UPTEC ES06010
Examensarbete 20 p September 2006
Future trading with regulating power
Magnus Brandberg
Niclas Broman
Teknisk- naturvetenskaplig fakultet UTH-enheten
Besöksadress:
Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0
Postadress:
Box 536 751 21 Uppsala Telefon:
018 – 471 30 03 Telefax:
018 – 471 30 00 Hemsida:
http://www.teknat.uu.se/student
Abstract
Future trading with regulating power
Magnus Brandberg & Niclas Broman
Regulating power is needed to compensate for unplanned deviations from production and consumption plans. The transmission system operator (TSO) in each country is responsible for keeping the electric system balanced, and activates regulation if needed to keep the frequency in the network stable. The need for regulations is assumed to grow if wind power is implemented in the power system, as it is difficult to forecast wind power production and make accurate production plans. This report investigates how trade with regulating power is affected by a large-scale installation of wind power in the Swedish electricity power system.
The report explains how energy and regulating power is traded today in the Nordic interconnected power system. By using data on wind power prognosis errors from the West Danish power system, the report predicts the need for regulating power in the Swedish system with 4000 MW of wind power installed. A model is used to foresee pricing on regulating power in Sweden based on the need for regulating power along with other aspects.
The results show that the regulating power market energy turnover increases along with the monetary turnover following the installation of a large amount of wind power. Bottlenecks have a large impact on the energy system and trade today and increase the need for regulations.
Sponsor: Vattenfall AB
ISSN: 1650-8300, UPTEC ES06 010 Examinator: Ulla Tengblad
Ämnesgranskare: Niklas Dahlbäck Handledare: Urban Axelsson
Sammanfattning
För att överföringen av el mellan producent och konsument skall fungera problemfritt måste elnätet hålla en frekvens på 50 Hz. Detta betyder i praktiken att produktionen av el i varje stund måste vara lika stor som konsumtionen för att elen ska bibehålla hög kvalitet. Den nationella systemoperatören, som i Sverige utgörs av Svenska Kraftnät, har ansvaret för att nätet hålls i balans. För att kunna fullfölja sitt ansvar tvingar man elproducenter och konsumenter som man ålagt så kallat balansansvar att i förväg att planera sin produktion och konsumtion så att dessa är i balans inför varje driftstimme.
Planer kan dock ändras, historiskt sett har elkonsumtionen alltid varit svår att beräkna exakt, men även ett plötsligt bortfall av ett kraftverk gör att de planerade produktions- och konsumtionsbalanserna förändras. Är produktionen större än konsumtionen ökar frekvensen i nätet, medan den sjunker då det motsatta inträffar. Vid de tillfällen det produceras eller konsumeras mer eller mindre el än beräknat måste andra enheter kompensera denna obalans för att elnätet skall återföras i balans. Denna kraft benämns reglerkraft och aktiveras av systemoperatören. I vissa kraftverk sker regleringen automatiskt då frekvensen förändras, medan annan reglering avropas manuellt.
Vattenkraft är den överlägset bästa reglerkraftsproducenten, eftersom man snabbt kan öka eller minska flödet genom turbinen. De nordiska länderna har en produktionsmix som till 50 % utgörs av vattenkraft.
Projektering och anläggningen av nya vindkraftverk har under de senaste åren ökat i omfattning. Till skillnad från traditionell energiproduktion, såsom vattenkraft och kärnkraft, är dock produktionen från vindkraftverk svår att planera, eftersom den beror på vindens hastighet, som kan vara svår att förutspå korrekt. Med en utbyggd kapacitet av vindkraft kan därför behovet av regleringar antas komma att öka.
Denna rapport utreder hur reglerkraftsmarknaden ser ut i dagsläget och hur den kan komma att påverkas av en storskalig svensk utbyggnad av vindkraft. Eftersom de nationella elnäten i Norge, Sverige, Finland och till viss del i Danmark i dagsläget är synkront sammanbundna, utreds hela den gemensamma nordiska reglermarknaden.
Rapporten beskriver både vilken energi och vilken ekonomisk omsättning som reglerkraften uppgått till under de senaste åren och hur dessa skulle förändras till följd av utbyggd vindkraft. För att kunna simulera en utbyggnad av vindkraften i Sverige har data från dansk vindkraft från 2003 använts. Västdanmark hade 2003 en
installerad kapacitet på 2400 MW, och reglerbehovet denna mängd vindkraft gav upphov till kan uppskattas genom att se hur mycket den verkliga produktionen avvek från den planerade. Genom att skala om reglerbehovet till 4000 MW, och överföra det till svenska förhållanden, uppskattas hur en storskalig vindkraft påverkar reglerbehovet i Sverige. Priset för dessa regleringar uppskattas med hjälp av en modell utvecklad av Klaus Skytte på Risö Institut.
I det Nordiska elnätet finns det flaskhalsar som begränsar hur mycket el som kan överföras genom vissa passager. Flaskhalsarna kan ha betydelse för vilka kraftverk som blir kontaktade att utföra regleringar. Om den enhet som lämnat ett erbjudande att reglera ligger på fel sida om en flaskhals kan man inte föra över reglerkraft från denna enhet. Man låter då någon annan enhet som ligger bättre till står för regleringen istället, trots att denna reglering kan vara dyrare att genomföra. I detta arbete utreds hur reglerkraftsproducenternas intäkter från reglerkraft skulle förändras om det inte fanns några flaskhalsar i elnätet.
Resultaten visar att den nordiska reglermarknaden under 2004 och 2005 uppgick till ca 5 TWh och hade en omsättning på ca 1500 mkr. Norge, med sin stora andel vattenkraft, står för den största andelen regleringar. Svenska aktörer står för cirka 20 procent av regleringarna. Slutsatserna är att utbyggd vindkraft ökar behovet av reglerkraft. Flaskhalsarna ökar också mängden regleringar i systemet och därmed reglerintäkterna.
Preface
For both of us, the spring at Vattenfall Utveckling AB has been an inspiring and instructive experience. Not only have we been given the opportunity to investigate the regulating power market and some of the factors affecting it. We have also been given the chance to enjoy the atmosphere at Vattenfall, and Vattenfall Utveckling AB in specific. This work could not have been concluded without the help of several people who we hereby would like to thank for their dear support:
Nils Andersson (Vattenfall PU) for entrusting us with this project and making it financially feasible. Urban Axelsson (VUAB), for being our supervisor and source of inspiration during the work. Viktoria Neimane and Robin Murray (VUAB), who with valuable comments and sincere interest have followed our work and helped us to choose what areas to concentrate on.
