-(7). The dynamic equaion for he sorage level is provided by (8). The iniial level for he air sorage cavern is specified by (9). B. Informaion-Gap Decision Theory The IGDT mehod is a non-probabilisic inerval opimizaion-based mehod ha formulaes robus and opporunisic formulaions under uncerainy [30], [31]. Since i makes no assumpion on he probabiliy disribuion of he uncerain variable, i makes i useful when high level of uncerainy exiss or no consisen probabiliy disribuion is available due o lack of informaion [30]. The mehod has been used for various decision-making problems under uncerainy in differen areas. Such applicaions include reserve neworks planning for biodiversiy conservaion [32], life cycle engineering design problems [33] and waer source planning [34]. In he area of power sysem, IGDT mehod has been applied o various decision making problems such as scheduling of elecric vehicle aggregaor [35], resoraion decision-making model for disribuion neworks [36], and also self-scheduling and bidding sraegy of hermal generaion companies [18] [20]. The reasons saed in hose sudies for choosing his mehod include severely deficien informaion, high level of uncerainy, no need for knowing he probabiliy disribuion funcion of uncerain parameers, significanly lower compuaional burden compared o probabilisic mehods, and simply managing financial risk wihou addiional compuaional cos. From he risk-aversion perspecive, he IGDT mehod maximizes he horizon of uncerainy and finds a soluion ha guaranees a cerain expecaion for he objecive. I is referred o as robusness funcion. In he conex of self-scheduling, assume a se of price forecass for he nex day is available. A pre-deermined level of profi is guaraneed by he IGDT-based self-scheduling soluion, if he observed marke prices fall ino a maximized price band cenered a he forecas prices. Furhermore, from he risk-seeking viewpoin, he IGDT mehod deermines a minimum favorable price variaion such ha a higher profi han expeced could be achieved. This is referred o as opporuniy funcion. The IGDT mehod consiss of hree componens, sysem model, uncerainy model and performance parameers [30], which are described in he following: 1) Sysem Model: Sysem model represens he srucure of inpu/oupu of he sysem. Consider P and Γ as decision variable and uncerain parameer, respecively. R(P, Γ) is he sysem model which is objecive o be maximized aking uncerainy ino accoun. In our problem, R(P, Γ) assesses he sorage profi wih respec o P as sorage charge/discharge schedule and Γ as day ahead price. 2) Uncerainy Model: Differen models for uncerainies are presens in [30]. The uncerainy model (Γ(α, π )) represens he informaion abou he uncerain parameer, which here is he day-ahead price. I basically shows he gap beween he known parameer π E and wha needs o be known π E. Envelope bound model wih π E as he surrounding funcion is used in his paper. I can be expressed as follows: Γ(α, π E ) = {π E : π E π E α π E } α 0, T (10) This model is a varian of he envelope-bound informaiongap model. In his mode, maximal variaion is proporional o he forecased value. In [30], α is he horizon of uncerain parameer. The larger α is, he wider possible variaion range for he uncerain parameer would be. The reason for selecing his uncerainy model is ha i represens he hourly relaive absolue error. In price forecasing lieraure, he mean of his error is calculaed over an operaion period (e.g. 24 hours) and commonly used as mean absolue percenage error (MAPE). Moreover, based on (10), since he range of uncerainy is deermined by α π E, wider error band is yield for high price hours, i.e., higher error in price forecas could happen for higher price hours. Noe ha, he funcion π E deermines

4 4 he shape of he envelope. One may use anoher funcion for he shape of he envelope, based on he naure of he price paerns and forecasing mehods o capure price volailiy. For insance, insead of using he price forecass π E for all he hours, he forecass can be scaled by a conrollable facor for expeced more price volaile hours, which reflecs he level of flucuaion of hose hours. 3) Performance Parameer: Based on he decision maker aiudes, wo differen performance funcions could be developed in an informaion-gap model i.e., robusness funcion corresponded o a risk-averse decision maker or opporuniy funcion corresponded o a risk-seeker decision maker. A robusness funcion ries o find he maximum permissible deviaion of price variaion so ha he minimum predeermined profi could be obained. This can be expressed as (11). α r (P, R r ) = max{α : min R(P, Γ) R r } (11) P π E Γ(αr, πe ) Equaion (11) can be inerpreed as deermining he charging/discharging schedule of he CAES by maximizing he possible range of unfavorable price variaion ensuring he predeermined profi. Thus, following he obained schedule could guaraneed he pre-defined profi if he uncerain price falls ino he maximized confidence inerval defined by α. An opporuniy funcion invesigaes he minimum favorable price variaion so ha a higher profi han ha of