Short-term electricity procurement: A rolling horizon stochastic programming approach

Applied Mathematical Modelling 35 (2011) 3980–3990

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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

Short-term electricity procurement: A rolling horizon stochastic programming approach Patrizia Beraldi ⇑, Antonio Violi, Nadia Scordino, Nicola Sorrentino Department of Electronics, Informatics and Systems, University of Calabria, Via P. Bucci Cubo 41/C, 87036 Rende (CS), Italy

a r t i c l e

i n f o

Article history: Received 4 August 2010 Received in revised form 20 January 2011 Accepted 2 February 2011 Available online 23 February 2011 Keywords: Electricity market Stochastic programming Risk management Rolling horizon framework

a b s t r a c t This paper addresses the problem faced by a large electricity consumer in determining the optimal procurement plan over a short-term time horizon. The inherent complexity of the problem, due to its dynamic and stochastic nature, is dealt by means of the stochastic programming modeling framework. In particular, a two-stage problem is formulated with the aim of establishing the optimal amount of electricity to be purchased through bilateral contracts and in the Day-Ahead Electricity Market. Recourse actions are used to hedge against uncertainty related to future electricity prices and consumer’s needs. The optimal plan is defined so to minimize the overall cost and to control risk, which is measured in the form of violation of budget constraints. The stochastic model is dynamically solved in a rolling horizon fashion by iteratively considering more and more recent information and a planning horizon of decreasing length. Extensive numerical experiments have been carried out to assess the performance of the proposed dynamic decision approach. The results collected considering a real test case are very encouraging and provide evidence of the superiority of the approach also in comparison with other alternative procurement strategies. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction Following the liberalization, electricity trading has become a key activity for the electric market operators, including producers, large consumers and retailers [1]. This paper addresses the crucial problem of determining the best electricity procurement plan from a consumer’s perspective. The two main alternatives available to cover electricity needs are represented by bilateral contracts, negotiated between parties, and participation in the Day-Ahead Electricity Market (DAEM). These procurement opportunities mainly differ in the deferment between trading and delivery time. In fact, while in the DAEM electricity, traded during the day-ahead, is actually delivered the day after, bilateral contracts are signed long time before delivery. Typically, bilateral contracts cover a time horizon spanning from several weeks to several years and are characterized by fixed tariffs and electricity amounts referred to blocks of hours (e.g. peak and off-peak hours). If compared with the participation in the DAEM, bilateral contracts provide a less risky alternative in terms of price volatility, but at the expense of higher average procurement costs. Looking at a given planning horizon, decisions related to bilateral contracts should be taken at a strategic level, also considering other possible alternatives, e.g. purchase in the DAEM and/or self-production [2]. Assuming known the electricity amount bilaterally contracted, and the corresponding time block

⇑ Corresponding author. E-mail addresses: [email protected] (P. Beraldi), [email protected] (A. Violi), [email protected] (N. Scordino), [email protected] (N. Sorrentino). 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.02.002