We would also like to thank Daniel Nordgren, Bo Wrang and Joakim Allenmark (Vattenfall PD) for sharing their profound knowledge about this subject and for taking the time to help us understand how regulating power is produced and traded. We are also thankful for the help we have received from Fredrik Wik at Svenska Kraftnät, who has helped us to understand the physical aspects of the transmission system and the regulating market. Björn Wetterborg (Vattenfall PDT) has also helped us to understand how hydropower resources are valued and used, which has been of great value. Finally we would like to thank Olof Nilsson (Vattenfall Trading Services) for providing us with data to work with.
Our wish is that some of the people that have helped us with our project will read this report and find something that can help them perform better in their own work.
Magnus Brandberg Niclas Broman
Executive summary
This report is an outcome of a degree project performed at Vattenfall Utveckling AB between January and June 2006. The purpose has been to investigate how the Nordic regulating power market will react on a large-scale development of wind power production. In addition to calculating the need for regulating power, the report also investigates the need for adjustment power from the time Nord pool closes until the end of the hour of operation in order to estimate the need for adjustments on a day- ahead time scale.
The current appearance of the Nordic regulating market
The physical and financial turnover of the Nordic regulating market and the revenues for the regulating market players have been calculated for 2004 and 2005 and beginning of 2006 to investigate the current appearance of the regulating market. The bottleneck’s effects for regulated volumes and revenues have also been investigated for the same years.
The results show that the regulated volume on the Nordic regulating market amounts to around 5 TWh annually and that most of the regulations are performed in Norway.
The Swedish share of the regulating market amounts to around 20 percent, which is displayed in Table 1.
NORDIC SUBSYSTEMS - TOTAL REGULATED VOLUME [MWh/h]
[MWh] [%] [MWh] [%] [MWh] [%] [MWh] [%] [MWh] [%] [MWh] [%] [MWh] [%]
Sweden 1172723 28% 1066108 22% 945778 21% 788886 17% 900440 18% 236317 20% 974787 21%
Norway 1 1600063 38% 1589608 33% 1310853 29% 1858142 41% 1949211 39% 526564 44% 1661575 36%
Norway 2 713556 17% 480138 10% 550662 12% 346574 8% 477124 10% 101684 9% 513611 11%
East Denmark 112637 3% 114406 2% 206445 5% 41026 1% 61853 1% 17474 1% 107273 2%
West Denmark 336738 8% 1220617 25% 1129527 25% 1197316 26% 1257689 25% 252120 21% 1028377 22%
Finland 233922 6% 325471 7% 358754 8% 323440 7% 368903 7% 58606 5% 322098 7%
Total 4169639 100% 4796348 100% 4502019 100% 4555384 100% 5015220 100% 1192765 100% 4607722 100%
Aver. 2001-2005 -March 2006
2005 2004
2003 2002
2001
Table 1: Total regulated volume in the Nordic subsystems 2001-March 2006.
Norway performs most of the regulations within the Nordic countries. Swedish regulating players have a market share of around 20 percent.
Regarding the financial aspects of the regulating market, Norway earns most of the revenues, but also receives the lowest marginal revenue of the Nordic subsystems, which is seen in Table 2.
NORDIC SUBSYSTEMS - REVENUES FROM REGULATING POWER
2004 2005
Total revenues from regulating power [SEK]
Extra revenues as compared to spot price [SEK]
Marginal revenues [SEK/MWh]
Total revenues from regulating power [SEK]
Extra revenues as compared to spot
price [SEK]
Marginal revenues [SEK/MWh]
Sweden 225 783 376 kr 28 963 862 kr 39 kr 323 444 578 kr 67 136 731 kr 75 kr Norway 1 559 226 232 kr 62 116 059 kr 33 kr 670 109 843 kr 138 606 624 kr 71 kr Norway 2 114 242 486 kr 22 259 461 kr 64 kr 170 799 320 kr 43 131 422 kr 90 kr East Denmark 12 122 929 kr 14 816 714 kr 36 kr 38 963 503 kr 38 088 581 kr 103 kr West Denmark 350 601 123 kr 22 259 461 kr 49 kr 518 725 241 kr 16 982 948 kr 275 kr Finland 93 622 873 kr 43 609 960 kr 49 kr 142 568 382 kr 94 532 422 kr 75 kr Total / Average 1 355 599 019 kr 194 025 516 kr 38 kr 1 864 610 867 kr 398 478 729 kr 79 kr
Table 2: Revenues from regulating power in the Nordic subsystems. As seen, one of Norway’s two subsystems earns most of the revenues, but also experiences the lowest marginal revenues.
Implementation of 4000 MW wind power in the Swedish power network
The starting point for the report has been a scenario where 4000 MW wind power is implemented in the Swedish power network, which is an substantial increase of installed power compared to the current amount of Swedish wind power of around 600 MW. To be able to calculate how the regulating power market would react on this implementation, the West Danish power system with an installed wind power capacity of 2400 MW has been studied. Using data from 2003, West Danish forecasts made 24 hours and 4 hours in advance of the hour of operation has been compared to real production to estimate the wind power forecast errors for 2003. The forecast errors have been scaled up and added to the Swedish need for regulating power for the same year in order to estimate how the Swedish regulating market would have been affected. The forecast errors from the forecasts made 4 hours in advance have been used to estimate the need for regulation while the 24 hours forecast errors have been used to estimate the additional need for adjustments in Sweden. That is, from the time the Nord Pool spot market closes until the end of the hour of operation.
Forecast errors can also be calculated using a method developed by Hannelle Holttinen, where the forecast error on a 1 hour perspective is defined by the change in production between the hour prior to the hour of operation and the hour of operation.
This forecast error has also been calculated for the Danish data and scaled up to 4000 MW. It has then been added on to the Swedish regulation need in the same manor as the other forecast errors mentioned above were. By calculating the effect on the Swedish regulating need in two ways, the hope was to give a broader picture of how large the effects would become.
A model initially developed by Klaus Skytte at the Risö Institute has been used to estimate how the changed volumes of regulation would affect the price of regulation power in Sweden.
The report shows that the Swedish need for regulating power and adjustments would increase due to a large implementation of wind power in the Swedish system as seen in Table 3. However, depending on what forecast error is used, the amount varies.
