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Robust optimization based price-taker retailer bidding strategy under pool market price uncertainty
Electrical Power and Energy Systems 73 (2015) 955–963
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Robust optimization based price-taker retailer bidding strategy under pool market price uncertainty Sayyad Nojavan, Behnam Mohammadi-Ivatloo ⇑, Kazem Zare Faculty of Electrical and Computer Engineering, University of Tabriz, P.O. Box: 51666-15813, Tabriz, Iran
a r t i c l e
i n f o
Article history: Received 17 May 2014 Received in revised form 3 December 2014 Accepted 16 June 2015
Keywords: Price-taker retailers Mixed-integer linear programming Optimal bidding strategy Robust optimization approach
a b s t r a c t In the restructured electricity markets, retailers purchase the required demand of its consumers from different energy resources such as self-generating facilities, bilateral contracts and pool market. In this process, the pool market price uncertainty modeling is important for obtaining the maximum profit. Therefore, in this paper, a robust optimization approach is proposed to obtain the optimal bidding strategy of retailer, which should be submitted to pool market. By the proposed method, a collection of robust mixed-integer linear programming problem (RMILP) is solved to build optimal bidding strategy for retailer. For pool market price uncertainty modeling, upper and lower limits of pool prices are considered instead of the forecasted prices. The range of pool prices are sequentially partitioned into a successive of nested subintervals, which permit formulating a collection of RMILP problems. The results of these problems give sufficient data to obtain optimal bidding strategy for submit to pool market by retailer. A detailed analysis is utilized to delineate the proposed method. Ó 2015 Elsevier Ltd. All rights reserved.
Introduction In an electricity market, the retailers purchase electricity from the wholesale market and sell it to the end-use customers [1]. The challenge for retailers is that they have to buy energy at a variable price on the different markets such as pool market, bilateral agreement and self-production facility and then sell it at a fixed price at the retail level to customers [2]. Therefore, the pool prices uncertainty modeling is important for retailers to purchase electric energy from different markets with the lowest cost to obtain maximum profit.
Retailers of electrical energy Consumers whose peak demand is at least a few hundred kilowatts may be able to save significant amounts of money by employing specialized personnel to forecast their demand and trade in the electricity markets to obtain lower prices. Such consumers can be expected to participate directly and actively in the markets. On the other hand, such active trading is not worthwhile for smaller consumers. These smaller consumers usually prefer to purchase at a constant price per kilowatt-hour that is adjusted at ⇑ Corresponding author. Tel./fax: +98 413 3300829. E-mail addresses: [email protected] (S. Nojavan), bmohammadi@ tabrizu.ac.ir (B. Mohammadi-Ivatloo), [email protected] (K. Zare). http://dx.doi.org/10.1016/j.ijepes.2015.06.025 0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.
most a few times per year. Electricity retailers are in business to bridge the gap between the wholesale market and these smaller consumers. The retailers are buying energy at a variable price from the pool market and selling it at a fixed price at the retail level. A retailer will typically lose money during periods of high prices because the price it has to pay for energy is higher than the price at which it resells this energy. On the other hand, during periods of low prices it makes a profit because its selling price is higher than its purchase price. Uncertainty of the available data brings some challenges for a retailer in making decisions for selection of the energy sources. In this paper, it is assumed that the retailer has sufficient data about the demand of the consumers, structure of the bilateral agreements, and cost function of its own producing facility. Likewise, the pool price is considered as an uncertain parameter and it is supposed that the retailer has information about the lower and upper bounds of the pool price (price confidence intervals).
Retailer model as client modeling According to worthy reference [2], a retailer must procure the required electricity demand of its clients in each period of the planning horizon. Usually, retailers sell energy to their clients at a price that remains stable during a pre-specified planning horizon. Every retailer fixes a selling price, and the potential clients select their
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Nomenclature Indices h i j l
Rdown j index of blocks of cost function for self-production facility index of period (h) index of the self-production facilities index of bilateral agreement
Parameters B total number of the bilateral agreements Di load at period i (MW) PMAX j;h N Ndg Pmax l;i Pmin l;i SDG j;h T kmin i kmax i ksi kl;i Rup j
size of block h of jth unit of the self-production facility output (MW h) number of the generation blocks of the self-production facility number of the self-production facility maximum capacity relating to agreement l at period i (MW) minimum capacity relating to agreement l at period i (MW) related costs of block h of jth unit of the self-production facility ($/MW h) total planning horizon (h) lower bound of the pool price at period i ($/MW h) upper bound of the pool price at period i ($/MW h) selling price to customers at period i ($/MW h) electricity price of agreement l at period i ($/MW h) ramping up limit of jth unit of the self-production facility (MW/h)
electricity supplier according to the prices offered by all available retailers. It is assumed that clients behave elastically with respect to the selling price offered by the retailer, which means that if the selling price is too high, clients will choose a rival retailer for their electricity supply. Literature review There is respectable literature describing retailer participation in electricity markets, particularly for their purchase of electrical energy. Information about the retailers is required to participate in electricity energy markets are conversed in [3]. In [4], the midterm benefit maximization problem with considering the risk analysis faced by a retailer is discussed, where the probability distribution functions is used for modeling uncertainty of price and load. A mathematical method based on mixed integer stochastic programming is proposed in [5] to decide the optimal contract price of retailer with consumers and the electricity procurement policy of a retailer for a specified period. A load profile clustering techniques for optimal price offering to customers for maximizing the profit of a retailer is proposed in [6]. In [7], an integrated inventory model with variable production rate and price-sensitive demand rate under two-level trade credit is investigated. The model considers two-level trade credit policy in which the retailer receives a full trade credit from its supplier, and offers partial trade credit to its customers. In [8], an optimal strategy for a retailer is presented that procures its electric energy in an electricity market, involving both pool and bilateral contract considering medium and short-term decisions. The medium-term decision making has been formulated as a stochastic optimization problem due to the uncertainties of pool price and demand of consumers and Monte Carlo
MUT j MDT j
ramping down limit of jth unit of the self-production facility (MW/h) minimum up time of jth unit of the self-production facility (h) minimum down time of jth unit of the self-production facility (h)
Variables U dg binary variable, which is equal to 1, if jth unit of the j;i self-production facility is selected at period i; otherwise, it is zero X on time duration for which the jth unit is ON at time i (h) j;i X off j;i
time duration for which the jth unit is OFF at time i (h)
PDG j;h;i
power relating to block h of jth unit of the self-production facility at period i (MW h) purchased power from the pool market at period i (MW) purchased power from the bilateral agreement l at period i (MW) produced power from the jth unit of the self-production facility at period i (MW) total purchased electricity from the bilateral agreements at period i (MW) binary variable, equivalent to 1 if bilateral agreement l is selected, and 0 otherwise pool price at period i ($/MW h)
PP;i Pl;i Pg;j;i Pb;i sl ki
simulation is adapted to solve it. A stochastic midterm framework is proposed in [9] for an electricity retailer including objective functions like expected value of the profit and expected downside risk allowing them to decide their optimal level of involvement in forward contracting and in the pool as well as deriving optimal selling prices for clients. In [10], a stochastic linear programming model is proposed for constructing piecewise-linear bidding curves to be submitted to Nord Pool, which is the Nordic power exchange. In [11], a bi-level programming approach is presented to solve the medium-term decision-making problem faced by a power retailer. A retailer decides its level of involvement in the pool and in the future markets in addition to the selling price submitted to its potential customer with the goal of maximizing the expected profit at a given risk level. When an electricity retailer faces volume risk in meeting load and spot price risk in purchasing from the wholesale market, conventional risk management optimization methods can be quite inefficient. For the management of an electricity contract portfolio in this context, [12] has developed a multistage stochastic optimization approach, which accounts for the uncertainties of both electricity prices and loads. The conditional-value-at-risk (CVaR) is used as a risk measure to optimize hedging across intermediate stages in the planning horizons. Also, in [13], a decision-making framework based on stochastic programming is proposed for a retailer: (1) to decide the sale price of electricity to the consumers based on time-of-use (TOU) rates and (2) to manage a portfolio of different contracts in order to procure its demand and to hedge against risks, within a medium-term period. In [14], a genetic algorithm is proposed for determining the optimal bidding strategy of a retailer who supplies electricity to end-users in the short-term electricity market. The aim is to minimize the cost of purchasing
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energy in the sequence of trading opportunities that provide the day-ahead and intraday markets. In this paper, we have considered a medium-term planning that includes four weeks (28 days). In order to attain tractability, every day of the planning horizon is separated into three eight-hour periods, in particular valley, shoulder and peak. This constitutes a proper tradeoff between result accuracy and computational burden. The considered time horizon contains 84 (28 3 = 84) periods. Also, the optimal offering strategy curve achieved using the proposed robust optimization approach considering the pool market price uncertainty. In this curves, the offering price and the quantity blocks achieved for the retailer to bid to the pool market. These curves make the essential information required by a retailer to bid prosperously consumers demand in a pool market.
