Optimal Bulk Energy Allocation of Hydroelectric Resources

Presented at the VII SEPOPE, (Symposium of Specialists in Electric Operational and Expansion Planning), Curitiba, Brazil, May 21st - 25th , 2000

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Optimal Bulk Energy Allocation of Hydroelectric Resources Egill B. Hreinsson, Department of Electrical and Computer Engineering, University of Iceland, Hjardarhagi 6, 107 Reykjavik, ICELAND E-mail: [email protected] Abstract -- A recursive optimization model is developed to determine the optimal allocation of hydroelectric resources for bulk energy sales. The modeling framework assumes that untapped resources are available as a set of hydroelectric projects to be developed in a sequence. The following quantities are assumed known: (a) energy price, (b) forecast for Basic Demand (BD), (c) the cost structure of the hydro-resources and (d) the unit cost of a long term energy supply after the hydro resources are exhausted. The model determines the maximum net revenue from energy sales and the optimal allocation of hydro-energy resources to industrial or bulk energy demand called Extra Demand (ED), such as energy intensive industries or energy export. The results should be of importance for long term planning of massive hydro-resource utilization. Keywords: Hydroelectric Energy Resources, Generation Expansion Planning, Recursive Modeling, Allocation of Resources, Cost Allocation

benefit is the production capacity as described above while the measure of cost is usually investment cost, since operations cost for hydro is often assumed either negligible or a fixed percentage of the investment cost. In previous papers, the hydroelectric capacity expansion process has been analyzed to determine the optimal sequence of projects [2] and the optimal production capacity (size) [3]. The demand function was always assumed known and a part of the fixed assumptions of the problem. The objective was to satisfy this demand at minimum cost. However, in this paper we expand on the previous results by decomposing the electrical energy demand into 2 parts: (1) The Basic Demand (BD) being part of the model's fixed assumptions and (2) Extra Demand (ED) which the planner has some control over and thus becomes a part of the model's decision variables.

I. INTRODUCTION The paper deals with hydroelectric power system expansion planning and optimal allocation of hydroelectric resources. With a specified basic load forecast, an attempt is made to satisfy this forecast at minimum discounted total cost by maximizing the net revenue from bulk energy sales. Such sales could in practice be to energy export or energy intensive industries. We assume a purely hydro-based power system. This means that thermal resources are only used intermittently and the system’s expansion potential consists of a set of hydroelectric projects. Each project has its unique characteristics due to diverse geographical and topographical conditions at each project’s site. This given set of projects thus comprises the hydroelectric resources of a region or country. The above capacity expansion framework is shown in Figure 1 (N=5 projects). While the total energy demand function increases gradually, the system capacity increases in a stepwise fashion. This leads to the period of excess capacity with a temporary underutilization of the project's output, as shown by the shaded areas of the figure. Each step in the expansion process corresponds to a hydroelectric project where the step size is the production capacity for that project. The definition of the production capacity is a given statistical measure of firm energy from the project rather than a simple figure of the project’s megawattage, as in the case of a thermal project, and we call this the project's production capacity. Each project has some measure of costs and benefits. The

z6

Capacity or demand Stepwise expansion of capacity

z5

x5

x4

z4

A nonlinear demand function D(t)

x3 z3 x2 z2

The period of excess capacity

x1 t1

Time t2

t3

t4

t5

t6

Figure 1 The Stepwise Expansion of Capacity to Meet a Non-linear Basic Energy Demand (BD).

This two-sided approach to the concept of the energy demand function, attempts to simulate conditions where the energy planners have the option to consider massive energy sales (ED) to energy intensive industries on top of the basic demand BD, which must be satisfied at all times, consisting of residential and commercial customers and “light” industry as exemplified by the case of Iceland. Thus we assume that the planner has some control over the timing and quantity of the energy sales ED while the BD is assumed fixed and known as part of the assumptions. In addition, we make the following assumption regarding unit cost or unit price of energy. • The unit price of energy in the category ED is assumed known. This reflects the assumption that the planner has the option to enter into a contract to sell, based on price, time and quantity. Different offers from investors in energy intensive industries or on the market in general may be available to the hydro system planner. • The unit cost (or price) for the category BD is however not known. This demand category must be satisfied at all times and the unit cost to satisfy this demand is


implicit in the cost of the resource development. Depending on the policy of the energy planner/authority certain cross subsidies could therefore take place between the above energy demand categories. The objective of this paper is the development of an analysis tool and model applicable and useful in the process of developing national policies for resource utilization in hydro-intensive regions and countries. Such a tool should be detailed enough for actual planning while sufficiently simple to reveal important trends and issues.

z6


z5

x5 x4

z4

A linear demand function D(t)

x3 z3 x2 z2

The period of excess capacity

x1

Time t2

t3

t4

t5

t6

Figure 2 The Stepwise Expansion of Capacity to meet a Linear General Energy Demand.

