Block Order Restrictions in Combinatorial Electric Energy Auctions

Transcription

1 Tis is te post-print version of tis article: Meeus, L., Veraegen, K., Belmans, R., Block order restrictions in combinatorial electric energy auctions. European Journal of Operational Researc, 196(3), pp Block Order Restrictions in Combinatorial Electric Energy Auctions Leonardo Meeus a, *, Karolien Veraegen a, Ronnie Belmans a a Department of Electrical Engineering, University of Leuven, Kasteelpark Arenberg 10, 3001 Leuven, Belgium Abstract In Europe, te auctions organized by power excanges one day aead of delivery are multi-unit, double-sided, uniformly priced combinatorial auctions. Generators, retailers, large consumers and traders participate at te demand as well as at te supply side, depending or weter tey are sort or long in electric energy. Because generators face nonconvex costs, in particular startup costs and minimum run levels, te excanges allow "block orders" tat are allor-noting orders of a given amount of electric energy in multiple consecutive ours, wile te standard order consists of an amount for a single our tat can be curtailed. All excanges restrict te size (MW/), te type (span in terms of ours) or te number (per participant per day) of blocks tat can be introduced. Tis paper discusses te rationale of block order restrictions. Based on simulations wit representative scenarios, it is argued tat te restrictions could be relaxed, wic some excanges ave already started doing. Keywords OR in energy, E-commerce, Combinatorial Auctions/bidding, Pricing, Integer programming 1. Introduction In Europe, te auctions organized by power excanges one day aead of delivery are an increasingly important part of te wolesale market (Meeus et al., 2005). Altoug participation is voluntary and te average traded volume is only about 10% of consumption, te ourly auction price is an important reference price for all contract negotiations. Generators, retailers, large consumers and traders increasingly participate at te demand as well as at te supply side, depending or weter tey are long or sort in electric energy. Te orders tat can be introduced at tese auctions are for te delivery or off-take of electric energy during an our of te next day. Te excanges also allow block orders tat are all-or-noting orders of a given amount of electric energy in multiple consecutive ours. An auction wit block orders can terefore be called a combinatorial auction. Combinatorial auctions ave in common tat orders can be placed on combinations of eterogeneous 1

2 items, called packages or bundles, rater tan just on individual items. An inspiring and compreensive work on tis topic is te book edited by Cramton, Soam and Steinberg (2005). Combinatorial auctions ave recently been employed in a variety of industries. De Vries and Vora (2003) provide a compreensive survey. Te advantage of combinatorial auctions is tat participants can more fully express teir preferences, suc as complementarities between eterogeneous items. In electricity markets, tere are complementarities between deliveries of electric energy in consecutive periods, for instance because of start-up costs of power plants. Block orders can indeed be seen as a combination of ourly orders. Blocks allow participants to provide an average price for a combination of ours. On average generators can offer ceaper prices for delivery in multiple consecutive ours as tis allows tem to spread out te start-up cost. Bot excanges and participants consider blocks as important. On some excanges up to 20% of total traded volume consists of block orders. Still, all excanges restrict te size (MW/), te type (span in terms of ours) or te number (per participant per day) of blocks tat can be introduced. Tis paper terefore analyses te rationale of block order restrictions. Limiting te allowable combinations is known to be effective in reducing computational complexity (Pekec and Rotkopf, 2003; Park and Rotkopf, 2005). Tis and oter reasons to restrict te use of block orders on excanges are investigated by solving to optimality representative scenarios, based on te istorical aggregated order curves of APX, to wic sets of block order are added wit various degrees of restrictions. Section 2 explains ow te representative scenarios ave been constructed. Section 3 introduces te model tat is used for te simulations. It terefore also introduces te auction optimization problem wit blocks and te pricing approac applied by excanges to clear teir markets. Section 4 ten discusses te effect of restrictions, based on te simulation results. Section 5 finally evaluates te restrictions imposed by excanges. 2. Representative scenarios Te power excanges wit blocks are APX (Neterlands), Belpex (Belgium), Borzen (Slovenia), EEX (Germany), EXAA (Austria), Nord Pool (Norway, Sweden, Denmark and Finland) and Powernext (France). As illustrated in Table 1, te kind of blocks tat can be introduced to tese excanges differ substantially. Table 1: Block order restrictions on APX, Belpex, Powernext and EEX Nr block types Max nr blocks / day / participant APX Powernext 10 INF EEX All combinations of consecutive periods are allowed 2 Per portfolio it is possible to submit every type once, but participants can submit several portfolios 3 Before 2005 it was 50 MW Max size (MW/) Powernext for instance does not restrict te number of block orders tat can be submitted per participant per day, wile te size is for instance more restricted on APX (50MW/) tan on EEX (250MW/). On APX, any combination of consecutive ours is allowed so tat 354 types of block orders can be traded. Powernext and EEX on te oter and restrict blocks to 10 or 11 types. Table 2 2