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articulation, the consumer must face the problem of defining the optimal procurement plan at a tactical level. Short-term scheduling must, thus, be carried out to allow the physical transaction of electricity between the involved parties. This means that on each hour of each day of the contract period, the consumer must decide the amount of energy to be consumed through long-term bilateral contracts previously signed. Within this decision process, the consumer can also decide to trade (buy and/or sell) power in the DAEM, during those hours in which DAEM prices are eventually lower/higher than the fixed prices bilaterally contracted. It is worthwhile noting that the idea of integrating the management of bilateral contracts with DAEM activities represents a current practice in some electricity markets. For example, in Europe, energy regulators are placing considerable effort in promoting the use of software platforms to embed bilateral contracts schedule within the DAEM mechanism [3]. Power Exchanges like N2EX in UK [4], European Energy Exchange in Germany [1], and GME in Italy [5] have developed dedicated webbased platforms in order to manage short period schedules relating to bilateral contracts, providing connectivity through Internet. Within each platform, a participant may own one or more injection or withdrawal accounts, on which purchases and sales are registered. In particular, a consumer must register the estimated consumption and the energy amount to acquire through bilateral contracts, according to an hourly basis or for blocks of hours. Depending on the specific platform, registrations are allowed up to a number of hours or days before the beginning of the corresponding physical delivery. Nonetheless, for each elementary time block, a deviation between purchases by bilateral contract and estimated consumption might be registered and converted into a selling and/or buying operations in the DAEM [3–5]. This paper aims at providing an electricity consumer with an effective decision approach to define the best procurement plan by an integrated management of bilateral contracts and participation in the DAEM. Despite its relevance, the problem addressed in this paper has been only partially investigated in the scientific literature, which, on the contrary, presents many different contributions for the producer’s side (see for example [6–11] and the references therein). As far as the optimal procurement problem is concerned, the contribution of Daryanian et al. [12], the recent papers of Kirschen [13], Su et al. [14], Carrión et al. [2,15] and Menniti et al. [16] deserve special attention. In particular, within a centralized decision framework, in [12] the optimal response of a large consumer to varying electricity spot prices has been derived in terms of consumptions and consumption rescheduling. A detailed analysis and a characterization of the main decision-making tools that consumers and retailers would need to participate in an electricity market have been presented in [13]. In [14] the authors proposed a new centralized complex-bid market-clearing mechanism to take into account the load shifting behavior of consumers submitting price-sensitive bids. Most of the contributions mentioned above, even if recognize the inherent uncertainty affecting the electricity market, deal only partially with the stochastic nature of the procurement problem. For example, in [16] the authors propose a systematic procedure based on a Monte Carlo technique for the definition of hourly aggregate step-wise demand curves of an electricity coalition. The model relies on a sensitivity analysis on the response of consumers to variable hourly electricity tariffs. In effect, future DAEM prices and consumer’s demand cannot be considered known in advance. This consideration suggests the limited effectiveness of traditional deterministic approaches calling for more sophisticated modeling paradigms, as the Stochastic Programming (SP) framework [17,18] . Both [2] and [15] have handled uncertainty of pool prices and load demand by treating them as stochastic variables, using a finite set of scenarios, whose number was opportunely reduced by means of proper reduction techniques, in order to make tractable the relating optimizations models. In [2], the authors propose a SP formulation for the optimal electricity procurement problem faced by a large consumer that also owns a limited self-production facility (e.g. a cogeneration unit). Moreover, a bi-level SP approach for solving the medium-term decision problem faced by a power retailer during the procurement process has been proposed in [15]. Similarly to the last two contributions mentioned above, we adopt the SP framework to address the short-term electricity procurement problem. In particular, uncertainty affecting the future values of both DAEM prices and consumer demands is explicitly dealt by the introduction of random variables defined on a given probability space. Under the assumption of a discrete distribution, several different realizations (scenarios) of the uncertain parameters are generated by a Monte Carlo procedure. The problem is modeled by adopting the two-stage SP paradigm: first-stage (here-and-now) decisions refer to the amount of electricity to procure through both (previously accepted) bilateral contracts and DAEM offers, whereas secondstage decisions refer to corrective actions undertaken to balance (in excess or shortage) the scenario-dependent consumer’s demands. The procurement plan is defined with the aim of minimizing the overall expected costs while keeping under control the risk, measured in terms of cost violation from a predefined budget. The two-stage problem is then dynamically solved on a rolling horizon fashion: as time progresses and new information become available, new updated scenarios are generated and new problem instances are formulated and solved over a planning horizon of decreasing length. The dynamic nature of the proposed approach represents one of the main distinctive features of our contribution. Instead of considering a multistage formulation as in [2], different two-stage models are solved within an iterative scheme. It is worthwhile noting that the proposed approach mathematically translates the proper use of any SP approach consisting in the implementation of the optimal values of the here-and-now decisions, that refer to the current time. The rest of the paper is organized as follows. Section 2 describes the decision problem and introduces the SP mathematical formulation. Section 3 shows the iterative decision approach in which the optimization model is integrated. The approach is then validated on a real test case referring to a large consumer of the Italian electricity market. Numerical results are provided and analyzed in Section 4. Section 5 reports some relevant conclusions. Finally, Appendix A illustrates the procedure used for generating scenarios of DAEM prices and consumer’s demand.