The paper is organized as follows: In section II, the model is presented, while in section III, the model is applied to data from the Icelandic energy system and a resulting case study presented. Section IV presents results and conclusions and finally appendices on references and notation conclude the paper.

through HVDC submarine cable to Scotland or continental Europe. Figure 3 shows a case when a "slice" of ED is added at the instance when the second project in the sequence is started up. However, additional "slices" of ED can be superimposed on the demand for each project in the sequence, as discussed below. Furthermore, it can be shown that this is the optimal timing and not in between projects, due to the discounting effect in the calculation of present worth of costs and benefits for the whole expansion program. A recursive type model that describes this expansion process in terms of discounted costs and benefits, timing of projects and different unit costs, will now be developed. We address the problem of how much of the hydro-resources should be allocated to ED, rather than postpone further development and use the resources domestically. We assume that the market energy price for these bulk sales is known, and when the resources are exhausted a long range energy supply with a given price is available to supply the demand. There exists a trade-off between selling too much and too little. • If too much energy is sold to ED, the most economical part of the hydro resources will be utilized for the bulk sales, leaving the expensive, perhaps small projects for satisfying the general demand. • If, on the other hand, the capacity sold is small, the potential benefit of the revenue from the bulk sales is ignored perhaps leading to higher prices to the general demand customers. Stepwise expansion of capacity

z6

Capacity or demand Bulk sales (ED) added at the startup of the 2 nd project

II. MODEL DEVELOPMENT Consider the expansion process for a purely hydroelectric power system, first without ED as shown in Figure 1. An appendix (Definitions and Notation) explains the symbols used. The general demand function D(t) for BD increases over time and capacity is expanded stepwise to satisfy this demand. The excess capacity (The difference shown by the shaded areas) is at all times positive and is at its maximum just after a new project has been started up. We now focus on a linear demand function and expand the concept by adding the ED on top of the BD. While a linear demand is an approximation, it often resembles closely a real situation [3] and results in a simpler more meaningful model as discussed below. Figure 2 shows conceptually the expansion process in the case of a linear demand function, while Figure 3 shows the same process with the ED added at the start up time of project #2. It should be noted that the BD consists of gradually increasing demand for energy, which increases smoothly in harmony with the general growth of the economy. However, bulk customers (ED) such as energy intensive industry or export of energy are assumed “large” compared to the size of the market. This would, for instance be the case for the Icelandic electricity market (See section III) with a new aluminum smelter or in the case of exporting electrical energy

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z5 z4

x5 A linar function for basic demand D(t) or BD,

x4

x3 Advancement in time of projects #3,#4,#5...

z3 x2 z2

x1

y2

The period of excess capacity for project #2

Time t3 t1 t4 t5 t2 t6 Figure 3 The Stepwise Expansion of Capacity to Meet a Linear Energy Demand with Bulk Sales Added.

When comparing Figures 2 and 3, it can be seen that the overall effect of adding the bulk sales will only be the advancement in time of all later projects in the sequence. It also has the effect that excess capacity is reduced for the project associated with the bulk sales (The second project) but not for later projects, unless "a slice" of ED is associated with these projects. In this paper we use a similar type of model as developed in [2] and [3] describing the expansion process of the hydroelectric system. The model is recursive in nature and has the property that the multidimensional non-constrained optimization problem decomposes into several one-dimensional problems.


The following equation describes the recursive nature of the expansion process and assumes no ED. It connects the discounted cost of the expansion sequence between adjacent projects in the sequence. (1) Pi = Ci + Pi+1 exp{- α (t i+1 - t i )} With a linear demand function equation (1) reduces to:  αx  Pi = Ci + Pi+1 exp − i  (2)  q  As discussed in [3], the above recursive equation can be restated as follows, replacing the discounted cost with the Long Range Marginal Cost (LRMC): kir = C i

 αx  α2 + kir+1 exp − i  q  q 

(3)