Procedure Due to uncertain nature of pool prices in pool market, uncertainty modeling is important for retailers. This uncertainty implies that the final procurement cost is unknown for the retailer. Whereas estimation are ‘‘always wrong’’, we rely on lower and upper limits for price estimated, and not on actual estimated [15]. In addition, we combine these lower and upper limits for price estimated with robust optimization approaches (as described in [16]) to obtain electricity procurement strategy. The electricity procurement strategy can be formulated as a MILP problem take into account price estimated. If, on the other side, lower and upper limits for pool price are regarded instead of price estimated, the electricity procurement strategy can be remodeled as an RMILP problem [16]. This approach, with considering the lower and upper limit definition for pool market price, leads to robust decision within the defined price bounds. This robustness decision is necessary and contribution for many retailers.
Model In this section, the developed electricity procurement strategy model is explained. Method description To achieve a robust electricity procurement strategy, a number of RMILP problems are solved consecutively, which are described in the following. (1) Firstly, the hourly prices are set to their minimum values (lower levels of the price bound) and the standard MILP problem is simulated to derive the hourly electricity procurement strategy from the pool market. (2) Then, starting from lower level to upper level, the related RMILP problem is solved for each interval within the price bound to achieve hourly electricity acquisition strategy from different resources (pool market, bilateral agreements and self-production facility). The electricity procurement from pool market is smaller than or equal to the maximum electricity procurement achieved solving the MILP problem in item (1) above without considering price bound. (3) Finally, the considered hourly price bounds are sequentially increased until the whole price bound are included; the resulting of some RMILP problems fined an optimal electricity procurement sequentially smaller and smaller that correspond to bigger and bigger prices at pool market. Finally, the considered hourly price intervals are successively enlarged until the whole confidence intervals are covered; the resulting collection of RMILP problems provides a sequence of optimal electricity procurement successively smaller and smaller that correspond to bigger and bigger prices at pool market. Using the sequence of prices and electricity procurement strategy from different resources allows the retailer for building hourly electricity procurement and optimal bidding strategy curves.
Contributions In this paper, a robust optimization based framework is proposed for optimal bidding and electricity procurement strategy of a retailer for first time. In comparison with exhausting stochastic programming methods, the robust optimization approach results in an effective, robust solution with low computational burden. With considering the above framework, the contributions of this paper are summarized as follows: (1) Presenting a method to achieve electricity procurement strategy for retailer considering lower and upper limit for prices, not price estimated; (2) To propose robust optimization approach for solving the problem in item 1) above; (3) The electricity procurement strategy could be formulated as a RMILP problem where the global optimal energy procurement is guaranteed.
Paper organization The following of the paper is categorized as following. The proposed robust formulation for retailer is presented in section ‘Model’. The used approach flowchart to achieve electricity procurement strategy is provided in section ‘The proposed algorithm to derive optimal bidding strategy’. Section ‘Case study’ contains case study, results, discussions and assessments in details. The conclusion of this paper is presented in section ‘Conclusion’.
Robust formulation In this section, the mathematical formulation of the acquisition problem for retailer using the robust optimization approach is presented. The profit function of retailer consists of the revenue from sales to the customers minus the electricity acquisition costs from different resources. These components are calculated as below.
Maximize P l;i ;P p;i ;PDG j;h;i
T X i¼1
ksi Di
( ) N dg N B X T T T X X X X X DG DG kl;i Pl;i þ ki Pp;i þ Sj;h Pj;h;i l
i¼1
i¼1
i¼1 j¼1 h¼1
ð1Þ The first item of Eq. (1) considers the revenue of retailer by selling purchased power to customers in fixed price in retail level. Also, the second item of Eq. (1) represents the cost of providing electricity from different resources such as bilateral agreements, pool market purchase and self-production facility. Energy purchased cost from bilateral agreements is composed in second term of Eq. (1). The electricity acquisition cost from the pool market is presented in third term of Eq. (1). The finally term of Eq. (1) is related to operation cost of the self-production facility. A three-block piecewise linear curve is used for operation cost modeling of the self-production facility, which is shown in Fig. 1. The following constraints should be considered in optimal operation of the retailer:
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Cost ($)
P2,DG i
P1,DG i
purchased power from pool market, bilateral contracts and self-production facility. Therefore, the maximizing objective function (1) is equivalent to minimizing objective function (11). Finally, the objective function of electricity acquisition problem for retailer can be reformulated as following.
P3,DG i
( !) N dg N T B X X X X DG s Minimize ki Pl;i þ P j;h;i þ Pp;i
S3DG
P l;i ;P p;i ;P DG j;h;i
S 2DG S
P1MAX
P2MAX
P3MAX
Power
Fig. 1. Operation cost model of self-generating facility.
