SHARING THE COSTS OF TRANSMISSION EXPANSION: A COOPERATIVE GAME THEORY APPROACH APPLIED ON THE NORDIC ELECTRICITY MARKET

Transcription

1 SHARING THE COSTS OF TRANSMISSION EXPANSION: A COOPERATIVE GAME THEORY APPROACH APPLIED ON THE NORDIC ELECTRICITY MARKET Hans Nylund, Division of Economics, Luleå University of Technology, SE Luleå, Sweden hans.nylund@ltu.se Abstract The electricity markets in the Nordic countries have during the last decade been integrated into a regional market. An important part of this market is the interconnected high-voltage transmission grids that form the infrastructure for trade. Development of the grids needs to be done with a regional perspective in order to facilitate further development of the regional market. To achieve this, the transmission system operators (TSO) need to cooperate on grid expansions. The purpose of this paper is to analyse how the costs of expanding the grids can be shared by the TSOs. The analysis is done within the cooperative game framework. Three allocation methods are evaluated; the Shapley value, the Nucleolus; and proportional division. Two case-studies of expansion projects are used to test the methods. The game-theory analysis shows that all methods produce acceptable solutions from a strategic perspective. When adding the properties of monotonicity and simplicity to the evaluation, the proportional method seems to be the most appealing choice for allocations. 1. Introduction The national electricity markets in the Nordic countries have during the last decade grown beyond national boarders into a common Nordic market for electricity. This has been a politically driven development originating in a vision shaped in 1995 to create A borderless [Nordic] electricity market with efficient trade with neighbouring markets (Norden, 2004). The benefits from a common market are economic and security of supply related, as well as environmental. Some building blocks of a regional market such as the Nordic electricity market are harmonized national markets, a common marketplace for the good and a sufficient infrastructure for the good to be delivered on. The focus of this paper is on the last one of these, the high-voltage transmission girds that form the infrastructure of the electricity market. Transmission grids have limitations on the amount of electricity that can be transferred. Where demand exceeds capacity, decisions need to be made on whether investments in new capacity are worth while. The answer to this can be found through cost-benefit analysis, but problems can arise when it comes to financing the investments. Benefits from an investment in new infrastructure at one place often spill-over across boarders, but costs largely remain in the investing country (or countries in the case of cross-boarder investments between two countries). This means that in the absence of shared financing, a common market will suffer the risk of underinvestment in infrastructure since local benefits do not always outweigh costs (Bougheas, et al 2003, Nordic Competition Authorities, 2007). The apparent solution is that if we share the benefits, we should share the costs. The purpose of this paper is to analyse how different cost-sharing methods could be applied to transmission investments on the Nordic electricity market. The problem is approached as a cost-allocation problem where the costs (and thereby benefits) of an investment are to be allocated among the beneficiaries. Cooperative game theory is used to model how independent countries are likely to respond to the allocations resulting from different allocation rules. To promote investments, the allocations need to be such that beneficiaries agree to take part in the financing. The results of different allocation rules are demonstrated using data from two cost-benefit analyses made for transmission investment projects. The following section gives a more detailed description of the Nordic electricity market and the transmission grids. The case-studies are presented and put into context. Section 3 describes cost- 1

2 allocation theory and the connection between cost allocation and cooperative game theory. In section 4 the results of the allocation rules applied on the case-studies are presented and evaluated. Finally, section 5 draws the conclusions. 2. The Nordic electricity market Although the national electricity markets have been integrated into a regional market on the wholesale side, and progress is being made towards an integrated retail market as well, the main institutional entities have remained national in scope and jurisdiction. These include regulatory authorities and transmission system operators. The regional market has been built on voluntary cooperation and consensus, without a legal basis (EMG, 2008). There is Nordic cooperation between regulatory authorities and between system operators, but decisions regarding the electricity market are still in essence national. It is in this context that the problem of financing transmission expansions has surfaced. The transmission system operator (TSO) The basic feature of any electricity transmission system is that, at all times, the amount of electricity fed into the system by producers must equal the amount taken out by consumers (minus losses on the transmission grid). The task of ensuring that the system always maintains this balance between supply and demand is carried out by the transmission system operator (TSO). On the Nordic market each country has one national TSO that is responsible for balancing the national system. The role of the TSO also includes planning and financing of necessary expansions of the national grid. A key priority in this work is to ensure security of supply, which basically means reducing the risk of blackouts. Another influential consideration in expansion planning is the economic benefits that increased trade can bring. Cost-benefit analyses are used to estimate the benefits and costs from expansion projects. Since the national grids are interconnected across the boarders into a common Nordic grid, an expansion in one part of the system will affect the whole system. To promote an efficient expansion of the national grids and their interconnections from a Nordic perspective, the TSOs have a cooperative organisation (Nordel 1 ), which gives recommendations on expansions that are beneficial for the common market. The decision on what to build is however decided by the respective national TSO. This scattering of decisions, costs and benefits creates interdependence between the Nordic countries. The problem of financing expansions under these conditions has been pointed out by several actors on the market (ETSO, 2006; Nordic Competition Authorities, 2007; Electricity Market Group, 2008; Svenska kraftnät, 2009; Energy market inspectorate, 2009). A report to the 2008 meeting of the Nordic Council of Ministers 2 states: The issues regarding allocation of costs and benefits of Nordic gird investments must now be addressed. Progress on this issue is critical, as projects may have costs in one country and benefits in other countries (EMG, 2008). The TSOs finance their national grid investments through tariffs on their own connected customers. Congestion rent 3 is also used for investments to varying extent. Cross-boarder lines between two countries, so called interconnections, are financed bilaterally by the two TSOs involved. The cost of the line is normally divided equally (Nordel, 2005b). Sharing of congestion rent in proportion to investment cost has been tried within Nordel as a way of inter-compensation between the TSOs. This was applied in Nordel s 2004 investment program called five prioritized cross-sections (Nord Pool Spot, 2009). 1 Nordel is as of closed down and all tasks have been transferred to the newly formed pan-european organisation European Network of Transmission System Operators for Electricity (ENTSO-E). 2 The Nordic Council of Ministers is a regional partnership for the Nordic countries. The formal cooperation on energy issues is done through the Council. 3 Congestion rent, also known as bottleneck-income, is generated from the trade between two price areas with different area prices. Se the next section on Nord Pool for a more detailed explanation of congestion rent. 2