No wind power
4000 MW Wind Power ELTRA
4000 MW Wind Power HOLTTINEN
Need for regulating power [GWh] 2 279 3566 2680
Turnover of the Elbas Market [GWh] 2490 4010 -
Adjustment market [GWh] 4 769 7 576 -
Need for regulation and adjustments due to implementation of 4000 MW wind power
Table 3: Need for regulating power and adjustments due to implementation of 4000 MW wind power. The need for regulating and adjustment power would increase. As seen, no calculation of need for adjustment power has been made with Holttinen’s method, as it only is applicable on a shorter time horizon.
Regulation resources in other subsystems, such as Norway 1, supply much of the need for regulation in Sweden. Therefore, the Swedish part of the regulating market would increase less than the need for regulation within Sweden would. The regulated volumes in Sweden would change from 942 GWh to 1108 GWh using Holttinen’s forecast errors and to 1646 GWh using Eltra’s forecast errors. This is displayed in Table 4.
No Wind Power
4000 MW Wind Power ELTRA
4000 MW Wind Power
HOLTTINEN
Volume Swedish regulating market [GWh] 942 1646 1108
Revenues Swedish regulating market [MSEK] 304 405 338
Extra revenues [SEK] 74 160 78
Change of volumes and revenues due to implementation of 4000 MW wind power in Swedish power network in 2003
Table 4: Change of volumes of regulating power and revenues from the same due to implementation of 4000 MW wind power in the Swedish power system.
The bottlenecks seem to increase the amount of regulations in the network. Without bottlenecks the extra revenues from regulating power in 2004 and 2005 would decrease in general. The extra revenues are defined as the additional revenues that are
obtained by selling the energy for regulating price rather than spot price. The results are shown in Table 5. Important to know is that no investigation has been carried out to describe the effect of the internal bottlenecks in Sweden.
2004 Norway 1 Norway 2 Finland East Denmark West Denmark Vattenfall's extra revenues 15 844 136 kr 15 844 136 kr 15 844 136 kr 15 844 136 kr 15 844 136 kr Difference if no bottleneck existed -1 012 071 kr -191 495 kr -2 388 212 kr -694 592 kr -5 000 671 kr 14 832 064 kr 15 652 641 kr 13 455 924 kr 15 149 544 kr 10 843 465 kr
Change [%] -6,4% -1,2% -15,1% -4,4% -31,6%
2005 Norway 1 Norway 2 Finland East Denmark West Denmark Vattenfall's extra revenues 28 540 112 kr 28 540 112 kr 28 540 112 kr 28 540 112 kr 28 540 112 kr Difference if no bottleneck existed 4 809 907 kr 4 269 155 kr -348 849 kr -5 617 946 kr -13 153 973 kr 33 350 019 kr 32 809 267 kr 28 191 263 kr 22 922 166 kr 15 386 139 kr
Change [%] 16,9% 15,0% -1,2% -19,7% -46,1%
Table 5: The bottlenecks’ effect on the Swedish regulating power producers’
extra revenues. The bottlenecks increase the amount of regulation in the system and thereby increase revenues for regulating power producers.
Data has been retrieved and compiled from several databases to enable the calculations. Data concerning Elspot prices, Elbas prices and transfer of regulating power between the Nordic subsystems have been retrieved from Nord Pool.
Regulating prices have been found in Vattenfall’s internal databases, while data concerning consumption forecasts has been retrieved from Svenska Kraftnät.
The work has been quantitative to a large extent, as much of the results have been calculated using the databases. However, interviews with Vattenfall, Svenska Kraftnät and Nord Pool have also been performed along with studying of literature to get a deeper understanding of the subject.
Table of Contents
PageThe current appearance of the Nordic regulating market 4 Implementation of 4000 MW wind power in the Swedish power network 5
1 INTRODUCTION AND BACKGROUND INFORMATION 1
1.1 Introduction 1
1.1.1 Intermittent production requires adjustments to keep the network balanced 1 1.1.2 The closure of the Nord Pool spot market creates the need for
predictions 2
1.1.3 Predictions contain mistakes 2
1.1.4 Regulation must be viewed from a Nordic perspective 3
2 PURPOSE AND PROBLEM FORMULATION 4
2.1 Problem Statement 4
2.2 Purpose 4
2.3 Problem Formulation 4
3 THEORY 6
3.1 Definitions of regulating and adjustment power 6
3.1.1 Defining regulating power 6
3.1.2 Defining adjustment power 6
3.2 Physical aspects 7
3.2.1 Production in the interconnected Nordic power system 7 3.2.2 The balance service and balance responsibility 10
3.2.3 Bottlenecks 11
3.2.4 Market Division 12
3.2.5 Counter Trade 13
3.2.6 Congestion management in practice 13
3.3 Financial aspects 14
3.3.1 The Nordic market place; Nord Pool 15
3.3.2 The Regulating market 16
3.3.3 Differences in market conditions between the Nordic countries 18 3.3.4 Demand and supply for adjustment power 19 3.3.5 Strategy for trading with energy and power 20
4 METHOD 22
4.1 Quantitative to a large extent but with qualitative elements 22
4.2 Statistical methods 22
5 IMPLEMENTATION 24
5.1 Physical aspects 24 5.1.1 Regulating power in the Nordic power network 24 5.1.2 4000 MW wind power – impact on volumes traded on the adjustment
market 25
5.2 Financial aspects 27
5.2.1 4000 MW wind power – impact on the adjustment power market turnover 27 5.2.2 The amount of regulating power in Sweden and in Denmark 36 5.2.3 The relation between wind power forecast error and adjustments in
Denmark 38 5.2.4 The turnover of the Nordic regulating market 41 5.2.5 The financial effects of bottlenecks 42
6 RESULTS 45
6.1 Physical aspects 45
6.1.1 Activated regulating power in the Nordic network 45 6.1.2 No form of regulation is more common than the other in Sweden 46 6.1.3 The need for regulating power is not the same as the activated
regulating power 47
6.1.4 Bottlenecks and how they affect the regulating market 48 6.1.5 Introducing wind power increases the volumes of regulating power 50 6.1.6 The need for adjustment power increases 51
6.2 Financial aspects 52
6.2.1 The turnover of the Nordic regulating market January 2004 – March 2006 52 6.2.2 A comparison between the Nordic subsystems 52
6.2.3 Impact of bottlenecks 53
6.2.4 Regulating market turnover rises with 4000 MW of wind power 54 6.2.5 Adjustment power market turnover rises with 4000 MW of wind power55
7 ANALYSIS 56
7.1 The time of closure at NordPool has a big impact on the amount of regulation56 7.2 Differences in regulation between the Nordic countries 56
7.3 Bottlenecks – Costs or benefits? 57
7.3.1 Bottlenecks in practice 57
7.3.2 Bottlenecks – good or bad? 58
7.4 The comparison with Denmark and the scenario with 4000 MW wind power in
the Swedish power system 58
7.4.1 Surprising results concerning the need for regulating power when
using a four-hour perspective 59
7.4.2 The Holttinen´s one-hour perspective gives us another view 59
8 DISCUSSION 60
8.1 The problem formulation 60
8.1.1 The regulating power market of today and in the future 60 8.1.2 How is the regulating market affected by bottlenecks? 61 8.1.3 What can be done to increase revenues from trading with regulating
power? 62
8.2 Other Issues 63
8.3 About using historical data 63
8.4 Further studies 63
9 CONCLUSIONS 65
9.1 Effects on the regulating market due to 4000 MW wind power 65
10 LIST OF REFERENCES 67
10.1 Internet 67
10.2 Literature 67
10.3 Oral sources 68
Appendices
PagesAPPENDIX 1 Volumes and share of upward and downward regulation
January 2001- March 2006
p. 66
APPENDIX 2 Examples of activated regulation and need for regulation