3 illustrates te 10 block types tat can be traded on Powernext. Table 2: Block products on Powernext Contract name Time interval Block Bid Block Bid Block Bid Block Bid Block Bid Block Bid Block Bid Block Bid Block Bid Block Bid Te scenarios used in tis paper are based on te istorical aggregated order curves of te Dutc power excange APX. Teir order curves are publicly available, wic is not te case for most oter excanges. Te 19 days illustrated in Table 3 ave been randomly selected. APX launced teir dayaead auction in 1999 and its liquidity as since steadily increased as can be seen from te table. Table 3: Days used for scenarios Date (DD/MM/YY) Average price ( /MW) Maximum price ( /MW) Total traded volume (MW) 15/01/ /03/ /05/ /07/ /11/ /02/ /04/ /06/ /08/ /10/ /12/ /01/ /02/ /03/ /04/ /05/ /05/ /06/ /07/ Tese days are from different years, seasons, week-weekend. Te ourly orders are extracted from tese curves. Every scenario includes te ourly orders of one of tese days. To simulate te effect of adding blocks to tese representative days, sets of blocks are generated wit various degrees of restrictions as follows: To study te effect of a type restriction, in alf of all scenarios blocks can be of any type, as on APX, wile in te oter alf, block are restricted to te 10 types found on Powernext (Table 2). Note tat te Powernext types ave been cosen because tey are most restrictive. To study te effect of a size restriction, every scenario as a maximum block size between 10 and 300MW/. Te blocks in a scenario can terefore ave different sizes, but all are smaller tan te determined scenario size limit. Note tat te size limit considered in te analysis is iger tan te largest allowed blocks of 250MW/ on EEX. Blocks larger tan 300MW/ are not considered because suc large capacity plants are base load and typically sceduled outside te excanges. To study te effect of an number restriction, te number of blocks in a scenario ranges between 0 and 200. Note tat if 200 blocks would be submitted, teir sare in total traded volume in te scenarios would be larger as it currently is on te excanges. As mentioned in te introduction, 3

4 blocks are said to represent up to 20% on some excanges. Given an average block size of 150MW/, 200 blocks correspond to 30000MW/. For a block tat on average spans 8 ours (1/3 of a day), tis corresponds to a total volume of 1000MW/day, wic is up to 35% of te total traded volume on te days used to construct scenarios (Table 3). Additionally, te following assumptions in line wit wat can observed on excanges, ave been made: Blocks are as likely to be introduced at te demand and supply side Blocks are price-setting orders, meaning tat teir prices are significantly different from zero and close to te market prices. Teir price limits ave been generated so tat tey deviate less tan 10%, from te average price of te day (Table 3). Te maximum admissible order price limit (Pmax) is 2500 /MW, as on APX. Note tat tis is not intended to be a price cap but rater to protect against uman error. A batc of 200 scenarios as been created in te manner explained above. Te results are presented in Section 4. Increasing te batc size to 200 as proved to be sufficient to present results tat are not batc specific. Te next Section explains ow te scenarios are solved to optimality. 3. Auction optimization problem wit blocks Combinatorial auctions are typically difficult to solve optimization problems (Xia et al., 2005). Tis is also te case for te auction problem wit blocks. Te all-or-noting constraint of block orders means tat binary variables are necessary to model te auction problem. Models wit binary variables for blocks and constrained continuous variables for ourly orders are Mixed Integer Linear Problems (MILP), wic are difficult to solve. Wit, ourly orders caracterized by te our () in wic tey are introduced, weter tey are supply (i) or demand (j) and by a price ( /MW) and quantity (MW) limit ( P, Q ) ; block orders caracterized by te ours included in te block ( H ), weter tey are supply (k) or demand (l) and by an average price ( /MW) and quantity (MW/) limit ( P, Q ); nh te number of ours included in a block; block orders aving a binary variable to implement te all-or-noting constraint ( b =1 if block is accepted; b =0 oterwise); block orders aving a quantity limit for every our to simplify te notation, wic is zero for te ours not included in te block ( Q = 0 if H ); te accepted order quantities ( q i, q, q k, q l ) as te decision variables; Te auction optimization problem wit blocks is as follows: maximize total gains from trade (or trade efficiency), Max qp + qlpl qipi qkpk j l i k (1) subject to market clearing constraints, equalizing demand and supply in every our: (2) q + q = q + q : i k l i k j l and te order constraints: q q q i Q (3) i Q (4) = bq (5) k k k 4