3 Regional markets are developing in continental Europe as well and there is a consensus within the European Union on the creation of an internal electricity market for the whole of Europe (EC, 2003). The problem of cost recovery for gird investments has been recognized by the European Commission and an inter-tso compensation mechanism (ITC) has been in place since 2002 as a voluntary agreement between the European TSOs including the Nordic TSOs. The ITC-mechanism is designed to compensate for costs incurred on a national grid as a result of hosting transit flows in-between two other neighbouring countries. The compensation consists of two parts; one part for increased losses in electricity transmission due to the higher pressure on the grid caused by the transit; one part for the capital cost of the transmission gird (ETSO, 2005). The ITC-mechanism has been criticised for several weaknesses that distort incentives for efficient transmission expansion (Elforsk, 2008). Nord Pool the market place for electricity The regional electricity market of the Nordic countries is organised into one common wholesale market and separate national retail markets. On the wholesale market about 2/3 of total electricity consumption is traded through the common market place Nord Pool, the rest are bilateral contracts between buyers and sellers. Nord Pool is a day-ahead power exchange where producers and buyers make price and quantity bids to buy or sell for each hour the following day. Nord Pool sums the bids into hourly supply and demand curves for the coming 24 hours. A common price for the Nordic market is set at the equilibrium of supply and demand for each hour, this is called the system price (also spot price). This price setting method is referred to as marginal cost pricing because the sell bids from producers will to some extent reflect the marginal cost of their production technique. This means that when demand is high, power plants with more expensive production techniques need to be utilised to balance the demand. Consumer and producers are spread out over a large geographical area in the Nordic market. Their ability to trade with each other will therefore be limited by the transmission capacity of the grids. To make trade through Nord Pool work under these limitations, the market is divided into price areas (which also constitute bidding areas). This method to handle trade constraints is called market splitting. On any given hour, supply and demand in the price areas can be either in balance or in surplus/deficit under the given system price. In areas with surplus supply, the export capacity will be utilised to transmit power to areas with deficit supply. When export (or import) demand exceeds transmission capacity for an area, the area price will differ from the system price. The situation is depicted for two areas in figure 1. P Area A Nord Pool Area B Supply surplus market place Supply deficit P S P S Pa S Psys Pb D D Q D Q Q Q export Q import Figure 1. Common market equilibrium and price area equilibriums for a surplus and a deficit area. At the system price Psys determined in figure 1, the sell bids made by producers in area A will exceed the buy bids from buyers in area A. This means that the surplus production on offer at system price in area A can be sold to the deficit area B, up to the limit set by the transmission capacity between A and B. In figure 1 the transmission capacity is not enough to fully equalize the area prices to the system 3

4 price. The resulting area prices are Pa and Pb. The price that buyers and sellers in an area pay or receive when importing or exporting is their own area price, as determined by the day-ahead bids given to Nord Pool (Energimyndigheten, 2004). This means that in figure 1 the exporting producers in area A receives the price Pa, while the importing buyers in area B pay the price Pb>Pa. This price difference generates an income known as congestion rent, which is collected by Nord Pool and distributed among the TSOs as owners of the transmission lines. Trade patterns Electricity production on the Nordic market consists of two main generation types; hydro power and thermal power. Hydro generation dominates the northern part of the system with the main resources located in Norway and northern Sweden. Thermal generation is located in the southern part with nuclear power in Sweden and Finland and coal plants in Denmark. These production types complement each other as hydro production varies over time with precipitation levels, while thermal power can be utilised on demand. The geographical dispersion of hydro and thermal generation, and the difference in marginal cost of production (hydro is low-cost), gives rise to certain trade patterns. Figure 2 displays the characteristic trade flows on the Nordic market. Norway Sweden Finland Denmark Crosssection 4 Great Belt connection Under construction German connections Poland connection Figure 2. Transport channels in the Nordic transmission system. Source: Own representation based on Nordel, The main direction of flow as described in figure 2 is along the north-south axis. The motivation behind this flow is the exchange between the hydro dominated areas in the north and the thermally dominated areas in the south. In times of normal or high precipitation, the low cost hydro power is exported to Denmark and the European continent. In dry years or periods of high demand in the Nordic countries, thermal power is imported from Denmark and the continent. The east-west channel is used to balance the national systems in Sweden and Finland and increase security of supply (Nordel, 2008a). It also provides a link to Russia and Estonia via Finland. A transport channel consists of several connected lines with varying capacity. The channels can therefore be constrained by capacity bottlenecks along the way (Nordel, 2003). Such bottlenecks are referred to as cross-sections. Figure 2 indicates an important bottleneck located internally in Sweden called cross-section 4. It is part of Nordel s five prioritized cross-sections and a reinforcement of this section has been decided by the 4