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2. Problem formulation We consider the problem faced by a large consumer who must define the optimal electricity procurement plan over a short-term planning horizon of G days. We assume that the hours (h = 1, . . . , 24) of each day g = 1, . . . , G can be classified in time blocks (for example, peak, intermediate and off peak hours). We denote by F the set of time blocks, indexed by f, and by BTgh the time block of the day-hour pair (g, h). In the remainder, we shall assume that the consumer is a price-taker, i.e. his choices cannot influence the market-clearing prices. This assumption is quite realistic, since the actual impact of demand-response programs on the market clearing process is negligible. We assume that the consumer’s needs can be covered through two main procurements ways: bilateral contracts and participation in the DAEM. We denote by K the set of active bilateral contracts already established, as explained in the previous section, at strategic level. For each contract k 2 K, we denote by Qgfk the electricity quantity available during the time window from g onwards and over the hours belonging to time block f. In addition, we denote by BCPfk the corresponding price. In contrast to the price of bilateral contracts, DAEM prices are stochastic since they refer to future time and result from market auction. Similarly, consumer’s needs can be difficulty estimated in advance, since different unpredictable situations may occur. In order to explicitly address the inherent stochastic nature of the optimal procurement problem, we have adopted the SP framework. In particular, the uncertain parameters are modeled as random variables defined on a given probability space ðX; F; PÞ. Under the assumption of discrete distributions, the future uncertain evolution of DAEM prices and electricity des mand is represented by a scenario set S. For each scenario s, occurring with probability ps, we denote by dgh the electricity s s demand on day g, at hour h, and by Pgh and W gh the demand-side and supply-side DAEM prices on day g at hour h, respectively. We observe that demand and supply prices could be different. For example, in the Italian electricity market, the demand-side price is the average of the zonal supply-side prices, weighted by the zonal consumption. Details on the scenario generation techniques are reported in Appendix A. Scenarios provide the input data for the SP formulation. Thus, in contrast to deterministic models, which allow to determine the optimal plan under a specific situation (for example represented by the expected value of uncertain parameters), the SP counterpart determines a more ‘‘robust’’ solution by explicitly taking into account all the possible situations that may occur. Within the SP framework, the optimal procurement problem has been formulated by adopting the two-stage paradigm. Here, the set of decisions is partitioned into two subsets, i.e. first-stage decisions, to take here and now in the face of uncertainty, and second-stage decisions, i.e. corrective actions which depend on the specific realization that occurs. According to the nature of the problem, we have considered as first stage all the decisions related to the procurement plan through both bilateral contacts and trading in the DAEM. In particular, for each hour h of each day g, we shall denote by xghk the amount of electricity to purchase through bilateral contract k, and by ygh and zgh the amount of energy to buy and sell in the DAEM, respectively. Second-stage decisions compensate the scenario dependent consumer’s needs. In particular, for each s hour h on every day g, we denote by Dsþ gh and Dgh the amounts of energy required to balance (excess/shortage) the consumer’s s demand under scenario s. In addition, we denote by V sþ gh and V gh the corresponding unbalance costs. The solution of the SP formulation will provide the optimal procurement plan over the considered time horizon, consisting of G days. From a practical standpoint, the decisions really implemented will refer to the upcoming day. This consideration suggests to dynamically redefine and solve a model by exploiting the progressive revealing of uncertain parameters and gradually considering a planning horizon of decreasing size fg; . . . ; Gg, where g denotes the generic upcoming day. In the remainder, we shall omit the dynamic feature of the proposed approach which will be described in the next section and we shall introduce the key ingredients (constraints and objective function) of the SP formulation. 2.1. Constraints The procurement plan should be defined so to satisfy some restrictions, modeled by the following constraints: G X

X

xghk ¼ Q gfk

8f 2 F;

8k 2 K;

ð1Þ

g¼g ðhjTBgh ¼f Þ

xghk 6 U gfk g ¼ g; . . . ; G; X

8f 2 F;

8hjTBgh ¼ f ; s

s xghk þ ygh  zgh þ Dsþ gh  Dgh ¼ dgh

8k 2 K;

g ¼ g; . . . ; G;

8f 2 F;

ð2Þ

8hjTBgh ¼ f ;

8s 2 S;

ð3Þ

k2K

xghk P 0 g ¼ g; . . . ; G;

8f 2 F;

ygh ; zgh P 0 g ¼ g; . . . ; G; s Dsþ g ¼ g; . . . ; G; gh ; Dgh P 0

8hjTBgh ¼ f ;

8f 2 F; 8f 2 F;

8k 2 K;

8hjTBgh ¼ f ; 8hjTBgh ¼ f ;

ð4Þ ð5Þ

8s 2 S:

ð6Þ

Constraints (1) state that the sum of the electricity purchased through bilateral contract k must match the residual electricity amount available by contract ðQ gfk Þ, within hours of the same time block f, for all the days from g onwards.

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Constraints (2) impose an upper bound on the hourly quantity, xghk, to be purchased through each bilateral contract. This bound can be set according to the residual quantity Q gfk for each contract k and to the number of residual hours, Nr gf , belong to G. A possible expression for this bound is represented by (7): ing to time block f, for days from g

/ Q ; Nrgf gfk

U gfk ¼

ð7Þ

where / is a user-defined parameter, which may depend on contractual conditions. Constraints (3) enforce the demand satisfaction under each scenario s by the total energy amount scheduled by bilateral contracts, the quantity traded in the DAEM and the excess/shortage consumption. Constraints (4)–(6) are classical nonnegative conditions. 2.2. Objective function With the aim of defining the optimal procurement plan, the consumer is typically concerned with the minimization of the overall costs, simultaneously keeping risk exposure under control. In the last decade, a growing attention has been devoted by the scientific literature see e.g. [2,15,16,19–21] to the definition of suitable measures to control and manage risk, in different fields including the electricity one. In this paper, risk is associated with the violation of a predetermined budget, typically available for the entire planning horizon. In order to accommodate these two conflicting objectives, an objective function (8) with a mean-risk structure has been defined:

minð1  kÞC þ kD;