expeced could be achieved which formulaed as (12). β(p, R op ) = min P {α op : max π E Γ(αop, πe ) R(P, Γ) R op } (12) If he uncerain parameer (e.g., price in his paper) favorably deviaed from he forecased value by a leas β, he higher profi R op han wha expeced could be gained. C. Characerisics of he IGDT mehod A key feaure of he IGDT mehod is ha i can ie forecas error inervals o opimal scheduling. If he user has a forecasing mehod wih a reasonably consisen pas performance, he IGDT mehod enables he user o find schedules ha are opimal considering he performance of he forecasing sysem. Furhermore, in he IGDT mehod, no assumpion is made abou he probabiliy disribuion funcion of he uncerain parameers. This is imporan when he level of uncerainy is high and finding a probabiliy disribuion for he uncerain variable is challenging. Given he high volailiy of elecriciy prices and exisence of price spikes [11], deermining a price probabiliy disribuion over ime may no always be possible. In probabilisic mehods, on he oher hand, i is necessary o assume a probabiliy disribuion funcion for he uncerain parameers o be able o generae scenarios. Compared o probabilisic mehods where he compuaional burden could someimes be problemaic [37], [38], he IGDT mehod can handle high uncerainy levels wih lower compuaional burden. Furhermore, unlike probabilisic mehods where he risk is modeled using merics such as value a risk (Var) [39] and condiional value a risk (Cvar) [26] by adding some addiional consrains o he model, he risk is modeled in IGDT along wih uncerainy by seing a guaraneed level of profi wihou adding compuaional cos o he problem. Thus, probabilisic mehods give a probabilisic risk merics (e.g., Var or Cvar), whereas IGDT gives a confidence inerval and guaranees o achieve a predefined profi level if he uncerain parameer falls ino he maximized confidence inerval. Moreover, he IGDT mehod enables he decision maker o develop boh opimisic and pessimisic soluions ha guaranee a cerain value for he objecive. Hence, he IGDT mehod covers he decision making problems under uncerainy from risk-averse and also risk-seeking viewpoins. Robus Opimizaion and minmax mehods, on he oher hand, only look a wors case scenarios of he uncerain parameers. Furher, IGDT could be seen as more undersandable from a decision making poin of view. This is because in IGDT mehod, he user ses he level of expeced profi, whereas in Robus and mimmax opimizaion mehods, he user ses he uncerainy level; i is perhaps easier for a high-level decision maker o deal wih profi deerminaion decisions han uncerainy quanificaion decisions [19]. Thus, IGDT is a reasonable and suiable approach for deermining bidding and offering sraegies aking ino accoun price forecasing errors. III. THE PROPOSED METHODOLOGY AND FORMULATION In his secion, an IGDT-based self-scheduling is presened for a risk-averse as well as a risk-seeker operaor of a merchan CAES. In he developed approach, he decision variables are he amoun of power o be purchased or generaed a each inerval, and he uncerain parameer is he wholesale elecriciy price. I is assumed ha naural gas prices are known in advance. A. IGDT-Based Operaion Scheduling Formulaion for a CAES Plan For a risk-averse CAES plan operaor, he IGDT model objecive is o maximize an uncerainy parameer, referred o as α here onward, while a minimum pre-deermined profi is guaraneed. To develop he formulaion, we sar wih he risk-neural model of (1)-(9). Since price is he source of uncerainy, a price deviaion, say π E, will be added o he forecased price values, i.e., π E, as follows: P i,d max,p i,s,p c α r (13) subjec o: R R r = R 0 (1 σ) (14) (2) (9) R = min π E subjec o: T =1 [(P i,d P c ) ( π E + π E ) OC ] (15) α π E π E α π E T (16) The above opimizaion consiss of wo levels, i.e., he upper level maximizaion and he lower level minimizaion. The upper level of he proposed opimizaion is o find he maximum price deviaion ha would saisfy he pre-specified profi. The lower level deermines he wors case price deviaions. Observe ha R 0 is he risk-neural profi obained by solving he riskneural self-scheduling of (1)-(9). Pre-deermined profi R r is

5 5 he facor of R 0 defined by σ. The level of R r shows he level of risk ha he risk-averse operaor would be willing o ake. In oher words, i conrols he risk level of he operaor in he riskaverse case. Lower value for he R r means he decision maker is more conservaive and does no wan o ake much risk and prefer o guaranee a lower level of profi by considering more pessimisic price forecas. Changing he defined parameer σ changes he level of predefined profi, which implies changing he risk-aversion level. For a risk-seeker plan operaor, a favorable price deviaion could lead o a higher profi han wha is expeced. Thus, he IGDT model objecive is o find he minimum favorable price flucuaion, referred o as α op here, while hoping o earn a higher profi. P i,d min,p i,s,p c α op (17) subjec o: R R op = R 0 (1 + δ) (18) (2) (9) R = max π E subjec o: T =1 [(P i,d P c ) ( π E + π E ) OC ] (19) α π E π E α π E T (20) The upper level opimizaion is o find he minimum price deviaion ha could lead o he argeed pre-defined profi. The lower level explores he bes case of price deviaions. Predefined profi R op is he facor of R 0 defined by δ. The level of R op shows he level of risk he risk-seeking decision maker is willing o ake. The higher he level of R op is, he riskier he decision maker is and more opimisic price forecass he is expecing. The defined parameer δ is used o change he level of risk-seeking. B. The Equivalen Single-Level Opimizaion The bi-level opimizaion problems described earlier are convered o heir single-level equivalens in order o be solved wih convenional solvers. In he following, he proposed approach is presened for robus and opporunisic cases. 1) Robusness funcion: As menioned before, hrough (15)- (16), he objecive is o find he wors case scenario of price flucuaion ha saisfies a minimum profi. Thus, in he lower level, he decision variable is π E and he objecive is o minimize he uiliy s profi. Since he variables P i,d, P i,s, and P c are decision variables a upper level opimizaion, hey can be considered as consan parameer a lower level [19]. Hence, he objecive (15) is a linear opimizaion and consequenly, he minimum profi of he objecive (15), i.e., he wors case scenario, occurs in one of he bounds of he permied variaion horizon. Mahemaically: { π E α r π E = α r π E if P i,d if P i,d P c 0 P c, T (21) 0 In plain language, (21) saes ha he wors case for charging is when he price is higher han he forecass; he wors case for discharging is when he price declines from he forecased value. The wo erms of (21) can be expressed as follows: (P i,s (P i,s + P i,g P c )( π E + α π E ) 0 T (22) + P i,g P c )( π E α π E ) 0 T (23) When he sorage is in charging mode, he firs erm of (23) is negaive. In addiion, since α π E π E α π E, he second erm of (23) is equal or less han zero. Thus, in order o saisfy (23), he second erm mus be zero, which leads o π E = α π E. In his case, (22) is neural. Similarly, i can be proved ha during discharging or simple cycle mode, (22) forces π E = α π E. Using (22) and (23), he single level opimizaion for he robusness funcion can be derived as follow: P i,d max,p i,s,p c α r (24) subjec o: R R r = R 0 (1 σ) (25) T R = [(P i,d P c ) ( π E + π E ) OC ] (26) =1 (2) (9), (22) (23) 2) Opporuniy funcion: hrough (19)-(20), he objecive is o find he bes scenario of price flucuaion ha could lead o a higher profi han expeced. In he following, he approach o conver he bi-level opimizaion of (17)-(20) o a single-level opimizaion is proposed. Similar o wha described for he robus case, he objecive funcion (19) is linear wih respec o he only decision variable π E and hus, he objecive happens in one of he bounds of he uncerainy horizon. However, in conras wih he previous case, since we are looking for he maximum profi, i.e., he bes case scenario, i occurs in he opposie price bound compared o ha of he robusness funcion. Mahemaically: { π E α op π E = α op π E if P i,d if P i,d P c 0 P c, T (27) 0 (27) shows ha he bes case for charging is when he price is lower han he forecass. In conras, he bes case for discharging is when he price is higher han he forecass. The wo erms of (27) can be express as follows: (P i,s (P i,s + P i,g P c )( π E + α π E ) 0 T (28) + P i,g P c )( π E α π E ) 0 T (29) When charging, (28) is bounded which leads o π E = α π E. Similarly, when generaing elecriciy, (29) is bounded, i.e., π E = α π E. Accordingly, he single level opimizaion for he risk-seeker case can be formulaed as he following: P i,d min,p i,s,p c α op (30) subjec o: R R op = R 0 (1 + δ) (31) T R = [(P i,d P c ) ( π E + π E ) OC ] (32) =1

6 6 Fig. 1: The process of consrucing a 4-sep bidding curve Fig. 2: The process of consrucing a 4-sep offering curve (2) (9), (28) (29) The opimizaion problem of boh robusness and opporuniy funcions are mixed ineger non-linear programming (MINLP) which can be solved using commercial MINLP solvers, such as SBB [40]. For all he case sudies in his work, he SBB solver was able o find he soluion in less han a few seconds using he GAMS plaform [41]. I should be menioned he nonlinear IGDT-based proposed model wih muliplicaion nonlineariy, known as bi-linear in he lieraure, can be linearized using reformulaion-linearizaion echnique [42] or using he linear cuing plane algorihms [43] wih he cos of some over simplificaions. However, he focus of his paper is no linearizing he IGDT-based model. IV. THE PROPOSED METHOD FOR BIDDING AND OFFERING STRATEGY In order o sell and purchase energy, he sorage plan needs o submi is hourly offers and bids o he marke. Thus, an appropriae bidding and offering sraegy is required. The proposed IGDT-based robus and opporunisic scheduling formulaions are applied o build up he sraegy, i.e., consruc offering and bidding curves. The mehod for consrucing he offering curve is in line wih he approach used in [19]. However, he approach in [19] is exended o incorporae opporunisic acions described