N dg N B X X X DG Pl;i þ Pj;h;i þ P p;i ¼ Di
i ¼ 1; . . . ; T
ð2Þ
j¼1 h¼1
max Pmin i ¼ 1; . . . ; T; l ¼ 1; . . . ; B l;i sl < P l;i < P l;i sl ; 8
0 6 PDG j;h;i 6
ð3Þ
n o dg MAX PMAX j;h P j;h1 U j;i ; 8 i ¼ 1; . . . ; T;
i¼1
i¼1
ð11Þ
i¼1 j¼1 h¼1
In this optimization problem, it is important to attention that objective function (11) and constraints (2)–(10) as optimization (2)–(11) is meaningful provided that pool market prices ki ði ¼ 1; . . . ; TÞ are input parameter data and the corresponding values are constant. But, the pool market prices ki ði ¼ 1; . . . ; TÞ are random variables, therefore, the optimization problem requires to be replaced by a cost distribution function, for example with the expected amount. In this research, pool market prices are assumed to be unknown variables within known confidence intervals. The objective function (11) with its related constraints (2)–(10) is an MILP formulation, which can be stated as standard form
Minimize xi ;8j
h ¼ 2; . . . ; N; j ¼ 1; . . . ; Ndg
j¼1 h¼1
l¼1
( ) N dg N B X T T T X X X X X DG DG þ kl;i Pl;i þ ki P p;i þ Sj;h Pj;h;i l
DG 1
l¼1
i¼1
ð4Þ
n X ej xj
ð12Þ
j¼1
Subject to MAX dg 0 6 PDG j;1;i 6 P j;1 U j;i ; 8 i ¼ 1; . . . ; T; j ¼ 1; . . . ; N dg
ð5Þ
ð6Þ
j¼1 h¼1 N N X X up dg PDG PDG j;h;i j;h;i1 6 Rj U j;i ; 8 i ¼ 1; . . . ; T; j ¼ 1; . . . ; N dg
ð7Þ
h¼1
N N X X down PDG P DG U dg j;h;i1 j;h;i 6 Rj j;i ; 8 i ¼ 1; . . . ; T; j ¼ 1; . . . ; N dg h¼1
m ¼ 1; . . . ; M
ð13Þ
j¼1
Ndg N X X DG Pj;h;i þ Pp;i P 0; 8 i ¼ 1; . . . ; T
h¼1
n X amj xj 6 bm ;
h¼1
ð8Þ dg dg ½X on j;i1 MUT j ½U j;i1 U j;i P 0; 8 i ¼ 1; . . . ; T; j ¼ 1; . . . ; N dg
ð9Þ dg dg ½X off j;i1 MDT j ½U j;i U j;i1 P 0; 8 i ¼ 1; . . . ; T; j ¼ 1; . . . ; N dg
ð10Þ In these constraints, constraint (2) represented the load power balance with the energy electricity from different resources. In constraint (3), the lower and upper electrical energy limit procured from bilateral agreements should be between the acceptable bounds as presented. The operation constraints related to self-production facility is expressed in Eqs. (4)–(10). Also, constraint (6) declares that the sum of the electricity procurement by the self-production facility and the electricity purchased from the pool market must be positive for all time periods. It should be mentioned that the maximum power of electrical energy that the retailer can sell in the pool market is limited by the maximum capacity of the self-production facility. Furthermore, constraints (7) and (8) describe the ramping up/down rate limits of self-production facility. Finally, the minimum up/down time constraints are expressed as (9) and (10), respectively. Finally, it should be mentioned that in objective function (1), Di in first item could be replaced based on Eq. (2) that it is the sum of
xj P 0;
j ¼ 1; . . . ; n
ð14Þ
xj 2 f0; 1g for some j ¼ 1; . . . ; n
ð15Þ
In Eq. (12), ej are coefficients of the objective function and are assumed to be known. If they be unknown parameters with known lower and upper bounds, a significant RMILP optimization related to (12)–(15) can be reformulated. To perform so, if it is assumed that dj shows the deviance from the nominal coefficient ej ; therefore, all coefficient ej takes amount in the interval ½ej ; ej þ dj . Moreover, for building an RMILP formulation, it is essential to define C0 as an integer parameter for controlling the level of robustness in the objective function, which considers values in ½0; jJ 0 j, where J 0 ¼ fjjdj > 0g. For ignoring the effect of the cost deviations in the objective function, C0 should be set zero, while if C0 ¼ jJ 0 j, we considered all deviations of cost function. The form of the RMILP optimization related to optimization (12)–(15) is
Minimize xi ;qoj ;yj ; 8j;z0
n n X X ej xj þ z0 C0 þ qoj j¼1
ð16Þ
j¼1
Subject to
Constraints ð9Þ—ð11Þ
ð17Þ
z0 þ qoj P dj yj ;
ð18Þ
qoj P 0; yj P 0;
j 2 J0
j ¼ 1; . . . ; n j ¼ 1; . . . ; n
z0 P 0 xj 6 y j ;
ð19Þ ð20Þ ð21Þ
j ¼ 1; . . . ; n
ð22Þ
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The objective function (16) and related constraints (17)–(22) above is achieved considering duality theorem and linearizing technique, for more details on the obtaining robust counterpart of an MILP problem please see [16]. Variables z0 and qoj are dual variables of the optimization (12)–(15) used to consider the known bounds of coefficients ej . Also, to achieve the corresponding linear declarations, yj is used as auxiliary variable. Most information about the robust optimization approach and explanation about of how to achieve these equations are clarified in [16]. Eventually, the RMILP problem of electricity procurement strategy for retailer can be formulated as
Minimize Pl;i ;P p;i ;P DG j;h;i
( !) Ndg N T B X X X X DG ksi Pl;i þ Pj;h;i þ Pp;i i¼1
þ
( B X T X l
l¼1
i¼1
þ z 0 C0 þ
j¼1 h¼1
N dg N T T X X X X DG DG kl;i P l;i þ ki Pp;i þ Sj;h Pj;h;i i¼1
T X qoi
) Fig. 2. Nested price intervals to derive electricity procurement strategy problems.
i¼1 j¼1 h¼1
P kp;i ; P kl;i ; PDG;k j;h;i ði ¼ 1; . . . ; TÞ. The electricity procurement and optimal bidding strategy for retailer in all periods are computed using the prices in all iterations ðkki ; 8kÞ and the electricity procurement in all iterations ðPkp;i ; P kl;i ; PDG;k j;h;i ; 8kÞ results.
ð23Þ
i¼1
Subject to
Constraints ð2Þ ð10Þ
ð24Þ
z0 þ qoi P di yi ;
ð25Þ
Fig. 3 shows the flowchart of the proposed algorithm.
qoi P 0; yi P 0;
i ¼ 1; . . . ; T
i ¼ 1; . . . ; T i ¼ 1; . . . ; T
z0 P 0 Pp;i 6 yi ;
ð26Þ ð27Þ ð28Þ
i ¼ 1; . . . ; T
ð29Þ
Once the robust electricity procurement strategy problem (23)–(29) has been formulated, the algorithm described in the following subsection will be used to extract electricity procurement strategy of the retailer.
The proposed algorithm to derive optimal bidding strategy The following algorithm is used to derive electricity procurement and optimal bidding strategy of the retailer. (1) Set prices ki ¼ kmin ði ¼ 1; . . . ; TÞ, and C0 ¼ T (all periods, i.e., i 84) to consider all possible deviation of pool market prices. k
(2) Set di ¼ Gk ðkmax kmin Þ; ði ¼ 1; . . . ; TÞ, where Gk is a coeffii i cient that uses increasing values in ½0; 1 and k shows the counter of iteration. It should be mentioned that parameter Gk enables deriving a succession of nested sub-bounds of the ; kmin þ di . For clarifying purposes, Fig. 2 illustrates form ½kmin i i a pattern including three various nested sub-bounds, namely (a), (b), and (c). (3) RMILP optimization (23)–(29) is simulated to achieve the hourly electricity procurement strategy at iteration k; Pkp;i ; Pkl;i ; PDG;k j;h;i ði ¼ 1; . . . ; TÞ. (4) For covering the all range of coefficient Gk , it should repeated iteratively (categorized byk) steps 2–3 above. The increasing step is d > 0 as illustrated in Fig. 3. (5) Construct the hourly electricity procurement strategy using the achieved results. An offering prices, kki ¼ kmin þ i k di ði ¼ 1; . . . ; TÞ are computed using each iteration k , and an electricity procurement level per time period,
Case study In this part, a case study is used to show the application of the robust optimization approach for building electricity procurement and optimal bidding strategy curves. Data In this paper, a single day is considered with its demand deconstructed into three load levels marked as peak, shoulder, and valley, as categorized in Table 1. The time horizon of this study contains 84 periods (four weeks). In this study, twelve bilateral agreements are considered; Table 2 presents data for bilateral agreements, which include the maximum and minimum amount of electric power with related prices. The selection decision of these bilateral agreements should be made at the beginning of the study horizon. There are four available bilateral agreements for the whole month and two contracts for each week. Six contracts are available for all load levels and six contracts only for the peak load levels. Table 3 presents the periods of the bilateral agreements. The self-production facility data are presented in Table 4. It should be mentioned that the single self-facility generation is considered in this simulation. The load profile for the study horizon is shown in Fig. 4. Also, the lower and upper bound of pool market price data for the study horizon is shown in Fig. 5. Results and discussion The electricity energy procurement strategy problem is formulated for retailer as a RMILP problem and solved using CPLEX solver [17] in GAMS software [18] on Intel(R) Celeron(R) CPU at 2 GHz and 1 GB of RAM. The 20 RMILP problems required to construct the electricity procurement strategy curves. The computing time for solving the 20 RMILP problems is 5.42 s. The simulation result of minimizing Eq. (11) with considering the lower level price of pool market price values is $8,057,616.00. This value is the minimum acquisition cost. The simulation results show that the retailer should purchase 73% of
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Fig. 3. Algorithm flowchart.
