Marginal costs in electricity production

Marginal costs in electricity production

16 July 2007

Marginal costs in electricity production... 0

1. Introduction ... 3

2. Problem formulation ... 5

2.1. The data... 5

2.1.1. Supply & demand... 5

2.1.2. Fuel ... 6

2.1.3. Ramping... 6

2.1.4. Start up ... 6

2.1.5. Operation & Maintenance ... 6

2.2. The model... 7

3. The market and the company... 8

3.1. The electricity market for supply ... 8

3.2. The markets for trading electricity... 9

3.3. Electricity production... 10

3.4. Electricity suppliers ... 11

3.5. Green energy... 12

3.6. The company ... 13

4. The data set and definitions ... 15

4.1. Demand ... 15

4.2. The plants ... 16

4.3. Fuel prices and plant capacities ... 16

4.4. Starts ... 17

4.5. Efficiency ... 18

4.6. Values for various other data ... 18

4.7. History... 18

5. The model ... 20

5.1. The indices ... 20

5.2. The parameters and independent variables ... 20

5.2.1. Supply & demand... 20

5.2.2. Fuel ... 20

5.2.3. Ramping... 21

5.2.4. Start up ... 21

5.2.5. Operation & maintenance ... 22

5.3. The dependent variables and the constraints ... 22

5.3.1. Supply & demand... 22

5.3.2. Fuel ... 23

5.3.3. Ramping... 23

5.3.4. Start up ... 24

5.3.5. Operation & maintenance ... 25

5.4. The objective ... 26

5.5. Model size... 26

6. The solver ... 27

7. The results ... 28

8. Sensitivity analysis & scenario analysis ... 31

8.1. Sensitivity analysis... 31

8.2. Scenario analysis... 32

8.2.1. The results without decentralised production capacity ... 32

8.2.2. The results with low gas prices ... 33

8.2.3. The results with high demand ... 33

8.2.4. The results without the nuclear plant ... 34

9. Conclusions and recommendations... 35

References ... 36

1. Introduction

In a perfect market the electricity price per hour depends on supply and demand. But what influences supply and demand?

Supply is influenced by variable costs and technical aspects of the plants. For example most plants can’t go from zero to maximum capacity within an hour. Further there are plants that have to run all the time, because they have to produce heat for city heating or steam for industrial processes. Demand is influenced by the day of the week, the hour of the day and the weather conditions. The weather conditions include in this definition not only if the sun is shining or if it’s raining, but also at what time the sun rises and sets. The effect of this can be seen in figure 1. In winter the demand on weekday evenings is almost as high as the weekdays during the day. The demand on weekend evenings is higher than the demand on weekends during the day. In summer the demand in the evening is lower than the demand during the day, with the Sunday as an exception. 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0 12 0 12 0 12 0 12 0 12 0 12 0 12 Hours Monday - Sunday No rm al is ed v o lu m e s Winter Summer

Figure 1. Demand: summer profile compared to winter profile.

For NUON it is interesting to know what, based on marginal costs, the expected electricity prices per hour will be. To know this, I was asked to build a model for calculating marginal costs of producing electricity.

The composition of this paper is as follows. In this first chapter, the introduction, I’ll give a small introduction on the factors that influence supply and demand. Further this chapter contains a summary of this paper. In the second chapter I will formulate the problem in more detail. In the third chapter I will write about a few aspects of the energy markets, about the company NUON and about the division Energy Sourcing in particular.

Chapter 4 describes the data that I had available for building this model. Next to this I will give some definitions in this chapter. In chapter 5 I will give a mathematical description of the parameters, variables and constraints that together form the model that I build. This includes sub paragraphs to group parts of the model that belong to different aspects of the cost structure. In the next chapter, chapter 6, I will explain which programming language and solver I used and why I used these. In chapter 7 I will give some definitions that I will use for presenting the results. Then I will explain which results I expect to see. Finally I will present the results the model gave. The next step would be performing a sensitivity analysis. In chapter 8 I explain why this is very difficult for a mixed integer model with a lot of binary variables. Instead of this sensitivity analysis, I chose to do a scenario analysis. I used common sense to choose some scenarios from which I will present the results in this chapter.

My conclusions and recommendations will follow in chapter 9, which is the last chapter of this paper.