ð8Þ

where C and D stand, respectively, for the expected overall cost and a risk measure, and k is a real parameter ranging from 0 to 1, which accounts for the consumer’s risk aversion attitude. The choice of the k value is up to the decision maker, with higher values privileging less risky strategies at the expense of more costly electricity procurements. By varying the k value, different strategies can be evaluated in terms of risk and cost savings trade-off. The expected overall cost C, reported in Eq. (9), is determined as the weighted sum (by scenario probabilities) of the scenario-dependent overall cost Cs (10), this last computed as the sum of cost for bilateral contracts (11), of the market operations net value (12) and of the unbalance cost (13):

C¼

X

ps C s ;

ð9Þ

s2S

C s ¼ C bil þ C smkt þ C serr C bil ¼

XXX

8s 2 S;

X

ð10Þ

BCPfk xghk ;

ð11Þ

k2K gPg f 2F ðhjTBgh ¼f Þ

C smkt ¼

XX

X

Psgh ygh 

gPg f 2F ðhjTBgh ¼f Þ

C serr ¼

XX

X



XX X

W sgh zgh

8s 2 S;

ð12Þ

gPg f 2F ðg;hÞ2Hf

sþ s s V sþ gh Dgh  V gh Dgh



8s 2 S:

ð13Þ

gPg f 2F ðhjTBgh ¼f Þ

The risk measure D is defined as the semi-deviation from a target [22,23]. In particular, D penalizes the expected value to exceed the residual budget, Bg , available on day g:

D¼

X

 r ps C s  Bg þ

!1=r ð14Þ

s2S

with r a non-negative integer parameter. This risk measure enjoys the nice property to be convexity preserving [24] and, if included in a mean-risk approach like the proposed one, makes the overall formulation consistent with the stochastic dominance criterion [22,23]. The semi-deviation also satisfies the axioms of ‘‘coherent’’ risk measures, a property which is becoming a standard in both academic and operative settings [25,26]. Depending on the value of r, the objective function has a linear or non-linear convex (if r > 1) structure. Assuming r = 1 in (14), the corresponding term of the risk measure

 s  C  Bg þ ¼ maxf0; C s  Bg g;

ð15Þ s

can be rewritten by adding a set of additional variables, L , which represent the losses under each scenario s:

Ls P C s  Bg

8s 2 S;

Ls P 0 8s 2 S:

ð16Þ

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The objective function (8) thus becomes:

minð1  kÞ

X

ps C s þ k

X

s2S

ps Ls

ð17Þ

s2S

Alternative choices of risk measure could be suitable as well, considering, for example, recent popular measures like the Conditional Value at Risk [24–26]. The overall model belongs to the class of two-stage stochastic programming problems with simple recourse. The size of the model is affected by the number of residual days in the planning horizon, Nrgf, the number of bilateral contracts, jKj, the number of time blocks, jFj, and the number of scenarios, jSj, as shown in Table 1, where we report the number of variables and constraints. It should be emphasized that the number of scenarios critically affects the size of the model. Though larger number of scenarios might lead to a more accurate representation of the underlying random process, nonetheless the corresponding increased size might even prevent the use of general-purpose solution software. This computational drawback only slightly affects our decision approach because of the problem nature (linear or convex programming model). In addition, the proposed rolling horizon approach relies on the solution of problems defined on a time horizon of decreasing length. 3. Decision approach The SP model presented in the previous section represents the core of the proposed dynamic approach. The basic idea is to encapsulate the day-by-day SP electricity procurement model within a rolling framework: as time progresses, an updated SP model is defined over a planning horizon of decreasing length. The advantage of this approach is to operate with more and more recent information, by updating the scenario set day-by-day. The proposed decision approach can be sketched as follows:  and the last day of the planning horizon, G. (Initialization) Set the first upcoming day g ; . . . ; G. (Scenario generation) Generate the set of scenarios for the whole time horizon g ; . . . ; G. (Problem solution) Solve the SP model over the planning horizon g (Implementation) Implement the suggested first-stage decisions for day g. (Updating) Update model parameters, that is DAEM hourly prices and consumer demands, by exploiting the new market conditions. If g ¼ G, then go to Step 5. Else set g ¼ g þ 1 and go to Step 1. Step 5 (Termination) STOP.

Step Step Step Step Step

0 1 2 3 4

Within the previous scheme, the scenario generation phase (Step 1) plays a crucial role, since scenarios quality affects model recommendations. During the last decades, many different scenario generation techniques have been proposed in the scientific literature. For a recent survey, we remind to [27,28] and references therein. In our case, a Monte Carlo procedure based on a Mean Reverting Process [29] has been considered, for the representation of electricity prices evolution. This choice has been motivated by the need to take into account both seasonal effects and the tendency of spot prices to fluctuate around and drift over time to a sort of equilibrium level. As for the consumer’s demand, a heuristic method which exploits historical information has been implemented. Details on scenarios generation are reported in Appendix A. Scenarios provide the input data for the optimization model solved at Step 2. The only recommendations implemented refer to the up-coming day g. Then, before the next problem formulation and solution, some input parameters must be consequently updated. In particular, the electricity quantity already procured through the kth bilateral contract within hours on P day g belonging to time block f ;  hk , is subtracted to the residual electricity quantity defined by Q gfk : ðhjTBgh ¼f Þ xg

