Convex Combinatorial Auction of Pipeline Network Capacities
aa r X i v : . [ ec on . T H ] F e b Convex Combinatorial Auction of Pipeline NetworkCapacities
Dávid Csercsik Faculty of Information Technology and Bionincs, Pázmány Péter Catholic University, 1083 Práter u.50/A, Budapest, Hungary
February 18, 2020
Abstract
In this paper we propose a mechanism for the allocation of pipeline capacities, assum-ing that the participants bidding for capacities do have subjective evaluation of variousnetwork routes. The proposed mechanism is based on the concept of bidding for route-quantity pairs. Each participant defines a limited number of routes and places multiplebids, corresponding to various quantities, on each of these routes. The proposed mech-anism assigns a convex combination of the submitted bids to each participant, thus itscalled convex combinatorial auction. The capacity payments in the proposed model aredetermined according to the Vickrey-Clarke-Groves principle. We compare the efficiencyof the proposed algorithm with a simplified model of the method currently used for pipelinecapacity allocation in the EU (simultaneous ascending clock auction of pipeline capaci-ties) via simulation, according to various measures, such as resulting utility of players,utilization of network capacities, total income of the auctioneer and fairness.
The European natural gas network represents an enormous infrastructure system, whichis also constantly in the focus of geopolitics [1, 6]. In the traditional model, national ormultinational energy companies built their own pipelines requiring huge investments, andexpected that their latter trade transactions using the pipeline will provide them withsufficient returns. Nowadays, exclusive ownership is not the general institutional setting.Many pipelines within the European Union are subject to regulated third party access(TPA). Since the early 1990s, the EU have adopted a number of increasingly assertivedirectives and regulations to develop the common market for gas by ensuring fair TPAaccess to the transportation system within the Union — see [7, 8, 9, 2, 11]. According tothis scheme, the member countries have established a system of transport fees overseenby a regulatory authority. Under such a regime, the owner of a pipeline no longer enjoysexclusive right over the transport capacities. Instead, he has to grant access, provided heis compensated according to the regulated tariff. Cooperative game theoretic analysis ofTPA and the implied transfer profits in natural gas networks has been proposed in [3]. .1 Motivation: Current practice of pipeline network capac-ity allocation in the EU A decade ago, regulations of the third energy package [10] basically separated the networkoperation from the trading and supply, expanded the rights of regulation authorities, andcreated the Agency for the Cooperation of Energy Regulators (ACER), and the EuropeanNetwork of Transmission System Operators for Gas (ENTSOG). As a result, in the last 10years, the bias of trading already significantly shifted from long term (usually fixed-price)contracts to more liquid trading platforms (markets corresponding to so called gas hubs ).In this framework, to provide infrastructure for such increasingly interactive trading, thetransmission system operators (TSOs) market the transfer capacities of pipelines as stan-dardized products of variable time-frames (from yearly to intra-day intervals). Accordingto the reports of [4], the volume of engagements corresponding to short-time productsconstantly increases. The efficiency of capacity bookings in Germany has been discussedby [14].As long as the capacities required for the planned trade transactions do not exceedthe pipeline capacities, allocation is simple, and it practically means only administration.However if the available capacities are not enough to satisfy all participants aiming toallocate capacities in the network, some kind of capacity-allocation method must be usedto distribute the available pipeline capacities among participants (players) who apply forthem. The first auction, which coordinated the long-term bookings of available existingand future pipeline capacities on the EU-level has been held in 2017 March on the PRISMAauction platform. During this auction, yearly, quarterly and monthly pipeline capacityproducts have been auctioned simultaneously using an ascending clock auction (ACA).Altogether 2165 unique auctions took place on 6 March for each point and each year. Aspointed out by [17], in most of the cases no real competition emerged, and as the resultof this auction, the dominant market player (GAZPROM) was able to acquire the greatmajority of high-importance capacity licenses for in some cases as long as 20 years (forexample, all interconnection capacities on the border of Slovakia have been booked for20-25 years by GAZPROM).Several factors may be identified as underlying causes for this result. First, Russia,unlike other suppliers of Europe, like Algeria or Norway, typically delivers gas to theborder of the importer country, thus countries which import gas from Russia do havemodest interest in acquiring transport routes. The reasons for this are partially historical– in deals of the former decades, the market power of GAZPROM was very high, soimporters were compelled to agree with such details of bargains.Second, if a large producer supplying a significant number of clients aims to buy ca-pacities for his deliveries, all the delivery paths in question originate from the productionsite, and they have potentially large overlaps (see e.g. the Nord Stream I and II and theirconnected pipelines, which will practically supply the majority of Europe). In this case, itis easy to identify pipelines and interconnection points which are critical for these deliveryprojects, and thus represent high value for the player. In other words, the optimal biddingstrategy of such large producers is quite straightforward in the current framework, whilethey typically also have the resources to obtain capacity licenses for long periods.In contrast, the optimal bidding strategy in the current framework is not trivial forsmaller consumers. Consumers typically buy gas the on established hubs, the prices ofwhich may be different and also uncertain regarding longer periods (e.g. years) for whichcapacity may be booked in the present system. Such hubs correspond to market areas , nside which the physical transportation of gas is the responsibility of the TSO. Thearticle of [14] discusses the implications of these market areas in Germany, and analyzesthe efficiency of inter-area capacity bookings.Overall, it can be said that the current allocation practice and the respective algorithmsdo have their pros and cons, but in general it is reasonable to ask if there’s any alternative tothe current method of capacity allocation. In the current paper we propose exactly such analternative approach, called convex combinatorial auction (CCA) of pipeline capacities. Asa first step, we define and test this method on an abstract model under several simplifyingassumptions (see subsection 2.1).We consider a scenario where, under the principle of regulated third party access,local (national) TSOs have the right to determine transfer fees for their pipelines, but thepipeline capacity licenses are allocated by a central authority via auction. We compare thenewly proposed CCA-based allocation to the allocation based on the simultaneous ACA,assuming a simple but reasonable optimal bidding strategy of the participants of the ACA(see subsection 2.3.2). We use various measures for the comparison, such as resultingutility of players, utilization efficiency of network infrastructure, total amount of paymentfor the capacity rights (income of the auctioneer) and fairness. The structure of the paperis as follows. In section 2 we define the principles of the used model of the network, and thealgorithms modelling the ACA and CCA based allocations, and demonstrate the conceptson a simple, small example. In section 3, we perform simulations on high numbers ofrandomized scenarios to get statistical data about the performance of the two methodsand summarize the respective results. Section 4 discusses the results of the simulationsand the properties of the two different allocation mechanisms, while section 5 concludes. In the following, an example with a simple network will be introduced to demonstratethe modelling and simulation assumptions corresponding to the two allocation methodsconsidered.After defining the network and related parameters, we will first discuss the simulationof the ACA for the allocation of pipeline capacities, and determine the resulting capacityrights, payments and utilities in this framework, which is motivated by the current practiceof the EU for capacity allocation [12]. Following this, we introduce the proposed alternativemethod, the CCA, demonstrate how it can be applied to the proposed example, andcompare the results with the outcome of the ACA.In this proposed simple example, we will focus on consumers, and we will assume thatthey are the only participants of the capacity auction. Regarding realistic scenarios, atleast in addition to local gas distribution companies, who may be considered as consumers,multinational energy companies and traders are also present as typical bidders of suchauctions. The benefit of considering only consumers as bidders is however that, as we willsee, according to their explicit demand elasticity characterization, their rational biddingstrategy (under a few additional assumptions in the case of ACA) may be easily derived– this task would be much more harder in the case of agents representing multinationalcompanies with more complex incentives.In the current work we focus on long-term capacity rights. The regulation [12] definesyearly, quarterly, monthly, daily and within-day capacity products, from which the firstthree are sold via the ACA algorithm. In other words this means that if one is willing to llocate capacities for the first month of the year, he/she has 3 opportunities to do it.Motivated by this, our model of the ACA process will have three steps. In the first stepwe assume that all capacities of the network in question are subject to auction. After thefirst step has finished, remaining (not-allocated capacities) are subject to the second roundof ACA auctions, and so similarly, the remaining not-allocated capacities are subject tothe last round of ACA auctions . As discussed by [14], market areas (MAs) are sets of physical network nodes, betweenwhich the transportation of gas is the responsibility of the TSOs. Network users are ableto inject gas at any entry point of the MA and withdraw gas at any exit point that belongsto the same MA, if they have the capacity rights for the respective entry and exit points.Some entry points of a MA may correspond to production sites, while others may representincoming pipelines. Let us note, that multiple such pipelines exists (see Fig. 3 in [14]).In the terms of our model, the nodes represent MAs, and the edges represent thecapacities connecting them. According to the above considerations, it is possible thatmore than one edge is present between two nodes. Although we do not consider such casesin the paper, the model is capable of handling these scenarios.