Since equations (2) and (7) above involving the discounted cost are completely analogous, as well as equations (3) and (9) involving the LRMC there is not a significant difference between the model structure with or without ED. s Now, with a given market price for bulk sales, k i it is possible to find the optimum amount of ED from each project. By differentiating equation (6), and setting the derivative equal to zero, we get: k s αP  αx  y  dPsi = − i + i+1 exp − i  exp  i  = 0 α dyi q q   q

kir = Pi

α = q

Pi ∞

[ ]

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∫ qt exp − αt dt

d 2 Psi = α 2 Pi+1 exp − αxi  exp  yi  ≥ 0  q  q dyi2 q2    

α Ci (5)   α xi   q 1 − exp  q   Until now we have assumed no bulk sales (ED). Assume that bulk sales are associated with project number i. For this project the above equation for net cost (cost minus income from bulk sales) now changes to the following form:

2  α ( xi − yi )  kis = α Pi+1 exp −  q q  

 α ( xi − yi )  yi kis Psi = C i − α + Pi+1 exp −  q  

 α ( xi − yi )  kis = kir+1 exp −  q  

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This expression leads to the following conclusion: The optimal amount of energy sold by bulk sales is the amount (in GWh/year e.g.) where the power price equals the long-range marginal cost (LRMC). This involves advancing all future projects an amount of time until equality is reached. The optimal amount for each project can therefore be calculated from the following expression: q  kir+1  ln α  kis  0 ≤ yi ≤ xi

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where the second factor on the right hand side is the discounted income from the ED over an indefinite horizon. The index “s” indicates net discounted cost with ED sales. The above equation can be restated as:  α ( xi − y i )  (7) P si = C si + P s (i+1) exp −  q  

(13)

which reduces to the following expression:

2

AUC = k ia =

(12)

is positive, we have a minimum rather than a maximum and get the following conditions for a minimum:

0

Furthermore [3], the Actual Unit Cost (AUC) of each project, representing the unit cost of energy from the project and accounting for the period of excess capacity, can be stated as follows:

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Since the second derivative:

where 2

(3)

yi = xi −

(15)

(16) An alternative form for the above optimality condition is: yi = xi −

q  α 2 Pi +1  ln α  qkis  0 ≤ yi ≤ xi

(17)

or

where s i

yi k (8) α is the net investment cost of the project (Investment cost minus the discounted income from sales) We assume that projects later in the sequence have bulk energy sales associated with them. Similarly, the net discounted cost can be replaced by the LRMC as follows: C si = C i −

k sir = C si

 αu  α2 + k sr(i +1) exp − i  q  q 

(9)

where ui = xi − yi

is the net output of the project for the basic demand. We must require that: 0 ≤ ui ≤ xi 0 ≤ yi ≤ xi

(10)

2 q  α Ps (i +1)  ln   α  qkis    0 ≤ yi ≤ xi

yi = xi −

(18)

(19) depending on whether we think in terms of LRMC or discounted costs - or whether we assume bulk sales associated with projects later in the sequence or only associated with the current project. In summary therefore, starting with the price for the long term supply, it is possible to recursively calculate backwards in time the optimal amount of energy allocated to bulk sales, if the market price for the bulk sales is known, using the expressions (15), (16) and (7) or (9).


III. RESULTS OF A CASE STUDY WITH THE MODEL The above recursive mathematical model has been implemented using spreadsheet software and data from the Icelandic hydroelectric power system [7]. Table 1 shows these data as a set of 25 projects, where we assume that the projects are to be developed in the sequence indicated by the Table. The sequence is fairly arbitrary, but it reflects to some extent the actual conditions in the Icelandic system. The total capacity of all projects is 25000 GWh/year and the sum of the investment cost is 410,000 Icelandic kr. (krónur) with the approximate rate of exchange such that 1 U.S.$ equals approximately 70 kr. (krónur). All figures are at a 1990 price level. The long range supply is assumed to cost 2.50 kr/kWh and the growth rate of the basic demand or BD is 150 GWh/year/year which gives a total time of 25000/150=166.7 years for development. The Table shows the AUC and LRMC as defined in sections II and IV.

