ARTICLE IN PRESS. Energy Policy

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1 Energy Policy 38 (2010) Contents lists available at ScienceDirect Energy Policy journal omepage: Electricity procurement for large consumers based on Information Gap Decision Teory Kazem Zare, Mosen Parsa Mogaddam, Moammad Kazem Seik El Eslami Tarbiat Modares University, P.O. Box , Teran, Iran article info Article istory: Received 2 May 2009 Accepted 11 September 2009 Available online 30 October 2009 Keywords: Energy procurement Information Gap Decision Teory (IGDT) Large consumer abstract In te competitive electricity market, consumers seek strategies to meet teir electricity needs at minimum cost and risk. Tis paper provides a tecniue based on Information Gap Decision Teory (IGDT) to assess different procurement strategies for large consumers. Supply sources include bilateral contracts, a limited self-generating facility, and te pool. It is considered tat te pool price is uncertain and its volatility around te estimated value is modeled using an IGDT model. Te proposed metod does not minimize te procurement cost but assesses te risk aversion or risk-taking nature of some procurement strategies wit regard to te minimum cost. Using tis metod, te robustness of experiencing costs iger tan te expected one is optimized and te related strategy is determined. Te proposed metod deals wit optimizing te opportunities to take advantage of low procurement costs or low pool prices. A case study is used to illustrate te proposed tecniue. & 2009 Elsevier Ltd. All rigts reserved. 1. Introduction One of te main reasons for a large consumer to take part in an electricity market is to meet its electricity demand at minimum cost, buying from different trading floors via bilateral contracts and even utilizing its own generating facilities. Because of te uncertainty in pool prices, te main problem facing any consumer is ow to procure electric energy from different sources wit te lowest cost and risk (Yan and Yan, 2000). Because te consumer cannot know te bidding strategies of te rest of te electricity market agents and because te cost of te energy is uncertain and volatile, uncertainty modeling is necessary for tis optimization problem. Te consumer needs appropriate data to make informed decisions in electricity markets: Demand data for all ours of te time orizon in uestion. Pool price forecast for all sceduling ours. Te price and amount of energy associated wit all contracts. Te economic and tecnological specifications of te generating facility. In te process of decision-making regarding energy sources, lack of information brings some callenges for a large consumer. In tis paper, it is assumed tat te consumer as sufficient Corresponding autor. Tel./fax: addresses: k_zare@modares.ac.ir (K. Zare), parsa@modares.ac.ir (M.P. Mogaddam), aleslam@ut.ac.ir (M.K. Seik El Eslami). information about its energy demand, te structure of bilateral contracts, and te specification of its own generating facility (Conejo and Carrión, 2006; Carrión et al., 2007a). In addition, te pool price is considered uncertain and tat te consumer as an estimated value for te pool price. Electricity procurement decisions are related to variations in pool price. Tis paper addresses te electricity procurement problem faced by a large consumer and analyzes te appropriateness of various procurement strategies against market-price volatility. We propose ere tat te consumer implement a procurement strategy wit ig immunity against ig pool prices and low immunity against low pool prices. For tis purpose, te proposed metod evaluates different strategies wit different cost levels for procurement of electricity, and te results elp te large consumer coose an immune strategy. Tere is considerable literature addressing consumer participation in markets, specifically teir procurement of electrical energy. An analysis of te tools tat consumers and retailers need to participate in electricity markets is discussed in Kirscen Daniel (2003). Te optimal response of a consumer to pool prices is caracterized in Daryanian et al. (1998) in terms of load elasticity. Demand-side bidding and purcase allocation in two markets are discussed in Yan and Yan (2000) and Lin and Guan (2003), were price volatility is modeled using a stocastic process. Te medium term risk-constrained profit maximization problem faced by a retailer is analyzed in Gabriel et al. (2002), were te uncertainty of load and price are modeled by probability distributions. Available options for industrial customers in a competitive electricity market are discussed in Talati and Bednarz (1998). Electricity procurement by large consumers /$ - see front matter & 2009 Elsevier Ltd. All rigts reserved. doi: /j.enpol