Similar to transfer fees, capacity products in the practical European system (PRISMA - see ) are also considered corresponding to the entry andexit points of MAs. The basic reason for this is that the transfer capacities are managedlocally by the TSO’s of the respective price zone. To make the life of bidders easier, thecapacity allocation platform defines so called bundled products, composed of an exit andan entry capacity. This means that if I’d like to transfer from node A to node B, I havethe possibility to bid for a bundled AB product, in which the exit capacity of A and theentry capacity of B are included. These bundled products are handled in the PRISMAsystem in a way, which ensures that the total entry and exit capacities of price zones arerespected.For the aim of simplicity, in the used modelling framework we consider only suchbundled capacity products. Let us note however, that the used methodology may beeasily generalized to a more detailed scenario. The current model takes capacity from Ato B into account as a product, if node A is connected to node B. If one would like toconsider entry and exit capacities distinctly, an intermedier node X may be introducedon the edge A-B. In this case the edge A-X represents the entry and exit capacities of A,while the edge X-B represents the entry and exit capacities of zone B.
In the proposed model framework it isassumed that natural gas is available at distinguished nodes (representing market areas), Let us note that according to the current practice, 10 % of the available transfer capacity is reserved forshort term trading. rom where consumers must ensure themselves routes to transport it to consumption sites.In the current model we assume that these sources are able to provide arbitrary quantitieson prices, which are fixed for the period in question (for which we consider the allocationof transfer capacities). Let us note that in the case of realistic scenarios, the price ofnatural gas at the trading hubs may be significantly volatile, and depends on the natureof the source as well (obtained e.g. from actual transports, gas reservoirs or from LNGterminals). According with the recent line of EU regulations, we assume that no pricedifferentiation is allowed at the market.Let us consider the network depicted in Fig. 1. Based on the above considerations, thesource S in the model (at node 1) provides gas for every consumer at the same price (23EUR/MMBtu – EUR per Million Metric British unit) Figure 1: Example network. Each node and edge is labeled with its ID. S denotes the onlysource present in this simple example network, while C , C , and C denote the consumers. c ti denotes the transfer cost on line i , while q i denotes the maximal capacity of the line. We assume one source ( S ) located in node 1, and three consumers ( C − C ) locatedin nodes 2, 3 and 4 respectively. As in the current study we focus on network capacityallocation, we assume that the sources can provide arbitrary quantities (we assume noinlet limits).We assign a direction to each edge, as denoted in the Figure, to account for positiveand negative flows, but we assume that the pipelines corresponding to the edges are bi-directional, and their maximal transfer capacity ( q j for pipeline j ) is the same in bothdirections. We also assume that the transfer cost for pipeline ( c tj for pipeline j ) is also thesame in the positive and in the negative direction. In practice, local TSOs set entry andexit fees at interconnection points, but from these values transfer fees of a certain line i n the context of the model may be easily derived as the exit fee of the source point andentry fee of the destination point. ¯ q c t ¯ q ) and transfer cost( c t ). Consumer demand
We use piecewise constant inverse demand curves for the de-scription of demand elasticity as depicted in Fig. 2. Each piecewise constant part has twoparameters: A price ( P ) and a consumption quantity ( Q ). In this formalism P ij denotesthe price level of the j -th step of the inverse demand function of player i . The parametersof the demand functions used in the example and depicted ion Fig. 2 are summarized inTable 2. quantity [MMBtu] E UR / MM B t u Demand curve of consumer 1 quantity [MMBtu] E UR / MM B t u Demand curve of consumer 2 quantity [MMBtu] E UR / MM B t u Demand curve of consumer 3
Figure 2: Inverse demand functions of consumers assumed in Example 1. MMBtu stands formillion metric British unit. P P P Q Q Q P P P Q Q Q P P P Q Q Q Table 2: parameters of the inverse demand functions considered in the example.
We assume a central regulatory authority who has the exclusive right to sell pipelinecapacity licenses for market participants, who, according to their individual positions nd demand functions, have different evaluations for particular routes in the network.Participants do have a strategy space – they decide which bids they would like to submit,thus they can be considered as players of the game. For the clarification of the terminology,we will use ’participants’ and ’players’ as synonyms in the rest of the paper. In thispaper we will assume that the exclusive participants of the capacity allocation game areconsumers of the model, since, as mentioned earlier, their optimal bidding strategy maybe plausibly derived from the modelling assumptions via simple computations in both ofthe mechanisms analyzed. In the following we summarize the assumptions by which the bidding behavior in our modelis described, and evaluate the ACC auction based on the principles laid down in [12]. TheACA auction process is carried out simultaneously for each line. Let us summarize thecritical points which are defined in the regulation, regarding our auction model. • Ascending clock auctions shall enable network users to place volume bids againstescalating prices announced in consecutive bidding rounds, starting at the reserveprice P . • The volume bid per network user at a specific price shall be equal to or less than thevolume bid placed by this network user in the previous round. • If the aggregate demand across all network users is less than or equal to the capacityoffered at the end of the first bidding round, the auction shall close. • If the aggregate demand across all network users is greater than the capacity offeredat the end of the first bidding round or a subsequent bidding round, a further biddinground shall be opened with a price equal to the price in the previous bidding round,plus the large price step. • If a first-time undersell occurs, a price reduction shall take place and a further biddinground shall be opened. The further bidding round will have a price equal to the priceapplicable in the bidding round preceding the first-time undersell, plus the small pricestep. Further bidding rounds with increments of the small price step shall then beopened until the aggregate demand across all network users is less than or equal tothe capacity offered, at which point the auction shall close.
