Allocation of Hydroelectric Economic Rent Using a Cooperative Game - PowerPoint PPT Presentation

The 44th International Universities’ Power Engineering Conference, September, 1-4, 2009, Glasgow, Scotland Allocation of Hydroelectric Economic Rent Using a Cooperative Game Theoretic Approach by Egill Benedikt Hreinsson Department of Electrical and Computer Engineering, University of Iceland, Hjardarhagi 6, Reykjavik, (Iceland) Email: egill@hi.is 1

Presentation overview • Introduction • Model for allocating economic rent – HAM1. ~ to the gross energy flow (Kárahnjúkar) – Cooperative Game Theory-HAM2-Nucleolus • Linear Programming • Simple computational examples • Discussion and conclusion 2

Introduction • Assume N independent riparian owners of water rights. Their benefit measure is the project economic rent • Should they build each their own “small” projects or join a coalition, in particular the grand coalition to construct a larger project ? • How should the benefits of coalitions be allocated among the N owners ? • This lends itself to a cooperative game theoretic approach 3

A general model for a valley N Territories and • a k-1 a N-2 a N N-2 estimate points at a k+1 boundaries a k k+1 a N-1 We have lateral inflow • k N-1 ( a i ) in each territory a 5 a 4 Head differences are a 3 • 5 4 a 2 between boundaries Legend: 3 Boundary with a 1 2 flow estimate point (estimate points) Theoretical Lateral x1 1 inflow, an potential Owner # n n Ocean Boundary with flow estimate point 4

A simplified linear model for a valley w 2 w 1 Lateral energy inflows: a 2 a 1 Lateral inflows: a N River and flow Ocean estimate points: N 2 1 Owner #: e 1 e 2 e N Elevation: u 1 u 2 Energy contribution v N v 2 v 1 Stream-flow series 5

Energy contributions a e = w k 1 2 Energy inflow to zone 1: g 1 2 a e a e � � = + w k 2 2 2 3 Energy inflow to zone 2 : � � g 2 � � 2 2 = w k a e Energy inflow to zone 3 : g 3 3 3 The energy contribution of each owner is given by, where e = 0 : 1 a = + u k v e 1 Energy contribution of zone 1: ( ) g 1 2 2 2 a = + − u k v e e 2 Energy contribution of zone 2: ( )( ) g 2 3 3 2 2 Zone 3 is “dummy”. It is easy to verify that the total energy in + = + + = u u w w w W this simple 3 zone case is . 1 2 1 2 3 6

Sub/superadditive costs and capacities Costs : = S N Grand coalition: {1,2,3,..., } is the set of all projects N { } ⊂ = S S S Any coalition: is a subset, for instance 2,3 N { } = S c S c The cost of the project built by is ( ) ( 2,3 ) subadditive ∪ ≤ + c S T c S c T Costs are assumed ( ) ( ) ( ) for ⊂ ⊂ ∩ = ∅ S S T S S T any and and N N Capacities x S S ( ) is the capacity of the project built by coalition , { } = x S x for instance ( ) ( 2,3 ) { } = S x S x The capacity of the project for is ( ) ( 2,3 ) superadditive ∪ ≥ + x S T x S x T Capacities are assumed , or ( ) ( ) ( ) 7

Economic rent Economic rent r S ( ) : = ⋅ − r S p x S c S p ( ) ( ) ( ); The energy price is The rent is therefore the annual income in the electricity market, minus the annual cost. superadditive ∪ ≥ + r S T r S r T The rent is therefore ( ) ( ) ( ) benefit b S S The , ( ) in joining coalition is � � = − = − b S x S x i b S r S r i ( ) ( ) ({ }) ( ) ( ) ({ }) x r i S ∈ i S ∈ � = − b S c i c S ( ) ({ }) ( ) c ∈ i S 8

Perfectly additive energy inflows Note that the energy inflow in each zone is perfectly additive , i.e. ∪ = + u u u ⊂ ∩ = ∅ . S T S with S T for all disjoint sets , S T S T N 9

Allocation problem LP formulation The allocation of Economic rent , Z implies a vector of allocations, = � ≥ ∈ ∀ so Z z z i N i . Where 0 , , i i ∈ i N Z might be total benefit of the grand coalition The core of the game = + + r z z z ({1,2,3}) 1 2 3 ≥ + ≥ z r z z r ({1}) ({1,2}) 1 1 2 ≥ + ≥ z r z z r ({2}) ({1,3}) 2 1 3 ≥ + ≥ z r z z r ({3}) ({2,3}) 3 2 3 10

Allocation problem LP formulation δ Max The objective is δ ≤ − z r ({1}) to maximize the 1 δ ≤ − minimum benefit , z r ({2}) 2 δ for each owner δ ≤ − z r ({3}) to join any 3 δ ≤ + − z z r coalition. The LP ({1,2}) 1 2 problem is: δ ≤ + − z z r ({1,3}) 1 3 δ ≤ + − z z r ({2,3}) 2 3 = + + r z z z ({1,2,3}) 1 2 3 δ ≥ 0 11