Table 3 Bilateral contracts usage periods.
Table 1 Classification of daily load levels. Level
Hours of the day
Contract #
Usage period
Validity level
Valley (V) Shoulder (S) Peak (P)
1, 2, 3, 4, 5, 6, 7, 8 9, 10, 15, 16, 17, 18, 23, 24 11, 12, 13, 14, 19, 20, 21, 22
1 2 3 4 5 6 7 8 9 10 11 12
Month Month Month Month Week one Week one Week two Week two Week three Week three Week four Week four
V, P V, P V, P V, P V, P V, P
Table 2 Bilateral contracts specification. Contract number
Min. (MW)
Max. (MW)
Price ($/MW h)
1 2 3 4 5 6 7 8 9 10 11 12
15 10 15 10 25 20 25 20 25 20 25 20
30 20 30 20 50 40 50 40 50 40 50 40
39.96 43.29 49.86 48.69 32.31 41.40 51.03 47.88 42.66 57.69 52.02 57.96
its electricity needs from the pool market, 7.2% from bilateral agreements, and 19.8% from the self-production facility. Considering the input parameter data, the RMILP problem (23)– (29) is solved 20 for iterations (Gk replaces in steps of amount d ¼ 0:05) to produce the needed data to construct optimal bidding strategy curves using the presented algorithm in section ‘The proposed algorithm to derive optimal bidding strategy’. The robust objective function (23) with constraints (24)–(29) is solved considering iteration value of Gk with a fixed step that
S, P S, P S, P S, P S, P S, P
V: valley; S: shoulder; P: peak.
actual pool market prices are increases related to the lower pool market prices. Figs. 6–8 are shown the results of simulation. These figures are indicated the level of robustness and percentage of acquisition from various resources such as limited self-production facility, bilateral agreements and pool market purchase. In Figs. 6–8, the minimum robust cost is related to total procurement cost considering the minimum pool market prices (lower level). Then, starting from lower level to upper level, the related RMILP problem is solved for each interval within the price bound to achieve robust cost. The robust cost is bigger than or equal to the minimum robust cost. Also, Figs. 6–8 features that the electricity procurement percent decreases from the pool market with increasing robust cost function, while acquisition from bilateral agreements and the self-production facility increase with increasing robust cost function. The results of robust cost function are interesting, because retailer purchases more of its electricity
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Table 4 Data for the self-generating facility. Capacity Minimum power output SDG 1
100 0 33
MW MW $/MW h
SDG 2
36
$/MW h
SDG 3
39
$/MW h
P MAX 1
40
MW
P MAX 2
75
MW
P MAX 3 MUT j MDT j
100
MW
Rup j
2 2 50
h h MW
Rdown j
50
MW
Fig. 6. Optimal procurement from the pool as a function of robust cost.
Fig. 4. Load profile of the consumer for study horizon.
Fig. 7. Optimal procurement from bilateral contracts as a function of robust cost.
Fig. 5. Confidence intervals for pool price at study horizon.
required from the markets whose prices have no uncertainty. In other words, retailer for management of pool prices uncertainty purchases more of its electricity required from bilateral agreements and self-production facility. Note that we consider a price-taker retailer, believing that his energy bid cannot influence the market clearing price. In other words, if the retailer offers either at zero or at $1000/MW h, market
Fig. 8. Optimal procurement from the self-generating facility as a function of robust cost.
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prices remain unaltered. In the bidding problem, retailer should determine two decisions as the offered price and the offered quantity (power). Hence, the objective of this study is determining the offered priced and power for obtaining the maximum profit and winner in power market based on step-wise bidding strategy. Therefore, the step-wise bidding strategy help the retailer for obtaining the more profit under pool market price uncertainty. Therefore, Figs. 9–11 show the selected optimal offering curves achieved using the proposed approach. In this curves, the x-axis considered the offering price whereas the y-axis shows the quantity blocks that the retailer bids to the pool market. These curves make the essential information required by a retailer to bid prosperously consumers demand in a pool market. It should be mentioned, these curves change meaningfully from time period to time period. Furthermore, Fig. 9 is related to optimal bidding curve for peak load level in 25th day. Also, Figs. 10 and 11 are related to optimal bidding curve for shoulder and valley load levels in 12th and 6th days, receptivity. Fig. 9. Optimal bidding curve for peak load level in 25th day.
Conclusion
Fig. 10. Optimal bidding curve for shoulder load level in 12th day.
In this research, we studied the electricity acquisition problem faced by a retailer from alternative resource such as self-production, bilateral agreements, and the pool market. Also, we recognized a medium-term planning that may include from one month (four weeks), and, in order to attain tractability, every day of the planning horizon is separated into three eight-hour periods, in particular valley, shoulder and peak. For considering the uncertainty nature of pool market prices, this research presents an application of robust optimization technique to appraise value choice acquisition strategies for retailers with multiple procurement options. Upper and lower limits of pool prices are considered for uncertainty modeling, rather than using forecast prices as input data. The range of pool prices are sequentially partitioned into a successive of nested subintervals, which permit formulating a collection of RMILP problems. The results of these problems give sufficient data to obtain electricity procurement strategy and optimal bidding strategy to pool market for retailers. Also, these curves make the essential information required by a retailer to bid prosperously consumers demand in a pool market. Finally, retailer for management of pool market prices uncertainty purchases more of its electricity required from bilateral agreements and self-production facility. References
Fig. 11. Optimal bidding curve for valley load level in 6th day.
[1] Shahidehpour M, Yamin H, Li Z. Market operations in electric power systems. New York: Wiley; 2002. [2] Conejo AJ, Carrión M, Morales JM. Decision making under uncertainty in electricity markets, vol. 153. Springer; 2010. [3] Kirschen Daniel S. Demand-side view of electricity markets. IEEE Trans Power Syst 2003;18(2):520–7. [4] Gabriel SA, Conejo AJ, Plazas MA, Balakrishnan S. Optimal price and quantity determination for retail electric power contracts. IEEE Trans Power Syst 2006;21(1):180–7. [5] Hatami AR, Seifi H, Sheikh-El-Eslami MK. Optimal selling price and energy procurement strategies for a retailer in an electricity market. Electr Power Syst Res 2009;79:246–54. [6] Mahmoudi-Kohan N, Parsa Moghaddam M, Sheikh-El-Eslami MK, Shayesteh E. A three-stage strategy for optimal price offering by a retailer based on clustering techniques. Int J Electr Power Energy Syst 2010;32:1135–42. [7] Hardik NS, Kamlesh AP. Optimal strategy for an integrated inventory system involving variable production and defective items under retailer partial trade credit policy. Decis Support Syst 2012;54:235–47. [8] Nazari M, Akbari Foroud A. Optimal strategy planning for a retailer considering medium and short-term decisions. Int J Electr Power Energy Syst 2013;45:107–16. [9] Ahmadi A, Charwand M, Aghaei J. Risk-constrained optimal strategy for retailer forward contract portfolio. Int J Electr Power Energy Syst 2013;53:704–13.