In the above mechanism the clear aim of the large and small price steps is to reduce thenumber of bidding rounds (the auction switches to small price steps before the undersell).For the aim of simplicity, in our simulations we use only one step size, which is smallenough to capture the details of the change of individual evaluations as the price of acertain line increases: It is easy to see that as every source cost, transfer cost, and demandfunction parameters are defined as integers, a step size smaller than the unit will notmake any difference to the bidders in their evaluations (if the price started from an integervalue). Thus in our simulations, we will assume that the price of every capacity productis increased by 1 in every step, if no undersell occurs.In our simulation we assume that in the beginning, no capacities are allocated, everytransfer capacity is subject to the auction. This means that we will have 12 products:capacities corresponding to positive and negative directions for each edge. According tothe principles described in [12], the initial price of capacity products are set to cover nly the expanses of the TSO. As the proposed model considers transfer prices separately(not included in capacity prices), this implies that we assume that the initial price of thecapacity products is 0.According to the considerations discussed before in subsection 1, we simulate 3 roundsof ACA auctions, each with the starting price of one unit for each active product.To clarify the terminology, each auction round begins with the declaration of capacitiesunder auction (and the initial prices of them, which is assumed to be 0), and after thesubmission of bids to every active product, the prices of overbidded products increase inevery step of the actual auction by 1 unit. If the total amount of submitted bids for acertain capacity are less than the volume of the particular capacity, the product becomespassive, and capacities are allocated according to the last submitted bids. The auctionround ends, if all products become passive. The bid format in the ACA is simple. In each bidding round, we have active products, forwhich bids may be still placed and closed products, for which the capacity is over. In theinitial round all products are active. In our case the products are the + and - directionaltransfer capacities of the pipelines, namely [ q +1 q +2 q +3 q +4 q +5 q +6 q − q − q − q − q − q − ] Each player must define the bid quantities he/she places on the active products in eachround, in other words the overall bid of a player is defined by a vector.In the next subsection we describe how we model the optimal bidding strategy ofparticipants in these framework.
Let us first emphasize that in thecurrent model we assume that players have perfect information about source and transfercosts of the network (and of course about the actual prices of capacities in the auction),but they have no information about other player’s utility functions. In other words, in thecurrent simulation framework player do not make any speculations of other player’s bidsto optimize their own bidding strategy, they just consider the network parameters, theactual capacity prices in the particular step of the actual auction, and their own inversedemand function to determine their bids for the actual bidding step.In the proposed modelling formalism we distinguish between already allocated capac-ities , and potential capacities . In a general case (e.g. if we are not considering the firstbidding round), players may hold already allocated capacities. To be more precise, weassume that after the yearly auction finishes, each player assigns flows to the capacities,he/she obtained at the end of the auction to maximize its utility via fulfilling the demandin the respective consumer node. However, after the determination of these flows, somecapacities may remain unused – these are considered as free-to-use already allocated capac-ities in the following (since the payment for them has already been completed). A simpleexample for such a scenario may be considered, if a player bids for several components(e.g. 2 line capacities) of a route, receives one of them in the early steps of the auction,but during the following steps the price of the other one increases so much that it doesnot make sense for the player to maintain its bid anymore. This way the player will be robably not able to assign any flow to this single capacity, thus he/she will have unusedcapacity at the end of the first round, which will be considered as already allocated capac-ity in the second bidding round. Capacities still under auction are considered differently,since the payment for them has not been completed yet (using them implies additionalcost, since they must be acquired first at their actual price).To exactly determine the actual bids of a player in a given auction step, we use theprinciple of optimal potential flows . This approach means the following. In each step ofany auction round (1 2 or 3), players determine their optimal flows, which maximizes theirresulting utility U , assuming that they will receive the capacities on which they place bids.The resulting utility may be calculated as the utility of consumption at the consumer node U C , minus the cost of transfers ( C T ) and inlets ( C I ) , minus the payment for the capacityrights ( C C ). In this calculation the player takes into account that flows planned on alreadyallocated capacities do not imply furthers costs in addition to the transfer cost, in contrastflows planned on potential capacities imply additional cost, namely the actual capacityprice.In our model, players submit bids according to these optimal actual flows, namelywe assume that they submit a bid vector, which is able to ensure the flows on potentialcapacities calculated in the optimal actual flows problem.The formalism of the approach is the following. Let us consider a linear programmingproblem, as max U ( x ) where x = f aac f cau IY s.t ≤ f aac ≤ AAC ≤ f cua ≤ CU A ≤ f cua ≤ P B ≤ I ≤ Y ≤ Y A eq x = 0 (1)In the above notation, f aac denotes the vector of flows on already allocated capacities, andmay be decomposed as f aac = (cid:18) f + aac f − aac (cid:19) (2)where f + aac stands for the positive directional flows (according to the directions of edges)and f − aac denotes the negative directional flows. f cau denotes the flows on capacities underauction (potential capacities), and it has the same structure as f − aac . f + aac , f − aac , f + cau , f − cau ∈R m , where m is the number of edges (6 in the case of the proposed example). I ∈ R n s denotes the vector of inlet values ( n s stands for the number of source-nodes,and equals to 1 in the defined example), while Y describes the consumption. To eachpiecewise constant part of the inverse demand function of the respective player (these aredepicted in Fig. 2), we assign a variable y k , so Y ∈ R n p , where n p is the number of thesepiecewise constant parts ( n p = 3 for all 3 players in our case). As discussed previously, I is the maximum inlet, while Y = Q i Q i Q i so Y = y y y ≤ Q i Q i Q i (3) here the Q i values depend on the actual player i .The parameter vector AAC holds the already allocated capacity values for each edge-direction pair.
CU A stands for the vector of capacities currently under auction.
P B stands for the value of the previous bid vector submitted by the player. This constant isnecessary, since the rules of the ACA discussed in subsection 2.3 state that ’volume bidper network user at a specific price shall be equal to or less than the volume bid placed bythis network user in the previous round’ .The equation A eq x = 0 formalizes the nodal balances. For each node, the inlets plusthe inflows must be equal to the outflows plus the consumption. Using the variables in x ,and the network topology, the n equations corresponding to the rows of the A eq matrixmay be easily derived ( n is the number of nodes).Finally U ( x ) stands for the resulting utility of the actual player, and may be decom-posed as U = U C − C T − C I − C C , where, as discussed previously, U C denotes the utilityof consumption, C T and C I denote the cost of transfers and inlets, and C C denotes thepayment for the capacity rights. It is clear that U C , C T and C I are linear functions of thevariables of x – the coefficients of the linear functions may be derived from the P ij values,the c t transfer costs (defined for the edges in Table 1), the inlet cost per unit (23 in ourcase). Finally, C C may be derived from the actual prices present in the auction step.After each step, according to the potential allocations (if for any product, if there’s nooverbidding, the auction for that product is finished, it will be allocated at the actual priceto the actual bidders), the values of already allocated capacities ( AAC ) and the capacitiesunder auction (CUA) are updated.