Q ðgþ1Þfk ¼ Q gfk 

X

xghk

f 2 F;

k 2 K:

ð18Þ

ðhjTBgh ¼f Þ

 can Subsequent to the realization of the uncertain DAEM prices and consumer’s demand, the actual overall cost on day g be determined and the residual budget must be updated (19):

Bðgþ1Þ ¼ Bg 

XX k2K

X

BCP fk xghk 

f 2F ðhjTBgh ¼f Þ

24 X h¼1

PRgh ygh þ

24 X h¼1

W Rgh zgh 

24 X

Rþ V gRþ h Dgh 

h¼1

24 X

R V gR h Dgh

8f 2 F;

8k 2 K;

ð19Þ

h¼1

R where PRgh and W Rgh are the actual values of demand and supply-side DAEM prices respectively, V Rþ h the actual values gh and V g R of unbalance costs, and DRþ and D the actual realizations of unbalance electricity quantities. h h g g

Table 1 Model size. Variables

Constraints

24Nrgf(2jSj + jKj + 2) + jSj

24Nrgf(jKj + jSj) + jKjjFj + jSj)

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4. Computational experiments In this section, we report on the computational experiments carried out to validate the proposed decision approach. The implementation has been performed by using the algebraic modeling system AIMMS [30] and CPLEX 10.1 as solution kernel. Numerical results have been collected on a Pentium 4, 1.8 GHz with 1056 MB of RAM. In the remainder, we briefly describe the test case and we present and analyze the computational results, providing also a comparison of our approach with respect to other alternative procurement strategies. 4.1. Test case The test case is represented by a large electricity consumer located in the south of Italy, the University of Calabria. It presents an overall energy demand of about 20 GWh per year and covers its electricity needs through a bilateral contract, negotiated by a consortium [31]. Historical hourly consumption diagrams have been considered as starting basis for generating scenarios referring to the future demand. As for the DAEM prices, we have adopted the Italian market structure. In particular, zonal prices have been used for sales [5], while the Unified National Price (PUN in Italy), which is computed as the average of zonal selling prices weighted on the consumption in each zone, has been considered on the purchase side. In order to penalize excess and shortage errors on the estimated average demand, consumer is supposed to pay the electricity surplus at a price 20% higher than PUN at the same hour, whereas electricity shortage is considered like a sale, but at a s s s price 20% lower than the zonal price, at the same hour (that is, V sþ gh ¼ 120%P gh and V gh ¼ 80%W gh ). This assumption is motivated by the consideration that real-time demand unbalances must be modeled as undesired behaviors and not as a trading option. For the clarity of presentation, in the remainder, we shall consider a planning horizon of one week (February 22–28 2009), but a longer time horizon might be considered as well. At each run of the proposed approach, scenario prices and demand are generated by the procedure reported in Appendix A. However, other scenario generation procedures could be used as well. Fig. 1 shows the evolution of the mean value of estimated and observed hourly PUNs, averaged on the considered planning horizon. As evident, price forecasts obtained by the proposed scenario generation procedure seem to be quite accurate, with an average error computed over the 24 h far below 10%. We assume that information related to bilateral contracts (electricity amount and prices) are known in advance, since they result from previous long-term decisions. In particular, on December 2008, as a result of a public tender, a bilateral contract has been signed between a retailer and consortium CRETA [31] for a delivering period of one year, from February 1 2009 to January 31 2010. Table 2 reports the main details. In particular, prices and amounts refer to the considered time blocks structure, i.e. peak, intermediate and off peak hours (in the remainder denoted by F1, F2 and F3, respectively) defined by the Italian Electricity and Gas Authority. 4.2. Numerical results A first set of numerical experiments has been carried out with the aim of validating the proposed decision approach. We assume that the amount of electricity related to the bilateral contract is equal to 60% of the expected weekly demand reported in Table 2. This percentage provides a realistic value for a long-term strategic planning, since it represents a low-risk behavior in the procurement planning process to cover future electricity needs.

Fig. 1. Mean value of observed and estimated hourly PUN over the week February 22–28 2009.

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P. Beraldi et al. / Applied Mathematical Modelling 35 (2011) 3980–3990 Table 2 Electricity price and quantity bilaterally contracted by University of Calabria.