2 K. Zare et al. / Energy Policy 38 (2010) Nomenclatures Indices i l Constants index for te generating blocks of te self-generating facility owned by te consumer time () index bilateral contract index B number of bilateral contracts D i load at time i (MW) E MAX size of production block (MW) N number of production blocks of te generating facility P max l;i maximum power pertaining to contract l at time i (MW) P min l;i minimum power pertaining to contract l at time i (MW) r c cost level for te robustness function ($) r w cost level for te opportunity function ($) S cost associated wit block of te self-generating facility ($/MW) T number of time periods () a price uncertainty parameter ~l i estimation of te pool price at time i ($/MW) Variables E ;i P P,i P l,i P g,i P b,i i s l l i l l l,i Functions ^aðr c Þ ^bðr w Þ Uða; ~uþ energy pertaining to block of te self-generating facility at time i (MW) procured power from te pool at time i (MW) procured power from bilateral contract l at time i (MW) procured power from te self-generating facility at time i (MW) total procured electricity from bilateral contracts at time period i (MW) set of decision variables at time i set of decision variables, =( 1, y, T ) T binary variable, wic is eual to 1 if bilateral contract l is selected, and 0 oterwise pool price at time i ($/MW) set of pool prices ($/MW), l=(l 1, y, l T ) T energy price of contract l at time i ($/MW) robustness function opportunity function uncertainty model is addressed in Conejo et al. (2005), were it is assumed tat all reuired data is available witout considering any uncertainty in market prices or consumer load. Te same problem is solved in Conejo and Carrión (2006) using a mean variance metodology in wic bot forecast and istorical price data are used to construct te price covariance matrix. Te problem was addressed again in Carrión et al. (2007a) troug a scenario generation algoritm and a stocastic programming metod. Electricity procurement cost minimization for a local electricity distribution company (LDC) is addressed in Woo et al. (2004a), using a mean variance tecniue subject to a cost-exposure constraint from pool and bilateral contracts. Woo et al. (2006) also solved te problem by considering te tolling agreement as anoter energy procurement resource. In Woo et al. (2004b), a teoretical framework is developed to determine wic forward-contract purcase minimizes te expected procurement cost of an LDC, subject to a cost-exposure constraint. Different types of electricity-specific financial instruments suc as future contracts, call/ put options, and interruptible contracts are reviewed in Deng and Oren (2006). In particular, te roles of tese options in mitigating te market risks and structuring edging strategies for different agents are analyzed. A multi-period, value-at-risk (VaR) constrained portfolio optimization of real and contractual assets is proposed in Kleindorfer and Li (2005). A stocastic optimization model for determining optimal forward loads and selling prices for a single retailer is proposed in Gabriel et al. (2006). Hatami et al. (2009) and Carrión et al. (2007b) bot propose a stocastic programming metodology to determine te optimal sale price from retailers to customers based on fixed pricing and on te amounts of power purcased from te pool and from forward contracts. In addition, strategies suc as call options and selfproduction facilities are considered in Hatami et al. (2009). In Oum and Oren (2009), a self-financed edging portfolio consisting of a risk-free bond, a forward contract, and a spectrum of call/put options wit different strike prices is selected to maximize te expected profit, subject to a VaR constraint. Tis paper proposes te use of Information Gap Decision Teory (IGDT) as a tool to develop a decision strategy for electricity procurement by a large consumer. Te proposed metod enables te large consumer to evaluate te robustness of its decisions against ig pool prices, as well as te opportunity associated wit low pool prices and determines te strategy for procurement from alternative resources. Tis metod ensures tat te decisions taken by te consumer are robust against ig procurement costs and favorable for