Long Range Marginal Cost

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42.6 Billion Icel. kr. (Column #7, line #1 in Table 1) The resources as indicated by Table 1 reflect the different unit costs for different projects ranging from projects early in the expansion sequence, where the most favorable sites are exploited to more expensive projects later in the sequence. This cost structure is plotted in Figure 4, which shows the rising long range marginal cost (LRMC) as we move through the development sequence from left to right in the figure. Each rectangular block in the figure shows a project; the width of the block represents the project's capacity, while the height is the LRMC. The optimal allocation of the total resources is plotted in Figures 5 and 6 with different energy prices. Figure 5 shows how the net discounted cost of the expansion program with optimal allocation depends on the market price for bulk ED sales. The net discounted cost for the base case is 42654 Mkr with the energy price as indicated in Figure 5 and in Table 1. The figure shows how the net cost of the development is reduced as price increases and with income from and sales to ED. With a price of 1.02 kr/kWh, the net cost is relatively close to zero being 3812 Mkr. 25,000

Total Optimal Sales (GWh/year)

20,000 15,000

"Base Case"

5,000 0

Smávirkjanir

Sigölduvirkjun Hrauneyjafossvirkjun Blönduvirkjun Fljótsdalsvirkjun Vatnsfell Bjallavirkjun Búðarhálsvirkjun Norðl+Gljúfurl Sultartangi Núpsvirkjun Búðarfossvirkjun Urriðafossvirkjun Abóti+stóraver Haukholt Stafnsvatnavirkjun Hestvatnsvirkjun Villinganesvirkjun Ishólsvatnsvirkjun Hafrahv+Brúarvirkjun

Sogsvirkjun Lagarfossvirkjun Búrfellsvirkjun

10,000

Accumulated Capacity Figure 4 The LRMC of the Expansion Process with Data from the Icelandic Power System.

The Table shows the optimal allocation of resources with the ED price of 0.90 kr/kWh with the results that 14480 GWh/year are allocated to ED leaving 10520 GWh/year allocated to the BD. This will be referred to as the “Base Case”. In this case the discounted cost of the development is

Net Discounted Cost (Mkr) 60.000 50.000 40.000 30.000

"Base Case" 20.000 10.000 0 0,77 0,79 0,81 0,83 0,85 0,87 0,89 0,91 0,93 0,95 0,97 0,99 1,01

Energy Price (kr/kWh)

Figure 5 The Net Discounted Cost with Optimal Allocation. Data from the Icelandic Power System

0.77 0.79 0.81 0.83 0.85 0.87 0.89 0.91 0.93 0.95 0.97 0.99 1.01

Energy Price (kr/kWh) Figure 6 The Optimal ED Sales with Data from the Icelandic Power System

Figure 6, however, illustrates how the optimal amount of sold ED energy depends on the market price. Therefore, in Figure 6, the optimal amount of energy increases rapidly where the market price is similar to, or exceeds the system LRMC. In particular the "Base Case" (0.90 kr/kWh) is pointed out in Figure 6 and the optimal sales for this case is 14480 GWh/year. Figure 7 shows graphically the expansion process for the base case with the ED "slices" and the stepwise expansion of capacity, while Figure 8 illustrates how the stepwise expansion of capacity is affected by different ED prices. This figure therefore shows how the projects are advanced in time with increasing sales to ED. It should be pointed out that all these figures, and the present study, assume a fixed sequence, while it is possible that a different sequence might be more economical, for instance when a large project is associated with ED. Such could be the case with the present debate around the Fljotsdalur project in eastern Iceland. There, a certain large hydroelectric project is "associated with" a large aluminum smelter. This large project would probably not be possible without the large industrial customer since small projects would have been more economical to serve the BD, [4].


Figure 9 shows how flat the discounted cost function is around the optimal value of the ED. The optimum seems rather flat but its sharpness increases with increasing energy prices to ED.

25000 GWh /year


this amount cannot exceed the total resources. For this unconstrained problem the sales are allocated to individual projects using the recursive optimality conditions, as set forth above. Further research and investigation could be directed towards finding the optimal allocation for the constrained problem, as well as investigating the incremental cost of ED and the interactive problem of sequencing, sizing and bulk sales of energy from hydroelectric projects for instance to aluminum smelters or other energy intensive industry. V. REFERENCES

A linear function for basic demand or BD, D(t)

[1]

Erlenkotter, D.: "Coordinating Scale and Sequencing Decisions for Water Resources Projects", from Economic Modelling for Water Policy Evaluation, North Holland /American Elsivier, New York (1976).

Time

[2]

Hreinsson, E.B.: "Hydroelectric Project Sequencing using Heuristic Techniques and Dynamic Programming". Proc at the 9th Power Systems Computation Conference (PSCC), August 30 - Sept. 4, 1987, Cascais, Portugal.

[3]

Hreinsson, E.B. “Optimal sizing of Projects in a Hydro-based System”. IEEE Transactions on Energy Conversion, Vol 5, number 1, March 1990, pp 32-38.