Calculations between ACA rounds
After a round of ACA auction has finished,each player calculates the optimal flows on the capacities actually allocated to him/her.This can be done easily by solving 1, under the assumption that there are no capacities un-der auction, only already allocated capacities. According to the results of this calculation,every player performs the two following operations:1. First, the player in question determines the quantity ensured by the calculated flows.As this quantity is ensured for him/her in the following, the player updates its inversedemand function for the remaining auction steps accordingly (demand is reduced bythe already accessible quantity).2. Second, the player divides the capacities allocated to him/her into two groups: Ca-pacities which are used by the flows are considered ’out of the game’ in the following,thus they are allocated to the flows fixed after this round. Allocated capacities onthe other hand which are not used in the flows are considered as already allocatedcapacities in the next auction round (payment for them has been already completed,thus they may be used in the design of potential optimal flows in the following).
According to the principles discussed above, the three rounds of ACA auctions may besimulated for the proposed simple example. The details of the calculations can be foundin Appendix A.The resulting capacity allocation of the ACA process is described by the matrix detailedin E.q. (4) (where rows correspond to players 1-3) and results in the resulting utility vector U ACA U ACA U ACA ] = [720 400 1325] . Thus the total utility of players ( U TACA ) equals2445 in this case. AC ACA = (4)Let us introduce some further characteristic values as bases for latter comparison. Thetotal payment received by the auctioneer ( P C C ) for the auctioned capacities is 700 unitsin this case. We can furthermore characterize how much the available infrastructure isutilized. If we consider the total transfer capacity of the network versus the amount ofcapacity allocated via the process and versus the capacity actually used by the playersat the end in their resulting optimal flows on their available capacities, we can calculatethat the ratio of allocated network capacities ( r ANC ) is 0.321 while the ratio of networkcapacities which are used in the end ( r UNC ) is 0.2963 (let us note that one edge is typicallyused in one direction, but capacity products for both directions are present in the auction,thus total usage ratios over 0.5 are very unlikely).In the following we introduce the convex combinatorial auction and demonstrate itsfunctioning and results in the case of the introduced example.
As we will see, the CCA framework uses a route-centered formalism. Our first task in thisapproach is to define the routes of players, via which they are potentially able to transportthe gas for themselves. We consider the following routes for player 1 2 and 3 respectively: R -1 R -2, 4 R -3, 5 R -2, -6, 5 R -3, 6, 4 R -2 R -1, -4 R -3, 5,-4 R -3, 6 R -3 R -1, -5 R -2, -6 R -2, 4,-5Table 3: Routes of players considered in the proposed example. Every route is a set of edgesleading from the source to the consumer. The signs are positive if the direction of the edgecoincides with the route. In this setup we suppose that players may submit bids for route-quantity pairs, ac-cording to the principle that in the outcome of the auction a convex combination of theirsubmitted bids will be assigned to them. This assumption allows bidding for alternativeroutes: If a consumer needs 1 unit of gas and there are 2 alternative sources in the network,corresponding to two different routes, he/she can submit two bids for the capacity licensesof the two distinct routes, both with the quantity of 1 unit. At the end of the auction aconvex combination of the two bids will be assigned to him, which means that he/she willnot get more network capacity towards the sources than 1 unit, but this maximally 1 unit ay be composed of arbitrary proportion of the two routes. Of course this line of thoughtapplies for a single source with multiple access routes as well. As mentioned earlier, in the CCA framework participants of the auction may submitbids for route-quantity pairs. Let us denote the k -th bid for route j of player i with B ij,k = ( q ij,k , p ij,k ) where q ij,k is the quantity of the bid, and p ij,k is the price offered for theroute-quantity pair in question. x ij,k ∈ [0 , denotes the acceptance indicator of the bid B ij,k . To formulate the constraints which describe the limited ca-pacity of pipelines, we need to decompose the routes considered in the auction to theircomponents – to edges which correspond to pipelines. Furthermore we take into accountthe possibility that counter-directed flows cancel each other.Let us denote the set of (directed) edges in the network with E , while e ∈ E denotes asingle edge. Each route j (of player i ) may be represented as an R ij ⊆ E subset of edges,where each element is signed, according to whether the direction of the route coincideswith the direction of the included edge or not.Let us suppose furthermore that edge e m has different maximal capacity in the pos-itive and negative direction (think of one-directional pipelines), denoted by ¯ q + m and ¯ q − m respectively.In this case the maximal capacity constraints may be formulated as X i,j,k e m ∈ R ij s ij,m x ij,k q ij,k ≤ ¯ q + m X i,j,k e m ∈ R ij − s ij,m x ij,k q ij,k ≤ ¯ q − m ∀ m (5)where s ij,m is an indicator variable, which equals to 1 if edge e m has positive sign inroute j of player i , and -1 otherwise.Maximal output limitations of sources in the network may be derived very similarly byconstraining the total outflow of the edges connected to the source in question. Convexity constraint:
By definition, the auction assigns to each player a convexcombination of his/her submitted bids. This consideration is formalized as X j,k x ij,k ≤ ∀ i (6) The objective of the optimization process is to maximize the nominal income from thebids, under the previously detailed constraints. ax x x ij,k p ij,k s.t. P i,j,k e m ∈ R ij s ij,m x ij,k q ij,k ≤ ¯ q + m ∀ m P i,j,k e m ∈ R ij − s ij,m x ij,k q ij,k ≤ ¯ q − m ∀ m P j,k x ij,k ≤ ∀ i (7)The above problem falls into the class of linear programming problems. Let us recallthat regarding the ACA framework, we only used linear programming to model optimalbidding behavior, but the allocation itself in that case has been performed by a logicalalgorithm described in subsection 2.3. In contrast, in the case of CCA, the allocationprocess itself relies on solving a linear programming problem. Let us point out herehowever that in other auction framework related to energy economics as electricity auctionslinear programming, and even more computationally demanding programming problems(as integer and quadratic programming) are routinely used in practice (see e.g. [15]). After the optimization process has been completed and the bid acceptance ratios have beendetermined, the payments of the players have to be completed. To determine paymentsin the proposed framework, we use the Vickrey-Clarke-Groves (VCG) mechanism [18,13, 16], which charges each individual the harm they cause to other bidders with theirparticipation. The VCG mechanism gives bidders an incentive to bid their true valuations,by ensuring that the optimal strategy for each bidder is to bid their true valuations of theitems. It is a generalization of a Vickrey auction [18] for multiple items. In the following,we apply the proposed CCA framework for the simple example, and also detail how theVCG-payments are calculated.