low procurement costs. Te results of te proposed metodology elp te consumer analyze its electricity procurement strategies. Uncertainties are generally uantified by matematical abstractions suc as probability density functions or fuzzy logic membersip. IGDT, wic as been developed in Ben-Haim (2001), is a newly developed alternative for decision-making under uncertainty tat makes minor assumptions about te structure of te uncertainty. Info-gap models involve no measure functions neiter probabilistic density nor fuzzy membersip functions. Te distinct feature of IGDT-based models is tat tey do not reuire any assumption of te nature of uncertainty. In modeling te uncertainty using te probabilistic formulation, one often cooses te form of te model, e.g. Gaussian, and ten determines te coefficients of tat model (e.g. mean and variance), wile, te IGDT metod focuses on te disparity between wat is known and wat could be known. In tis metod, te coice of uncertainty parameters is based on a maximizing robustness or a minimizing opportunity rule. Te fact tat te uncertainty parameters are initially unspecified makes IGDT different from oter decision-making approaces suc as stocastic programming. It sould be noted tat in te cance-constrained problem, some constraints are tied to some probability or reliability levels; owever, in te metod addressed in tis paper we do not consider any probability levels. Te mean variance metodology is based on minimizing te variance subject to a cost-exposure constraint, wile te IGDT metodology optimizes te robustness

3 236 K. Zare et al. / Energy Policy 38 (2010) or opportunity functions for a given cost level. Te scenario-based optimization metods suc as stocastic programming metodology explicitly reuire a procedure to generate scenarios based on some uncertainty assumptions, wile IGDT modeling does not. Also, VaR approac and IGDT are different in te way tey include uncertainty in te objective function. In te VaR approac te maximum cost (worst case) is determined for a given confidence level, wile wit IGDT te uncertainty parameter is determined suc tat te maximum cost (worst case) may not exceed a given cost level witout considering any confidence level. Te main contributions of tis paper are: (1) In te proposed metod, te risk factor is not initially determined as it is wit VaR and Conditional VaR metods, and te robustness and opportunity functions are derived based on uncertainty parameters to explain te risk-aversion and risk-taking strategies, respectively. (2) Te proposed metod provides a robust procurement strategy to avoid ig procurement costs or to take advantage of low procurement costs. (3) Te proposed metod as a practical advantage tat finds te optimum procurement strategy for any cost level, not for a given price scenario. Te remainder of te paper is organized as follows: Section 2 provides background about IGDT using a simple example. Section 3 is devoted to analyzing uncertainty issues and introducing te IGDT tecniue for te problem of electricity procurement by a large consumer. Numerical studies are provided in Section 4 to sow te validity of te proposed metod and Section 5 presents te conclusions. 2. IGDT background IGDT is based on uantitative models and provides numerical decision-support assessment; owever, its underlying teory is not a closed computational metodology. Rater, te uantitative assessments assist te decision maker to evaluate options, develop strategies, and evolve preferences in ligt of te analyzed uncertainties, expectations, and demands. Tus it can elp te decision maker to recognize priorities, evaluate risks and opportunities, and ultimately make more informed decisions. Uncertainties can be pernicious and lead to ig cost or propitious and lead to windfall benefits. IGDT addresses tese two conflicting issues using two immunity functions: robustness and opportunity. For a set of decision variables and uncertainty parameter a, we can verbally express te robustness and opportunity functions ^a and ^b as follows: ^a ¼ maxfa : minimal reuirements are always satisfiedg ð1þ a ^b ¼ maxfa : sweeping success is sometimes enabledg ð2þ a Te robustness function addresses te pernicious face of uncertainty and expresses te greatest level of uncertainty at wic te specified minimal reuirements are always satisfied. In tis paper, te robustness function measures te degree of resistance to uncertainty and immunity against ig procurement cost; tus a large ^a value is desirable. Te