Price [€/MWh] Annual demand (2009) [MWh] Weekly demand (February 22–28) [MWh]

F1

F2

F3

119.7 10,081 200.59

91.3 4167 83.75

58.7 6678 119.11

The effectiveness of the decision approach has been evaluated by means of a back-testing analysis, based on the real observed values of market prices and demands. Roughly speaking, this analysis allows to determine the real performance which would be obtained by implementing the energy procurement plan suggested by the model solution on a real setting. More specifically, for each execution of Steps 1–4 of the rolling horizon approach, the analysis has been carried out by imple  menting first-stage decisions suggested by the solution of the optimization problem, i.e. bilateral contract scheduling xghk   and market operations ygh ; zgh for each hour of the upcoming day ðgÞ. Costs (or profits) related to these decisions are economically evaluated by means of real observed values of PUNs, zonal selling prices and consumer demand, whereas the daily overall cost is thus computed and the residual budget is determined according to (19). Similarly, the residual quantity of the bilateral contract can be derived from (18). The process is iterated for each day of the planning horizon and the real weekly procurement cost is calculated as the sum of the daily overall costs. In what follows, we report the results obtained by setting the risk aversion parameter k at 0.75. The choice of this value reflects a quite conservative behavior, very likely adopted in current practice. Fig. 2 shows the expected demand and the electricity procurement plan over each hour of the planning horizon, in terms of electricity amount purchased through the bilateral contract. The difference between these quantities determines the energy amount bought (if positive) or sold (if negative) in the DAEM. The figure shows that the energy amount bought by bilateral contract is not uniformly spread over the hours of the planning horizon, rather it is concentrated within those hours when PUN is estimated to be higher than bilateral contract price. For the sake of clarity, Fig. 3 gives a zoom on February 26 and reports the procurement plan in terms of bilateral contract and DAEM operations. As it can be noticed, within some hours (see hours 10, 11 of Fig. 3) the consumer buys an energy amount higher than the expected demand, reselling the difference in the DAEM. This opportunity allows to optimize the electricity procurement, and to implement a trading strategy as well, as explained below. The consumer has signed a long term bilateral contract which partially cover the electricity need. Since the bilateral contract price is known in advance, the consumer is preserved from the risk related to market price fluctuations. On the other hand, the consumer may pay the energy at a higher price than the DAEM price. The aforementioned trading strategy moves in the direction of recovering this disadvantage. In fact, Fig. 4 shows that the consumer may buy through the contract as much energy as possible, in the respect of constraint (2), within those hours when the contracted price is lower than the expected DAEM price (as an example see hours 11–14, 19–24 in Fig. 4). On the other hand, during some hours when the contracted price is higher than the expected DAEM price (hours 34–35, 57–58 and so on in Fig. 4), the consumer might choose to buy through bilateral contract rather than in the more convenient DAEM. The reason of this strange and apparently wrong behavior is due to the consumer’s obligation to satisfy the signed bilateral contract by the end of the contractual period. He thus prefers to comply this obligation by concentrating some of the purchases through contract over those hours when DAEM price is maximum. But even though the consumer purchases at a fixed price higher than DAEM price, he could resell during the hours with higher expected DAEM prices. 4.3. Risk management A second set of computational experiments has been carried out to evaluate the impact produced by the consumer’s risk aversion on the procurement plan. As explained above, the stochastic model accounts for risk aversion by means of the k

Fig. 2. Procurement through bilateral contract.

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Fig. 3. Procurement plan on February 26.

Fig. 4. Procurement plan through bilateral contract.

parameter, which weights risk versus the overall cost minimization. Fig. 5 shows solutions, in terms of weekly overall cost and semi-deviation, obtained for different values of the k parameter. Each solution corresponds to a ‘‘non dominated’’ procurement plan, i.e. no other alternative can be more convenient and less risky at the same time. The analysis of the results proves that a conservative attitude (high values of k) corresponds to a potentially more expensive procurement plan, while a more risky strategy can lead to more convenient solutions, but can expose the consumer to higher losses. This result is coherent with operator’s policy. For example, a ‘‘low-risk’’ consumer might wish to avoid speculative trading operations and rather schedule procurements through bilateral contracts following the expected demand.

Fig. 5. Weekly expected cost versus risk for different values of k.

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Fig. 6. Procurement plan for Case A and Case B.

Table 3 Monthly procurement cost (€) for different strategies and electricity quantity bilaterally contracted. Approach