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[14] Herranz R, San Roque AM, Villar J, Campos FA. Optimal demand-side bidding strategies in electricity spot markets. IEEE Trans Power Syst 2012;27(3): 1204–13. [15] Conejo AJ, Contreras J, Espinola R, Plazas MA. Forecasting electricity prices for a day-ahead pool-based electric energy market. Int J Forecast 2005;21(3): 435–62. [16] Bertsimas D, Sim M. Robust discrete optimization and network flows. Math. Prog. Ser. B 2003;98:49–71. [17] The GAMS Software Website; 2014.. [18] Brooke A, Kendrick D, Meeraus A. GAMS user’s guide. Redwood City (CA): The Scientific Press; 1990.
Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Robust optimization based price-taker retailer bidding strategy under pool market price uncertainty Sayyad Nojavan, Behnam Mohammadi-Ivatloo ⇑, Kazem Zare Faculty of Electrical and Computer Engineering, University of Tabriz, P.O. Box: 51666-15813, Tabriz, Iran
a r t i c l e
i n f o
Article history: Received 17 May 2014 Received in revised form 3 December 2014 Accepted 16 June 2015
Keywords: Price-taker retailers Mixed-integer linear programming Optimal bidding strategy Robust optimization approach
a b s t r a c t In the restructured electricity markets, retailers purchase the required demand of its consumers from different energy resources such as self-generating facilities, bilateral contracts and pool market. In this process, the pool market price uncertainty modeling is important for obtaining the maximum profit. Therefore, in this paper, a robust optimization approach is proposed to obtain the optimal bidding strategy of retailer, which should be submitted to pool market. By the proposed method, a collection of robust mixed-integer linear programming problem (RMILP) is solved to build optimal bidding strategy for retailer. For pool market price uncertainty modeling, upper and lower limits of pool prices are considered instead of the forecasted prices. The range of pool prices are sequentially partitioned into a successive of nested subintervals, which permit formulating a collection of RMILP problems. The results of these problems give sufficient data to obtain optimal bidding strategy for submit to pool market by retailer. A detailed analysis is utilized to delineate the proposed method. Ó 2015 Elsevier Ltd. All rights reserved.
Introduction In an electricity market, the retailers purchase electricity from the wholesale market and sell it to the end-use customers [1]. The challenge for retailers is that they have to buy energy at a variable price on the different markets such as pool market, bilateral agreement and self-production facility and then sell it at a fixed price at the retail level to customers [2]. Therefore, the pool prices uncertainty modeling is important for retailers to purchase electric energy from different markets with the lowest cost to obtain maximum profit.
Retailers of electrical energy Consumers whose peak demand is at least a few hundred kilowatts may be able to save significant amounts of money by employing specialized personnel to forecast their demand and trade in the electricity markets to obtain lower prices. Such consumers can be expected to participate directly and actively in the markets. On the other hand, such active trading is not worthwhile for smaller consumers. These smaller consumers usually prefer to purchase at a constant price per kilowatt-hour that is adjusted at ⇑ Corresponding author. Tel./fax: +98 413 3300829. E-mail addresses: [email protected] (S. Nojavan), bmohammadi@ tabrizu.ac.ir (B. Mohammadi-Ivatloo), [email protected] (K. Zare). http://dx.doi.org/10.1016/j.ijepes.2015.06.025 0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.
most a few times per year. Electricity retailers are in business to bridge the gap between the wholesale market and these smaller consumers. The retailers are buying energy at a variable price from the pool market and selling it at a fixed price at the retail level. A retailer will typically lose money during periods of high prices because the price it has to pay for energy is higher than the price at which it resells this energy. On the other hand, during periods of low prices it makes a profit because its selling price is higher than its purchase price. Uncertainty of the available data brings some challenges for a retailer in making decisions for selection of the energy sources. In this paper, it is assumed that the retailer has sufficient data about the demand of the consumers, structure of the bilateral agreements, and cost function of its own producing facility. Likewise, the pool price is considered as an uncertain parameter and it is supposed that the retailer has information about the lower and upper bounds of the pool price (price confidence intervals).
Retailer model as client modeling According to worthy reference [2], a retailer must procure the required electricity demand of its clients in each period of the planning horizon. Usually, retailers sell energy to their clients at a price that remains stable during a pre-specified planning horizon. Every retailer fixes a selling price, and the potential clients select their
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Nomenclature Indices h i j l
Rdown j index of blocks of cost function for self-production facility index of period (h) index of the self-production facilities index of bilateral agreement
Parameters B total number of the bilateral agreements Di load at period i (MW) PMAX j;h N Ndg Pmax l;i Pmin l;i SDG j;h T kmin i kmax i ksi kl;i Rup j
size of block h of jth unit of the self-production facility output (MW h) number of the generation blocks of the self-production facility number of the self-production facility maximum capacity relating to agreement l at period i (MW) minimum capacity relating to agreement l at period i (MW) related costs of block h of jth unit of the self-production facility ($/MW h) total planning horizon (h) lower bound of the pool price at period i ($/MW h) upper bound of the pool price at period i ($/MW h) selling price to customers at period i ($/MW h) electricity price of agreement l at period i ($/MW h) ramping up limit of jth unit of the self-production facility (MW/h)
electricity supplier according to the prices offered by all available retailers. It is assumed that clients behave elastically with respect to the selling price offered by the retailer, which means that if the selling price is too high, clients will choose a rival retailer for their electricity supply. Literature review There is respectable literature describing retailer participation in electricity markets, particularly for their purchase of electrical energy. Information about the retailers is required to participate in electricity energy markets are conversed in [3]. In [4], the midterm benefit maximization problem with considering the risk analysis faced by a retailer is discussed, where the probability distribution functions is used for modeling uncertainty of price and load. A mathematical method based on mixed integer stochastic programming is proposed in [5] to decide the optimal contract price of retailer with consumers and the electricity procurement policy of a retailer for a specified period. A load profile clustering techniques for optimal price offering to customers for maximizing the profit of a retailer is proposed in [6]. In [7], an integrated inventory model with variable production rate and price-sensitive demand rate under two-level trade credit is investigated. The model considers two-level trade credit policy in which the retailer receives a full trade credit from its supplier, and offers partial trade credit to its customers. In [8], an optimal strategy for a retailer is presented that procures its electric energy in an electricity market, involving both pool and bilateral contract considering medium and short-term decisions. The medium-term decision making has been formulated as a stochastic optimization problem due to the uncertainties of pool price and demand of consumers and Monte Carlo
MUT j MDT j
ramping down limit of jth unit of the self-production facility (MW/h) minimum up time of jth unit of the self-production facility (h) minimum down time of jth unit of the self-production facility (h)
Variables U dg binary variable, which is equal to 1, if jth unit of the j;i self-production facility is selected at period i; otherwise, it is zero X on time duration for which the jth unit is ON at time i (h) j;i X off j;i
time duration for which the jth unit is OFF at time i (h)
PDG j;h;i
power relating to block h of jth unit of the self-production facility at period i (MW h) purchased power from the pool market at period i (MW) purchased power from the bilateral agreement l at period i (MW) produced power from the jth unit of the self-production facility at period i (MW) total purchased electricity from the bilateral agreements at period i (MW) binary variable, equivalent to 1 if bilateral agreement l is selected, and 0 otherwise pool price at period i ($/MW h)