opportunity function addresses te propitious face of uncertainty and evaluates te possibility of benefits from low pool prices. In te case of tis problem, ^b is te minimum value of a tat can be tolerated tat still enables te possibility of low procurement cost as a result of decision variables. Expressions (1) and (2) sould be tailored to te specific problem being addressed. An IGDT decision problem is specified by tree components tat will be described below in more detail: system model, performance reuirements, and uncertainty model System model Te system model R(,l) expresses te input/output structure of te system to wic te decision is applied. Tis expression assesses te response of te system, in terms of reward or uality of performance, to te decision maker s coices and uncertainty parameter l. In tis paper, te uncertainty parameter is te pool price l, te decision variables are te amount of electricity procurement from alternative sources, and te system model is te procurement cost function of a large consumer Performance reuirements Te performance reuirements describe te reuirements or anticipated values from te system or problem, and tey can be expressed as cost or oter functions. Tese reuirements are evaluated based on te robustness and opportunity functions tat are defined according to Es. (1) and (2). In tis paper, we express tese procurement problem functions as follows: ^a ¼ maxfa : maximum procurement cost ðworst costþ is not a iger tan a given cost levelg ð3þ ^b ¼ min a fa : te minimum procurement cost ðleast costþ is less tan a given cost levelg Te robustness function expresses te greatest level of uncertainty at wic te maximum procurement cost cannot be greater tan a given cost level r c. In oter words, tis function describes te risk-aversion potential of a procurement strategy. Terefore, we can define it matematically troug an optimization problem: ^aðr c Þ¼maxfa : maxðrð; lþþrr c g ð5þ a If ^aðr c Þ is large te decision is robust, risk averse, and insensitive to uncertainties. On te oter and, if ^aðr c Þ is small, te decision is fragile and does not yield generally consistent decisions. Te opportunity function is te lowest value of a suc tat te minimum procurement cost could potentially be as small as a given cost level r w. Tis function is related to determining risk-taking procurement strategies. Te corresponding matematical function can be represented by following minimization problem: ^bðr w Þ¼minfa : minðrð; lþþrr w g ð6þ a were, r w is generally smaller tan r c. In oter words, tis function describes te immunity against windfall benefits; tus, a small value of ^b is desirable. A small value of ^bðr w Þ reflects te situation in wic a benefit is possible even witout te presence of low prices. Te robustness and opportunity functions are uantitative, but numbers alone are not sufficient to solidify a decision. Te decision maker must make value judgments: ow muc robustness is needed for te perniciously uncertain parameter and ow muc sould opportunities from te uncertainty be facilitated? Te answers to tese uestions cannot be uniue or algoritmic; ð4þ

4 K. Zare et al. / Energy Policy 38 (2010) at best, tey are ualitative and imprecise. But te responsible decision maker must make a connection between uantitative decision analysis and ualitative, linguistic, and even subjective values dependent on te context of te decision (Ben-Haim, 2001) Uncertainty model Te uncertainty can be modeled using an info-gap model. Tis info-gap uncertainty model Uða; lþis ~ based on prior information about te uncertainty vector l. Here we ave considered te fractional error model for uncertainty parameter l. Te following example sows te application of tis metod (Ben-Haim, 2001). Consider a static production situation in wic te manufacturer must coose te uantity to be produced, and let us set tis to eual te uantity tat will be sold. Let p() be te known price per item, wic may depend on te uantity produced, and let p() be an uncertain cost function of te production volume. Te profit R(,c) depends on bot te manufacturer s decision and te uncertain production costs incurred. Te profit euation can terefore be expressed as Rð; cþ¼pðþ cðþ ð7þ Let us suppose tat p() is known exactly and tat we know a nominal or typical value of te production cost function ~cðþ, but we also know tat te actual product cost function can deviate from ~cðþ in some unknown way. Tis variability of cost function is represented by a fractional error info-gap model: Uða; ~cþ¼fcðþ : jcðþ ~cðþjra~cðþg; az0 ð8þ Te minimum profit (up to uncertainty a) in te evaluation of te E. (1) robustness function is readily seen to occur for te greatest product cost function allowed by te info-gap model, wic is simply ~cðþþa~cðþ: min Rðc; Þ¼pðÞ ½~cðÞþa~cðÞŠ ð9þ cðþ A Uða;~cðÞÞ For a fixed production volume, te robustness function is te greatest value of te uncertainty parameter a for wic tis minimum profit is at least te predetermined value r c. Euating te minimum profit in (9) to te profit value r c and solving for a results in and self-generating facility): i ¼½P b;i ; P p;i ; P g;i Š; 2;...; T ð12þ Procurement cost model Te electricity procurement cost function faced by a large consumer is (Fig. 1) Rð; lþ¼ XB l X T l l;i P l;i þ XT l i P p;i þ XT X N S E ;i ¼ 1 ð13þ Te cost function (13) is composed of te cost of electricity purcased from bilateral contracts and te pool along wit te operation cost of te available self-generating facility. Te first term of E. (13) represents te cost of energy purcased from bilateral contracts. Te second term provides te procurement cost of electricity from te pool. Te traded energy in te pool can be positive or negative; a positive value means tat te consumer buys from te pool, and a negative value means tat te consumer sells to te pool. Note tat te sold power cannot be larger tan te available self-generating capacity. Te tird term provides te operation cost of te available generating unit. Te operation cost model of te generating facility is depicted in Fig. 2 by a treeblock piecewise linear function (Carrión et al., 2007a) Uncertainty model Here we suppose tat te pool price of electricity is uncertain and consider te following fractional error info gap model: ( Uða; l ~ jl i Þ¼ l i : i l ~ ) i j ra ; az0 ð14þ ~l i GENCO GENCO Wolesale electricity market GENCO að; r c Þ¼ pðþ ~cðþ r c ð10þ ~cðþ For simplicity, let te price function be constant p()=p 0, te nominal cost function be linear ~cðþ¼~c 0, and c 0 be constant. We also consider tat p 0 4 ~c 0. Based on te definition of te robustness function in (1), we sould find te maximum value of a for wic te minimum profit is not less tan a given profit value, r c. Using E. (10), te robustness function can tus be computed as ^að; r c Þ¼max p 0 ~c 0 r c ~c 0 1 ð11þ Te production volume can be defined as te result of tis maximization procedure for given profit values. DSB,, etc. Large Consumer Retailer DISCO Retail electricity market DISCO Retailer 3. Matematical formulation Te formulation of te procurement problem by te IGDT metod is provided below Decision variables Te variables considered are te amounts of electricity procurement from alternative resources (pool, bilateral contracts, C C C C C C Consumers GENCO: Generation company, DISCO: Distribution company, : Distributed generation, DSB: Demand side bidding, C: consumer Fig. 1. Large consumer relationsips wit oter participants in a competitive market.

5 238 K. Zare et al. / Energy Policy 38 (2010) were a is te uncertainty parameter tat models te size of te gap between te known and te unknown. Tis model states tat te fractional deviation of te pool prices from estimated values ~ l i is not greater tan a. In oter words, an envelope-bound model is used in wic te magnitude of deviation is proportional to te forecasted value ~ l i Robustness function Te robustness function ^aðr c Þ is related to ig procurement costs and represents te largest value of te uncertainty parameter tat still precludes te procurement cost exceeding a desired cost level, r c. Tis function expresses te degree of resistance to uncertainty and provides a measure of immunity against ig pool prices or ig procurement cost. Note tat a large value of ^aðr c Þ is desirable. We can compute ^aðr c Þ by solving te optimization problem below: ^aðr c Þ¼max aðr c ; Þ ð15þ Based on E. (3), to compute te robustness function we sould find te maximum procurement cost. Using te uncertainty model in E. (14), if we express ig pool prices as l i ¼ l ~ i þa l ~ i ð16þ substitute (16) into (13), and solve for a, ten based on (15) te robustness function can be computed as ^aðr c Þ¼maxffr c XB l s.t. X B l ¼ 1 P l;i þ XN ¼ 1 X T l l;i P l;i XT ~l i P p;i XT E ;i þp p;i ¼ D i ; ;...; T X N S XT E ;i g= ¼ 1 ~l i P p;i g ð17þ ð18þ P min l;i s l op l;i op max l;i s l ; ;...; T; l ¼ 1;...; B ð19þ 0rE ;i remax E MAX 1 ; ;...; T; ¼ 2;...; N ð20þ 0rE 1;i remax 1 ; ;...; T ð21þ X N ¼ 1 Cost ($) E 1,i S 1 E ;i þp p;i Z0; ;...; T