in Secion III. Moreover, an algorihm for consrucing he bidding curves for purchasing elecriciy is also presened. The approach no only guaranees a minimum level of profi, bu also akes advanage of desirable price flucuaions. In he following, he process of simulaneously building offering and bidding curves is presened. The CAES operaor ends o submi descending bidding for purchasing elecriciy and ascending offering curves for selling elecriciy. The lower he price is, he more power he operaor is willing o purchase. The operaor is also willing o sell more power for higher prices. Boh robus and opporunisic acions are aken ino accoun when forming he bidding and offering curves. As an example, Figs. 1 and 2 illusrae he process of consrucing a 4-sep bidding and a 4-sep offering curve. To do so, we selec four levels of profi below and above he expeced profi (R ex ), i.e., R r1 R r2 R ex R op1 R op2. Then, he proposed IGDT-based scheduling is applied sequenially. For each level of profi, i deermines he opimum confidence level, he corresponding price profile, and he corresponding hourly charging and discharging quaniy. Using he obained resuls, he hourly bidding and offering curves are consruced. In he following, he procedure for consrucing each sep of he bidding curve for a specific hour of charging period is described. 1) Sep 1: According o he resuls of he firs profi level, for each hour of charging period, he obained price level is considered as he bid price. This level of price is acually he upper bound of he confidence inerval, i.e., he wors case. The corresponding charging quaniy, i.e., P c,1, is also considered as he charging power. If he marke price is lower han he bid price, he CAES would purchase he submied amoun of elecriciy. 2) Sep 2: Afer defining he firs sep, he resuls of he second profi level is employed. Higher level of profi compared o he firs level leads o a lower value of α r2 α r1. The marke price needs o be lower o jusify purchasing more power. The difference beween he charging quaniy of his sep and he previous one, i.e., P c,2 P c,1 is considered as he bidding quaniy. The corresponding prices is chosen as he bid price. This level of price is acually higher han he forecased value by erm of α r2. 3) Sep 3: For he hird level of profi, he proposed IGDTbased scheduling deermines he minimum favorable price deviaion, i.e., α op1, he corresponding scheduling and price profile. Thereafer, for charging hour, he defined price level is he bid price. I is he lower bound of he confidence inerval, i.e., he bes case as deermined by he proposed mehod. Furhermore, he difference beween he quaniy of his sep and he previous one, P c,3 P c,2, is considered as he bid quaniy for his sep of hourly curves. The nex block of hourly bidding curves is deermined in a similar way. Similar process is employed o consruc he hourly offering curves. The difference is ha, during discharging hours, for he robus cases, i.e., R r1 and R r2, he IGDT-based opimizaion deermines he price level as he lower bound of he confidence inerval, which is he wors case scenario. For he opporunisic par, i.e., R op1 and R op2, he price level is he higher bound of confidence inerval, i.e., he bes case scenario of discharging. A. Sequenial Consrains As depiced in Figs. 1 and 2, since he quaniy submied o he marke for eiher purchasing or selling a each sep mus be greaer or equal han ha of submied in he previous sep, consrains (33)-(34) are added o boh robus as well as opporunisic problems. P c (R 1 ) P c (R 2 ), T, R 1 R 2 (33) [P i,d ](R 1 ) [P i,d ](R 2 ), T, R 1 R 2 (34)

7 7 Fig. 4: Forecased price, he wors case of price for robus case and he bes case of price for opporunisic case Fig. 3: The sequences of defining seps of he bid and offer curves Saring from he lowes profi level, consrains (33)-(34) are updaed sequenially for higher level of profi. Fig. 3 shows he flowchar summarizing he process of sequenially consrucing he seps of bid and offer curves. I should be menioned ha wih he proposed bidding/offering sraegy, he sorage plan migh face an infeasible schedule in he case when charging bids are acceped while sorage reservoir is full. In such case, he sorage faciliy would be in a siuaion where i is commied o buy for he marke bu has no room o sore he energy. Such siuaions could happen for oher markes paricipans oo. For example, a wind producer paricipaing in he marke may forecas a level of available power ha is less han wha is acually produced. Or a reailer may bid in day-ahead marke o buy power bu he real-ime demand urns ou o be less han expeced. In such cases, he marke paricipans need o adjus heir schedules in he real-ime marke buy selling or buying he exra/defici energy. V. NUMERICAL EXAMPLE Numerical simulaions are performed for a CAES faciliy wih 150 MW of discharging power, 100 MW of charging power, and 20 hours of full discharging capabiliy as he sorage capaciy. Hea rae, energy raio, and VOM of expander and compressor are aken from [10]. The required hea rae for he simple cycle mode is assumed