The first step is to derive the bids of the CCA approach. This may be done via simplecalculations, which are detailed in Appendix B. Considering the bids detailed in in Ap-pendix B and calculating the optimum of the auction problem summarized in (7), we getthe following results. • Regarding player 1, the nonzero acceptance indicators are x , = 0 . and x , = 0 . resulting in the total capacity of 70 units on route R of player 1. • Regarding player 2, the nonzero acceptance indicators are x , = 0 . and x , =0 . , resulting in the capacity of 55 on route R of player 2. • Regarding player 3, the relevant indicators are x , = 0 . and x , = 0 . ,resulting in the capacities of 70 and 15 on routes R and R .From the resulting acceptance indicators, and from the bid and route data, we candetermined the allocated edge capacities C CCA = (8)If we compare the resulting capacity allocation of the CCA in eq. (8) to the resultingallocation of the ACA in eq. (4), we can already see that the two methods result indifferent allocations. As detailed in subsection 2.5, we use the VCG mechanism for the determination of capacitypayments. The payment of any player is equal to the harm its participation in the auctionimplies for other players. The ’harm’ in our case is measures in the cumulative nominalvalue of accepted bids for each player. Let us consider player 1.In this case we have to first calculate the total nominal value of (at least partially)accepted bids for players 2 and 3. If we consider the bid acceptance indicators detailedabove, and the CCA bid values detailed in Appendix B, we can see that the total nominalbid value of Player 2 (
T N BV ) may be calculated as T N BV = 23 600 + 13 915 = 705 Similarly for Player 3,
T N BV = 0 . · . · , thus the total nominal bid value of Players 2 and 3 is 2150 in this case.If we would like to know how much harm is implied by Player 1 to Players 2 and 3 byits participation, we have to simply re-run the CCA allocation process, assuming no bidsfor player 1. In this case we get the following results. x , = 0 . x , = 0 . (9) x , = 0 . x , = 0 . , (10)implying the quantity of 70 on routes R and R and the quantity of 15 on routes R and R . The reason for this symmetry is that the if we consider the sum of the quantities ofthe first two steps in the demand curves, we get 85 for both players 2 and 3 (40+45 vs 50+ 35).Let us now calculate the modified total nominal bid values (denoted by T N BV m ) T N BV m = 0 . ·
915 + 0 . ·
490 = 839 . (11) T N BV m = 0 . · . · . . , (12)thus the total nominal bid value of Players 2 and 3 is appr. 2277.5 in this case.According to the VCG principles described in subsection 2.5, the capacity payment ofPlayer 1 is 2277.5-2150=127.5.This process may be straightforwardly repeated for players 2 and 3 to determine thecapacity payments. In this case we get the result [ C C C C C C ] = [127 . .6.2 Results of the CCA process for the simple example After the allocated capacities have been determined, the optimal flows on the available ca-pacities may be calculated for each player. This can be done similarly to the case describedin Appendix A in the ACA case, but in this case the optimal flows will correspond exactlyto the (possibly partially) accepted bid-quantity pairs on the respective routes. Knowingthe inlet costs, the transfer costs, the capacity costs and the resulting consumption values,utility calculation may be carried out the same way as in the case of the ACA, describedin Appendix A. U CCA = U C − C T − C I − C C = 3130 − − − . . U CCA = U C − C T − C I − C C = 2410 − − −
80 = 625 U CCA = U C − C T − C I − C C = 4365 − − −
115 = 1330 (13)In this case the total utility of players is U TCCA = 2717 . in contrast to the ACA casewhere, as we have seen, U TACA = 2320 .The total payments for the capacity rights in this case is 322.5 in contrast to the valueof 825 in the ACA case, while the ratios of allocated network capacities and used networkcapacities ( r ANCCCA and r UNCCCA ) are both 0.2778, in contrast to the values r ANCACA = 0 . and r UNCACA = 0 . calculated in the ACA case (there are no unused capacities in the CCAcase).Naturally, this example is not a sufficient basis for reaching general conclusions aboutthe properties of the two methods, but is has been useful to demonstrate our models indetail. in the next section we provide a simulation-based computational analysis for thestatistical comparison of the two methods. We used a computational approach to compare the performance of the ACA end CCAmethods. We generated random setups and simulated the capacity allocation processes todetermine the resulting capacity allocations and payments of the two methods.Each setup was generated as follows. Input parameters were: • The number of vertices (nodes) n v • The number of edges n e • The number of sources n s • Upper and lower bounds for edge capacities q max and q min • Upper and lower bounds for transfer costs C maxT and C minT • Upper and lower bounds for source costs C maxS and C minS In the first step, graph of the network was generated. The first edge was placedrandomly, the second was placed randomly among unconnected node-pairs and so on, untilall n e edges have been placed (see Erdős-Rényi graphs [5]). At the end, connectedness andplanarity of the resulting graph was checked, and if any property did fail, the process wasstarted over. Once the graph proved to be appropriate, n s nodes were picked at randomfrom the set of nodes, and they were defined as source nodes (the rest are considered asconsumer nodes in the following). n the second step, the parameters of edges were determined. Transfer costs for edgeswere assumed to be identical in any direction, thus n e random values from a uniform dis-tribution between q min and q max were picked, and rounded to the closest integer value todetermine maximal edge capacities, and similarly, random values from a uniform distri-bution between C minT and C maxT were picked, and rounded to the closest integer value todetermine edge transfer costs.Following this step, the (maximum) 10 cheapest source-consumer routes were deter-mined for every consumer, considering source and transfer costs as well. The minimal andmaximal values of these routes ( C minRoute and C maxRoute ) were calculated from the results.In the following, supply and demand parameters were set as follows. n s random valuesfrom a uniform distribution between C minS and C maxS were picked, and rounded to theclosest integer value to determine the source costs.Inverse demand functions were determined as follows. We assumed the three-steppiecewise constant form as in subsection 2.1.3, where the quantity of each step was deter-mined by picking a random integer value from the interval [10, 50]. The price of each stepwas determined by picking a random integer value from the interval [ C minRoute + C minS , . · ( C maxRoute + C maxS ) ].We considered various network sizes each with different node, edge and source nodenumbers ( n v , n e , n s ), but the other parameters were fixed as summarized in Table 4. Foreach network size, 1000 setups were generated, on which the ACA and CCA methods havebeen evaluated. par. value par. value q min q max C minT C maxT C minS C maxS In this case n v = 6 , n e = 8 , n s = 1 were assumed. This resulted in an average routelength of 1.56 between consumers and sources in these networks.Regarding the total resulting utility values and their difference ( U TCCA − U TACA ), Fig.3 depicts the distributions over the 1000 simulated cases.The first thing we may notice is that in the ACA case there are scenarios, where U TACA is negative. The reason for this is that in the ACA framework, it is possible that in theprocess of capacity allocation such capacities will be allocated to players, which will beuseless for them as later they are determining their optimal flows on the capacities assignedto them. We have seen this phenomenon in the simple example, where at the end of theACA process a capacity of 10 units on edge 3 has been assigned (see the matrix in eq.(4)). Capacity payments for these unused capacities which do not form a full route at theend of the process imply negative utility components for these players. If these negativecomponents outweigh the positive ones in the context of all players, U TACA may be negativeas well. This happens in the 19 % of the cases when we apply ACA. Figure 3: Distributions of U TACA , U TCCA and of U TCCA − U TACA over the 1000 simulations donewith 6-node networks.