P l;i Z0; :::T; l ¼ 1;...; B E 2,i S 2 E 3,i S 3 MAX MAX MAX E 1 E 2 E 3 Fig. 2. Operation cost model of generating facility. Energy ð22þ ð23þ Expression (17) represents te robustness function to be maximized. Es. (18) (23) are te constraints. E. (18) expresses te power balance between te load and te procured electricity from alternative sources. Te amount of electrical energy purcased from bilateral contracts must be between te allowable bounds as defined by E. (19). Constraints (20) (22) are related to te operation of te generating facility. E. (22) expresses tat te sum of te traded electricity in te pool and te electricity produced by te generating facility sould be non-negative for all time periods. Eac of tese constraints expresses tat te maximum amount of electrical energy tat te consumer can sell in te pool is bounded by te capacity of te generating unit. Constraint (23) expresses tat te power procured from bilateral contracts must be positive. Te function ^aðr c Þ is related to costs iger tan te minimum cost, serving as a risk aversion mecanism. Tus a ig value of tis function, wic is related to a ig cost level r c, means tat te associated decision as ig robustness against ig pool prices. In te oter words, ^aðr c Þ measures te decision s level of protection against ig pool prices. It is expected tat as r c increases, te percentage of procurement from resources wit volatile prices (pool) decreases, and inversely procurement from resources wit stable prices increases Opportunity function Te opportunity function ^bðr w Þ is related to low procurement costs and it evaluates te potential benefits of experiencing low procurement cost due to low prices. Here te benefit is eual to te difference between te minimum cost (resulting from E. (13)) and te actual procurement cost level. In oter words, tis is te immunity against windfall benefit ; tus a small value of ^bðr w Þ is desirable. Te opportunity function is te smallest value of a suc tat te minimum procurement cost could be as small as r w. Te corresponding matematical function can be represented by ^bðr w Þ¼min aðr w ; Þ ð24þ were r w is generally smaller tan r c. Similar to te robustness function, we define te following relationsip: l i ¼ ~ l i a ~ l i ð25þ By replacing (25) in (13), solving for a and using te definition of opportunity function in E. (24), we can conclude tat ^bðr w Þ¼minff XB l X T l l;i P l;i þ XT ~l i P p;i þ XT X N ¼ 1 S E ;i r w g= XT ~l i P p;i g ð26þ Note tat a small value of ^bðr w Þ indicates tat te corresponding decision as low resistance against experiencing low procurement cost levels, so tat a windfall benefit can be acieved. Te opportunity function ^bðr w Þ relates to low procurement cost and indicates tat if te consumer makes a decision related to a given value of r w and if te pool prices are eual to or smaller tan 100 ð1 ^bðr w ÞÞ% of te estimated values, ten te actual cost is eual to or less tan r w. Making any decision wit a value of r w ten allows te consumer to take advantage of low pool prices. It is expected tat ^bðr w Þ increases as r w decreases. Also, te percentage of electricity procurement from te pool as a direct relationsip wit ^bðr w Þ, wile te procurement from oter resources as te inverse beavior. Te robustness and opportunity functions are defined in antisymmetric sense; tus big is good for ^aðr c Þ wile big is bad for ^bðr w Þ. Te consumer must make te following judgments: ow muc robustness is needed versus ow muc opportunities from

6 K. Zare et al. / Energy Policy 38 (2010) uncertainty sould be exploited and at wat sacrifice in robustness. Te proposed metod enables a large consumer to assess te costs near minimum cost via robustness and opportunity functions and to identify te related procurement strategy. Te robustness and opportunity problems are mixed-integer nonlinear programming problems tat can be solved using SBB (GAMS, 2008a) under GAMS (Brooke et al., 1998). Table 1 provides te size of tis problem, wic is expressed as te number of binary variables, real variables, and constraints. 