o be wice as ha of discharging mode [10]. The price of gas is assumed o be 3.5 $/GJ. A 24-hour scheduling period is considered and he iniial sorage level is se o zero. Fig. 4 depics he forecas prices for a ypical 24 hours period. A. Risk-consrained Self-scheduling: A Demonsraive Case This secion demonsraes he applicaion of he proposed robus and opporunisic self-scheduling approach proposed in Fig. 5: Scheduling of sorage for risk-neural, robus and opporunisic cases secion III for one ypical day. A firs, risk-neural case is considered as he reference, in which he forecased price is applied o schedule he plan, i.e., deerminisic scheduling. Then, he CAES self-scheduling problem is solved from boh risk-averse and risk-seeking perspecives by applying he robus and opporunisic IGDT-based scheduling mehods, respecively. Solving he risk-neural self-scheduling leads o $31, 905 profi, which is considered as he expeced profi, i.e., R 0. Fig. 5 shows he corresponding risk-neural scheduling. In his case, he robusness and opporuniy parameer are se o 0.25, i.e., σ = 0.25 and δ = In oher words, 25% of risk aversion and 25% of risk-seeking is chosen as he risk levels for he robus and opporunisic cases, respecively. The robus case finds he maximum unfavourable deviaion in price ha limis he reducion in operaing profi o wihin 25% of he he risk neural case. The opporunisic case finds he minimum favourable deviaion in price required o reurn a 25% increase in profi. Fig. 4 depics he price forecas, he wors case and he bes case of price flucuaion for hose wo pre-deermined robus and opporunisic profi level. Fig. 5 also illusraes he corresponding hourly scheduling plans of he CAES plan for wo robus and opporunisic cases. In he following, he resuls of each case are discussed in more deails. 1) Robus scheduling: The robus opimizaion is solved wih he predeermined profi level R r1 = R 0 (1 0.25)=$23, Fig. 4 shows he wors case of price deviaions ha guaranees R r1. I indicaes ha R r1 could be achieved if he unfavourable hourly price deviaion is less han α r = 8.9% for charging and discharging hours. In anoher word, if he hourly cleared prices during charging (discharging) hours deviae unfavorably no more han 8.9% from forecased value, i.e., go above (below) he forecass no more han 8.9%, he predefined level of profi could be achieved. Fig. 5 shows he corresponding robus scheduling. The comparison of hese wo figures show ha during charging

8 8 hours, i.e., 1-7, and 9, higher price han he forecas is considered as he wors case. Conversely, during he generaion hours, i.e., 12-14, and 21-24, lower price han he forecass is considered as he wors case scenario. Since, from a risk-averse decision maker s view poin, he wors case of price deviaion is considered as a higher and lower price han forecass for charging and discharging periods, respecively. The resuls in he figure verify ha he proposed mehod is able o effecively find he wors case of price deviaion and hen deermine he opimal scheduling ha could guaranee he predeermined profi level. Figure 5 shows ha for eiher cases, he scheduling plan follows he price paern. In oher words, i purchases elecriciy o sore he compressed air during off peak when he prices are low and in conras, sells he energy during peak hours which coincides high price periods. However, comparing he riskaverse and risk-neural scheduling indicaes ha aking lower level of risks, i.e., σ = 0.25, would lead o lower hours of charging and discharging or lower charging and discharging power values compared o ha of he risk-neural case, and consequenly gaining lower profis. In conras, in he riskneural case, he sorage plan is commied o charge and discharge for more hours han ha of σ = 0.25; since in order o earn a higher profi, higher level of risk should be aken. Thus, as depiced in Figs. 4 and 5, he operaor would no consider any unfavourable price deviaions and consequenly decides o charge more during off peak hours in order o generae more power during peak period. 2) Opporunisic scheduling: In his case, δ is considered o be 25% as he level of risk-seeking, which means he plan is looking o make 25% more profi by aking risk and looking for some favourable deviaions in he acual prices wih respec o he price forecass. Fig. 4 and Fig. 5 show he bes cases of price deviaion and he corresponding opporunisic scheduling, respecively.the resuls show ha he favourable price deviaion o gain addiional 25% profi should be a leas 7.95% for charging and discharging hours. In anoher word, if he hourly cleared prices during charging (discharging) hours flucuae favorably a leas by 7.95% from forecased value, i.e., go below (above) he forecass a leas by 7.95%, he predefined level of profi could be gained. Moreover, he comparison of hese wo figures shows ha during charging hours, lower prices, and during discharging hours, higher prices han he forecass are of ineres. I can also be observed from Fig. 5 ha in he opporunisic scheduling, he charging or discharging power of he sorage uni is higher han ha of