[4]

Hreinsson, E.B.: "Economies of Scale and Optimal Selection of Hydroelectric Projects". A paper presented at the IEEE/IEE DRPT2000 Conference, City University, London, U.K. April 4th 7th, 2000.

[5]

Manne, A.S., “Capacity Expansion and Probabilistic Growth”, Econometrica, Vol. 29, No. 4, pp. 632-649, (October 1961).

[6]

Manne, A.S., ed., Investments for Capacity Expansion; Size, Location and Time Phasing, MIT Press, Cambridge, Mass., (1967)

[7]

Elíasson, E.B., “Allowable Marginal Cost”, Internal Report, Landsvirkjun, The National Power Co., Reykjavík, July, 1992. (In Icelandic)

Slices of ED demand 0 GWh /year 0 years

166.7 years

Figure 7 The Stepwise Expansion for the "Base Case" with Data from the Icelandic Power System

Finally Table 2 shows values for optimal sales to ED and optimal discounted cost for different energy prices, reflecting numerically the information shown in Figure 5 and Figure 6. The conclusion from these figures and Tables 1 and 2 can be drawn, that the optimal sales rise very sharply when the market price exceeds the LRMC of power in the system. 25000 GWh /year

Capacity or demand 0.95 kr/kWh

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0.90 0.85 0.80 0.70 kr/kWh kr/kWh kr/ kr/kWh kWh

A linar function for basic demand or BD, D(t)

180,000

Discounted cost (Million Icelandic kr)

160,000

0 GWh /year 0 years

Stepwise expansion of capacity

140,000

166.7 years

Energy price (kr/kWh):

Minimum value

120,000

Time

0.9

100,000

0.8

Figure 8 The Optimal Stepwise Expansion with Different ED Energy Prices

80,000

1.0

60,000

IV. CONCLUSIONS AND FURTHER STUDIES

40,000

This paper has discussed the expansion process for purely hydroelectric power systems. The question has been addressed how much of the resources should be sold or exported massively. Obviously this amount would depend on the price offered in relation to the cost of the resource. It also depends on the domestic demand for the resource and what the cost of the replacement is, that is the long-range supply. This replacement, the long-range supply, is uncertain, but can be conceptually visualized as oil, coal or nuclear stations to be constructed when all economical hydroelectric resources have been exhausted. The paper has discussed how much energy should optimally be sold, given its price. This is basically an unconstrained optimization problem in the sense, that no limit is set on how much can be sold, although obviously, however

20,000 0 0

5,000

10,000

15,000

20,000

25,000

30,000

ED (GWh/year)

Figure 9 The Discounted Cost as a Function of Extra Demand (ED)

VI. APPENDIX: DEFINITIONS AND NOTATION BD

Basic Demand or market related demand which must be satisfied at all times, consisting of residential and commercial customers and “light” industry, as exemplified by the case of Iceland

ED

Extra Demand, step-function shaped, superimposed on the basic demand, BD. In many cases the decision maker has the option to serve the ED which consists of massive energy sales to EII or bulk energy export.


Name of Project

P #

i Sogsvirkjun Laxárvirkjun Lagarfossvirkjun Búrfellsvirkjun Sigölduvirkjun Hrauneyjafoss Blönduvirkjun Fljótsdalsvirkjun Vatnsfell Bjallavirkjun Búðarhálsvirkjun Norðl+Gljúfurl Sultartangi Núpsvirkjun Búðarfossvirkjun Urriðafossvirkjun Ábóti+Stóraver Haukholt Hestvatnsvirkjun Stafnsvatnav, Villinganesvirkjun Íshólsvatnsv, Hafrahv+Brúarv, Fjarðarárvirkjun Smávirkjanir Long range supply TOTAL

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Inv. cost Capac. ED Net Cost Net Prw AUC LRMC (Mkr) (GWh/year) (Mkr) (kr/kWh) a r

Ci

xi

yi

9,778 4,560 7,419 28,032 15,010 13,500 12,070 20,292 6,111 7,832 10,859 28,286 17,290 11,482 11,069 17,727 21,893 6,490 7,099 12,101 3,825 8,240 119,667 3,042 6,326

490 155 385 2,345 795 1,085 780 1,300 385 330 545 1,265 795 585 520 975 1,280 340 265 665 225 465 8,775 100 150

490 155 385 2,345 795 837 396 691 0 0 0 727 350 143 125 540 719 0 0 86 0 0 5,696 0 0