in Norway, Finland, Denmark and Sweden October 2004
pp. 67-68
APPENDIX 3 Key numbers from the Nordic regulating market January
2004-March 2006
p. 69
APPENDIX 4 Examples of how regulating bids are activated in the
interconnected Nordic power system
p. 70
APPENDIX 5 List of the two author’s individual contributions to this
report
pp. 71-72
1 Introduction and background information
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1.1 Introduction
Right now, there are strong winds blowing in favour for wind power both in the energy sector as well as among politicians. This is demonstrated by the ambitious goals set for the building of new wind power parks. A political planning scenario aims at production levels of 17 TWh by the end of 2016, corresponding to more than 4000 MW of installed wind power in the Swedish network. Compared to the present capacity of about 500 MW, these are high numbers.
Balance between production and consumption is a basic and important condition to keep the electric frequency stable in all electric networks. It is not possible to control output from wind power in the same way as in a thermal power plant or a hydropower plant. Production from wind power is intermittent and varies in cube with sometimes very fluctuating wind speeds. The introduction of intermittent production from wind power can add to imbalances in the power system, which requires other energy production units to instantaneously compensate for change in production. Thus, wind power production brings new challenges into the power system and to the players acting within the network.
1.1.1 Intermittent production requires adjustments to keep the network balanced
The electricity market regulations state that production and consumption must be reported in two separate balances that should cancel each other out. Traditionally, the difficulty in keeping the balance in the network has originated from variations in consumption. This is natural since it is impossible for system operators and producers to control and foresee how every individual user consumes electricity. Production on the other hand is easily controlled. The variations in consumption cause moments where the real consumption diverges from the foreseen. This means that production, which is planned to match the forecasted consumption, is in imbalance with real consumption. If the produced power is less than the consumed power the frequency in the network drops. The opposite situation results in an increase in frequency.
The problem with intermittent power production is that it possesses the same varying characteristics as the consumption. In some cases unplanned variations in wind power production will cancel out imbalances in consumption to various extents, but in other cases it will enhance the imbalance in the network. This means that the transmission system operators (TSOs) that are responsible for keeping the network in balance suddenly have two factors affecting the frequency.
There is one factor that has been left out of the discussion so far, a factor of great importance, time. Variations neither in load nor production need to cause any imbalances as long as they are foreseen. For example, if variations in production is correctly foreseen a producer will compensate for a possible decrease in one production unit by increasing production in another, and thereby keeping its reported balance. Nevertheless, the closer to the hour of operation, the more expensive it gets to adjust the imbalances.
1.1.2 The closure of the Nord Pool spot market creates the need for predictions
The Nordic electricity spot market, Nord Pool, allows trading until 12 o’clock (noon) the day prior to the actual day of operation. Then players on the market have a responsibility to be in balance, or contract someone else to do it for them. This means that those player’s which in the end are responsible for the balance have to make sure that, for every hour the following day, they produce or make available enough production to match own consumption and that of their customers. In order to make this possible the market players rely on predictions of consumption to plan their production.
1.1.3 Predictions contain mistakes
The fact that the Nord Pool spot market closes 12 to 36 hours before actual hour of operation requires wind power producers to use precise wind forecasts to plan how to best fulfil their production liabilities towards their buyers and the market. The problem with predictions is that they are incorrect from time to another. Power production predictions for wind power are dependant on wind speed but also temperature forecasts to some extent. Something that every one of us has experienced is the difficulty to foresee these parameters. How many times have the meteorologists ruined a nice picnic-day by promising calm and warm weather on the forecast the day before
when reality brought wind and rain? At times when the predictions fail other production units need to change their production to compensate the imbalance.
1.1.4 Regulation must be viewed from a Nordic perspective
More and stronger connections between the Nordic countries and the implementation of Nord Pool have increased the stability of the network, and provided more efficient use of the existing power plants, as power can be produced where it is most cost- efficient and then transported over national borders. However, inter-connections have also increased the need for communication and co-ordination between the national system operators. Regulation of imbalances in the network must therefore be examined from a Nordic perspective.
2 Purpose and problem formulation
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2.1 Problem Statement
The need for regulating power is believed to grow if the ambitious plans for installing wind power parks within the Nordic countries are fulfilled. Due to the wind power’s intermittence and the difficulty to estimate wind power production regulating power will play an important role in the future energy system and should consequently be thoroughly examined. More specifically, is interesting to observe how the value of hydropower as a system regulator is affected by an increased wind power penetration.
This report has its starting point in a scenario where 4000 MW wind power is installed in the Swedish part of the interconnected Nordic power system.
To be able to investigate this scenario properly, we have identified several important topics that have to be examined together. Firstly, the physical properties of the power system in the Nordic countries have to be understood, as placement of production resources and bottlenecks are of great importance. Secondly, the financial properties of the electricity markets in general and Nord Pool in particular have to be examined in order to understand how electricity is traded. The economical and business possibilities that trade with regulating power may create must also be investigated in order to fulfil the purpose of this report.
2.2 Purpose
This report will investigate how the regulating power market will react on a large- scale development of wind power production, both from a physical and a financial perspective.
2.3 Problem Formulation
• What is the current situation on the regulating power market and how will prices and traded volumes develop in the future?