risk-neural or risk-averse. Since, in order o gain higher profi, he schedule should be opimisic o price deviaions. Thus, as a general observaion, increasing he profi level would require an increase in he charging and discharging power of he uni compared o he risk-neural one (e.g., a hour 10 for charging, and a hour 16 for discharging) or a leas, no change (e.g., a hours 1-9 for charging, and hours 12-14, and for discharging). B. Consrucing Biding/Offering Curves Based on he Obained Resuls from IGDT-based Scheduling Cases The resuls obained in Secion V-A are employed o consruc he seps of he bidding and offering curves. Wihou loss of generaliy, hree sep curves are seleced for his sudy. To Fig. 6: Bid curve for hour 7. Fig. 7: Offer curve for hour 15. consruc he curves, a firs, hree levels of profi are seleced. For his case sudy, one robus level i.e., σ = 0.25, one riskneural level, and one opporunisic level i.e., δ = 0.25 are chosen. Then, he proposed IGDT-based scheduling approach is applied for each of he profi levels, which are presened and discussed in he previous subsecion hrough Figs Thereafer, based on he obained resuls, he seps of he bidding and offering curves, i.e., he amoun of power and he corresponding price, are deermined. The algorihm proposed in secion IV is employed o consruc he bidding and offering curves. According o Fig. 4, charging periods is beween hours 1 o 10, i.e., he off peak hours. As an example, Fig. 6 shows he consruced bid curve for hour 7. Based on Fig. 5, in his hour, he sorage uni is commied o compress he air wih 87.5 MW of power corresponded o he firs profi level. Thus, his amoun of power is submied for purchasing wih he corresponding price of he firs profi level i.e., $32.6/MWh; since in his hour, for second profi levels, he uni charges wih 100 MW, he difference i.e., = 12.5 MW, is submied wih he price deermined for he second profi level, which can be found in Fig. 4, i.e., $29.9/MWh. The hird sep of he curve is zero, since, based on Fig. 5, for he opporunisic case, he uni also charges wih 100 MW in his hour. Thus, he opporunisic sep of Fig. 6 is aligned wih risk neural sep. Similarly, for offering curves, according o Fig. 4, discharging periods is beween hours 12 o 24. For insance, Fig. 7 illusraes he consruced offer curve for hour 15. In his hour, and 16.5 MW are submied o he marke for selling wih he prices deermined for he second and hird profi levels, respecively. This is because 0, and 150 MW of discharging is considered for he menioned hour, as can be seen in Fig. 5, for he firs, second, and hird profi levels, respecively. Noe ha, he offer curve has wo sep insead of hree sep, since as shown in Fig. 5, in robus case, he uni does no discharge. Hence, in Fig. 7, he robus sep has zero value. Similarly, he bidding/offering seps for oher hours are creaed.

9 9 C. Afer-he-Fac Analysis Based on Consruced Bidding and Offering Curves and Simulaed Prices In his secion, differen simulaed marke prices are generaed. The simulaed prices are disribued randomly around he forecased values wih uniform densiy funcion by differen range of variaion. Then, he submied offers and bids, consruced in Secion V-B, are applied o invesigae how he bidding and offering sraegy, proposed in Secion IV, would work. The effecs of price flucuaion on he operaion of he CAES uni and he gained profi are also explored. As a reminder, he guaraneed level of profi is $23,929.0 considering he risk level σ = 0.25 for he risk averse case. The would-be profi of he merchan CAES faciliy is calculaed assuming ha he faciliy paricipae sraegically and he simulaed prices are observed in he marke. In each scenario, based on he bid and offer curves consruced in secion V-B and he simulaed prices, he acceped bids and offers are deermined. The gained profis are calculaed accordingly, which are presened in able I. According o his able, as he price forecasing e, For he sake of comparison, anoher case is also considered, in which he CAES faciliy operaes as a self-scheduling plan considering forecased price for is scheduling. A selfscheduling generaor submis is schedule (hourly generaion quaniy) o he marke and hen, follows i in real ime regardless of marke price [44]. The marke operaor does no send his kind of generaors dispach insrucions [44]. In our case, i is assumed ha he self-scheduling of CAES faciliy is defined in a deerminisic way based on he forecased price depiced in Fig. 4. In his case, he expeced profi based on he forecas prices is $31, The resuled deerminisic selfscheduling is he same as risk-neural self-scheduling shown in Fig. 5. Hence, he CAES uni operaes based on he resuled scheduling in his figure, regardless of marke price. Then, based on he charge/discharge scheduling and he simulaed marke price in each scenario, he would-be gained profi in ha scenario is calculaed, which are shown in Table I. Observe from he able ha he gained profi wih deerminisic self-scheduling is lower han ha of IGDT-based sraegy in he hree scenarios. This is because of aking he risk of