4. Case study In tis section, numerical simulations are conducted to illustrate te application of te proposed metod. A single day is considered wit its demand deconstructed into tree load levels denoted as peak, soulder, and valley, as indicated in Table 2. Several contracts are considered wose selection decisions would need to be made at te beginning of te study orizon containing 84 periods (4 weeks). Twelve bilateral contracts are considered; Table 3 provides data for te bilateral contracts, including te minimum and maximum uantity of energy along wit energy price. Tere are 4 contracts available for te entire mont and two contracts for eac week. Table 4 provides te usage periods of te contracts. Six contracts are available for all load levels, and six contracts are only for peak load levels. Te generating facility data are provided in Table 5. Te load profile of te consumer and te best estimated price data for te study orizon are depicted in Figs. 3 and 4, respectively. Te energy procurement problem is formulated as a mixedinteger nonlinear problem and solved using SBB/CONOPT (GAMS, Table 4 Bilateral contracts usage periods. Contract # Usage period Validity level 1 Mont V, S, P 2 Mont P 3 Mont V, S, P 4 Mont P 5 Week one V, S, P 6 Week one P 7 Week two V, S, P 8 Week two P 9 Week tree V, S, P 10 Week tree P 11 Week four V, S, P 12 Week four P V: valley; S: soulder; P: peak. Table 5 Data for te self-generating facility. Capacity 100 MW Minimum power output 0 MW S 1 33 $/MW S 2 36 $/MW S 3 39 $/MW E MAX 1 40 MW E MAX 2 75 MW E MAX MW 3000 Table 1 Computational size of te problem. # of binary variables B # of real variables T(B+N+1) # of constraints T(3B+2(N 1)+4) Table 2 Classification of daily load levels. Load (MW) Level Hours of te day 2300 Valley (V) 1, 2, 3, 4, 5, 6, 7, 8 Soulder (S) 9, 10, 15, 16, 17, 18, 23, 24 Peak (P) 11, 12, 13, 14, 19, 20, 21, 22 Table 3 Bilateral contracts specification. Contract number Min. (MW) Max. (MW) Price ($/ MW) Period Fig. 3. Load profile of te consumer for study orizon. 2008b) wit GAMS software. Te simulation procedure and derivation of te results are as follows: 1. Te cost minimization problem was simulated based on E. (13), considering te forecasted price data. 2. For some values of r c wit a fixed step tat are iger tan te minimum cost, te robustness function (17) wit constraints (18) (23) is simulated, and te results are depicted in Figs. (5) (9). Tese results represent te robustness level and percentage of procurement from alternative resources.

7 240 K. Zare et al. / Energy Policy 38 (2010) Estimated pool price ($/MW) Bilateral contracts procurement (%) Period Fig. 4. Best estimated price data for study orizon. 5 Fig. 7. Optimal procurement from bilateral contracts as a function of r c. Robustness value, α (r c ) Generating facility procurement (%) Fig. 8. Optimal procurement from te self-generating facility as a function of r c. Fig. 5. Robustness curve x 106 Pool procurement (%) R (λ, (α (r c ))) Fig. 6. Optimal procurement from te pool as a function of r c. 8 Fig. 9. Expected cost, Rð l; ~ ð ^aðr cþþþ, versus r c.

8 K. Zare et al. / Energy Policy 38 (2010) For some values of r w wit a fixed step tat are less tan te minimum cost, te opportunity function (26) wit constraints (18) (23) is simulated, and te results are depicted in Figs. (10) (14). Tese results represent te opportunity level and percentage of procurement from alternative resources. In tese simulations, te cost step for r c and r w is considered to be $20,000. Considering te price forecasts, te minimum procurement cost is calculated as $8,057,616. Tis value is te result of minimizing E. (13) wit te best estimate of pool price values. Te results sow tat te consumer sould procure 73% of its electricity needs from te pool, 7.2% from bilateral contracts, and 19.8% from te self-generating facility. From an uncertainty perspective, we can say tat ^að8; 057; 616Þ¼ ^bð8; 057; 616Þ¼0. Te results of simulations related to te robustness function will be useful if te large consumer decides on a risk-averse procurement strategy and te opportunity function-related procurement strategies can be used wen te large consumer cooses a risk-taking decision making mode. Te robustness function ^aðr c Þ is depicted in Fig. 5, were it is clear tat robustness increases wit r c as expected. If te consumer desires ig robustness it sould pay a ig cost; inversely, if te consumer pays a ig cost tis indicates its Bilateral contracts procurement (%) Fig. 12. Optimal procurement from bilateral contracts as a function of r w. 20 Opportunity value, β (r w ) Fig. 10. Opportunity function. Generating facility procurement (%) Fig. 13. Optimal procurement from te self-generating facility as a function of r w x Pool procurement (%) R (λ, (β (r w ))) Fig. 11. Optimal procurement from te pool as a function of r w. Fig. 14. Expected cost, Rð ~ l; ð ^bðr wþþþ, versus r w.