PP;i Pl;i Pg;j;i Pb;i sl ki
simulation is adapted to solve it. A stochastic midterm framework is proposed in [9] for an electricity retailer including objective functions like expected value of the profit and expected downside risk allowing them to decide their optimal level of involvement in forward contracting and in the pool as well as deriving optimal selling prices for clients. In [10], a stochastic linear programming model is proposed for constructing piecewise-linear bidding curves to be submitted to Nord Pool, which is the Nordic power exchange. In [11], a bi-level programming approach is presented to solve the medium-term decision-making problem faced by a power retailer. A retailer decides its level of involvement in the pool and in the future markets in addition to the selling price submitted to its potential customer with the goal of maximizing the expected profit at a given risk level. When an electricity retailer faces volume risk in meeting load and spot price risk in purchasing from the wholesale market, conventional risk management optimization methods can be quite inefficient. For the management of an electricity contract portfolio in this context, [12] has developed a multistage stochastic optimization approach, which accounts for the uncertainties of both electricity prices and loads. The conditional-value-at-risk (CVaR) is used as a risk measure to optimize hedging across intermediate stages in the planning horizons. Also, in [13], a decision-making framework based on stochastic programming is proposed for a retailer: (1) to decide the sale price of electricity to the consumers based on time-of-use (TOU) rates and (2) to manage a portfolio of different contracts in order to procure its demand and to hedge against risks, within a medium-term period. In [14], a genetic algorithm is proposed for determining the optimal bidding strategy of a retailer who supplies electricity to end-users in the short-term electricity market. The aim is to minimize the cost of purchasing
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energy in the sequence of trading opportunities that provide the day-ahead and intraday markets. In this paper, we have considered a medium-term planning that includes four weeks (28 days). In order to attain tractability, every day of the planning horizon is separated into three eight-hour periods, in particular valley, shoulder and peak. This constitutes a proper tradeoff between result accuracy and computational burden. The considered time horizon contains 84 (28 3 = 84) periods. Also, the optimal offering strategy curve achieved using the proposed robust optimization approach considering the pool market price uncertainty. In this curves, the offering price and the quantity blocks achieved for the retailer to bid to the pool market. These curves make the essential information required by a retailer to bid prosperously consumers demand in a pool market.
Procedure Due to uncertain nature of pool prices in pool market, uncertainty modeling is important for retailers. This uncertainty implies that the final procurement cost is unknown for the retailer. Whereas estimation are ‘‘always wrong’’, we rely on lower and upper limits for price estimated, and not on actual estimated [15]. In addition, we combine these lower and upper limits for price estimated with robust optimization approaches (as described in [16]) to obtain electricity procurement strategy. The electricity procurement strategy can be formulated as a MILP problem take into account price estimated. If, on the other side, lower and upper limits for pool price are regarded instead of price estimated, the electricity procurement strategy can be remodeled as an RMILP problem [16]. This approach, with considering the lower and upper limit definition for pool market price, leads to robust decision within the defined price bounds. This robustness decision is necessary and contribution for many retailers.
Model In this section, the developed electricity procurement strategy model is explained. Method description To achieve a robust electricity procurement strategy, a number of RMILP problems are solved consecutively, which are described in the following. (1) Firstly, the hourly prices are set to their minimum values (lower levels of the price bound) and the standard MILP problem is simulated to derive the hourly electricity procurement strategy from the pool market. (2) Then, starting from lower level to upper level, the related RMILP problem is solved for each interval within the price bound to achieve hourly electricity acquisition strategy from different resources (pool market, bilateral agreements and self-production facility). The electricity procurement from pool market is smaller than or equal to the maximum electricity procurement achieved solving the MILP problem in item (1) above without considering price bound. (3) Finally, the considered hourly price bounds are sequentially increased until the whole price bound are included; the resulting of some RMILP problems fined an optimal electricity procurement sequentially smaller and smaller that correspond to bigger and bigger prices at pool market. Finally, the considered hourly price intervals are successively enlarged until the whole confidence intervals are covered; the resulting collection of RMILP problems provides a sequence of optimal electricity procurement successively smaller and smaller that correspond to bigger and bigger prices at pool market. Using the sequence of prices and electricity procurement strategy from different resources allows the retailer for building hourly electricity procurement and optimal bidding strategy curves.
Contributions In this paper, a robust optimization based framework is proposed for optimal bidding and electricity procurement strategy of a retailer for first time. In comparison with exhausting stochastic programming methods, the robust optimization approach results in an effective, robust solution with low computational burden. With considering the above framework, the contributions of this paper are summarized as follows: (1) Presenting a method to achieve electricity procurement strategy for retailer considering lower and upper limit for prices, not price estimated; (2) To propose robust optimization approach for solving the problem in item 1) above; (3) The electricity procurement strategy could be formulated as a RMILP problem where the global optimal energy procurement is guaranteed.
Paper organization The following of the paper is categorized as following. The proposed robust formulation for retailer is presented in section ‘Model’. The used approach flowchart to achieve electricity procurement strategy is provided in section ‘The proposed algorithm to derive optimal bidding strategy’. Section ‘Case study’ contains case study, results, discussions and assessments in details. The conclusion of this paper is presented in section ‘Conclusion’.
Robust formulation In this section, the mathematical formulation of the acquisition problem for retailer using the robust optimization approach is presented. The profit function of retailer consists of the revenue from sales to the customers minus the electricity acquisition costs from different resources. These components are calculated as below.
Maximize P l;i ;P p;i ;PDG j;h;i
T X i¼1
ksi Di
( ) N dg N B X T T T X X X X X DG DG kl;i Pl;i þ ki Pp;i þ Sj;h Pj;h;i l
i¼1
i¼1
i¼1 j¼1 h¼1
ð1Þ The first item of Eq. (1) considers the revenue of retailer by selling purchased power to customers in fixed price in retail level. Also, the second item of Eq. (1) represents the cost of providing electricity from different resources such as bilateral agreements, pool market purchase and self-production facility. Energy purchased cost from bilateral agreements is composed in second term of Eq. (1). The electricity acquisition cost from the pool market is presented in third term of Eq. (1). The finally term of Eq. (1) is related to operation cost of the self-production facility. A three-block piecewise linear curve is used for operation cost modeling of the self-production facility, which is shown in Fig. 1. The following constraints should be considered in optimal operation of the retailer:
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Cost ($)
P2,DG i
P1,DG i
purchased power from pool market, bilateral contracts and self-production facility. Therefore, the maximizing objective function (1) is equivalent to minimizing objective function (11). Finally, the objective function of electricity acquisition problem for retailer can be reformulated as following.
P3,DG i
( !) N dg N T B X X X X DG s Minimize ki Pl;i þ P j;h;i þ Pp;i
S3DG
P l;i ;P p;i ;P DG j;h;i
S 2DG S
P1MAX
P2MAX
P3MAX
Power
Fig. 1. Operation cost model of self-generating facility.