2. Problem formulation

The purpose of this model is, to determine on an hourly basis which is the marginal plant and what are the variable costs of this plant at this hour. Marginal plant means the plant with the highest variable costs that is dispatched at this hour. Because most plants with a must run status have high variable costs, this would result in a must run plant being marginal at almost all hours. But must run plants do in reality not dictate electricity prices. Therefore I have to change this definition slightly into a definition where a plant is marginal if it has highest variable costs of all plants that are not producing at their must run capacity. This means that a must run plant can only be marginal if it is producing at a capacity level that is higher than minimum capacity. The marginal costs for a plant at each hour are defined as the sum of the variable costs for fuel, ramping and operation & maintenance. The start and stop costs do influence if a plant is running, therefore these costs are taken into account in the calculations. These costs are high for the first hour that a plant is running, after this first hour, these costs are 0.

2.1. The data

There are 2 different types of data available. The first type is data that can differ for each hour. The second type of data can differ per plant, but is the same for each hour. Throughout this whole paper five categories of data and variables that influence the production of electricity will appear. These categories are:

o Supply and Demand o Fuel

o Ramping o Start up

o Operation and Maintenance

In this chapter I will start with giving the kind of data available. In chapter 4 I will look into this data with more details.

2.1.1. Supply & demand

The demand for each hour is the first item in this category. The other three items in this category are part of the plant information. The first one is the fact if a plant must run or not. For example plants that provide city heating must run 24 hours a day, every day of the week. Based upon contracts with owners of gas-engines and upon import contracts the minimum production per week and the minimum number of production hours per day is sometimes fixed.

2.1.2. Fuel

The first item in this category is the fuel price. This item contains a matrix of prices for the different fuel types for each hour. Examples of the fuel type are coal or gas or import contract. The first items of the plant information that belong to this category are the minimum capacity and the nominal capacity of the plant. Another example of plant information is the type of the plant. Examples are conventional boiler and combined cycle power plant.

At this stage the number of operation levels is chosen. With these two tables are created, one for the capacity and one for the efficiency of the plants. In the capacity table are the nominal capacity, 80 percent of the nominal capacity and the minimum capacity present for each plant. The remaining capacity levels depend on the capacity of the plants. For calculation purposes the change in capacity should be at least 25 MW. For each plant type it is possible to create a formula to determine the heat rate that corresponds to the capacity at each operation level. The efficiency is defined as 3.6 divided by the heat rate.

2.1.3. Ramping

There are only two items available in this category, both of which are part of the plant information. These items are the maximum amount the plant is allowed to ramp up or down and the costs for ramping up and ramping down.

2.1.4. Start up

The costs for making a cold start or a hot start are the first two items in this category. The other items are the number of hours after which the plant is cooled down and the number of hours that the plant needs to start up when a start is a cold start.

2.1.5. Operation & Maintenance

There is only one item in the category “Operation & maintenance”. This item is the variable operating and maintenance costs.

2.2. The model

The electricity production park has a certain number of fixed costs. These fixed costs have to be paid, but they don’t have influence on the electricity produced by these plants. The total variable costs for producing electricity have to be as low as possible to determine which plant is producing electricity.

The variable costs for each plant are the sum of fuel costs, costs for ramping up and down, costs for starting the plant and operation and maintenance costs. The objective of this model is to minimize the sum of these variable costs under certain constraints.

The fuel costs are a function of the type of fuel, the efficiency of the plant and the dispatch level of the plant. The dispatch level of the plant is the capacity that is produced by this plant in MW. The ramping costs are a function of the unit costs for ramping up or down and the change in dispatch level. The costs for starting a plant are a function of the maximum capacity of that plant and the unit costs for making a start. There is a difference in costs for making a start when the plant is cooled down and when a plant is not cooled down yet. I will define this more precisely in a later stage. Finally the operating and maintenance costs are a function of the dispatch level and the unit costs for operating and maintenance.

There are constraints for the minimum and maximum number of hours that a plant has to produce electricity. Further there are constraints on how much a plant can ramp up or down. The next group of constraints are to determine if a plant can start or not. And of course maybe the most import constraint is that the supply has to meet the demand, within a certain range of accuracy.