5 Swedish TSO and is currently under construction. The figure also displays a new connection under construction in Denmark, called the Great Belt. This is a sea cable linking the systems in eastern and western Denmark, which has previously been separate. Case studies To demonstrate the benefits and costs from transmission investments, two case studies are used. These are the connections pointed out in figure 2; Cross-section 4 and the Great Belt. The case studies illustrate two different investment situations that both lead to the issues of spill-over benefits and difficulties in financing. The basis for an analysis of this problem is a regional cost-benefit analysis that estimates the effects of a project in all countries in the region in monetary terms. The cost-benefit analyses are done through simulation models that predict the changes in electricity flow and the market effects resulting from an expansion in a part of the system. The changes that an expansion can bring includes; gains in producer and/or consumer surplus; increased security of supply; changes in congestion rent and grid losses; cost-savings from trade in regulating power and increased competition (Nordel, 2003). Costs include the investment cost of the line, auxiliary parts and operation and maintenance costs (Nordel, 2008a). Cost-benefit analyses are available for the case studies chosen and they are described below. The Great Belt The costs and benefits of a connection between eastern and western Denmark have been studied by the Danish TSO Energinet.dk. The Nordic benefit from this connection comes from reduced bottlenecks in the north-south transmission channels, resulting from the connection of the eastern and western channels (Nordel, 2008). The connection will also have specific benefits for Denmark arising from the linking of the two separate systems in eastern and western Denmark. Table 1 displays the benefit estimates for the Nordic region and the neighbouring countries in the UCTE 4 system; Holland, Germany and Poland. The figures give the effects on consumer and producer surplus and congestion rent in million Euro per year 5. Table 1. Benefit estimates for the Great Belt project in Denmark. Source: Energinet.dk, M / year Norway Sweden Finland Denmark UCTE Sum Consumers 40,69 51,22 24,44 0,65 10,14 127,14 Producers -36,4-47,58-27,17 3,25-9,62-117,52 Cong.rent -5,85-1,82-0,65 0,39 1,56-6,37 Sum -1,56 1,82-3,38 4,29 2,08 3,25 The benefit estimates in table 1 show that Denmark, Sweden and UCTE have a positive total benefit and Norway and Finland have a negative benefit. The investment cost of the connection is estimated to 160 M (Nordel, 2004). Operation and maintenance is estimated to 1,04 M per year (Energinet.dk, 2005). Besides the effects in table 1, Energinet.dk has made monetary estimates of benefits arising from shared power-reserves, trade in regulating power and increased competition. These estimates are only carried out for Denmark. The value of shared reserves and trade of regulating power is estimated to 13 M /year. Increased competition is estimated to yield up to 20 M /year in socio-economic benefit. This effect is however uncertain and Energinet.dk have therefore chosen not to include it in the cost-benefit summation, although they do suggest that a careful alternative is to include 10-20% of the benefit. Energinet.dk s recommendation is to build a cable of 600 MW capacity, from which 300 MW will be reserved for regulating power. Because of the high degree of interdependence between the parts in an electricity system and the continuous expansion of the different parts, it is difficult to isolate the effects that a specific expansion will have over a longer period of time. Energinet.dk has for this reason chosen not to calculate the costs and benefits as present values over the lifetime of the investment. Instead they have made cost- 4 UCTE=Union for the coordination of transmission of electricity. A cooperative organisation of European TSOs, from they are a part of the new pan-european ENTSO-E organisation. 5 The figurs are recalculated from the Danish currency DKK to Euro at the exchange rate 1 DKK=0,13. 5