• How do physical constraints in the power transmission system affect the regulating market and the revenues from trade with regulating power?
• What is needed in order to increase revenues from trade with regulating power?
3 Theory
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3.1 Definitions of regulating and adjustment power
In order to determine how the trade with regulating power is affected by a large amount of wind power we have defined two aspects that need to be investigated thoroughly.
The physical aspects have to be regarded due to the fact that we are dealing with a physical phenomenon – electricity. The properties of the power system must also be investigated, as there are physical limitations in how much power that can be transferred through the system.
The financial aspects must be investigated to understand how electricity is traded and how the physical aspects are handled economically. Physical properties are in this way translated into monetary terms, to hopefully give a wider view of the investigated subject.
3.1.1 Defining regulating power
Regulating power is defined as regulating bids activated by transmission system operators (TSO) to maintain the balance (frequency) in the network. The bids consist of upward and downward regulation of production as well as consumption, which producers and consumers offer in exchange for compensation. Regulating power is activated during the hour of operation.[17]
3.1.2 Defining adjustment power
Production plans and consumption predictions are settled 12-36 hours prior to the hour of operation, i.e. when Nord Pools spot market closes. If something happens that have not been considered during planning the need for additional or less power during the hour of operation evolves. The imbalance that occurs from the time Nord Pool closes until the hour of operation, together with the balancing that occurs during the
time of operation will be described as adjustment power in this report, as it describes how the bids must be adjusted as compared to the time Nord Pool closes.
Figure 1: Definitions of regulating and adjustment power [22]. Regulating power is used to regulate imbalances during each hour of operation. Adjustment power can be described as adjustments made to reported plans from the closure of Nord Pool until the end of the hour of operation.
3.2 Physical aspects
3.2.1 Production in the interconnected Nordic power system
The national power networks within Norway, Sweden, Finland and East Denmark are connected synchronically and are therefore called the synchronous system. Together with the network in West Denmark, which is connected to Norway and Sweden with high voltage direct current (HVDC) cables in the north, they build up the interconnected Nordic power system (INPS). West Denmark is synchronically connected to the continental European power system, called UCTE, in the south. In each Nordic country there is a TSO that is responsible for the operation of the national network and for keeping it in balance. The TSOs within the Nordic power system are Svenska Kraftnät (SvK) in Sweden, Fingrid in Finland, Statnett in Norway, and Energinet.dk in Denmark. As the national power systems are connected, the TSOs are required to coordinate their work. This is carried out within the Nordel organization on a strategic level and agreements regulate how transmission of energy between the countries should be carried out. On a daily operational basis, SvK and Statnett share the overall responsibility of the entire network in cooperation with the other TSOs.[17]
The electricity consumption in the Nordic countries amounts to around 390 TWh annually[11], which in absolute numbers is small compared to the UCTE system that has an annual energy turnover of 2 300 TWh[8]. However, considering the relatively small population in the Nordic countries, the electricity consumption per capita is substantial. Hydropower delivers around half of the energy consumed in the Nordic region, even though the production mix varies considerably between the countries in the region. Norway produces 99 percent of its electricity from hydropower, Sweden 50 percent, Finland 15 percent and Denmark 0 percent.[11] The maximum amount of power needed in the INPS is about 60 000 – 70 000 MW, depending on timing and climate [7].
The interconnected Nordic power system is sorted in subsystems rather than countries, although the borders often coincide by historical reasons. Sweden and Finland both consist of one subsystem each, while Denmark and Norway are divided into several, mainly due to frequently occurring internal bottlenecks. When comparing the Nordic countries, wind power in larger scale currently only exists in Denmark. The penetration of wind power in West Denmark is among the highest in the world. Wind power contributed to nearly 32 percent of the installed power capacity and 24 percent of the energy production in 2004[4]. Due to the intermittent nature of wind power and the need for combined heat and power (CHP) plants to deliver heat the western region as a whole produces 25 – 30 percent more energy than consumed[4]. West Denmark exports excessive energy to neighbouring subsystems. Figure 2 shows the interconnected Nordic power system with the transmission network marked.
Norway 1
Norway 2
Sweden
Finland
West Denmark
East Denmark
Figure 2: The Nordic interconnected power transmission network. The map shows the Nordic countries and their corresponding subsystems. Most of the hydropower production within the system is found in Norway and in northern Sweden, while wind power mostly is found in West Denmark. Some of the connections that is more likely to give rise to bottlenecks than other connections are marked with blue lines. [6]
3.2.2 The balance service and balance responsibility
Every market player on the electricity market has a responsibility for keeping the network in balance. This means that for every hour the amount of produced and bought power must be equal to sold and consumed power. If successful, the player is said to be in balance. Players, such as individual consumers or companies, that lack own production, let other companies take care of their balance in exchange for a fee.
Today, there are around thirty of these balance responsible companies in Sweden.[22]
The TSO is responsible for coordinating all production and consumption and keeping the frequency in the power system stable at 50 Hz, within an interval of +/- 0.1 Hz.
This is done by coordinating production and consumption within the subsystem with export and import from neighbouring subsystems. The TSOs within the synchronous system have agreed to use available regulating resource in the most efficient way to regulate imbalances. For every hour, the TSOs gather bids containing information about what amount of power and energy the different players are ready to deliver within fifteen minutes time, and to what price. The bids contain capability to regulate upwards or downwards. The bids are arranged according to price in a bid list, which is updated every hour. The bid list is the same for the whole Nord Pool area, which means that regulating bids in neighbouring subsystems can be activated to regulate imbalances in Sweden and vice versa. At occasions where regulation is needed the most beneficial bid in the Nordic network is activated, assuming it is not locked in by physical constraints. The goal is that the regulation price should be the same within the whole INPS and that regulation should be carried out where the regulation costs are lowest.[22]
The national TSO activates bids within its own country but SvK and Statnett have the overall responsibility for all regulations within the Nordic grid. Fingrid and Energinet.dk therefore activate regulation bids only after consultation with SvK or Statnett. In Sweden, this is called the Balancing service, and means that SvK has the right to activate regulation bids within Sweden, if this is needed to keep the balance in the network.[22]
The manual regulation described above (also called secondary regulation) is combined with an automatic frequency activated regulation (primary regulation) that regulates
the output of generators within seconds. The regulation is automatically initiated by changes in the network frequency. Variations in production from wind power on a time scale of seconds are normally so small that it is not necessary to evaluate how it affects the primary regulation[15]. Instead, the time scale of 15 minutes is of greater importance when studying wind power, as it mainly is within this time span production varies. Thus, regulating power in this paper refers to manually activated secondary regulation.