3. The market and the company

Until 1998 the electricity market was a closed market. The market at that time was neatly distributed over several companies. Each region had its own electricity company. The European Union has created directives for the opening of the electricity market. These directives required a market opening of 33% in 2003 and of 100% in 2007. In the Dutch Electricity Law of 1998 is stated in what rate the Dutch electricity market has to change from a closed to an open market. In figure 2 the demands of the Dutch Electricity Law and the directives of the European Union are displayed.

Demands of the Dutch Electriciy Law Directive of the European Union Market for green electricity

1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 33% 26,5% 28% 66% 33% 100% 100% 100% 1/7 1/7 19/2 19/2 1/7

Figure 2. Phased opening of the electricity markets.

According to the Dutch Electricity Law there are three groups of consumers. The first group of consumers with a contracted capacity of more than 2 MW, the industrial bulk consumers, was free to choose its provider immediately after the electricity law was introduced. The second group of customers, the bulk consumers, has freedom of choice since the first of January 2002. These customers have a connection of more than 3 × 80 Amperes, with a contracted capacity of less than 2 MW. The small customers, they have electricity contracts with a connection of less than 3 × 80 Amperes, form the last group. They have freedom of choice as of the first of July 2004. (See figure 2.) All customers are free to choose their supplier of renewable energy since the first of July 2001, provided that they use it for all their electricity needs.

The free electricity market has not only an effect on the consumers, but also on the electricity companies. These companies can offer their products and services nationally and internationally instead of regionally, with a real market as a consequence. Furthermore in a free market the supply of electricity and the conduct of the electricity system will be performed by separate companies. With approval of the supervisor DTe, the office of Energy Regulation, the system owner assigns the system administrator. The system administrator has a geographic monopoly, which means that for each region there can be only one system administrator.

Companies that delivered electricity to tied customers before the liberalization have a delivery license for the region they were delivering. When all customers in the electricity market are free

to choose a supplier, the delivery company still needs a license to deliver electricity. At that moment there are no regional restrictions anymore [1].

The separation of the grid companies from the production- and distribution companies is currently a hot topic. In November 2006 the decision was made to allow for the possibility of such a break up. This was motivated by the fact that the European Legislation would require this and the fear that the electricity companies would take too many risks outside the Netherlands. Based on the proposed 60% take-over of the Belgian waste-burner Indaver by Delta and the international aspirations of NUON and Essent after their proposed merger, the minister is planning on issuing a “Koninklijk Besluit” to pass a law requiring this break up.

3.2. The markets for trading electricity

The markets where electricity is traded can be split-up in 3 different kinds of markets. The first one is the Over-The-Counter market (OTC). In this market capacity blocks are traded. The most commonly traded blocks are baseload and peakload. Baseload is defined as delivery during 24 hours for 7 days a week. Peakload is defined as delivery during working days from 7:00 until 23:00; all other hours are called Offpeak. Blocks can be traded for calendar years, quarters, months, weeks and days. Occasionally other blocks are traded, like Offpeak, Superpeak (delivery during working days from 8:00 until 20:00) and weekend peak (delivery during weekends and holidays from 8:00 until 20:00). In the OTC markets there are different prices for buying and selling.

The second market is the Amsterdam Power Exchange (APX). On this day-ahead spot market hourly volumes are traded. In general the baseload and peakload blocks will be stripped out of a profile to be purchased on the OTC market. After this the remaining volumes will be purchased and sold on the APX on an hourly basis. In figure 3 a demand profile is split up in baseload OTC blocks, peakload OTC blocks and APX volumes. Prices are the same for buying and for selling and are the result of an auction.

0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 16,000 0 12 0 12 0 12 0 12 0 12 0 12 0 12 Hours Monday - Sunday N o rm al is ed vo lu m e s Buy on APX Sell on APX Peakload Bas eload Profile

Figure 3. Demand profile split up in market blocks.

The third market is the Tennet imbalance market. Tennet is responsible for balancing the Dutch grid. It’s possible, but not necessary, that the prices for surpluses and shortages are different in this market. This is the only market where prices can be negative. This is the result from the fact that electricity can not be stored. Prices can differ for each programming time unit (PTU); a PTU has a length of 15 minutes.

3.3. Electricity production

In the Dutch electricity market large-scale electricity plants cover over 50% of the electricity demand. Import covers about 18% and decentralized production covers the rest [2]. Figure 4 shows that the four large-scale producers (E.ON Benelux, NUON, Electrabel and EPZ/Essent) cover two-thirds of the domestic electricity production. The total capacity of the production in 2005 was about 21,800 MW [3].