N dg N B X X X DG Pl;i þ Pj;h;i þ P p;i ¼ Di
i ¼ 1; . . . ; T
ð2Þ
j¼1 h¼1
max Pmin i ¼ 1; . . . ; T; l ¼ 1; . . . ; B l;i sl < P l;i < P l;i sl ; 8
0 6 PDG j;h;i 6
ð3Þ
n o dg MAX PMAX j;h P j;h1 U j;i ; 8 i ¼ 1; . . . ; T;
i¼1
i¼1
ð11Þ
i¼1 j¼1 h¼1
In this optimization problem, it is important to attention that objective function (11) and constraints (2)–(10) as optimization (2)–(11) is meaningful provided that pool market prices ki ði ¼ 1; . . . ; TÞ are input parameter data and the corresponding values are constant. But, the pool market prices ki ði ¼ 1; . . . ; TÞ are random variables, therefore, the optimization problem requires to be replaced by a cost distribution function, for example with the expected amount. In this research, pool market prices are assumed to be unknown variables within known confidence intervals. The objective function (11) with its related constraints (2)–(10) is an MILP formulation, which can be stated as standard form
Minimize xi ;8j
h ¼ 2; . . . ; N; j ¼ 1; . . . ; Ndg
j¼1 h¼1
l¼1
( ) N dg N B X T T T X X X X X DG DG þ kl;i Pl;i þ ki P p;i þ Sj;h Pj;h;i l
DG 1
l¼1
i¼1
ð4Þ
n X ej xj
ð12Þ
j¼1
Subject to MAX dg 0 6 PDG j;1;i 6 P j;1 U j;i ; 8 i ¼ 1; . . . ; T; j ¼ 1; . . . ; N dg
ð5Þ
ð6Þ
j¼1 h¼1 N N X X up dg PDG PDG j;h;i j;h;i1 6 Rj U j;i ; 8 i ¼ 1; . . . ; T; j ¼ 1; . . . ; N dg
ð7Þ
h¼1
N N X X down PDG P DG U dg j;h;i1 j;h;i 6 Rj j;i ; 8 i ¼ 1; . . . ; T; j ¼ 1; . . . ; N dg h¼1
m ¼ 1; . . . ; M
ð13Þ
j¼1
Ndg N X X DG Pj;h;i þ Pp;i P 0; 8 i ¼ 1; . . . ; T
h¼1
n X amj xj 6 bm ;
h¼1
ð8Þ dg dg ½X on j;i1 MUT j ½U j;i1 U j;i P 0; 8 i ¼ 1; . . . ; T; j ¼ 1; . . . ; N dg
ð9Þ dg dg ½X off j;i1 MDT j ½U j;i U j;i1 P 0; 8 i ¼ 1; . . . ; T; j ¼ 1; . . . ; N dg
ð10Þ In these constraints, constraint (2) represented the load power balance with the energy electricity from different resources. In constraint (3), the lower and upper electrical energy limit procured from bilateral agreements should be between the acceptable bounds as presented. The operation constraints related to self-production facility is expressed in Eqs. (4)–(10). Also, constraint (6) declares that the sum of the electricity procurement by the self-production facility and the electricity purchased from the pool market must be positive for all time periods. It should be mentioned that the maximum power of electrical energy that the retailer can sell in the pool market is limited by the maximum capacity of the self-production facility. Furthermore, constraints (7) and (8) describe the ramping up/down rate limits of self-production facility. Finally, the minimum up/down time constraints are expressed as (9) and (10), respectively. Finally, it should be mentioned that in objective function (1), Di in first item could be replaced based on Eq. (2) that it is the sum of
xj P 0;
j ¼ 1; . . . ; n
ð14Þ
xj 2 f0; 1g for some j ¼ 1; . . . ; n
ð15Þ
In Eq. (12), ej are coefficients of the objective function and are assumed to be known. If they be unknown parameters with known lower and upper bounds, a significant RMILP optimization related to (12)–(15) can be reformulated. To perform so, if it is assumed that dj shows the deviance from the nominal coefficient ej ; therefore, all coefficient ej takes amount in the interval ½ej ; ej þ dj . Moreover, for building an RMILP formulation, it is essential to define C0 as an integer parameter for controlling the level of robustness in the objective function, which considers values in ½0; jJ 0 j, where J 0 ¼ fjjdj > 0g. For ignoring the effect of the cost deviations in the objective function, C0 should be set zero, while if C0 ¼ jJ 0 j, we considered all deviations of cost function. The form of the RMILP optimization related to optimization (12)–(15) is
Minimize xi ;qoj ;yj ; 8j;z0
n n X X ej xj þ z0 C0 þ qoj j¼1
ð16Þ
j¼1
Subject to
Constraints ð9Þ—ð11Þ
ð17Þ
z0 þ qoj P dj yj ;
ð18Þ
qoj P 0; yj P 0;
j 2 J0
j ¼ 1; . . . ; n j ¼ 1; . . . ; n
z0 P 0 xj 6 y j ;
ð19Þ ð20Þ ð21Þ
j ¼ 1; . . . ; n
ð22Þ
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The objective function (16) and related constraints (17)–(22) above is achieved considering duality theorem and linearizing technique, for more details on the obtaining robust counterpart of an MILP problem please see [16]. Variables z0 and qoj are dual variables of the optimization (12)–(15) used to consider the known bounds of coefficients ej . Also, to achieve the corresponding linear declarations, yj is used as auxiliary variable. Most information about the robust optimization approach and explanation about of how to achieve these equations are clarified in [16]. Eventually, the RMILP problem of electricity procurement strategy for retailer can be formulated as
Minimize Pl;i ;P p;i ;P DG j;h;i
( !) Ndg N T B X X X X DG ksi Pl;i þ Pj;h;i þ Pp;i i¼1
þ
( B X T X l
l¼1
i¼1
þ z 0 C0 þ
j¼1 h¼1
N dg N T T X X X X DG DG kl;i P l;i þ ki Pp;i þ Sj;h Pj;h;i i¼1
T X qoi
) Fig. 2. Nested price intervals to derive electricity procurement strategy problems.
i¼1 j¼1 h¼1
P kp;i ; P kl;i ; PDG;k j;h;i ði ¼ 1; . . . ; TÞ. The electricity procurement and optimal bidding strategy for retailer in all periods are computed using the prices in all iterations ðkki ; 8kÞ and the electricity procurement in all iterations ðPkp;i ; P kl;i ; PDG;k j;h;i ; 8kÞ results.
ð23Þ
i¼1
Subject to
Constraints ð2Þ ð10Þ
ð24Þ
z0 þ qoi P di yi ;
ð25Þ
Fig. 3 shows the flowchart of the proposed algorithm.
qoi P 0; yi P 0;
i ¼ 1; . . . ; T
i ¼ 1; . . . ; T i ¼ 1; . . . ; T
z0 P 0 Pp;i 6 yi ;
ð26Þ ð27Þ ð28Þ
i ¼ 1; . . . ; T
ð29Þ
Once the robust electricity procurement strategy problem (23)–(29) has been formulated, the algorithm described in the following subsection will be used to extract electricity procurement strategy of the retailer.