6 benefit comparisons for the years 2010 and 2015 as a basis for decision (Energinet.dk, 2005). Nordel however calculates the costs and benefits for each year during a 30 year lifetime in their Nordic grid master plan (Nordel, 2008a). The choice of this study is a middle course where the benefits and costs are calculated as present values over 10 years. The discount rate is set at 5%, which is the same as in the Nordel grid master plan. Table 2 displays the calculated present values of Energinet.dk s cost and benefit estimations for In the Danish benefit the value of shared reserves and trade of regulating power is included. Also included is 30% of the benefit from increased competition in Denmark. Without this inclusion the investment would not show a positive present value for Denmark when calculated at 10 years, it is therefore included to preserve the analogy with Energinet.dk s calculations. Since the focus of this study is on the Nordic countries the UCTE-area is not included in table 2. Table 2. Present values of costs and benefits for the Great Belt connection. Costs: M Investment cost 160 Operation and Maintenance 8,03 Benefits: Denmark 180,23 Norway -12,05 Sweden 14,05 Finland -26,10 DK surplus: 12,20 Nordic surplus: -11,90 The present value calculation gives a positive surplus for Denmark when carrying out the investment on its own, which also will generate a positive benefit for Sweden. However, on a Nordic scale the investment shows a welfare-economic loss due to the negative spill-over on Norway and Finland. Cross-section 4 This expansion will reinforce the north-south transport channel on the Nordic market. Congestion in this channel is related to large export of hydro power in the north to south direction (Energimyndigheten, 2006). An expansion of the cross-section gives increased flexibility in dry and wet years (Nordel, 2004). Also, an increased internal capacity in a country will likely increase the capacity made available by the TSO on the country s cross-boarder connections. This is due to the form of congestion management referred to as moving internal bottlenecks to the boarder (Nordic Competition Authority, 2007). Internal bottlenecks inside a price area such as Sweden create a problem when the area price given through Nord Pool leads to surplus and deficit areas within the price area. It is the TSO s responsibility to maintain balance in the system at all times and in this situation they can reduce the trading capacity on cross-boarder connections and/or use counter-trade 6. In this choice the TSO has an incentive to reduce trading capacity rather than to use counter-trade because it is less costly (Nordic Competition Authority, 2007). In the light of this it can be understood that an increased capacity internally in Sweden will have effects on neighbouring countries through increased trading capacities. A study done on behalf of the Nordic Council of Ministers show that during the capacity made available on the cross-boarder lines was on average 75% of the full capacity (Team Nord, 2007). In the same study a simulation is done to estimate the welfare-economic consequences of a 20% reduction in transmission capacities on all cross-boarder lines in the Nordic region. The reduction is simulated from a base scenario for 2015 where the five prioritized cross-sections have been built and 6 Counter-trade is done by the TSO through regulating production and sometimes consumption in the deficit and surplus areas. The TSO buys power in the deficit area and pays generators to not produce in the surplus area. This leads to a cost for the TSO. 6

7 there is 100% availability of transmission capacities. The benefit estimates from the Tema Nord study is presented in table 3. Table 3. Changes in benefit from a 20% reduction in cross-boarder transmission capacity in 2015, M. Source: Tema Nord, M Denmark Finland Norway Sweden Consumer surplus -4,4 2,4-72,1-35,2 Producer surplus 0,7-5,6 55,3 36,2 Congestion rent 3,8 1,4 7,6 7,2 External trade surplus* -11,3-0,3-14,9-5,2 Total -11,2-2,1-24,1 3,0 *The surplus from trade with countries outside the Nordic region. These are Russia, Estonia and the Continent. The figures in table 3 show that the reduced capacity will have a negative effect on the total benefits in all Nordic countries except Sweden. By reversing the signs on the figures in table 3, the results can likewise be used as an estimate of the welfare economic consequences of going from a situation of reduced capacity to full capacity. As previously explained, an increased internal capacity in Sweden would increase cross-boarder capacities as well. Based on this fact, the benefits in table 3 with reversed signs will be used as estimates of the benefit changes resulting from an increased transmission capacity at cross-section 4 in Sweden. This approach is chosen in lack of any other published cost-benefit analyses (other than the Great Belt) for the Nordic market. Nordel s recommendations on beneficial expansions given in the Nordic grid master plans do not convey the underlying estimations. In the referred Tema Nord study there is also a specific simulation of the effects of removed internal bottlenecks in Sweden. Unfortunately that analysis uses the assumption of full availability of capacities on all cross-boarder connections as a basis. Removed internal bottlenecks can therefore not give the benefits related to increased cross-boarder capacities, as in the example of table 3. Given the current practice of handling internal bottlenecks by reducing cross-boarder capacities with up to 25% on average (Tema Nord, 2007), it is likely that the capacities on lines bordering to Sweden would be increased as a result of an expansion of cross-section 4. For the purpose of illustrating the spill-over benefits related to this expansion, it is the choice of this study to use the results in table 3 as approximations of those benefits. As explained in the Great Belt example, there are large uncertainties related to the future development of the electricity system. The benefits and costs are therefore present value calculated for a 10-year period. The discount rate is set at 5%. The investment cost is assumed to be 190 M based on the estimates for the South Link expansion of Cross-section 4 described in Nordel 2008b. Operation and maintenance is assumed to be 1 M /year, based on the Great Belt estimation. Table 4 displays the present value calculation. Table 4. Present values of costs and benefits for the cross-section 4 expansion. Costs: M Investment cost 190 Operation and Maintenance 7,72 Benefits: Denmark 86,48 Finland 16,22 Norway 186,09 Sweden -23,17 Sum: 67,90 7