67% 24%

6% 3%

Electricity production companies

CHP

Wind, sun, waterpower, biomass

Other

Figure 4. Spread of domestic electricity production over production types in 2005.

A number of plants deliver next to electricity also heat. Combined Heat and Power systems (CHP) are exploited by electricity suppliers, electricity buyers or by joint ventures between suppliers and buyers. In 2005 they had jointly a share of 24% in the domestic electricity production. The rest of the produced electricity comes from waste burning installations and renewable sources like wind, sun, waterpower and biomass [3].

Figure 5 shows a spread of the electricity production capacity over the most important players in the Dutch electricity market. The figure shows that three electricity suppliers, Essent, NUON and Electrabel, own a considerable part of the production capacity. The position of Essent is based on the vertical integration with EPZ [3].

29% 23% 12% 28% 8% Electrabel NUON E.On Essent/EPZ Other

Figure 5. Spread of the electricity production capacity (total 14,836 MW)in 2005.

3.4. Electricity suppliers

In the year 2001 the majority of the customers were still tied, which means that a licensee delivers their electricity. The electricity supplier, with whom the customer has a relationship, is part of the same company as the system administrator. Because of this the market shares of electricity suppliers are approximately the same as the number of connections. Figure 6 shows the market shares based on the number of connections [4].

33% 36% 18% 8% 3% 2% Essent Nuon Eneco Endessa Delta

Other electricity companies

Figure 6. Division of market shares of the electricity suppliers based on number of customers.

Figure 6 shows that based on the number of customers the three largest electricity suppliers NUON, Essent and Eneco have a joint market share of 87%. Since the opening of the market no new numbers have become publicly available.

3.5. Green energy

Through an increase in the Regulating Energy Taxes (REB) and an exemption of these taxes for renewable electricity is the price of green energy almost equal to the price of conventional or grey electricity. The market for green energy has been free since the first of July 2001, meaning that customers can buy green electricity from other suppliers. Moreover new suppliers participated in the green energy market. Figure 7 shows an estimated division of the players on the market of green energy [5].

27% 26% 24% 7% 7% 1% 8% Essent Nuon Eneco Endessa Delta

Other electricity companies Real Energy / Energy Competitor

Figure 7. Division of market shares of the electricity suppliers on the green energy market based on number of customers (July 2001).

The division of the market is comparable to the market for conventional electricity (see also figure 6). New players, like Real Energy an Energy Competitor, only have a moderate market share. This is the case because the growth of the number of customers for green energy in the first half of the year 2001 has taken place under existing customers. Although this is already a long time ago, more recent numbers are not publicly available. In 2006 NUON had approximately 409.000 customers with a green contract.

3.6. The company

The present company NUON is evolved from a group of companies. The first merger was in 1999. The companies ENW, EWR, Gamog and NUON (old) together formed the company NUON. In the same year, NUON and Essent sold the production company EPON, in which both companies had a 50% share, to Electrabel. After this NUON became a grid company and a distribution company only.

Due to changed market circumstances NUON decided to buy the plant Demkolec of 250 MW in 2001. In February 2003 NUON announced the principal agreement of the takeover of the company Reliant Energy Europe BV. With this NUON would own an extra 3500 MW of production capacity. Next to this NUON had access to the 800 MW power plant of Intergen. The NMa approved the takeover under the condition that NUON would auction 900 MW for 5 years. With the sale of the 800 MW of Intergen to Eneco Energie, the NMa changed the capacity to be auctioned to 200 MW for the calendar year 2005 only.

NUON owns also 669 MW of renewable capacity. This exists mainly out of windmills. Approximately 60% of this volume is located in other countries. NUON is, together with Shell, building a new wind park in the North Sea, near the coast of Egmond aan Zee. This park will have a capacity of 108 MW. Further NUON delivers heat to about 130.000 households.

In the mid of 2006 NUON announced its intention to build a plant with a capacity of 1200 MW. This plant is supposed to start delivery in 2011. This plant will replace a few plants that will reach the end of their production life in the coming years and will be located in Eemshaven.