SvK has, to some extent, the possibility to regulate imbalances without using the regulating market. Instead of activating bids from the regulating power bid list SvK can take advantage of the power plants that are in use at the moment for regulating purposes. More specifically, SvK can order plants to start or stop their production up to 15 minutes earlier or later than planned. By doing this, SvK can reduce the sudden change of production that is likely to occur in the splice between two hours in an efficient way. The players that have been contracted are paid according to regulating market price for that hour or spot price with a ten percent mark up depending on which one is highest.[30]
3.2.3 Bottlenecks
Bottlenecks evolve when the physical capacity of a power line not is enough to supply the market’s demand for transmission of power. Within the systems there are bottlenecks that set limitations for available transmission and it is actually the bottlenecks that cause the division of the network into subsystems. The national networks were built up separately within the Nordic countries. Connections over the borders were built later and therefore the weakest links (the smallest transmission capacity) in the network are often found at national borders, although there are bottlenecks within the national countries as well.
The connections between the subsystems are used for transmission of electricity. For every hour, the TSOs that manage the subsystems the connection runs between together decide the transmission capacity, which describes the maximum amount of power that can be transferred in the connections between the two subsystems. A regulating margin is reserved from the maximum capacity, which gives room for instantaneous automatic regulation deviating from the planned transmission. What
then remains is called the trading capacity and is the active power capacity the TSOs let the market players trade with at the electricity market Nord Pool.[17]
The HVDC cables connecting Norway and Sweden to West Denmark are used intensely and often give rise to bottlenecks in the system. A large amount of energy is transited through West Denmark due to the differences in price in the Nordic countries as compared to the rest of Europe. Generally, energy is transferred northwards during night (hours of low-load) when the price is lower in Europe than in the Nordic countries and southwards during day (hours of high-load) when the price is lower in the Nordic countries than in Europe [31]. The HVDC-connections do not have any reserved regulating margin, which means that no regulating power can be transferred between West Denmark and the Nordic countries during the hours the HVDC- connections are run at maximum capacity [31]. This means that regulation bids obtained from the Nordic countries outside West Denmark not are available to regulate imbalances in West Denmark at some occasions. More specifically, when West Denmark is exporting electricity to Sweden and Norway at full capacity and downward regulation is needed in West Denmark, this has to be activated within the West Danish subsystem. Moreover, if West Denmark needs upward regulation and it at the same time imports electricity at full capacity, the bid has to be activated within the subsystem. This assumes that no regulating power is available from Germany.
Due to the occurrence of bottlenecks, bids delivered to the Nord Pool spot market must not only include information about price and volume, but also where the physical generation or consumption is situated. This information is used by the national TSOs to calculate if bottlenecks will occur. If bottlenecks occur there are two methods to handle the effects of bottlenecks in the Nordic electricity market, market division and counter trade. Both methods are used within the INPS and the dominating of the two is market division.[22]
3.2.4 Market Division
Market division is a measure performed during the planning phase of each hour of trade. By analysing the supply, the demand and the geographic location of bids for every hour it is possible to, in advance, identify potential bottlenecks within the INPS.
If the market driven need for electricity transmission is bigger than the maximum transmission capacity between two subsystems, the systems are divided into price
areas creating two separated markets. The subsystem with high demand requiring energy (downstream the bottleneck) experiences a higher price (due to less supply) than the area with high production capabilities (upstream the bottleneck). The division reduces demand downstream as well as make it profitable to start up more costly productions in the high-price area. At the same time, in the subsystem with lower demand and higher supply the price-level drops and supply decreases along with rise in demand. Hence, the market’s solution to the physical phenomenon of bottlenecks gives rise to different price areas and within each of these areas there is an equilibrium point established where supply and demand meet. Market division is considered to give incentives to place new investments in the most suitable area, lead to a more market based efficient market and increase the stability of the market.[19]
3.2.5 Counter Trade
Counter trade is, unlike market division, performed during the hour of operation. It is carried out if the transmission of electricity between two parts in the network is restricted by a bottleneck. In the case of upward regulation the TSO buys production in the area that experience deficit of power by accepting a bid on the regulating bid list. Historically, the TSO at the same time ordered a facility to down regulate on the upper side of the bottleneck by paying the producers not to produce as much as planned. However, SvK now leave the downward regulation to the regulating market, which means that the market will decide (by using the regulating power bid list) which facility that will regulate down. This action is called special regulation (Sv.
Specialreglering). The report will use counter trade instead of special regulation as this term is assumed to be more familiar among the readers of this report. Special regulation, or counter trade as we from now on refer to it leads to fewer and larger subsystems.[30] Large subsystems are considered to improve competition and reduce market concentration [19]. Within the Nordic grid counter trade is only performed in Sweden and Finland [22].
3.2.6 Congestion management in practice
In the best of worlds no bottlenecks would occur in the system. However, it is not economically feasible to expand the grid to the extent that no bottlenecks ever occurred. Nordel has however adopted a strategy that aims to reinforce the power system in so-called structural bottlenecks where congestion occurs frequently.
Temporary bottlenecks on the other hand, where congestion only occurs occasionally should be remedied with counter trade.[19]
As it takes time to invest in new power lines, not unusually up to ten years, it is important to have transparent rules for how bottlenecks should be handled. While Energinet.dk and Statnett prefer congestion management by use of market division, SvK and Fingrid advocate counter trade. Thus, this is why Norway and Denmark are divided into several subsystems while Sweden and Finland are not, even though all countries have internal bottlenecks. Nordel is currently investigating what method should be applied in the Nordic region and plans to present results from this investigation during the summer of 2006.[19]
As mentioned above, SvK aims at keeping Sweden as one price area. This means that no bottlenecks are allowed within the country. However, as most production occurs in the north of Sweden and much of the demand is found in the south of Sweden and in the neighbouring countries south of Sweden the maximum transmission capacity of certain part of the Swedish network is sometimes reached. These are referred to as cross-section 2 and cross-section 4. Instead of dividing Sweden into two price areas with a bottleneck as border, SvK restricts the transmission capacity on the connection to the neighbouring countries that is to Denmark, Germany and Poland. By doing so, the demand south of the bottleneck is reduced and no internal bottlenecks evolve.