8 The cost-benefit comparison in table 4 shows that there is a positive net benefit for the investment. All Nordic countries except Sweden have a positive benefit. The negative benefit for Sweden might seem peculiar, but can be understood when looking at the benefit distribution in table 3 (with reversed signs). Producer surplus will be reduced more than the gain in consumer surplus and the loss of congestion rent will be larger than the gain in trade surplus at connections to the continent. If limiting the analysis to these criteria, there would be no economic incentive for the Swedish TSO to pursue the investment. Although the Tema Nord example in table 3 is not ideal for the purpose, it does help in illustrating an important problem for the Nordic market. That is making investments that are beneficial for the Nordic market as a whole come into place, when the national benefit for the country of expansion is negative (or less than the investment cost). 3. Cost allocation and cooperative game theory There are no boarder-tariffs on electricity trade between the Nordic countries and non-discriminatory access to the grids is guarantied by the independent system operators (TSOs). In this sense the Nordic transmission gird can be viewed as a public good. Each of the four national TSOs are providers of this public good. There is no formal agreement on how the cost of providing the public good should be shared among all the users. This situation constitutes a cost allocation problem. The solution to an allocation problem is a rule that describes how an allocation shall be performed. Thomson (2007) lists three approaches to obtain such rules: (1) The direct approach that defines a formula or an algorithm that seems intuitively appealing. (2) The game-theoretic approach that associates a cooperative game to the problem and solves the game by identifying an allocation that appropriately reflects the powers and opportunities of each player. (3) The axiomatic approach that defines a rule based on mathematically expressed properties that capture the intuition behind how agents solve certain situation on their own. All these approaches are concerned with defining methods that solve allocation problems in an equitable and fair manner. Why then should equity and fairness be the criteria for allocations? The answer provided by Young (1985) is that these are the criteria that can encourage voluntary agreement among parties with diverse interests. To reach an acceptable solution to an allocation problem, it is important to understand what can be perceived as fair by different agents. Shubik (in Young, 1985) points out that cost allocation is an exercise in modelling: thus the full diversity of interested parties, their purposes, and the goals of allocation must be recognized before meaningful solutions can be obtained. Model choice The expansion of the Nordic transmission system can be described as a form of interactive decision making. The decisions of one TSO effects the others and they therefore attempt to coordinate their investments through regional expansion plans. As previously described, each TSO has its national concerns which sometimes deviate from the overall Nordic concern in expansion related issues. The game-theoretic approach seems to be a suitable choice to model this situation, because it is characterized by independent players with diverse interests who interact and produce an observable outcome. There are three conceptually different forms of modelling a game of interactive decision making (Young, 1985). These are the extensive, strategic, and cooperative forms of a game. The first two are part of the non-cooperative branch of game theory, which describes a game in great detail and focus on the procedure towards an outcome. The cooperative part abstracts from this level of detail and focuses instead on the value of the outcomes that different combinations of players can create if cooperating (Brandenburger, 2007). Another important difference between the two branches is that the non-cooperative form does not allow for binding agreements between players, whereas the cooperative form does. Solution concepts applicable to cost allocation problems have a well-developed theoretical basis within the cooperative form, with the established solutions of the Shapley value, the Core and the Nucleolus. In light of the previous description of the Nordic market, the model choice best suitable for 8

9 the context appears to be the cooperative form. The value of cooperation can be found in the Nordic cost-benefit analyses and binding agreements are to a relevant extent possible between the TSOs. The cooperative form The representation of an interaction as a game in cooperative form starts of with defining a finite set of players N and a function v that associates with each subset of players S N, a value denoted v(s) (Brandenburger, 2007). The overall value created is v(n). A cooperative game is thus defined as a pair (N, v) and the function v is called the characteristic function. A subset of players is simply a group of players that can come together and form a coalition through binding commitments. The value v of a coalition is assumed to be expressed in monetary units and transferable between the members of the coalition. A coalition can also consists of a single player, in which case the characteristic function will define the value that the player can get by acting alone. Defining the different coalitions that can form, and their respective values, gives a description of the strategic positions that different groups of players hold (Mas-Colell et.al., 1995). A solution to a cooperative game is an allocation of the overall value among the players, given as a collection of number (y 1, y 2,,y n ), where y 1 denotes the allocation to player 1 and so on (Brandenburger, 2007). An allocation that fully allocates the value in v(n) such that y v(n), is called an imputation. i N i = What will determine how the value v(n) is divided among the players? This will to a large extent depend on two aspects; a player s opportunity cost and the marginal contribution that a player makes to a coalitions value. The logic behind the first aspect is that a player will only join a coalition if he can get at least as high a reward as when acting alone, his opportunity cost. The second aspect can be interpreted as a player s bargaining power (Brandenburger, 2007). A player s marginal contribution to a coalition is defined as the amount by which the coalition s value would shrink if the player were to leave. For example, on the Nordic market the marginal contribution (MC) of Norway can be defined as MC Norway = v(norway, Denmark, Sweden, Finland) v(denmark, Sweden, Finland). The characteristic function can be used to describe either costs or benefits of coalitions. Which one to choose depends on the context of the problem (Young, 1985). An allocation of benefits will also result in a parallel allocation of costs, and vice versa. Let u i denote the initial benefit that player i has from participating in a project, let y i denote the benefit allocated to player i through an allocation rule, and let x i denote the cost allocated to player i. The relationship between benefit allocation and cost allocation is found in (1). x i = u i y i (1) An important aspect of a cooperative game is whether the value of the game v(n) is simply the sum of the values of its parts, or if it is larger than this. A game is called additive if the parts sum up to the total value, i.e. v(a)+v(b)=v(ab). If the value under cooperation is larger than the sum of the parts, then the game is super-additive, i.e. v(a)+v(b) v(ab). The value of an integrated electricity market is created through the possibility of trade between national markets. As shown in the cost-benefit analyses by Energinet.dk and Nordel, there are wide-ranging benefits form transmission investments. As mentioned, these include benefits from higher utilisation of low-cost production, increased competition, reduced costs for regulating power and increased security of supply. Based on this, the electricity market cooperation of the Nordic countries is in this study viewed as super-additive. This means that the Nordic countries can reach a higher benefit when cooperating on beneficial investments, compared with acting alone. This view is shared by several market actors and institutions, including Nordel; the industry association Nordenergi (Nordenergi, 2008); the Nordic Council of Ministers (Norden, 2004) and the European Commission (EC, 2007). Modelling the game as super-additive requires an estimate of how much additional benefit is created through cooperation. The electricity exchange between the Nordic countries amount to about 10% of the total Nordic demand (Tema Nord, 2007, Nordel, 2008c). On the basis of this it is assumed in this 9