In February 2007 NUON and Essent announced their intention to merge. Currently the NMa is looking into this plan. The decision is expected at the end of 2007.

The company NUON can be split in 4 divisions. Those are Energy Sourcing, Netwerk Services, Business Customers and Retail Customers. Next to this NUON has an office in Belgium and some offices in Germany. The grids are in hands of Continuon Netbeheer. Further there are some departments that provide services to the rest of the company.

The division Energy Sourcing has 3 operational departments; next to this are the departments with various staff functions and Risk Management. The operational departments are Generation, Technical & Project Development and Energy, Trade & Wholesale. The department Generation is NUON‘s producer of electricity and heat. Technical & Project Development is responsible for all technical projects, like building plants and wind mill parks. Finally Energy, Trade & Wholesale is responsible for the sourcing of electricity and gas for the sales customers.

4. The data set and definitions

As stated in the problem formulation the purpose of this model is to determine for each hour which plant is the one that has marginal variable costs. Which plant will be used at each hour

both depends on which plants were used in a recent history and on which plant needs to be used in the near future. For determining this I have to create a model in which the total variable costs are minimized during a certain time period.

4.1. Demand

The demand data that will be an input for this model will be estimated by using another model that is created at NUON Energy Sourcing. Since this data is not available on a longer term basis, I downloaded historical demand figures from the Tennet website [6]. To this figures 1015 MW is added for every hour. This is based on an assumption of the supply of owners of decentralized capacity for their own electricity needs. The decentralized capacity, which is capacity owned by companies that don’t have electricity production as their main task, is taken into account as existing capacity in the data, because the owners of this capacity will not only produce for their own electricity needs, but they will also deliver electricity to the grid.

To give an idea of the demand, I defined 6 different blocks. I will use these blocks again when I will present the results. These blocks are:

o Superpeak (working days 8-20)

o Shoulders (working days 7-8 and 20-23) o Saturday peak (Saturday 9-17)

o Sunday peak (Sunday and holidays 9-17)

o Weekend evenings (Saturday, Sunday and holidays 17-22) o Night (all other hours)

0 12 0 12 0 12 0 12 0 12 0 12 0 12 Hours Monday - Sunday 0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 16,000 MW D e m a nd Superpeak Shoulders Saturday peak Sunday peak Weekend evenings Night Demand

Figure 8. Demand against blocks.

In figure 8 I made a graph of the average demand in the first quarter of 2005, that I will use for my calculations. I graphed this against the defined blocks.

4.2. The plants

The total capacity in the Netherlands exists of centralized and decentralized capacity. For calculation purposes I had to minimize the number of plants, without losing too much information. To do this I combined all the plants with a capacity smaller than 25 MW and efficiency at nominal capacity of around 25%. I also made groups of the decentralized capacity, based upon the type of the plant and the efficiency rate. I assumed that all the decentralized plants must be in use all the time, since this capacity is mainly situated at industry sites, where the produced heat or steam is needed for the production process. All these plants are gas fuelled. There is also a small group of Diesel engines. These are often situated at hospitals, water companies, etc. for emergency power. These engines have to run about 4 hours per engine per week, for testing purposes.

The biggest part of the centralized capacity is fuelled with gas, followed by coal. There is one plant fuelled with uranium, one fuelled with blast-furnace gas and one fuelled with a combination of gas and blast furnace gas.

Besides centralized and decentralized capacity there is also the possibility to import electricity. There are two sorts of import, import contracts and auctioned import capacity. In the import contracts, which were closed by the “Samenwerkende Electriciteits Producenten” (SEP), the prices, maximum volumes and other subjects are fixed. The price for auctioned capacity is assumed to be equal to the price of the fuel that represents the largest proportion of the fuel mix in the country from where the capacity is imported.

Further there is some renewable capacity. This is for example driven by wind, sun or water.

4.3. Fuel prices and plant capacities

To compare the costs of the different fuels I looked at the average costs per group of plants at maximum capacity. The fuel price for durables is assumed to be 0, since the only input is from natural sources. This results in fuel costs of 0.00 €/MWh with a capacity of 51 MW. The nuclear plant has a capacity of 427 MW. The fuel costs are 10.33 €/MWh.