Average values of U TACA and U TCCA are 2267.5 and 3213.1 in this case, while theirstandard deviations are 2660.6 and 3224.7 respectively. As we see in Fig. 3, the CCAmethod does not perform better in every case. In the 34% of cases, the ACA allocationmethod results in higher U T values.Regarding the income of the auctioneer, the average total capacity payments are 5424.3and 4673 in the ACA and CCA cases respectively.In addition, let us compare the ratios of the allocated and used network capacities( r ANC and r UNC ). r ANCACA = 0 . r ANCCCA = 0 . r UNCACA = 0 . r UNCCCA = 0 . Regarding the ’fairness’ of the auction method, several approaches and measures canbe used. In this work we restrain ourselves to a very simple indicator regarding this aspect.For each simulation, we can calculate the difference of the maximal and minimal resultingutilities among players, and average this value over the simulations. These ’unfairness’indicator (
U F ) results in the following values in the case of 6-node networks.
U F
ACA = 1772 . U F
CCA = 1390 . Finally, let us note that the ACA method reached its final values in the 77.5% of thesimulated cases (in these cases no bids were submitted for the 3rd round).
In this case n v = 9 , n e = 12 , n s = 2 were assumed, resulting in an average route lengthof 1.8739 between sources and consumers. egarding the total resulting utility values, Fig. 4 depicts the distributions over the1000 simulated cases. -1 -0.5 0 0.5 1 1.5 2 2.510 Figure 4: Distributions of U TACA , U TCCA and their difference over the 1000 simulations done with9-node networks.
Average values of U TACA and U TCCA are 6231.9 and 7407.3 in this case, while their stan-dard deviations are 4440 and 6105.7 respectively. The ACA method resulted in negativetotal utilities in the 5.9 % of the simulated cases. Regarding this network size, the ACAallocation method results in higher U T values in the 40.6% of simulated cases.Regarding the income of the auctioneer, the average total capacity payments are 8240and 7221.1 in the ACA and CCA cases respectively.Let us note that as the network size increases, the average length, thus the averagetotal cost of routes is also increased. Since according to the simulation assumptions,the inverse demand functions are determined based partially on the route costs, they arealso affected. These considerations explain the increasing trend in the utility values andcapacity payments.The ratios of the allocated and used network capacities ( r ANC and r UNC ) are as follows. r ANCACA = 0 . r ANCCCA = 0 . r UNCACA = 0 . r UNCCCA = 0 . The ’unfairness’ indicators (
U F ) are as
U F
ACA = 3339 . U F
CCA = 2496 . . The ACA method reached its final values in the 47.3% of the simulated cases (in thesecases no bids were submitted for the 3rd round). .3 Results on 15-node networks In this case n v = 15 , n e = 20 , n s = 3 were assumed, resulting in an average route lengthof 2.503 between sources and consumers.Regarding the total resulting utility values, Fig. 5 depicts the distributions. -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.510 Figure 5: Distributions of U TACA , U TCCA and their difference over the 1000 simulations done with15-node networks.
Average values of U TACA and U TCCA are 9272 and 12223 in this case, while their stan-dard deviations are 7077 and 9353 respectively. The ACA method resulted in negativetotal utilities in the 9.6 % of the simulated cases. Regarding this network size, the ACAallocation method results in higher U T values in the 35.3% of simulated cases.Regarding the income of the auctioneer, the average total capacity payments are 16555and 14499 in the ACA and CCA cases respectively.The ratios of the allocated and used network capacities ( r ANC and r UNC ) are as follows. r ANCACA = 0 . r ANCCCA = 0 . r UNCACA = 0 . r UNCCCA = 0 . The ’unfairness’ indicators (
U F ) are as
U F
ACA = 5124 . U F
CCA = 3490 . . The ACA method reached its final values in the 24.2% of the simulated cases (in thesecases no bids were submitted for the 3rd round).
In this case n v = 20 , n e = 30 , n s = 4 were assumed, which resulted in an average routelength of 2.3514 between sources and consumers. Figure 6: Distributions of U TACA , U TCCA and their difference over the 1000 simulations done with20-node networks.
Regarding the total resulting utility values, Fig. 6 depicts the distributions.Average values of U TACA and U TCCA are 13635 and 17988 in this case, while their standarddeviations are 9149 and 13853 respectively. The ACA method resulted in negative totalutilities in the 5 % of the simulated cases. Regarding this network size, the ACA allocationmethod results in higher U T values in the 38.2% of simulated cases.Regarding the income of the auctioneer, the average total capacity payments are 23715and 21644 in the ACA and CCA cases respectively.The ratios of the allocated and used network capacities ( r ANC and r UNC ) are as follows. r ANCACA = 0 . r ANCCCA = 0 . r UNCACA = 0 . r UNCCCA = 0 . The ’unfairness’ indicators (
U F ) are as
U F
ACA = 5849 . U F
CCA = 3947 . . The ACA method reached its final values in the 8% of the simulated cases (in thesecases no bids were submitted for the 3rd round).
One of the the most importantly required characteristics of a capacity allocation methodis the efficiency in the terms of the resulting utility of players. This corresponds to the lessformal principle of ’capacities shall be allocated to those who value them most’. As thesimulations have shown, regarding this aspect, the proposed CCA method outperforms theACA in the majority of cases. The expected total utility of players over the analyzed high umber (1000) of random scenarios was 18-41 % higher in the case of CCA. As the CCAmethod always assigns network capacities to players in a way which ensures connectedpaths, and is able to consider multiple alternative routes, it seems reasonable to presumethat these properties of the method result in higher gains in the case of larger networksand more consumers.If we analyze the results in the terms of allocation efficiency, we can see that theACA method always produces a higher allocation rate ( r ANCACA > r
ANCCCA ), but the capacitiesallocated this way can be only partially utilized by the players (see the r UNCACA valuesand their relation to the r ANCACA values). In contrast, if we consider the utilized allocatedcapacities, the CCA method performs better in every case ( r UNCCCA > r
UNCACA ).Regarding the income of the auctioneer, the ACA method results in 8.73 - 13.85 %higher values. This shows that (based on the detailed simulation results) the CCA methodis not the best choice if one aims to maximize the auction incomes. Let us however em-phasize that in the case of ACA, the capacity payments partially correspond to products,which are of no use in the final evaluation for the player.Regarding the maximal difference between the maximal and the minimal utility amongplayers, simulation results show that the CCA method results in a more fair allocation.In addition to the evaluation of the above quantitative measures, let us point out somemore differences between the two analyzed methods. • In the proposed CCA framework the players place their bid for route-quantity pairs,in contrast to the ACA framework, where bids are placed on single capacities. • The ACA framework has usually multiple rounds (in the simulations, motivatedby the reality of the practical applications, 3 rounds were considered), and in eachround a significant number of steps are present. This means that players do haveto recalculate their evaluations in each of these steps. In contrast, in the CCAframework, players evaluate their respective route-quantity pairs only once, at thebeginning of the auction, which allocates capacities and determines payments in onesingle step. • As optimal bidding is not trivial under the ACA framework (see subsection 2.3.2), inother words, we may say that the computational burden is put to the participants inthe ACA case, where the clearing algorithm is simple, and in contrast, in the CCAframework bidding is simple, and the clearing algorithm is more complex.