10 study that the additional benefit from cooperation is 10%. This assumption means that each country can get 10% more benefit form a given amount of investment under cooperation compared with acting alone. Allocation rules There are plenty of allocation methods that could be applied to an allocation problem. An allocation method within the cooperative game framework is a solution to the game and is hence called a solution concept. By what criteria can an allocation method be judged? Young (1985) lists a couple of basic properties that all method should have, they are the following: (1) An efficiency constraint that require all costs to be fully allocated and that the full value that can be created, v(n), is in fact created. (2) A symmetry assumption requiring that the characteristic function v contains all the data relevant to the allocation problem, guaranteeing that allocations are not influenced by some outside information as well. (3) A dummy axiom that states that if a player contributes nothing to any coalition, the player will not receive any allocation of either costs or benefits. Apart from these basic properties there are also some more context-specific properties that are preferred depending on the problem at hand. These are additivity, monotonicity and consistency. Additivity is a decomposition property. This means that if an allocation problem can be split up into sub-problems and the solutions of the sub-problems are added, then these would add up to the total allocations of the non-split up problem. Monotonicity has to do with how allocations change when the value of the game, v, changes. For example, if the value of the game increases then no player should be allocated less than what they had before. Similarly, if costs increase (and the value thereby decrease) no one should receive more than what they had before. Consistency means that an overall allocation of v(n) should be viewed as fair by all possible subgroups of N. No subset of players should find incentive to change the allocation. As pointed out in Young (1985), these properties can not all be satisfied in the same allocation method but they do help in determining which methods are appropriate for a given problem. The Core This is a solution concept that defines a set of allocations in which no player or subgroup of players can on its own attain a better allocation (Mas-Colell et.al., 1995). A game with a non-empty core does in this sense contain allocations that can be voluntarily agreed to by all players and is thereby stable. An empty core on the other hand means that some of the options of subgroups of players are better than the options available for the whole group (the grand coalition N). This means that some players or groups of players are better off when acting alone than when cooperating with the other players. The existence of a non-empty core in a game can be found by defining the characteristic function for the grand coalition v(n) and for all sub-coalitions v(s), S N. A non-empty core will exist if there is at least one allocation y = y, y,..., y ) for which both of the following hold, y v(s) and i N y i ( 1 2 n v(n) (Mas-Colell et.al., 1995). The boundaries of the core will therefore be determined by the players opportunity costs and their marginal contributions. i S The Shapley value This famous solution concept was introduced by Lloyd Shapley in It is based on the earlier mentioned concept of marginal contribution. The Shapley solution assumes that players join a coalition in some random order. The marginal contribution that a player makes to the overall value of the coalition can therefore depend on at which point in coalition formation process that he joins. This process has a parallel interpretation in case of modelling costs instead of benefits. In that case a player pays the marginal cost of including him at the time of joining (Young, 1985). The idea of the Shapley value is that since the order of joining can be thought of as random, a player s expected marginal contribution (or cost) can be found as the average marginal contribution out of all possible orders of joining. This average calculated for each player is the Shapley value allocation. The Shapley value has been shown to fulfil several important properties including efficiency, symmetry, additivity and monotonicity (Young, 1985). i 10