The prices for the import contracts vary from 13.60 €/MWh up to 19.60 €/MWh, with a weighted average of 16.45 €/MWh. The total capacity of this group is 1650 MW. The prices for auctioned import are based on both coal prices and uranium prices, resulting in an average price of 19.68

€/MWh with a total capacity of 3400 MW. The average price of coal fired plants is 24.47 €/MWh with a capacity of 3949 MW.

The highest average costs are for gas fired plants, 35.40 €/MWh and for must run plants, 39.66 €/MWh, with capacities of 6490 MW and 6112 MW.

4.4. Starts

A start is defined as when the plant is in use at time t, but was not in use at time t – 1. There are a few conditions that have to be satisfied to determine if a start is allowed and if the start is a hot or a cold start. A start is defined as a hot start if the plant has not yet cooled down. A start is a cold start if the plant was not in use during the time the plant needs to start up. There is no start allowed if the plant was in use during the time the plant needs to start up, but was not in use during the time the plant needs to cool down.

There are four possible situations, for each of which I have created a timeline, with the conclusions if a start is allowed and what kind of a start it is. In these time intervals a box means that the plant is in use at that time. It takes (time – cool down) hours for a plant to cool down and it takes (time – start up) hours for a plant to start up.

start up cool down time

A start is allowed; it has to be a cold start.

start up cool down time

There is no start allowed.

start up cool down time

A start is allowed; it has to be a hot start.

start up cool down time

A start is allowed; it has to be a hot start.

The costs for a start vary with the type of the plant and the fuel type of the plant. Highest costs for a cold start are 25 €/MW, lowest costs are 10 €/MW. Exceptions for this are import volumes and must run plants, since these plants never have a start. Therefore start costs are assumed to be 0 €/MW. Costs for a hot start are assumed to be half of the costs for a cold start.

The start up time for a cold start varies from 1 hour up to 26 hours. The cool down period varies from half an hour up to 6 hours. Neither depends on type or fuel.

4.5. Efficiency

The efficiency of a plant depends on the type of the plant, the fuel type of the plant and the year that the plant is built or revised. The efficiencies used in this model vary for the decentralised capacity from 14% to 42%. The durables are assumed to have an efficiency of 100%. Import contracts have an efficiency of 100% and auctioned import is assumed to have the efficiency of the underlying fuel type. This is 30% for the import from Belgium/France which is based on uranium and 38% for the import from Germany which is based on brown coal.

The gas turbines that were build between 1971 and 1982 have an efficiency of 25% or 26%. These have a capacity of around 25 MW. The combined cycle plants have an efficiency percentage between 38% and 45% if they were build in 1993 or earlier. The fuel type does not have an influence on this. This type of plants has an efficiency percentage between 50% and 54% if they are fuelled with gas and build after 1993. The plant that is fuelled with blast furnace gas has an efficiency of 45%.

4.6. Values for various other data

Variable operating and maintenance costs are between 0 €/MWh and 6 €/MWh. Ramping costs are higher for ramping up than for ramping down. Ramping up costs between 0 €/MW and 2.70 €/MW. Ramping down costs between 0 €/MW and 0.80 €/MW. Most of the plants are able to ramp up or down about 95% of their total capacity within an hour. Exceptions are coal plants, which are able to ramp between 44% and 83% of total capacity and import contracts where ramping is allowed for 33%, 67% or 100% of total capacity.

The minimum of production per week is set for two import contracts and for the Diesel engines. Although Diesel engines can not run for a lot of hours per week, there is no need for putting a maximum on the number of hours that these engines are allowed to run, since these engines have high fuel costs, low efficiencies and high operating and maintenance costs. The minimum of production hours per day is set for two of the import contracts.

The minimum load varies between 25% and 40% of maximum capacity. Exceptions are durables and one of the import contracts, where the minimum load is equal to the maximum load.

4.7. History

The model that I have to build will not have a feasible solution if there is no history available of which plants are running in the past 12 hours, because all the plants together can not ramp from

0 MW to the necessary capacity within 1 hour. Further the start up costs would determine which plants would be used in the first hours. This could mean, that plants that normally wouldn’t be used will be used, just based on the start up costs, not based on the rest of the variable costs. Although the start up time of one of the plants is 26 hours, this plant is normally not used, therefore 12 hours history is sufficient. This plant is normally not used because the efficiency of this gas-fired plant is only 39%, making it one of the most expensive plants that is available. The history is for the first run determined by taking the cheapest plants that together come approximately to the necessary demand. After this first run, history is set for the first week of all of the future runs. It is necessary to know if a plant is used during the past twelve hours, at what level the plant was used during the last hour and what the dispatch level was during that last hour.