Sweden is kept as one price area, while the possibility for the neighbouring countries to use Swedish production units is restricted.[30]
The European commission is examining whether this is allowed or not, as it interferes with the principle of a free market. An example of what can happen when SvK aims at keeping Sweden as one price area was seen in Denmark in November 2005. SvK restricted the capacity on the cable to East Denmark. This resulted in a relatively low price in Sweden, where not all production capacity was activated. In East Denmark, which at the same time suffered from production capacity reduction, the price rose to the highest ever in Nord Pool’s history due to activation of an expensive production bid. As a result of this, Energinet.dk has initiated a discussion about SvK’s current operation of the connections abroad.[14]
3.3 Financial aspects
Trade with electricity is carried out in different ways depending on when trading is performed relatively to the hour of operation. The Nordic electricity market Nord Pool
spot market and over the counter trade• outside the market place are used for trading most of the energy. The Nord Pool Elbas market is used for buying or selling adjustment power after the spot market has closed, while trade with regulating power only is performed during the hour of operation with the TSO as the single buyer.
3.3.1 The Nordic market place; Nord Pool
In short, Nord Pool consists of a spot market and an adjustment market, Elbas. In addition to this, there is a market place for trade with financial derivatives, such as futures and options. The futures reduce the market player’s risk as electricity is bought at a future date to a price set in advance while options makes it possible for market player’s to choose whether they want to buy electricity to a certain price determined at the time of the signing of the option[22]. The volumes traded on the spot market amounts to around 20 000 - 30 000 MWh per hour [9]. Outside the financial and physical (spot-market) markets Nord Pool also has a clearing service for bilateral market trade [22].
3.3.1.1 The Spot Market
Norwegian, Swedish, Finnish and Danish market players use the spot market to trade electricity. Before noon, the market players make their bids for buying and selling a certain amount of energy each hour for the next-coming day. Every hour is traded individually, from midnight and twenty-four hours forward. At noon, the spot market closes and Nord Pool compiles all the bids from producers and consumers for each hour the next-coming day. Supply and demand curves are constructed for every hour and the point were they cross marks the system price and the amount of power traded at that price. Nord Pool informs the market players what the price eventually is set to each hour and what amount of power that will be produced and used at what hour.
Nord Pool also delivers the information about the trade to the system operators in the Nordic countries that use it as information for keeping the balance in the grid.[22]
The spot market had a turnover of 167 TWh in 2004 while the financial market had a turnover of 590 TWh the same year [22]. In 2004 the clearing service cleared a volume of 1207 TWh and the total value of traded and cleared volumes reached 1964 TWh in 2004[22]. The largest market players are Vattenfall (17 percent market share), Fortum (14 percent), Statkraft (9 percent) and E.ON (8 percent) [11].
• Over the counter refers to trade that is carried out bilaterally between two market players.
3.3.1.2 The Elbas Market
The Elbas market was established in 1999 for market players in Sweden and Finland, and in 2004 for East-Danish players. The Elbas market opens at 3 PM every day, thus after the spot market has closed and allows additional trade for production hours the following day. By using Elbas, the market players get a chance to correct their balances reported to the spot market and compensate for unexpected events that occur after the spot market has closed. Trade on Elbas is performed with hourly amounts until one hour before the hour of operation it concerns. Nevertheless, the market players can update their balances until ten minutes before the start of operation.[22]
3.3.2 The Regulating market
The Nord Pool’s spot-market closes at noon the day prior to production day and from 3 PM until one hour prior to production market players has the opportunity to trade on Elbas or by bilateral agreements in order to be in balance at production hour. The changes made on these markets are usually adjustments based on new updated forecast on consumption, weather, and production as the hour of operation is approaching.[10]
The regulating market differs from the spot market and the Elbas market in several ways. Firstly, it is not a market in the sense that everyone can trade with every other player. Instead, it is the TSOs role to carry out trade with regulating power to keep the network in balance. Moreover, the regulating power market is only open during the hour of operation.[22]
3.3.2.1 Regulation and balance power settlement
The balance responsible market players have undertaken to guarantee that the bought and produced power is equal to the sold and by their customers consumed power, on an hourly basis. If deviation from the balance occurs the TSO has to activate regulation bids to correct the imbalance. At the end of each hour, the TSO sum up the regulating events and if regulation has occurred a regulating price (called RK-price) is set to the highest activated regulation bid that has been activated in case of upward regulation and the lowest regulating bid in case of downward regulation. All bids used to regulate are then given this price [22]. This way of pricing is referred to as marginal pricing, and is explained in Figure 3.
By performing a balance power settlement, SvK can assign the costs and revenues for regulation among the balancing responsible players. After all players’ balances have
been calculated this is put down in the settlement balance and saved in SvK’s settlement system. Depending on if SvK has ordered regulation and depending on whether upward or downward regulation has been ordered, there are four different possibilities to calculate the price of the regulating power:
If only upward regulation has been carried out, the upward regulation price will apply for those players with a negative balance (having excess consumption or too little production), while players with excess production get paid according to spot price.[22]
If only downward regulation has been performed, all market players with a positive balance have to pay the downward regulating price, while the spot price holds for all the others.[22]
If no regulation has occurred, all market players pay or get paid according to spot price.[22]
In those cases both upward and downward regulation has occurred during the hour of operation, price is set according to the dominating imbalance. If they are equally big, the spot price is used.[22]
Figure 3: The Swedish regulating price model [22]. Bids are sorted according to price and activated in price order. The last activated bid sets the price for all players involved in the regulation.
The main purpose of this pricing model is to ensure that market players not get any positive incentive to deviate from their planned production or consumption. For deviations that passively contribute to the balance in the system, the spot price is used, while regulation prices are used for imbalances that add to the network’s imbalance.
Imbalances are in this way looked upon in different ways depending on if they contribute to increasing or decreasing the imbalance in the system.[22] Appendix 4 contains examples of how regulating prices are settled.