The proposed algorithm to derive optimal bidding strategy The following algorithm is used to derive electricity procurement and optimal bidding strategy of the retailer. (1) Set prices ki ¼ kmin ði ¼ 1; . . . ; TÞ, and C0 ¼ T (all periods, i.e., i 84) to consider all possible deviation of pool market prices. k
(2) Set di ¼ Gk ðkmax kmin Þ; ði ¼ 1; . . . ; TÞ, where Gk is a coeffii i cient that uses increasing values in ½0; 1 and k shows the counter of iteration. It should be mentioned that parameter Gk enables deriving a succession of nested sub-bounds of the ; kmin þ di . For clarifying purposes, Fig. 2 illustrates form ½kmin i i a pattern including three various nested sub-bounds, namely (a), (b), and (c). (3) RMILP optimization (23)–(29) is simulated to achieve the hourly electricity procurement strategy at iteration k; Pkp;i ; Pkl;i ; PDG;k j;h;i ði ¼ 1; . . . ; TÞ. (4) For covering the all range of coefficient Gk , it should repeated iteratively (categorized byk) steps 2–3 above. The increasing step is d > 0 as illustrated in Fig. 3. (5) Construct the hourly electricity procurement strategy using the achieved results. An offering prices, kki ¼ kmin þ i k di ði ¼ 1; . . . ; TÞ are computed using each iteration k , and an electricity procurement level per time period,
Case study In this part, a case study is used to show the application of the robust optimization approach for building electricity procurement and optimal bidding strategy curves. Data In this paper, a single day is considered with its demand deconstructed into three load levels marked as peak, shoulder, and valley, as categorized in Table 1. The time horizon of this study contains 84 periods (four weeks). In this study, twelve bilateral agreements are considered; Table 2 presents data for bilateral agreements, which include the maximum and minimum amount of electric power with related prices. The selection decision of these bilateral agreements should be made at the beginning of the study horizon. There are four available bilateral agreements for the whole month and two contracts for each week. Six contracts are available for all load levels and six contracts only for the peak load levels. Table 3 presents the periods of the bilateral agreements. The self-production facility data are presented in Table 4. It should be mentioned that the single self-facility generation is considered in this simulation. The load profile for the study horizon is shown in Fig. 4. Also, the lower and upper bound of pool market price data for the study horizon is shown in Fig. 5. Results and discussion The electricity energy procurement strategy problem is formulated for retailer as a RMILP problem and solved using CPLEX solver [17] in GAMS software [18] on Intel(R) Celeron(R) CPU at 2 GHz and 1 GB of RAM. The 20 RMILP problems required to construct the electricity procurement strategy curves. The computing time for solving the 20 RMILP problems is 5.42 s. The simulation result of minimizing Eq. (11) with considering the lower level price of pool market price values is $8,057,616.00. This value is the minimum acquisition cost. The simulation results show that the retailer should purchase 73% of
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Fig. 3. Algorithm flowchart.
Table 3 Bilateral contracts usage periods.
Table 1 Classification of daily load levels. Level
Hours of the day
Contract #
Usage period
Validity level
Valley (V) Shoulder (S) Peak (P)
1, 2, 3, 4, 5, 6, 7, 8 9, 10, 15, 16, 17, 18, 23, 24 11, 12, 13, 14, 19, 20, 21, 22
1 2 3 4 5 6 7 8 9 10 11 12
Month Month Month Month Week one Week one Week two Week two Week three Week three Week four Week four
V, P V, P V, P V, P V, P V, P
Table 2 Bilateral contracts specification. Contract number
Min. (MW)
Max. (MW)
Price ($/MW h)
1 2 3 4 5 6 7 8 9 10 11 12
15 10 15 10 25 20 25 20 25 20 25 20
30 20 30 20 50 40 50 40 50 40 50 40
39.96 43.29 49.86 48.69 32.31 41.40 51.03 47.88 42.66 57.69 52.02 57.96
its electricity needs from the pool market, 7.2% from bilateral agreements, and 19.8% from the self-production facility. Considering the input parameter data, the RMILP problem (23)– (29) is solved 20 for iterations (Gk replaces in steps of amount d ¼ 0:05) to produce the needed data to construct optimal bidding strategy curves using the presented algorithm in section ‘The proposed algorithm to derive optimal bidding strategy’. The robust objective function (23) with constraints (24)–(29) is solved considering iteration value of Gk with a fixed step that
S, P S, P S, P S, P S, P S, P
V: valley; S: shoulder; P: peak.
actual pool market prices are increases related to the lower pool market prices. Figs. 6–8 are shown the results of simulation. These figures are indicated the level of robustness and percentage of acquisition from various resources such as limited self-production facility, bilateral agreements and pool market purchase. In Figs. 6–8, the minimum robust cost is related to total procurement cost considering the minimum pool market prices (lower level). Then, starting from lower level to upper level, the related RMILP problem is solved for each interval within the price bound to achieve robust cost. The robust cost is bigger than or equal to the minimum robust cost. Also, Figs. 6–8 features that the electricity procurement percent decreases from the pool market with increasing robust cost function, while acquisition from bilateral agreements and the self-production facility increase with increasing robust cost function. The results of robust cost function are interesting, because retailer purchases more of its electricity
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Table 4 Data for the self-generating facility. Capacity Minimum power output SDG 1
100 0 33
MW MW $/MW h
SDG 2
36
$/MW h
SDG 3
39
$/MW h
P MAX 1
40
MW
P MAX 2
75
MW
P MAX 3 MUT j MDT j
100
MW
Rup j
2 2 50
h h MW
Rdown j
50
MW
Fig. 6. Optimal procurement from the pool as a function of robust cost.
Fig. 4. Load profile of the consumer for study horizon.
Fig. 7. Optimal procurement from bilateral contracts as a function of robust cost.
Fig. 5. Confidence intervals for pool price at study horizon.
required from the markets whose prices have no uncertainty. In other words, retailer for management of pool prices uncertainty purchases more of its electricity required from bilateral agreements and self-production facility. Note that we consider a price-taker retailer, believing that his energy bid cannot influence the market clearing price. In other words, if the retailer offers either at zero or at $1000/MW h, market
Fig. 8. Optimal procurement from the self-generating facility as a function of robust cost.
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prices remain unaltered. In the bidding problem, retailer should determine two decisions as the offered price and the offered quantity (power). Hence, the objective of this study is determining the offered priced and power for obtaining the maximum profit and winner in power market based on step-wise bidding strategy. Therefore, the step-wise bidding strategy help the retailer for obtaining the more profit under pool market price uncertainty. Therefore, Figs. 9–11 show the selected optimal offering curves achieved using the proposed approach. In this curves, the x-axis considered the offering price whereas the y-axis shows the quantity blocks that the retailer bids to the pool market. These curves make the essential information required by a retailer to bid prosperously consumers demand in a pool market. It should be mentioned, these curves change meaningfully from time period to time period. Furthermore, Fig. 9 is related to optimal bidding curve for peak load level in 25th day. Also, Figs. 10 and 11 are related to optimal bidding curve for shoulder and valley load levels in 12th and 6th days, receptivity. Fig. 9. Optimal bidding curve for peak load level in 25th day.
Conclusion
Fig. 10. Optimal bidding curve for shoulder load level in 12th day.
In this research, we studied the electricity acquisition problem faced by a retailer from alternative resource such as self-production, bilateral agreements, and the pool market. Also, we recognized a medium-term planning that may include from one month (four weeks), and, in order to attain tractability, every day of the planning horizon is separated into three eight-hour periods, in particular valley, shoulder and peak. For considering the uncertainty nature of pool market prices, this research presents an application of robust optimization technique to appraise value choice acquisition strategies for retailers with multiple procurement options. Upper and lower limits of pool prices are considered for uncertainty modeling, rather than using forecast prices as input data. The range of pool prices are sequentially partitioned into a successive of nested subintervals, which permit formulating a collection of RMILP problems. The results of these problems give sufficient data to obtain electricity procurement strategy and optimal bidding strategy to pool market for retailers. Also, these curves make the essential information required by a retailer to bid prosperously consumers demand in a pool market. Finally, retailer for management of pool market prices uncertainty purchases more of its electricity required from bilateral agreements and self-production facility. References
Fig. 11. Optimal bidding curve for valley load level in 6th day.
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