Case A

Case B

Myopic Deterministic Stochastic

180,697 156,407 142,369

174,703 153,470 138,228

This analysis on possible solutions does provide the consumer with an effective tool for evaluating the best trade-off between risk and overall cost, according to his requirements and strategies. 4.4. Comparison with other decision strategies A last set of numerical experiments has been carried out with the aim of comparing the economical effectiveness of the proposed decision approach with respect to other alternative strategies that a consumer might wish to adopt. In particular, we have considered two alternative policies. The first one (in the remainder ‘‘myopic’’ strategy) relies on the idea to uniformly spread the electricity amount of the bilateral contract over the hours of the same time block for the entire planning horizon. Fig. 6 illustrates the procurement plan related to this strategy for two levels of energy bilaterally contracted, respectively equal to 100% and 60% of the expected demand (in te remainder Case A and Case B). In the second case, the myopic strategy suggests the consumer to buy the remaining electricity quantity in the DAEM. The second benchmark decision strategy relies on the solution of a deterministic mathematical model obtained by replacing the random parameters (market prices and consumer demand) with the corresponding expected values. The comparative analysis has been carried out by a back-testing simulation, by considering the real data observed over the entire month of February 2009. The performance in terms of overall costs, obtained by the proposed stochastic approach and by the two described alternative strategies (myopic and deterministic), have been finally evaluated and compared. Table 3 reports the monthly procurement costs related to the three procurement plans for the two levels of energy bilaterally contracted (100% and 60% of the expected demand) and for a fixed value of k equal to 0.75. As said before, the choice of this value is related to a quite conservative behavior, which should reflect the current practice. As evident from the results, the proposed stochastic programming approach outperforms the other ones, since uncertainty is explicitly taken into account. In fact, though at a first glance there is no need to operate in the DAEM in the case of electricity bilaterally contracted equal to 100% of the expected demand, nonetheless the stochastic approach allows lucrative trading operations which reduce the relating overall cost. As mentioned above, consumer might actually decide to purchase through bilateral contracts more electricity than the expected hourly demand within those hours characterized by high expected DAEM prices and to sell the difference in the DAEM. Conversely, the consumer might choose to cover electricity consumptions during other hours of the same time block by buying in the DAEM at a lower price. 5. Conclusion This paper has addressed the problem faced by a large electricity consumer in determining the optimal procurement plan over a short-term time horizon. The two main procurement channels are represented by bilateral contracts and participation in the Day-Ahead Electricity Market (DAEM). The inherent stochastic nature of the problem, mainly related to the uncertain hourly DAEM prices and to the electricity consumption, has been dealt by the stochastic programming modeling framework. In particular, a two-stage problem has been formulated with the aim of establishing the optimal amounts of electricity to be

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purchased through bilateral contracts and in the DAEM. Recourse actions are used to hedge against uncertainty related to electricity demands. The optimal plan is defined so to minimize the overall cost and to control risk related to the violation of budget constraints. The defined model represents the core of a dynamic approach relying on the iterative solution of the problem by considering more and more recent information and a planning horizon of decreasing length. Extensive numerica experiments have been carried out to evaluate the performance of the proposed decisional approach by considering a real test case. Results are very encouraging and provide evidence of the superiority of the proposed approach also in comparison with other alternative strategies. Appendix A. Scenario generation procedure This section illustrates the simulation technique implemented for scenario generation. As regards the uncertain evolution of electricity prices, we have adopted a Mean Reverting model, similar to the one proposed in [29]. This choice is motivated by the fact that this model incorporates the tendency of energy prices to gravitate towards a ‘‘normal’’ equilibrium price level, that is usually governed by a set of heterogeneous issues (production cost, level of global demand, market operators size). As demonstrated in Section 4.1, price forecasts obtained by this model are quite accurate. The evolution model adopted to forecast the electricity price can be formulated as follows:

  Pgh ¼ Pðg1Þh þ agh Pgh  Pðg1Þh þ vgh gh ;