3.3.3 Differences in market conditions between the Nordic countries
Trade with adjustment power, i.e. trade after the closure of Nord Pool spot market, and pricing of balancing power is carried out in somewhat different ways in the each of the subsystems within the INPS. The markets in the Swedish, Finnish and East Danish subsystems have all implemented Elbas, which not yet has been done in Norway and West Denmark. However, Energinet.dk plans to implement it in West Denmark in April 2007[5], and there is also a dialogue in Norway between Statnett and market players concerning the same issue. According to the latest report from Nordel, this might be done before the end of 2007[19]. The regulating market is common for all Nordic countries in the way that all regulating bids are gathered in a common bid list, which are activated by one of the Nordic TSOs depending on where the resource is located. There is however some important differences concerning the balance power settlement between the Nordic countries.[19]
3.3.3.1 One or three balances
The first difference concerns the number of balances that the balance responsible players are required to report to the TSOs. Fingrid and Statnett allow balance responsible players to report only one balance (production added to bought power minus consumption added to sold power). Energinet.dk and SvK require the balance responsible players to report three balances (own production, own consumption, and trade). The reason for this change of reporting carried out around five years ago was to make it less attractive for balance responsible players to adjust imbalances within the company. In this way, the adjustment resources would instead reach the market and become available for all other players. The purpose was to make the market fairer to players lacking own production.[19] Nordel has proposed that two separate balances should be reported (production and consumption) in all Nordic countries by the end of 2007.[19]
3.3.3.2 One or two regulating prices
The other difference concerns the pricing of balancing power. Energinet.dk, SvK and Fingrid use a two-price model while Statnett uses a one-price model instead. Using the one-price model, balance responsible players passively contributing to the balance (by having an imbalance opposite to the dominating imbalance) get paid the same amount, as the players that contribute to the net imbalance have to pay. A passive imbalance in Norway is thus rewarded with a higher price than the spot price while the passive players in other countries get paid according to spot price. This makes it theoretically possible for a market player in Norway to speculate in being in imbalance, even though this not has been the case in practice.[19] Nordel aims to implement a two- price system for imbalance in the entire INPS by the end of 2007[19].
3.3.3.3 Balance power settlement
As mentioned earlier the balance power settlement aims to allocate the costs and revenues for regulating power among the players that have been in imbalance during each hour of operation. The power balance settlement differs somewhat in West Denmark as compared to the other subsystems. Firstly, balance responsible players have to pay for deviations from the plan they deliver to the Danish TSO one day in advance of the hour of operation like in the rest of the Nordic countries[13]. In addition to this, balance responsible players also have to pay for deviations in produced energy on a quarterly basis within the hour of operation[13]. The deviation is measured as difference between the power plan (Dan. effektplan) and measured production or consumption. The power plan describes production for every 5 minutes and can be updated until around 15 minutes before time of operation[31]. Only energy deviations higher than 2,5 MWh per quarter is penalised[13]. By giving the balance responsible players an incentive to keep their power plans or update them in advance Energinet.dk minimizes the need for primary regulating reserves as regulation can be carried out with manual reserves instead[13].
3.3.4 Demand and supply for adjustment power
The demand for adjustment power has traditionally been caused by variations in consumption since production of mainly hydropower and thermal power is quite easy to plan. However, when introducing wind power into the production system demand for adjustments will derive from the production side as well.
The price of adjustment power is dependant of the type of production used. The type of production that is considered to be most suitable for making adjustments is hydropower. The reason for hydropower being one of the best ways of making adjustments is the relative flat efficiency curve of certain types of hydropower plants.
Changes in production level therefore results in minor changes in efficiency. Another good reason for using hydropower as adjustment power is the high speed of which changes can be made. Gas powered turbines can also be used for regulating power production but they are usually more expensive to use than hydropower.[24]
3.3.5 Strategy for trading with energy and power
3.3.5.1 The EMPS model
Concerning power generated by hydropower the overall goal is to sell the electricity when the price is the highest possible. Several Nordic energy companies with hydropower resources use a modelling program called EMPS• to value the water in their hydropower reservoirs and to evaluate when it is as most profitable to use the water. The program has a starting point in the Water Value principle, where the value of the water is assumed to be dependant on the current water level in the reservoirs, the expected precipitation and forecasted future spot price. From this, the program optimizes the usage of water on a weekly basis for a period of up to ten years.
The results from the program can be used to sort the hydropower production from each power plant into a bid list, which then can be used to place bid on the spot market. The program takes seasonal variations into account, which means that water in general is saved during summer time to be used in wintertime when prices are higher.
The same reasoning is applied on a daily basis. Generally, the spot price is higher during daytime and therefore the best would be to only sell electricity at these hours.
However, if electricity only were to be produced during these hours, water would overflow the reservoirs. Therefore electricity is sold during night-time as well even though the price is lower.[29]
The EMPS model cannot be used to value regulating power, as the time frame of the program is one week. Also, the need for regulating power follows a stochastic pattern, so it is hard to correctly evaluate when the water would be most valuable to use. The
• Also referred to as Samkörningsmodellen
cost for producing regulating power and in the end the price that the producer should demand for the regulating power production is also affected by factors that change from hour to hour such as:
What hydropower plants are in operation at the moment
What hydropower plants are available for regulation
Current reservoir levels, both in the hydropower plant’s own reservoir and reservoirs downstream of the power plant
Costs for starting-up hydropower plants
What efficiency curve losses occur
All together, this means that the strategy for trade with regulating power must be more flexible than for trade with energy sold at the spot market. [26]
4 Method
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4.1 Quantitative to a large extent but with qualitative elements
This project is mainly based on quantitative investigations of statistical databases.
Qualitative elements appear in the form of interviews mainly with people inside Vattenfall but also with people from SvK and Nord Pool.
We have chosen to study data from the years 2000 to 2005 in order to get broad and yet updated input to our report. Focus has been on the most recent years at the times when it has been impossible to study the whole period from 2000 until 2005. When comparing data from different sources, we have put down great efforts to make them match in time so that relations between the two sets of data not only become apparent but also accurate.
4.2 Statistical methods
The major part of this work handles evaluation of databases containing parameters related to the behaviour of the power system and thereby the regulating market. The parameters themselves and their relations are investigated by the use of statistical methods such as regression analysis and statistical correlation.
When comparing modelled results with the actual outcome and evaluating the model a commonly used evaluation principle is the least square error method. This method is used to minimise the error of a function created to mimic an observed behaviour when performing a regression analysis.
When the observed data (x) and the modelled data (y) is:
) ( ) ( )
(
x1,y1 , x2,y2 ,..., xn,ynThe modelled curve is described by:
) , ( bx f y= where bis a coefficient vector.
The aim for the least square method is to minimize b for:
[ ]
∑
= n −i
i
i f x b
y
1
)2
, (
Moreover, the built in functions of Microsoft Excel has been used. The solver function for solving optimization processes has been the most useful for us. This function uses a numerical process that step-by-step evaluate possible solutions until it fulfils a criteria set by the user. It is also possible to set constraints to make the model fulfil certain demands. Some macro’s has been constructed to sort out data retrieved from databases into an easy-to-handle format.