5. The model

The model will be a mixed integer linear programming model, where all the integer variables will be binary. All costs are linearly related to the capacity at which the plants are operated.

5.1. The indices

In this model four indices will be used. The first one, p, will be used for the plants. In this case there are 64 plants, therefore p ∈ [1,…,64]. The second one, r, will be used for the level at which the plant is operated, I have chosen to use 10 levels, so r ∈ [1,…,10]. The last one, t, will be used for the time. This index is defined in hours. The total timeframe per run of this model is one week, so t ∈ [1,…,168]. In formulas where history is needed, t will be replaced by ht ∈ [-11,…,0].

5.2. The parameters and independent variables 5.2.1. Supply & demand

There are two parameters that belong in this category. These are the parameters that place bounds on the supply. Those parameters are δ for the lower bound and ε for the upper bound. Both δ and ε are an element of the range [0, 1]. Further there are five groups of independent variables .The first one is the demand at time t, in MW, that is Demand(t). The next independent variable that belongs in this category is Mustinuse(p). This variable has value 1 if plant p must be used at all times t and has value 0 if there are no such restrictions. The next two are the minimum production per week in MWh, called Minprodweek(p) and the minimum number of production hours per day, called Minprodhours(p).

The last two groups of independent variables will contain the history that is needed to be able to solve this model. These are HistInuseatanylevel (p,ht) and HistDispatchlevel(p,0). The first of these variables will have value 1 if plant p was used at time ht and value 0 if plant p was not used at time ht. Finally the last of these independent variables will contain the supply of plant p at time 0 in MW.

5.2.2. Fuel

One of the independent variable groups in this category is the fuel costs for plant p at time t and operation level r, in €. The variables in this group can be derived from the other independent variables in this category and have to be nonnegative. The price of fuel for plant p at time t, in €/MWh, that is Price(p,t) is the first of these other variable groups. Another group is the nominal capacity of plant p, in MW, that is Plantcapmax(p). The last two groups are the capacity at which

plant p is operated at level r, Operationcapacity(p,r) and the efficiency percentage of plant p at operation level r, Plantefficiency(p,r).

The fuel costs, Fcplant(p,r,t), are defined as

Fcplant(p,r,t) := Price(p,t) * Operationcapacity(p,r) / Plantefficiency(p,r) ∀ p, r, t.

5.2.3. Ramping

The maximum amount plant p is allowed to ramp up in MW/hour, Maxplantramplim(p), is the first group of independent variables in this category. The second group is the maximum amount plant p is allowed to ramp down in MW/hour. This variable group, Minplantramplim(p) is defined as

Minplantramplim(p) := -Maxplantramplim(p) ∀ p.

The other independent variable groups are Rampcostsup(p), that is the costs for plant p to ramp up and Rampcostsdown(p), the costs for plant p to ramp down, both in €/MW.

5.2.4. Start up

The first two independent variable groups in this category are Plantcoldstartcosts(p) and Planthotstartcosts(p). These are the start up costs if plant p makes a ‘cold’ or a 'hot' start respectively. These costs are in €/MW. Another variable group is Cooldownlength(p). This is the number of hours after which plant p is cooled down. Further there is a variable group called Coldstartlength(p). This is the number of hours that plant p needs to start up when a start is a ‘cold’ start.

The last two groups of independent variables have the names Relevanthotstartperiod(p,t,tt) and Relevantcoldstartperiod(p,t,tt). These variable groups are used to determine which periods are relevant to check if a 'hot' or a 'cold' start respectively are allowed. These independent variables will only get a value for the time between t and t – 7 or t and t – 27, since the maximum time that it takes for a plant to cool down is 7 hours and the maximum time to start up is 27 hours.

The definitions of these parameter groups are

⎩0 if (t – tt) ≥ Coldstartlength(p)

In these formulas tt is used as an index for time steps between t and t – 7 or t – 27 respectively.

5.2.5. Operation & maintenance

In this category there is only one group of independent variables. This group is PlantvarOM(p), which is the variable operation and maintenance costs for plant p in €/MWh. There are no fixed operation and maintenance costs taken into account, because we are only interested in the marginal costs for producing electricity.