forecasing error ino accoun in he developed sraegy. Thus, in one hand, i makes he sraegy robus agains undesirable forecasing errors and prevens unprofiable acions. On he oher hand, considering opimisic acions in he proposed sraegy enables he plan o ake advanages of desirable forecasing errors. According o Table I, as he forecasing error ges higher, he level of gained profi decreases. However, comparing he obained profi wih he guaraneed profi shows ha he gained profis in scenario 1 and 2 are higher han he guaraneed one. I shows he robusness of he proposed bidding and offering sraegy. In scenario 1, price forecasing error, 5%, is less han he maximum allowable price deviaion of firs profi level, 8.9%. As a resul, he gained profi is higher han he corresponding guaraneed level of profi, presened in Table I. For he second scenario, he variaion slighly exceeds he allowable inerval. However, due o considering opporunisic acions, he gained profi is higher han he guaraneed level. For he hird scenario, due o high forecasing error, he guaraneed profi level is no achieved. However, he TABLE I: AFTER-THE-FACT ANALYSIS USING SIMULATED PRICES Price($/MWh) Scenario Range of Sraegic Self-scheduling price variaion gained profi gained profi 1 ±5% $26,600 $25,926 2 ±10% $26064 $25,673 3 ±20% $22,860 $ 22, Acual Price Forecased Price Time (Hour) Fig. 8: Hourly forecased and acual prices performance of he developed bidding and offering sraegy is beer han he deerminisic self-scheduling. D. Afer-he-Fac Analysis Using Acual Marke Daa 1) Three-Day Analysis: In his secion, he same analysis as he previous secion is applied sequenially using he acual and forecased price of hree-day period in he New England elecriciy marke. Fig. 8 depics he acual and generaed forecased prices for he hree days. In his analysis, he arbirary values of σ = 0.25, σ = 0, i.e., risk-neural, and δ = 0.25 are used as he risk levels o consruc hourly offering and bidding curves. Thus, he profi level 25% lower han he deerminisic risk-neural profi is seleced as he guaraneed profi, i.e., R r = R 0 (1 0.25). In his way robus acions are aken ino accoun in he sraegy. The wo ohers profi levels are also chosen o incorporae more opimisic acions in he bidding and offering sraegy. The developed sraegy is applied for each day based on he price forecass. Then, according o he acual marke price, he acceped offers and bids are deermined. Accordingly, oher oucomes, such as he daily profi values, he sae of charge, are calculaed. The process is repeaed sequenially for all days of he week. In order o invesigae he performance of he proposed sraegy, Table II repors he resuls of IGDT-based scheduling for he hree days. For day 1, according o Fig. 8, he price is mosly overesimaed. Thus, all he bids for purchasing he elecriciy are acceped, as presened in Table II. On he oher hand, none of he 10 supply offers are cleared due o prices lower han he forecas during mos of he peak hours, as shown in Fig. 8. Thus, as provided in Table II, he uni does no make any profi and only purchases energy, a a oal cos of $29,805. Through his process i charges he air reservoir o he level of 1,062.0 TABLE II: AFTER-THE-FACT ANALYSIS USING ACTUAL MARKET PRICES Day Guaran. Profi [$] α r (%) No. Acc. Bids No. Acc. Offer SC Hr. Gained Profi [$] FSOC [MWh] /10 0/ /5 9/ /6 9/ Guaran.: Guaraneed, Acc.: Acceped, SC: Simple Cycle, FSOC: Final SOC

10 10 MWh a low prices. Consequenly in he second day, he proposed sraegy leads o significanly high profi. As shown in Fig. 8, due o lower prices han forecas during charging periods and high price spikes during discharging periods, 4 ou of 5 demand bids and 9 ou of 14 supply offers are acceped. As a resul of acceped bids and offers, he sorage faciliy no only makes high level of profi, $125,130, bu also reains 450 MWh of compressed air a he end of he day. During he hird day, he marke prices are mainly underesimaed, as seen in Fig. 8. Thus, as presened in Table II, none of he charging bids are acceped. However, during nine ou of he en hours which discharging offers are submied, marke prices are higher han forecas, hus nine supply offers are acceped. The acceped offers lead o $32807 profi which exceeds he guaraneed level of profi. Noe ha since he iniial sorage level is 450 MW and none of he charging bids are acceped in his day, afer four hours of discharging, he sorage reservoir is depleed. Hence he uni swiches o pure gas mode for he res of he discharging hours, in spie of is inefficiency, in order o follow he schedule. 