C si

P si k

i

739 42,654 1,053 1,701 41,914 1.472 317 40,213 1.000 -15,225 39,896 0.834 345 55,121 1.045 -1,935 54,776 0.720 4,769 61,480 0.855 7,549 64,260 0.934 6,111 69,139 0.824 7,832 71,436 1.221 10,859 70,811 1.061 14,869 71,580 1.331 10,838 67,549 1.204 8,837 65,548 1.052 8,760 65,471 1.129 7,771 64,483 1.035 8,626 65,337 1.020 6,490 68,059 0.984 7,099 68,769 1.364 10,510 67,221 0.987 3,825 68,458 0.860 8,240 69,540 0.932 14,599 71,310 2.015 3,042 154,394 1.508 6,326 156,356 2.108 157,531

k

410,000 25,000 14,480

Interest rate (%/year) Exp int, rate (%/year) Market growth (GWh/year/year) Long range supply (kr/kWh) Bulk price to SBED (kr/kWh)

5.00% 4.88% 150.0 2.50 0.90

Table 1 A Case Study with Data from the Icelandic Power System. i =1,2,...,N Index of individual hydroelectric projects. We assume that the projects are to be constructed in the sequence indicated by this index. Ci

Construction- or investment cost for project number i.

Csi

Net investment cost for project number I. i.e. cost minus income from bulk sales.

xi

Capacity or size of project number i (expressed for instance in GWh/year).

yi

Amount of capacity or size of project number i to energy intensive industry or bulk sales (in GWh/year).

s

Energy price to energy intensive industry or bulk sales.

ki k

a

Actual unit cost (AUC) of each project, representing the unit cost from this project accounting for the period of excess capacity.

i r

ki

Long Range Marginal Cost (LRMC) of the expansion sequence for project number i and all projects later in the sequence.

D(t)

A nondecreasing demand function of time (expressed for instance in GWh/year) an indefinite time into the future. We assume in particular when demand is assumed to increase linearly, i.e. D(t)=qt, where q is a constant growth in demand.

r

Annual interest rate.

q

Constant growth rate of linear demand function (GWh/year/year)

α

Continuous time discount rate. i −1

zi =

∑x

j

Total accumulated capacity of all projects preceding

j=1

project number i at the time when project number i is started.

-1 t i = D (zi ) Construction- and/or start-up date of project number i.

Pi

Discounted total cost of project number i and all following projects, the time reference of discounting being the start-up date for project number i. No bulk sales are assumed.

Psi

Discounted total cost of project number i and all following projects, the time reference of discounting being the start-up date for project number i. This is with bulk sales.

PN+1

Total discounted cost of operating the long range back stop supply to be used after all economically efficient hydroelectric resources have been exhausted. The discounting reference date is tN+1 .

i

0.953 0.936 0.909 0.896 0.968 0.945 1.040 1.093 1.177 1.225 1.225 1.257 1.219 1.224 1.260 1.284 1.377 1.561 1.628 1.652 1.812 1.885 2.040 2.450 2.481

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VII. BIOGRAPHY Egill B. Hreinsson received the Master’s Degree in Electrical Engineering from the University of Lund, Sweden in 1972 and the Master of Science Degree in Industrial Engineering and Operations Research from the Virginia Tech, Blacksburg, VA, USA in 1980. In 1972 he joined Landsvirkjun, the National Power Co., Reykjavik, Iceland where he worked on control SCADA/EMS systems, telecommunication and telemetering systems. From 1977 – 1982 he was a Division Engineer with the Engineering and Planning Department, Landsvirkjun, specializing in power system planning and analysis of hydroelectric system expansion alternatives and studies relating to the expansion of energy intensive industries. From 1982 he has been on the faculty of Department of Electrical and Computer Engineering, The University of Iceland, where he is currently a professor. His research interests are in the fields of power system analysis, economics and operations. He has worked as a consultant both internationally and in Iceland on issues relating to power system planning and computer and communications applications in power systems. Energy Discounted Optimal ED Price (kr/kWh) Cost (Mkr) (GWh/year) 0.800 53,260 9,401 0.825 51,252 10,394 0.850 48,901 11,587 0.875 46,087 12,896 0.900 42,654 14,480 0.925 38,513 16,263 0.950 33,026 19,017 0.975 24,083 21,864 1.000 12,840 21,995 1.025 1,551 22,071 Table 2 Optimal Discounted Cost and Optimal Extra Demand (ED) as a Function of Energy Price to ED