ðA:1Þ

where price P gh at hour h on day g depends on price P ðg1Þh , at the same hour on the preceding day, plus a mean reverting component, agh ðPgh  Pðg1Þh Þ, and a random component vgh gh . In particular, agh represents the mean reversion rate, Pgh is the mean reversion level or long run equilibrium price, vgh is the volatility estimated up to day g at hour h, and gh is a normal . The values of agh ; P gh and vgh are calculated according to the procedure destandard random shock to price from g  1 to g tailed below. 1. At each hour h, prices varying with days during a given time interval preceding the planning period are considered. 2. Absolute daily changes of prices are calculated. 3. Using classical estimation techniques, like linear regression, model parameters are estimated. The mean reversion rate, agh , is the negative of the slope, obtained by regressing absolute price changes on the previous price levels. Moreover, the long run mean, P gh , is the intercept estimate of that regression divided by the mean reversion rate. Finally, the volatility of price, vgh , is given by the residual standard deviation. In order to determine jSj scenario prices P sgh , the random shock gh is extracted jSj times from a standard normal distribution. It is worthwhile to note that the proposed procedure is applied grouping working days and non-working days separately. As far as scenarios on electricity demand, they are derived starting from the hourly expected demand plus increments/ decrements of a given size (±2%, ±5%, ±8%, . . .), according with the required number of simulated values. It is important to outline that no explicit price-based demand response behavior has been considered, that is demand of a single consumer is considered independent of DAEM prices, which instead depend on the bids submitted by all market agents. Due to the mutual independence of consumer demand and DAEM prices, the whole scenario set is obtained by merging the single scenario sets through the Cartesian product. References [1] EURELECTRIC Working Group Trading, Regulatory Aspects of Electricity Trading in Europe. Available at: . [2] M. Carrión, A.B. Philpott, A.J. Conejo, J.M. Arroyo, A Stochastic programming approach to electric energy procurement for large consumers, IEEE Trans. Power Syst. 22 (2) (2007) 744–754. [3] Available at: . [4] Available at: . [5] Available at: . [6] J.M. Arroyo, A.J. Conejo, Optimal response of a thermal unit to an electricity spot market, IEEE Trans. Power Syst. 15 (3) (2000) 1098–1104. [7] S. de la Torre, J.M. Arroyo, A.J. Conejo, J. Contreras, Price maker self-scheduling in a pool-based electricity market: A mixed-integer LP approach, IEEE Trans. Power Syst. 17 (4) (2002) 1037–1042. [8] G.B. Sheblé, Decision analysis tools for GENCO dispatchers, IEEE Trans. Power Syst. 14 (2) (1999) 745–750. [9] A.J. Conejo, R. García-Bertrand, M. Carrión, Á. Caballero, A. de Andrés, Optimal involvement in futures markets of a power producer, IEEE Trans. Power Syst. 23 (2) (2008) 703–711. [10] R. Musmanno, N. Scordino, C. Triki, A. Violi, A multistage formulation for GENCOs in a multi-auction electricity market, IMA J. Manag. Math. 21 (2) (2010) 165–181. [11] L.P. Garces, A.J. Conejo, Weekly self-scheduling, forward contracting, and offering strategy for a producer, IEEE Trans. Power Syst., in press. Available at: doi:10.1109/TPWRS.2009.2032658. [12] B. Daryanian, R.E. Bohn, R.D. Tabors, Optimal demand-side response to electricity spot prices for storage-type customers, IEEE Trans. Power Syst. 4 (3) (1989) 897–903. [13] D.S. Kirschen, Demand-side view of electricity markets, IEEE Trans. Power Syst. 18 (2) (2003) 520–527. [14] C.-L. Su, D. Kirschen, Quantifying the effect of demand response on electricity markets, IEEE Trans. Power Syst. 24 (2009) 1199–1207. [15] M. Carrión, J.M. Arroyo, A.J. Conejo, A bilevel stochastic programming approach for retailer futures market trading, IEEE Trans. Power Syst. 24 (3) (2009) 1513–1522.

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P. Beraldi et al. / Applied Mathematical Modelling 35 (2011) 3980–3990

[16] D. Menniti, F. Costanzo, N. Scordino, N. Sorrentino, Purchase-bidding strategies of an energy coalition with demand-response capabilities, IEEE Trans. Power Syst. 24 (3) (2009) 1241–1255. [17] J.R. Birge, F.V. Louveaux, Introduction to Stochastic Programming, Springer Verlag, New York, 1997. [18] S.W. Wallace, W.T. Ziemba (Eds.), Applications of Stochastic Programming, MPS-SIAM Series in Optimization, 2005. [19] R. Dahlgren, C.C. Liu, J. Lawarree, Risk assessment in energy trading, IEEE Trans. Power Syst. 18 (2) (2003) 503–511. [20] K.L. Lo, Y.K. Wu, Risk assessment due to local demand forecast uncertainty in the competitive supply industry, in: IEEE Proc. Gener. Transm. Distrib., 2001, pp. 573–581. [21] P. Fusaro, Energy Risk Management, McGraw-Hill, New York, 1998. [22] R.B. Porter, Semivariance and stochastic dominance, Am. Econ. Rev. 64 (1974) 200–204. [23] W. Ogryczak, A. Ruszczynski, On consistency of stochastic dominance and mean-semideviation models, Math. Prog. 89 (2) (2001) 217–232. [24] S. Ahmed, Convexity and decomposition of mean-risk stochastic programs, Math. Prog. 89 (2) (2001) 217–232. [25] P. Artzner, H. Delbaen, J.M. Eber, H. Heart, Coherent measures of risk, Math. Finan. 9 (1999) 203–228. [26] R. Rockafellar, S. Uryasev, Optimization of conditional value at risk, J. Risk 2 (2000) 21–41. [27] J. Dupacová, G. Consigli, S.W. Wallace, Scenarios for multistage stochastic programs, Ann. Oper. Res. 100 (2000) 25–53. [28] M. Kaut, S. Wallace, Evaluation of scenario generation methods for stochastic programming, Pac. J. Optim. 3 (2) (2007) 257–271. [29] C. Blanco, D. Soronow, Mean reverting processes – energy price processes used for derivatives pricing & risk management, Commod. Now (2001) 68– 72. [30] Aimms, Paragon decision Technology. Available at: . [31] D. Menniti, A. Burgio, A. Pinnarelli, N. Scordino, N. Sorrentino, G. Costanzo, The energy markets from a small scale consumer perspective: the CRETA experience, in: Proc. 5th Int. European Electricity Market Conf. Lisbon, 2008.