In this work we proposed a convex combinatorial auction method (CCA) for the allocationof capacities in capacity-constrained networks, where prior given transfer and source costsapply and the evaluation of routes is subjective by the players. We compared the proposedmethod with a 3-round ACA allocation method, which aims to model the current practiceof capacity allocation of natural gas networks in the EU. We performed simulations onrandom networks, assuming only consumers as participants of the auction, to evaluatethe characteristics of these allocation methods. We found that while the ACA method inaverage gives higher incomes for the auctioneer, the proposed ACA method in the majorityof cases results in higher and more fair resulting utility for the participants, and contributesto the more efficient utilization of network resources. .1 Future wwork The explanation of the obtained results include some plausible, but still hypotheticalassumptions (e.g. the ACA framework performs worse if the average length of the transportroutes is higher), which have to be validated in further computational studies. In addition,regarding the most important question, the possible future availability of the method,several other studies have to be performed as well.First of all, the current modelling studies considered only consumers as participantsof the auction, which is unrealistic. It is possible that under different assumptions, theresults will be significantly different. If we consider a setup where consumers may bidfor routes, but they are ready to take the gas at their ’doorstep’ as well (for reasonableprice), the behavior of producers wishing to market their gas for maximal profit may bealso included in the model. In this case, producers will also be present among participantsbidding for routes/capacities and more general results will be obtainable.Second, as we have seen, the CCA method does not always provide better utility resultsfor players. It is straightforward to ask how the efficiency of the CCA method dependson network topology and parameters, and more importantly, how would these methodsperform on more or less detailed models of the Eurasian gas network, the structure ofwhich is known, and its transfer and demand parameters are also known to some extent.Although the introduction of the CCA method was the first step in the process ofthis analysis, several more computational studies will be required to properly characterizethe practical applicability of the CCA method. In particular, the shortcomings of theEuropean capacity allocation methods might stem from other aspects as well in additionto the intrinsic properties of the capacity allocation method used. We must note, thatchanging the allocation mechanism might not be the easiest and most convenient way foraddressing the shortcomings.In addition, we must note that the congestion problem seems an issue of limited sig-nificance at the European gas networks nowadays, and might be even less of an issue asgas consumption might fall due to decarbonization goals.
The author thanks Borbála Takácsné Tóth and Péter Kotek for their valuable input andfor the fruitful discussions. The author also thanks Tim Roughgarden who suggested touse the VCG algorithm for the problem. This work has been supported by the FundsPD 123900, K 131 545 of the Hungarian National Research, Development and InnovationOffice. Dávid Csercsik is a grantee of the Bolyai scholarship program of the HungarianAcademy of Sciences. ppendix A: The 3 rounds of ACA for the proposedsimple example Round 1
As mentioned in the main text, in the initial step of the first round, all capacities of thenetwork are subject to the auction. This means that the initial vector of capacities underauction (
CaU (1) ) may be written as
CaU (1) = [80 75 70 60 60 60 80 75 70 60 60 60] (14)The initial price for every product is 0. In the first step, each player calculates itsoptimal potential flows. As at this point none of the players has any allocated capacity,straightforwardly, the f aac vectors for each player will be qual to 0. On the other hand,according to the concept proposed in subsection 2.3.2, the f cau vectors will be as (cid:0) f cua f cua f cua (cid:1) = (15)This will determine the bid quantities submitted ( SBQ ) in the first step
SBQ (1) = (16)where the first row corresponds to the bid quantities submitted by player 1, etc.The next step is to calculate the sum of columns of (16), and determine the set ofproducts, for which there is overbidding and the set for which there is underbidding. Ifwe compare the bid sums X SBQ (1) = [0 0 0 10 0 0 90 100 70 10 0 15] with the capacity volumes under auction, now equal to the maximal edge capacitiesdetailed in table 1, we find that there is overbidding for line 1 and 2 in the negativedirection (columns 7 and 8). For these products, the price will be increased by 1 unit inthe next step, while for the others, the auction ends, and capacities are allocated to the idders at their actual price. Thus, the matrix of allocated capacities ( AC ) after the firstauction step will be AC (1) = (17)where, similarly to SBQ , row correspond to players. As players received these capacitiesat their initial price (0), this step implies no capacity payments for the players.In step 2, the only remaining products are the negative directional capacities of lines 1and 2 (80 and 75 units respectively), at the price of 1 units. If we redo the calculations forthe optimal potential flows, considering the new prices, we get the same result as before– in other words, we may conclude that the previously determined flows are still desiredby the participants, also in the case of higher capacity prices of the overbidded lines. Thismeans that they will submit the same bids for these products in this step as before.If we continue this line of calculations, we find that that in step 4, after recalculatingoptimal potential flows, player 2 drops its bid corresponding to line 1 (as the potentialflow on it no longer benefits him/her at the price of 3 for the line), thus the correspondingproduct will be allocated to player 1 entirely at the price of 3, implying 240 units ofcapacity payment for player 1.Similarly, in step 6, player 1 also drops its bid corresponding to the negative directionof line 2, but considering the bids of player 2 and 3, the overbidding remains (75+15>75).This situation remains unresolved until step 9, when the recalculation of optimal potentialflows results in the following decision: Player 2 reduces its bid from 75 to 40. This can beexplained by the fact, that the increase of total cost (capacity + transfer + inlet payments)implies that a flow of 75 units on line 2 is no longer profitable for him/her, but a flow of40 is still is. The reason is that the quantity of the first step of the inverse demand curvedepicted in subsection 2.1.3 is equal to 40 units, and at this total price, this is the onlyremaining demand part for which is worth to import the gas at this price.This way the overbidding of line 2 in the negative direction resolves, and the resultingallocated capacities of round one of the ACA may be written as AC (8) = . (18)We may also summarize the final prices of the line capacities as [0 0 0 0 0 0 3 8 0 0 0 0] Considering the allocated amounts and the resulting prices above, the resulting capacitypayments in the first round are , and for players 1,2 and 3 respectively.Next we calculate the actual optimal flows regarding the above allocation. f aac f aac f aac (cid:1) = (19)We can see that the accessible quantities ensured after round 1 are , and forplayer 1,2 and 3 respectively (the flow of 15 units in the case of player 3 includes twoedges, while the remaining flows are one-edge transfers). According to these quantities,the inverse demand functions are updated as depicted in Fig. 7. The updated demandcurves will be used in the following ACA round. quantity [MMBtu] E UR / MM B t u Inverse demand function of player 1 updated after round 1 quantity [MMBtu] E UR / MM B t u Inverse demand function of player 2 updated after round 1 quantity [MMBtu] E UR / MM B t u Inverse demand function of player 3 updated after round 1
Figure 7: Updated inverse demand functions after the first round of ACA.
After the first round, player 1 has 10 units of unused capacity on line 4 in the positivedirection and player 2 has 10 units of unused capacity on line 4 in the negative direction.
Round 2
Following the first round, the second round is initialized via the determination of thecapacities under auction. This is done by subtracting the column sums of matrix (18)from the vector (14).