2) Four-week Analysis: In his secion, in order o show he performance of he developed bidding and offering sraegy, he sraegy is applied daily o a period of four arbirary weeks wih he same risk levels as hose of he previous secion. Then, he four-week simulaion is repeaed for several imes wih differen generaed daily price profiles; in each run, he daily acual prices are disribued randomly around he forecased values wih uniform densiy funcion by differen range of variaion which is chosen randomly beween %5 o %15. The cumulaive guaraneed profi level for his period is $700, 061 ± 23, 503 whereas he oal profi achieved using he proposed sraegy is $816, 243 ± 35, 979. For he sake of comparison, profi obained by deerminisic self-scheduling is also invesigaed and is $45,379±22, 164. As can be seen, he gained profi by he proposed sraegy is significanly higher han ha of deerminisic scheduling. This comparison demonsraes beer performance of he proposed sraegy compared o he deerminisic scheduling when high level of price uncerainy exiss. Furhermore, he resuls show ha overall, he proposed sraegy is capable of guaraneeing a minimum level of profi by aking pessimisic acions ino accoun and prevening uneconomical charging or discharging acions. By incorporaing opimisic acions, he decision maker is also able o benefi from unforeseen price drops or spikes. These facors combine o provide a higher profi han ha obained by deerminisic self-scheduling. VI. CONCLUSION This paper develops an IGDT-based risk-consrained bidding/offering sraegy for a merchan CAES faciliy aking price uncerainy ino accoun. Robus acions in he proposed bidding/offering sraegy guaranees a minimum criical profi if he fuure prices fall wihin a maximized robusness region. The opporunisic acions enable he plan o benefi from favorable price deviaion and poenially earn a higher profi. The numerical resuls verify he applicabiliy of he developed sraegic scheduling approach. In he robus scheduling case, he IGDT-based opimizaion mehod is able o find he wors case of price deviaion and hen, deermine he corresponding opimal scheduling which guaranees he predefined profi. For he opporunisic case, he desirable price variaion and he corresponding opimal scheduling are effecively defined for any level of profi by he proposed opporunisic formulaion. The obained resuls from robus, risk-neural, and opporunisic cases are employed o consruc he hourly offering and bidding curves. The demonsraion clearly shows ha in differen price scenarios, he proposed sraegic scheduling leads o a more profiable scheduling han ha of deerminisic one. In oher words, he sraegy avoids uneconomic acions. A he same ime, i benefis from desirable price deviaion and hus, higher profis are achieved. I should be noed ha, alhough consan efficiency parameers are considered in he CAES self-scheduling model, he efficiency of he CAES sysem somehow depends on is operaion condiion. Considering his issue is no in he scope of his paper, and is lef o fuure work. ACKNOWLEDGMENT The auhors would like o hank he NRGSream company for providing us wih he sofware by which we could exrac he daa used in our sudy. REFERENCES [1] P. Denholm, E. Ela, B. Kirby, and M. Milligan, The role of energy sorage wih renewable elecriciy generaion, Naional Renewable Energy Laboraory, NREL/TP-6A , [2] Technology roadmap energy sorage, Inernaional Energy Agency, [3] R. Sioshansi, P. Denholm, T. Jenkin, and J. Weiss, Esimaing he value of elecriciy sorage in pjm: Arbirage and some welfare effecs, Energy Economics, vol. 31, no. 2, pp , [4] Office of Elecriciy Delivery & Energy Reliabiliy, Global Energy Sorage Daabase, Deparmen of Energy. [Online]. Available: hp:// [5] Apex Behel Energy Cener, Apex.CAES. [Online]. Available: hp://hp:// [6] E. Ferig and J. Ap, Economics of compressed air energy sorage o inegrae wind power: A case sudy in ERCOT, Energy Policy, vol. 39, no. 5, pp , [7] R. Sioshansi, P. Denholm, and T. Jenkin, A comparaive analysis of he value of pure and hybrid elecriciy sorage, Energy Economics, vol. 33, no. 1, pp , [8] R. Walawalkar, J. Ap, and R. Mancini, Economics of elecric energy sorage for energy arbirage and regulaion in new york, Energy Policy, vol. 35, no. 4, pp , [9] E. Drury, P. Denholm, and R. Sioshansi, The value of compressed air energy sorage in energy and reserve markes, Energy, vol. 36, no. 8, pp , 2011, {PRES} [10] H. Safaei and D. W. Keih, Compressed air energy sorage wih wase hea expor: An albera case sudy, Energy Conversion and Managemen, vol. 78, no. 0, pp , [11] H. Zareipour, A. Janjani, H. Leung, A. Moamedi, and A. Schellenberg, Classificaion of fuure elecriciy marke prices, IEEE Trans. Power Sys., vol. 26, no. 1, pp , Feb [12] J. Arroyo and A. Conejo, Opimal response of a hermal uni o an elecriciy spo marke, IEEE Trans. Power Sys., vol. 15, no. 3, pp , Aug [13] H. Yamin and S. Shahidehpour, Self-scheduling and energy bidding in compeiive elecriciy markes, Elec. Pwr Sys. Res., vol. 71, no. 3, pp , [14] A. Al-Awami and M. El-Sharkawi, Coordinaed rading of wind and hermal energy, IEEE Trans. Susainable Energy, vol. 2, no. 3, pp , July [15] T. Dai and W. Qiao, Trading wind power in a compeiive elecriciy marke using sochasic programing and game heory, IEEE Trans. Susainable Energy, vol. 4, no. 3, pp , July [16] R. Jabr, Robus self-scheduling under price uncerainy using condiional value-a-risk, IEEE Trans. Power Sys., vol. 20, no. 4, pp , Nov [17] A. Soroudi, Robus opimizaion based self scheduling of hydro-hermal genco in smar grids, Energy, vol. 61, no. 0, pp , 2013.