5 q = bq (6) l l l Combinatorial auctions are non-convex. Tis means tat linear market clearing prices do not necessarily exist (see for instance Scarf, 1994 and Elmagraby, 2004). If tere are no ourly prices at wic demand equals supply, one possibility is to resort to nonlinear pricing (see O'Neill et al., 2005 for a discussion on ow sadow prices can be used to implement nonlinear pricing). Nonlinear pricing means tat te optimal solution to (1)-(6) in terms of traded volumes (q, MW) would be settled at ourly prices (p, /MW) in combination wit a side payment (A, ) wic can be different for all orders, i.e. resulting in a pq + A settlement. Excanges in Europe owever ave in common tat tey do not use side payments to clear teir dayaead auction markets (A=0). Instead, tey equalize demand and supply at ourly prices by rejecting blocks tat sould be accepted looking at te ourly prices, i.e. Paradoxically Rejected Blocks (PRB). Note tat blocks are owever only accepted wen tey sould be and ourly orders are cleared (accepted and rejected) completely in accordance wit te ourly prices. To get te optimal solution wit te above caracteristics, te following constraints including te ourly prices ( p ) need to be added to te auction problem (1)-(6): First, if a supply block is accepted ( b k = 1), te average market price sould be at least as ig as te price limit of te block, wit nh te number of ours included in a block: k : b nh P p (7) k k k k k Hk Equally, if a demand block is accepted ( b l = 1 ), te average market price sould not be iger tan te price limit of te block, wit P max te maximum admissible price for an order: l : p nhl( Pl + Pmax (1 bl)) (8) l Hl Second, if an ourly supply order or offer is accepted ( b i = 1), te ourly price ( p ) needs to be at least as ig as te price limit of te offer ( P i ), wit b a binary variable equal to one if te ourly order is accepted: i, : bp i i p (9) Equally, if an ourly demand order or bid is accepted ( b = 1 ), te ourly price ( p ) cannot be iger tan te price limit of te bid ( P ):, : p P + P (1 b ) (10) max Tird, partially rejected or curtailed ourly orders sould set te price. Terefore, if an offer is partially rejected ( b = d = 1 ) or completely ( b = d = 0 ), i i te ourly price cannot be iger tan te price limit of te offer, wit d a binary variable equal to one if te ourly order is partially rejected: i, : p P + P ( b d ) (11) i max i i Equally, if a bid is partially rejected ( b = d = 1) or completely ( b = d = 0 ), te ourly price needs to be at least as ig as te price limit of te bid:, : P P ( b d ) p (12) max All excanges impose linear prices, wic means tat every day tey solve te optimization problem (1)-(12). If tey would drop constraints (7)-(12), tey would increase gains from trade (and avoid PRBs), but trade would ave to be settled by using sidepayments. As mentioned earlier, excanges ave owever cosen to avoid te complexities of a settlement wit side payments. Simplicity can indeed be considered as i i 5