9 242 K. Zare et al. / Energy Policy 38 (2010) procurement strategy is igly robust and more risk-averse. Figs. 6 8 sow tat te procurement percentage from te pool decreases wit r c, wile procurement from te self-generating facility and bilateral contracts increase wit r c. Tis is an interesting result indicating tat wen te consumer cooses a more robust procurement strategy, it is better to procure more of its electricity need from te sources wose prices ave no uncertainty. Fig. 9 sows te Rð ~ l; ð ^aðr c ÞÞÞ curve, wic provides te expected cost of procurement wit te best estimated pool prices along wit te procurement strategy related to a given cost level r c. Here it is interesting to see tat te Rð ~ l; ð ^aðr c ÞÞÞ cost is smaller tan r c. For example, we assume tat te consumer decides to pay $8,500,000 to procure its electricity demand, wic is 5.5% over te minimum procurement cost. Te robustness of tis procurement decision is about 9.5%, wic means tat tis decision is robust against up to a 9.5% pool price increase. Tis robustness means tat te maximum cost of procurement will be eual to r c wen pool price increases by as muc as 9.5% of te estimated price. Te expected cost Rð ~ l; ð ^aðr c ÞÞÞ for tis example is $8,100,236, wic is 0.53% iger tan te minimum cost. Tis decision is tus considered robust against up to a 9.5% increase in pool prices. Te opportunity value ^bðr w Þ, sown in Fig. 10, indicates tat if te pool prices are eual to or less tan 100ð1 ^bðr w ÞÞ% of te best estimated values, te minimum benefit is eual to te difference between te minimum cost and te related r w. Te increase in benefit means corresponds to increased risk. Figs sow tat te percentage of procurement from te pool decreases wit r w and increases from bilateral contracts and utilization of te self-generating facility. Tis is a logical beavior because coosing a low procurement cost level means deciding to take ig risk; tus, a comparatively iger portion of procurement comes from sources wit uncertain prices. Fig. 14 sows Rð ~ l; ð ^bðr w ÞÞÞ, wic expresses te expected cost of procurement for te pool price eual to its best estimation along wit procurement strategies related to r w as a function of r w. It is very interesting to note tat te cost Rð ~ l; ð ^bðr w ÞÞÞ increases as r w decreases. Tis figure provides te cost of te risk, wic is eual to te difference between minimum cost and Rð ~ l; ð ^bðr w ÞÞÞ. For example, consider tat te large consumer is willing to spend $7,500,000 and decides on te related procurement strategy. Te opportunity value of tis strategy will be about wit 77.87%, 7.2%, and 14.93% procurement from te pool, bilateral contracts, and self-generating facility, respectively. Tis means tat if pool prices decrease by 8.9% or more, te minimum benefit is eual to $557,616 (8,057,616 7,500,000), wic is 6.92% of te minimum cost. Tis value is te benefit of accepting risk; te cost of te risk is eual to $8,077,705 $8,057,616=$20,089, or is 0.25% of te minimum cost. It sould be noted tat benefit of accepting risk is iger tan te cost of risk. 5. Conclusion In tis paper, te procurement strategy for a large consumer is modeled based on IGDT metodology. Bilateral contracts, te energy pool, and self-generating facilities are considered as procurement sources. Te proposed metod elps te consumer identify its procurement strategy and evaluate its robustness against ig pool prices or ig procurement costs. Also, te proposed metod addresses opportunities for taking advantage of low procurement costs or low pool prices. Based on our analyses in tis paper, as well as oter uantitative and ualitative preferences, and te preference to be risk-averse or risk-seeking, te consumer can decide te most appropriate strategy. Te proposed metod as been evaluated using a realistic example and te results sow tat te strategies related to a iger procurement cost is more robust and risk averse, and tat procurement from sources wit uncertain prices decreases as te cost increases. It is also sown tat a buying strategy wen procurement costs are lower tan te expected minimum cost, is a risk seeking strategy tat provides opportunities for windfall benefits. Acknowledgment Te autors would like to tank Prof. Antonio J. Conejo and Dr. Miguel Carrión from Castilla-La Manca University in Ciudad Real, Spain, for teir valuable comments and relevant observations. References Ben-Haim, Y., Information Gap Decision Teory, Designs under Severe Uncertainty. Academic Press, California. 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