TABLE I: Energy Flow in a River Basin With 3 Zones Life Elevation Lateral Accumu- Energy Energy Energy cycle Energy Lateral Zone # inflow inflow Inflow lated Inflow Contri- Contri- Cost Flow bution bution (m) (Gl/Year) (GWh/year) (%) (M$/yr) e i a i v i w i u i a i w i f i c i i 487 32,1 1 0 1300 3450 113 38% 10% 43% 313 23,8 2 80 700 2150 183 20% 16% 28% 333 28,6 3 160 1100 1450 586 32% 52% 29% 330 350 350 251 10% 22% Total 3450 1133 1133 100% 100% 84,5 100% 12

A Matlab model using function linprog A=[-1 -1 0 0;-1 0 -1 0;-1 0 0 -1; -1 -1 -1 0;-1 -1 0 -1;-1 0 -1 -1] b = -xlsread('likan.xls', 1, 'ak17:ak22') lb=[-1000;0;0;0] ub=[0;1000;1000;1000] Aeq=[0 1 1 1] beq=xlsread('likan.xls', 1, 'ak23') f=[1 0 0 0] z = linprog(f,A,b,Aeq,beq,lb,ub) xlswrite('likan.xls', z, 1,'an17:an19') 13

Matlab computation A = b = -1 -1 0 0 -23.8609 -1 0 -1 0 -9.0783 -1 0 0 -1 -18.7924 -1 -1 -1 0 -61.1033 -1 -1 0 -1 -72.9709 -1 0 -1 -1 -60.9484 lb = ub = -1000 0 0 1000 0 1000 0 1000 Aeq = beq = 0 1 1 1 145.5817 f = z = 1 0 0 0 -30.3862 Optimization terminated. 54.2471 39.9023 51.4323 14

TABLE II: COALITIONS COSTS AND OUTPUTS IN THE 3 ZONE EXAMPLE Coalition Coalition energy Coalition Cost Number of projects in coal. Coa- Indi Benefit Indi Benefit Energy factor lition vidual in Cost Coa- vidual in firm project joining factor lition project joining Unit cost energy firm coa- cost cost coa- Projects in S Coalition # energy lition lition $/M Wh (%) (GWh/year) (%) (M$/year) α ( S ) β ( S ) c ( S ) x ( S ) � x i b x (S) � c i b c (S) S 224 224 0 100% 32.1 32.1 0.0 143 1 {1} 1 46% 131 131 0 100% 23.8 23.8 0.0 181 2 {2} 1 42% 190 190 0 100% 28.6 28.6 0.0 151 3 {3} 1 57% 448 355 93 50.9 55.9 5.0 114 4 {1,2} 2 56% 91% 508 414 95 54.1 60.7 6.7 106 5 {1,3} 2 62% 89% 426 321 105 45.6 52.4 6.8 107 6 {2,3} 2 66% 87% 849 545 304 66.8 84.5 17.8 7 {1,2,3} 3 75% 79% 79 15

TABLE III: The Calculation of Economic Rent and Results From the HAM2 Allocation AG AH AI AJ AK AL AM AN AO 9 Coalition Coalition Economic Rent Allocation 10 Projects Coa- Indi Benefit Benefit Number of projects 11 in lition vidual in Allocated joining 12 Coa- Coa- Economic project joining Economic grand Market price 13 in coalition lition lition Rent Economic coa- Rent coa- ($/MWh) 14 S # Rent lition lition 15 (M$/year) (M$/year) r ( S ) � r i b r (S) z i z i -r i 16 S 17 23.9 23.9 0.0 54.2 30.4 1 {1} 1 250 18 9.1 9.1 0.0 39.9 30.8 2 {2} 1 250 19 18.8 18.8 0.0 51.4 32.6 3 {3} 1 250 20 61.1 32.9 28.2 94.1 33.0 4 {1,2} 2 250 21 73.0 42.7 30.3 105.7 32.7 5 {1,3} 2 250 22 60.9 27.9 33.1 91.3 30.4 6 {2,3} 2 250 23 145.6 51.7 93.9 145.6 0.0 7 {1,2,3} 3 250 16

Discussions and conclusions • The independent ownership of private owners of water rights assumes that the owners have and interest in developing their resources. • By definition they cannot compare all options of joining various coalitions unless cost and firm output estimates are available and updated with the development of the river basin. • These possibilities should be weighted against the grand coalition whether or not imposed by the government on the private owners. • This methodology should be beneficial in the basic debate whether water rights are public or private goods and how private ownership plays a role in the borderline between the deregulated market environment and government imposing public interest on these owners. 17

Thank you! 18