6 an important design feature of te excanges in teir role of fine tuning market of wic te reference price is more important tan te volume tey clear directly. 4. Effect of block order restrictions A batc of 200 scenarios as been solved to optimality according to te MILP model (1)-(12) on a Pentium IV, using te CPLEX v11.0 solver software called from Matlab using te Tomlab interface. In two scenarios, te optimal solution was not yet found after 2.5 days so tat te solver was stopped. For all oter scenarios, te solver calculation time is 4 minutes on average. Te minimum and maximum calculation time is respectively a few seconds and 3.5 ours. 50% of te scenarios solve in less tan one minute and 95% less tan 10 minutes. Tis is typical for te performance of commercial MILP solvers. Te optimal solution to te MILP model (1)-(12) yields 4.15 PRBs per day on average, wit a maximum of 27 in a day. In total, tere are 829 PRBs for blocks in tese scenarios. Terefore, te likeliood of blocks to be paradoxically rejected is only 4.36%. It is important to note tat almost 40% of tese PRBs are actually not loosing any money, i.e. teir price limit is equal to te average market price, but oter blocks loose up to 18 /MW/. In te remainder of tis Section, te effects of restricting te use of blocks on calculation time, te number of PRBs and trade efficiency are considered based on te simulation results. described as an effective way to reduce te computational complexity of combinatorial auctions. Park and Rotkopf (2005) even propose an auction wit bidder-determined allowable combinations. Also in combinatorial electric energy auctions tis is true. As discussed in te Section 2, in 50% of te scenarios every combination of consecutive ours is allowed, wile in te oter 50% of scenarios only ave te 10 combinations tat are allowed at Powernext. Te difference in calculation time between tese scenarios is illustrated in Figure 1. Figure 1: Calculation time MILP model (1)-(12) in minutes wit and witout a block type restriction As illustrated in te figure, te group of scenarios in wic te allowed combinations or block types are not restricted as more extreme outliers. Indeed, also te two scenarios not indicated in te figure tat were stopped after 2.5 days of calculation are scenarios witout a type restriction. Significant coerence between calculation time 4.1 Calculation time and te number or size of blocks in te scenarios Pekec and Rotkopf (2003) discuss noncomputational could not be found. One could expect a correlation approaces to mitigating computational between te number of blocks and te solver problems in combinatorial auctions. Limiting te calculation time, as te number of blocks increases te combinations participants are allowed to bid is problem size in terms of binary decision variables, but 6

7 suc a correlation could not be found. Te correlation in te batc of 200 scenarios is only and not significant. Tis can be partly explained by te fact tat binary variables are also assigned to ourly orders and te number of ourly orders differs more between scenarios tan te number of blocks. Note tat if linear prices are not imposed on te clearing, te calculation time significantly reduces to 0.6 seconds on average wit a maximum of 1.4 seconds. Tis clearly indicates tat te most significant computational complexity comes from constraints (7)-(12) and te binary variables tat need to be assigned to te ourly orders to implement tese constraints and terefore not from te number of blocks. 4.2 Paradoxically Rejected Blocks (PRB) On average 4.36% of te blocks are paradoxically rejected. Tis indicates tat it is not tat big of an issue for te auction participants, wic as been confirmed by talking to traders. Still, tis paragrap will respectively consider weter block type, size and number restrictions are an effective way of reducing te number or likeliood of PRBs. Table 4 compares te PRBs of te scenarios wit and witout a type restriction. Tere is no significant difference in te number of PRBs between tese categories of scenarios. Te null ypotesis tat te means are equal, assuming a normal distribution for bot samples and equal standard deviations cannot be rejected for a 5% significance (p-value is ). Table 4: Effect block type restriction on PRB From te combinatorial nature of blocks, it can be expected tat small blocks are less likely to become paradoxically rejected. Indeed, for instance only 1% of blocks smaller tan 50MW/ are paradoxically rejected, wic is four time less tan te average for blocks. However, as indicated in Table 5, tere is no significant correlation between te likeliood of PRB and te maximum block size. Suc a correlation would appear if all blocks in te scenarios are taken equal to te maximum block size, but wat tese results indicate is te presence of large blocks does not increase te likeliood tat small blocks are paradoxically rejected. It can also be expected tat te number of PRBs increases wit te number of blocks. Te results in Table 5 confirm tis, but also indicate tat te increase is more or less proportional, as tere is no significant correlation between te likeliood of PRB and te number of blocks in a scenario. Table 5: Linear effect size and number of blocks on PRB trougout te wole range of tat data Correlations Nr blocks (linear regression R 2 ) Nr PRB (41.4%) Likeliood PRB (Illustrated in Figure 2) Likeliood PRB blocks < 50MWH/ (0.13%) (1%) Maximum block size (9.3%) (4.6%) (2.2%) Nr PRB All types Powernext types Mean Standard deviation 7