11 Determining the marginal contributions of the countries on the Nordic electricity market presents a difficulty. One approach could be to interpret the estimates of each country s benefit from the costbenefit analyses as the marginal contributions. However, such an approach would ignore the way in which benefits are created on an integrated electricity market. A country s benefit from an expansion will to a large extent come from the effects the expansion has on the trade patterns between countries. The effects on producer and consumer surplus and congestion rent largely come from changes in trade. This means that although a country s net benefit might be low, it may still have contributed a lot to the benefit of the other countries. To determine how benefits are created in this way would however be very complex to evaluate and it would be different for each specific expansion. Another way, suggested in this study, is to assume that each of the four Nordic countries contributes a certain percentage to the net benefit of each of the other countries. As previously addressed in the section on super-additive games, the additional benefit from cooperation is in this study assumed to be 10%. This can be used to calculate the marginal contribution of benefit that each player makes to the coalition. In a three-player game, each player contributes 5% to each of the other players present before him when joining. In a four-player game the contribution is 3,33%. In this way the full value of the coalition is always the same and is attained only when all players have joined. The marginal contribution of the players will differ depending on the order of joining. As explained above, the Shapley value is the average of a player s marginal contributions calculated in this way. Table 5 below gives an example of how the Shapley calculation is performed for three players. In the example, full compensation has been paid to Sweden and the remaining surplus is to be shared by the benefiting countries. A more detailed description is found in appendix A. Table 5. Shapley value calculation for cross-section 4. Order of joining: A Denmark B Norway C Finland ABC 18,30 42,59 7,02 ACB 18,30 44,96 4,64 BAC 21,51 39,38 7,02 BCA 22,71 39,38 5,81 CAB 19,51 44,96 3,43 CBA 22,71 41,76 3,43 Avarage marginal contribution M : 20,51 42,17 5,22 Cost allocation M : 65,98 143,92 10,99 The Nucleolus If the core of a game is non-empty, the Nucleolus solution to the game will always be in the core. This property makes the Nucleolus a consistent solution, meaning that the allocation is viewed as fair by all subgroups of N (Young, 1985). The solution is found by maximizing the surplus that each subcoalition gets in the grand coalition, in relation to what the sub-coalition could get if acting alone. The surplus is maximized in order from smallest to largest surplus. The nucleolus allocation will be as far away as possible from the boundaries of the core (Young, 1985). To find the nucleolus allocation in the cooperative game framework requires linear programming. There is however another way as shown by Aumann and Maschler (1985). By defining the allocation problem as a bankruptcy problem, the nucleolus allocation can be found by applying the ancient contested garment rule 7 to the problem. 7 The contested garment rule is a two-person allocation rule that can be extended to n-persons. The rule is based on the split the difference principle. It has it s origin in the Babylonian Talmud where it is formulated as follows: Two cling to a garment, one take as much as his grasp reaches and the other take as much as his grasp reaches and the rest is divided equally between them. (Moulin, 2001) 11

12 A bankruptcy problem is defined as an estate E that is to be divided among a number of players with claims d=(d 1,,d n ), ordered from smallest to largest 0 d 1 d n. In the cooperative game context the claims can constitute the players initial benefit u i and the estate E is the value of the characteristic function v(n). Aumann and Machler showed that by transforming the n-player bankruptcy problem into a series of 2-person problems and then solving these with the contested garment rule, the resulting allocations are the same as the nucleolus allocations of the corresponding cooperative game. The Proportional method This is the simple method of allocating resources in proportion to some single criterion such as population or usage. In the transmission investment problem the criterion could be the initial benefit assigned to each player through the cost-benefit analyses. As opposed to the other methods described, the proportional method is not based on game theory concepts such as individual rationality and marginal contribution. The advantages of the proportional method are its simplicity and the fact that it is monotonic. The Nordic market as a cooperative game In the standard version of a cooperative game it is assumed that the value of a coalition, as specified by the characteristic function v, is independent of whatever other coalitions form at the same time (Maskin, 2003). In other words, coalitions do not exert externalities or spill-overs on each other. As described in section 2, an integrated electricity market organized as the Nordic will exhibit spill-overs between countries and in case of positive spill-overs there is to some extent an incentive to free-ride on the investments of another country. The Great Belt example from section 2 is a demonstration of such a case. Denmark has enough benefit to make the investment on its own and Sweden and UCTE can hence benefit from the investment without joining in a coalition with Denmark. Norway and Finland will however see a negative spill-over from the investment and they would be better off by joining a coalition if they thereby could receive compensation for this. This situation can in essence only be solved by a binding agreement among the Nordic countries to cooperate on transmission expansion plans and financing. The point of applying the cooperative game approach to this problem is that the solution concepts available herein can provide an important part of such an agreement. This study will model a cooperative game with transferable utility where spill-overs between players are handled through side-payments. The players who benefit from an investment, directly or through the spill-overs, take part in the financing and compensation to the player with negative spill-overs. This model is used in Volij and Wenzhuo 2006 and is described below in (2). Cooperative game (N, v) v ( S) = max{0, ui { C + min{0, ui}}} S N (2) i S i S The set of imputations to the game is the set: yi = max{ 0, ui C} Every imputation induces a cost allocation: x i N i = u i The model (2) says that the value of a coalition S who cooperates on an investment project is the players combined benefit minus the investment cost C and the compensation to players outside coalition S who receives a negative spill-over from the investment. A misfortune of this model is that since it requires full compensation to sufferers it can in some cases hinder investments that would otherwise have been undertaken. An example of this is the Great Belt expansion. Using model (2) this investment would not be beneficial for the Nordic market as a whole because the benefit to Denmark and Sweden is not enough to cover both the investment cost and the compensation to Norway and Finland. This is a misfortunate result because without a compensation agreement, Denmark could have financed the connection on its own. An agreement requiring full compensation to sufferers would thus risk hindering expansions. A way to overcome this could be to modify the agreement so that sufferers y i i N 12