CaU (2) =
CaU (1) − X AC (8) = [80 75 70 50 60 60 0 20 0 50 60 45] (20)In addition all prices are reset to 0.Before the first step of the second round, players determine their optimal potentialflows. IOn this case this results in the quantities summarized in eq. (21). f aac f aac f aac f cua f cua f cua (cid:1) = (21)It can be seen in (21) that player 1 uses the already allocated free capacity of 10 units onedge 1 (positive direction).Thus, the submitted bid quantities ( SBQ ) in the first step of round 2 will be
SBQ (1) = (22)We proceed similar to round 1. After the first step we can see that there is overbiddingon product 8, namely the negative directional capacity of edge 2 (for all other productsthe auction ends after the first step, and no capacities are allocated). In the second step,as the price of the overbidded capacity is increased to 1, player 1 drops his/her bid, thusthe capacity is allocated to player 2 at the price of 1 at the end of step 2. The resultingallocated capacities of round two of the ACA may be written as detailed in (23). AC (8) = . (23)Player 2 gets its capacity of 20 units at the price of 1, thus the capacity payment forhim/her is 20 units in this round. Round 3
After the usual update of the
CuA vector and the resetting of the prices, the optimalpotential flows are zero vectors for each player in this case. This means that no bids aresubmitted in round 3, the final results of round 2 are the final results of the ACA processin this case.
Evaluation of the results
Player 1
The total allocated capacity of Player 1 ( AC T ) is C T = (cid:0) (cid:1) . (24)As it is also naturally reflected in the optimal flow calculations, player 1 is not able touse the 10 units of capacity allocated on line 4 (+ direction), its final optimal flow f ACA F on the available capacities is described by the vector (25). f ACA F = [0 0 0 0 0 0 80 0 0 0 0 0] T (25)The utility of consumption may be calculated from the original inverse demand functiondepicted in Fig. 2: U C = 50 ·
47 + 30 ·
39 = 3520 . The optimal flow implies the transfercost C T = 720 and the inlet cost C I = 1840 . Player 1 paid a total amount of C C = 250 during the ACA process, thus its resulting utility is U ACA = U C − C T − C I − C C = 710 (26) Player 2
The total allocated capacity of Player 2 ( AC T ) is AC T = (cid:0) (cid:1) . (27)The optimal flows of player 2 f ACA F on the available capacities are described by thevector (28). f ACA F = [0 0 0 0 0 0 0 60 0 0 0 0] T (28)Considering that player 2 paid 330 units as capacity payment in the first round, and40 in the second, its resulting utility may be calculated as U ACA = U C − C T − C I − C C = 2600 − − −
370 = 370 (29)
Player 3
The total allocated capacity of Player 3 ( AC T ) is AC T = (cid:0) (cid:1) . (30)The optimal flows of player 3 f ACA F on the available capacities are described by thevector (31). ACA F = [0 0 0 0 0 0 0 15 70 0 0 15] T (31)The resulting utility of player 3 may be calculated as U ACA = U C − C T − C I − C C = 4365 − − −
205 = 1240 (32)
Appendix B: CCA calculation details for the pro-posed simple example
Derivation of CCA bids
In this subsection, we show how the parameters of the network and of the inverse demandfunctions clearly define a set of bids according to the format described in subsection 2.4.1.For the derivation of the plausible bid set in the case of the proposed example, let usfirst consider the value of route 1 of player 1 ( R ) as an example in the case of the amountof 50 MMBtu. The source cost is 23, while the cost of edge 1 (the only edge of R is 9),resulting in the total cost of 32 for route R , which means a cost of 1600 in the case of 50units.On the other hand, considering the inverse demand function of consumer 1, depictedin Fig. 2 we can see that 50 units of gas results in the consumption utility of 2350 units(=47 · R produces a valueof 2350-1600=750 units for player 1. As in the current study we assume truthful bidding,we assume that player 1 bid its true evaluation (750 units) for the route-quantity pair of ( R , . Let us emphasize at this point that Players do not pay the nominal bidding valuefor their accepted bids in the CCA framework (see subsection 2.5) – if player 1 had its bidfully accepted, and he/she would pay 750 units for the capacity rights, its resulting utilitywould be 0.Let us now consider the same route in the case of 90 MMBtu (corresponding to thesecond step of the inverse demand function of player 1). The total cost of the transfer andthe source may be derived similarly (90 · ·
47 +40 ·
39= 3910 in this case. Thus importing 70 units of gas via R can be evaluated to1030 EUR by player 1. Regarding the third step of the inverse demand function, it canbe easily calculated that importing 95 units is not profitable via this route.Based on the above calculations, let us thus assume that player 1 submits these twobids for its first route: B , = (50 , B , = (90 , (33)What happens if both bids are partially accepted? As we stated before, the outcomeof the auction assigns a convex combination of the submitted bids to each player, so thisis a plausible scenario.If only the first bid is partially accepted ( < x , < ), the case is trivial – both theprice paid for the bid and the utility implied by the transport are multiplied by x , , whichis acceptable to player 1 (the net utility will be still 0 if the nominal price is paid). f the second bid is partially accepted ( < x , < ), the situation may be even better.Let us e.g. consider x , = 0 . , meaning the import of 54 units via route R , . The nominalpayment of Player 1 (not the realized payment, as emphasized before) for the route licenseis . · units in this case. Importing 54 units implies a cost of ·
32 = 1728 ,and the utility of ·
47 + 4 ·
39 = 2506 which leaves player one with a surplus of net utilityof 160 units (=2506-(618+1728)).This simple calculation demonstrates that because of the non-increasing inverse de-mand functions, partial acceptance of bids corresponding to the second or further steps ofdemand curves may be only beneficial for the players.Similar to the evaluation of R all other routes of the players may be evaluated, andfor each step of the respective demand function, bids may be derived.We consider only those bids, for which it is true that even its full acceptance is stillacceptable for the player (as we have seen this is not necessarily true for all the threesteps of the inverse demand function in every case). The numbers of bids correspondingto various players and routes are summarized in the Table 5. player R i R i R i R i R i The quantities and prices of the submitted bids are summarized in Tables 6 and 7respectively.route ( j ) k=1 k=2 k=31 50 90 02 50 90 03 50 90 04 50 0 05 50 0 0 route ( j ) k=1 k=2 k=31 40 85 1202 40 85 03 40 0 04 40 0 0 route ( j ) k=1 k=2 k=31 50 85 1302 50 85 03 50 85 1304 50 85 0 Table 6: The quantities of bids ( q ij,k ) corresponding to players 1, 2 and 3 in the proposedexample. route ( j ) k=1 k=2 k=31 750 1030 02 600 760 03 425 445 04 325 0 05 200 0 0 route ( j ) k=1 k=2 k=31 600 915 9502 400 490 03 140 0 04 280 0 0 route ( j ) k=1 k=2 k=31 950 1475 15652 825 1263 03 850 1305 13054 675 1008 0 Table 7: The prices of bids ( p ij,k ) corresponding to players 1, 2 and 3 in the proposed example.29 eferences [1] Mert Bilgin. Geopolitics of european natural gas demand: Supplies from russia,caspian and the middle east. Energy Policy , 37(11):4482–4492, 2009.[2] European Comission. Council regulation (EU) no 1775/2005 of the european parli-ment and of the council of 28 september 2005 on conditions for access to the naturalgas transmission networks.
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