8 side payments. Note tat only blocks would receive side payments, te average payment being 502. Likeliood PRB MILP (%) Max size blocks (MW/) Nr blocks Figure 2: Likeliood PRB in MILP model (1)-(12) 4.3 Trade efficiency Te value of te objective function (1) is largely driven by te ourly orders because tere are many price taking ourly orders. Tis does not mean tat power excanges sould simply stop using block orders and tereby avoid te complexity of dealing wit tem. On te contrary, blocks are important for market parties and represent up to 20% of traded volume on te excanges. Tis does owever explain wy restricting te number, size or types does not ave a statistically significant effect on te total gains from trade. Tis also explains wy imposing linear prices only results in a loss of.0.05% in terms of gains from trade. Note tat te lost value is linked to paradoxically rejected blocks and can terefore be avoided by applying nonlinear pricing. However, tis would also mean tat side payments would ave to be made. Applying te nonlinear pricing approac introduced in O'Neill et al. (2005) to te 200 scenarios, would for instance mean tat side payments need to be made in total. Tis is almost 9 times more tan te total gains from trade tat can be won by making tese Evaluation of restrictions From te previous section can be concluded tat a block type restriction is an interesting option to consider. Te results indicate tat a type restriction as a clear effect on te solver calculation time and reducing tis time can be of interest to excanges tat typically ave only between 15 and 30 minutes to clear teir day-aead auctions. A type restriction is also not necessarily binding for te auction participants as blocks are mainly introduced for base load, peak load, etc and te allowed combinations typically matc tese periods. From te previous section could also be concluded tat te number of blocks and teir size sould not be restricted. Te simulations clearly indicate tat tese restrictions ave no significant impact on calculation time, te likeliood of PRB or trade efficiency. Still, it can be explained wy all excanges ave suc restrictions. One possible explanation is tat participants were not used to trade blocks under te linear pricing regime introduced by power excanges, wic as been introduced in tis paper and wic is very different from te pricing approaces in oter combinatorial auctions, so tat every PRB is a potential complaint for starting excanges. Note owever tat restricting te use of blocks is an artificial way of reducing PRBs. Te real solution would be to avoid PRBs by resorting to nonlinear pricing. It is also sometimes said tat te unrestricted use of blocks would increase price volatility. For immature or illiquid markets wit a lack of ourly orders, te lumpiness of blocks can indeed be an issue for te formation of prices. Te scenarios used in tis 8

9 paper are based on APX from 2003 to 2005, wic is more tan 4 years after te excange started in Te results indicate tat for mature markets te impact on prices of adding blocks is limited. In oter words, tere are ways to explain wy excanges ave introduced tese restrictions, but as tese markets ave matured it is time for tem to omit or at least relax tem. Note tat te size restrictions are currently clearly binding for traders. Generation units are easily larger tan 50 MW and even larger tan 250 MW. Because blocks can be paradoxically rejected, submitting 5 blocks of 50 MW/ is not te same as submitting a block of 250 MW/. 7. Conclusions Te simulation results presented in tis paper argue against restricting te use of blocks in te dayaead auctions organized by excanges. It is in te benefit of excanges and auction participants to omit or at least relax tese restrictions. Some excanges ave already starting doing tat. Te Frenc Powernext as for instance doubled te allowed block size from 50 to 100 MW/ and more recently also allows more combinations of ours in a block order. Te simulations are based on representative scenarios using actual order data from te Dutc excange APX. Block sets wit various degrees of block restrictions are added to tese scenarios to study te rationale of tese restrictions. Te results clearly argue against block size restrictions and also against restrictions on te number of blocks a participant can submit per day. Inline wit existing combinatorial auction literature (Pekec and Rotkopf, 2003; Park and Rotkopf, 2005), te results owever do confirm tat limiting te allowable combinations tat can be time. Tis could terefore justify a block type restriction. It as also been explained tat order restrictions in general can be justified for starting or illiquid excanges. For instance te Austrian excange EXAA introduced blocks in 2003 after one year of operation wen te market ad somewat matured. More recently also te Belgian excange BELPEX started witout blocks in 2006, but introduced tem after a few monts of operation. Apart from providing guidelines to excanges on ow to deal wit blocks, tis paper also discusses teir particular approac of imposing linear prices in a nonconvex auction. An interesting extension to tis work could terefore be to consider tis pricing approac for oter combinatorial auction settings (see Xia et al for an overview of pricing approaces in combinatorial auctions). Specifically towards power excanges, tis work could be extended by considering oter combinatorial products. A block in itself is also a restricted product. Te auction participants migt for instance be interested to combine ours witout aving to offer te same amount of electric energy in every our. Note tat some excanges ave already started to introduce more flexible combinatorial products and oter are looking into tis issue. Acknowledgements Te autors gratefully acknowledge te support and contributions received from APX, Belpex, Elia and Powernext. Te autors also tank te anonymous referees and te editor for teir comments. Te analysis and conclusions are of course te sole responsibility of te autors. included in a block reduces te solver calculation 9

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