13 receive some level of compensation that lies within the constraint of the surplus available. Such an agreement could be modelled as the bankruptcy problem previously described in the section on the Nucleolus solution. The model is outlined in (3) based on Volij and Wenzhuo (2006) and Aumann and Maschler (1985). The bankruptcy problem (E; d) E = Estate d = Claims n E = ui C i=1 n d = (max{ 0, u i }) i= 1 (3) In a situation where the surplus of an expansion is not enough to cover the compensations, the players can be thought of as having claims on the surplus. The allocation methods in this study do not naturally deal with claims in the form of both benefits and losses at the same time. To overcome this problem a modification to the set-up is suggested as follows. Each player s benefit or loss is interpreted as a claim on the surplus. Losses are treated as benefits and added to the overall surplus, while keeping the compensations as part of the cost. In this way the surplus and cost allocations can be calculated for all players. 4. Results and Discussion The cooperative game model and the presented allocation rules are in this section applied to the case studies of the Great Belt and cross-section 4 expansions. The analysis is limited to the Nordic countries and the grand coalition of players N therefore consists of Denmark, Norway, Sweden and Finland. This choice is made because the Nordic market can be viewed as an entity in terms of integration and trade patterns. A binding agreement between countries would therefore be more likely of success if applied on the Nordic market to begin with. The Great Belt As shown in table 2, the Nordic countries benefiting from this expansion are Denmark and Sweden. Norway and Finland will suffer a loss. The surplus is not enough to meet the compensations in full. In this case the Bankruptcy model (3) is the model of choice. The cost allocations to Norway and Finland are in this case the losses they suffer after receiving their share of the surplus (in the form of sidepayments). The investment cost and the cost of operation and maintenance are shared by Denmark and Sweden. Table 6 gives the allocations of the Nucleolus, the Shapley value and the proportional method. Table 6. Allocations and side-payments for the Great Belt, M. Initial Benefit Surplus allocation y i u i Cost allocation x i =u i -y i Side-payments M Nucleo. Shapley Prop. Nucleo. Shapley Prop. Nucleo. Shapley Prop. Denmark 180,23 18,97 19,43 20,35 161,26 160,79 159,87-4,72-3,79-4,00 Norway -12,05 1,88 1,71 1,36 10,17 10,34 10,69 +1,88 +1,71 +1,36 Sweden 14,05 2,09 1,92 1,59 11,96 12,13 12,47-0,48-1,10-0,31 Finland -26,10 3,31 3,19 2,95 22,79 22,91 23,15 +3,31 +3,19 +2,95 Total compensation: 5,2 4,9 4,31 The results show that the Nucleolus allocates relatively more surplus to the smaller beneficiaries compared with the proportional method which allocates relatively more to the largest beneficiary (Denmark). The Shapley allocations lie in-between. The size of the side-payments varies between the 13

14 methods. The Nucleolus solution contains the largest side-payments and the proportional method the smallest with a difference of 0,89 M. Overall the compensations can appear to be small amounting to only about 11-15% of the claims from the losing players. However, this surplus allocation is in the same range as the allocations to the benefiting players. Cross-section 4 The figures in table 4 show that this investment gives a Nordic welfare-economic surplus. The only country that doesn t show a positive net benefit from it is Sweden. The Nordic surplus is large enough to compensate Sweden fully. The remaining surplus can thus be divided among Denmark, Norway and Finland. Table 7 presents the results of the three allocation methods. Table 7. Allocations for Cross-section 4 with full compensation to Sweden, M. Initial Benefit u i Surplus allocation y i Cost allocation x i =u i -y i Side-payments M Nucleo. Shapley Prop. Nucleo. Shapley Prop. Nucleo. Shapley Prop. Denmark 86,48 20,84 20,51 20,34 65,64 65,98 66,15-7,10-7,0-6,94 Norway 186,09 41,92 42,17 43,76 144,17 143,92 142,34-14,30-14,39-14,93 Sweden -23,17 +23,17 +23,17 +23,17 Finland 16,22 5,14 5,22 3,81 11,08 10,99 12,40-1,77-1,78-1,30 Total compensation: 23,17 23,17 23,17 As in the Great Belt example, the differences are largest between the Nucleolus and the proportional allocations. The Nucleolus solution premiers the smaller beneficiaries at the expense of the largest (Norway). Finland s allocation in the proportional method stands out as the biggest difference between the three methods. An alternative to using the full compensation model (2) is to use the bankruptcy model (3), which was used in the Great Belt example. In model (3) Sweden will not receive full compensation for its loss. Instead the surplus from the investment is shared between all four countries according to the allocation methods applied. The resulting allocations are presented in Table 8 below. Table 8. Allocations for Cross section 4 with less than full compensation, M. Initial Benefit u i Surplus allocation y i Cost allocation x i =u i -y i Side-payments M Nucleo. Shapley Prop. Nucleo. Shapley Prop. Nucleo. Shapley Prop. Denmark 86,48 25,0 25,08 25,25 61,48 61,40 61,24-2,95-2,42-2,03 Norway 186,09 51,17 52,22 54,33 134,92 133,87 131,77-5,04-4,34-4,36 Sweden -23,17 8,36 7,83 6,77 14,81 15,34 16,40 +8,36 +7,83 +6,76 Finland 16,22 6,53 5,94 4,73 9,69 10,28 11,48-0,37-1,07-0,38 Total compensation: 8,36 7,83 6,77 The compensation to Sweden decreases between 64-70%, compared with full compensation. The reduction in compensation is largest with the proportional method and smallest with the Nucleolus. The surplus allocations to the other countries show a general increase in the range of 22-27%, compared with the full compensation results in table 7. Discussion of results The two case studies represent different types of allocation problems. The cross-section 4 example has a positive overall Nordic surplus which enables full compensation to sufferers. In the Great Belt example there is an overall Nordic loss from the investment in terms of welfare-economic consequences. An allocation can in this case be formulated along the lines of a bankruptcy problem 14