5.3. The dependent variables and the constraints 5.3.1. Supply & demand

The variable Inuse(p,r,t) is a binary variable that has value 1 if plant p is used at operation percentage r and time t and value 0 if plant p is not used at operation percentage r and time t. The variable Inuseatanylevel(p,t) is defined as

Inuseatanylevel(p,t) := ∑rInuse(p,r,t) ∀ p, t,

where

0 ≤ Inuseatanylevel(p,t) ≤ 1 ∀ p, t.

According to its definition this is a variable that states if plant p is used at time t or not. This variable must satisfy the following constraint.

Inuseatanylevel(p,t) ≥ Mustinuse(p) ∀ p, t.

The supply of plant p at time t in MW is called the dispatch level. The definition is

Dispatchlevel(p,t) := ∑r [Operationcapacity(p, r) * Inuse(p,r,t)] ∀ p, t.

The definition of the change in dispatch level from period t – 1 to period t in MW is

Dispatchlevelchange(p,t) := Dispatchlevel(p,t) – Dispatchlevel(p,t – 1) ∀ p, t. At t=1 the variable Dispatchlevel(p,0) will be replaced with the independent variable HistDipatchlevel(p,0).

The minimum production per week in MWh has to satisfy the constraint

∑t Dispatchlevel(p,t) ≥ Minprodweek(p) ∀ p.

The minimum number of production hours per day has to satisfy the constraint

∑t | day Inuseatanylevel (p,t) ≥ Minprodhours(p) ∀ p.

In this constraint “t | day” means for every t that is an element of the set day, where day has the elements {d1,…,d7}. The element d1 corresponds to t = 1,…,24, d2 corresponds to t = 25,…,48, etc..

Finally the total supply in MW at time t is defined as

Totsupp(t) := ∑p Dispatchlevel(p,t) ∀ t.

The total supply at time t must suffice the following lower and upper boundaries:

(1-δ) * Demand(t) ≤ Totsupp(t) ≤ (1+ε) * Demand(t) ∀ t.

5.3.2. Fuel

The costs for fuel for plant p at time t in € are defined as

Fuelcosts(p,t) := ∑r [Fcplant(p,r,t) * Inuse(p,r,t)] ∀ p, t.

5.3.3. Ramping

The costs for ramping up or down for plant p at time t can be divided in costs for ramping up, Rampingcostsup(p,t), and costs for ramping down, Rampingcostsdown(p,t). From this follows the following relationship.

Rampingcosts(p,t) := Rampingcostsup(p,t) + Rampingcostsdown(p,t) ∀ p, t. These costs are all in € and must suffice the following constraints.

Rampingcostsup(p,t) ≥ 0 ∀ p, t,

Rampingcostsdown(p,t) ≥ 0 ∀ p, t,

Rampingcosts(p,t) ≥ 0 ∀ p, t.

The variables Rampingcostsup(p,t) and Rampingcostsdown(p,t) get their value based on the following constraints.

Rampingcostsup(p,t) ≥ Rampcostsup(p) * Dispatchlevelchange(p,t) ∀ p, t and

Rampingcostsdown(p,t) ≥ Rampcostsdown(p) * {-Dispatchlevelchange(p,t)} ∀ p, t. There is a limit to the amount plant p is allowed to ramp at time t. The following constraint makes sure that plant p does not ramp more than this amount.

Minplantramplim(p) ≤ Dispatchlevelchange(p,t) ≤ Maxplantramplim(p) ∀ p, t.

5.3.4. Start up

The costs for starting up plant p at time t, Startupcosts(p,t), are defined as

Startupcosts(p,t) := Plantcapmax(p) * [Plantcoldstartcosts(p) * Coldstart(p,t)

+ Planthotstartcosts(p) * Hotstart(p,t)] ∀ p, t.

These costs are in € and must suffice the following constraint.

Startupcosts(p,t) ≥ 0 ∀ p, t.

Two new variables are used in the definition of Startupcosts(p,t). Coldstart(p,t) is a binary variable that has value 1 if plant p is in use at time t, but was not in use during earlier relevant time steps. This variable has value 0 if plant p is not in use at time t or if plant p is in use at time t and plant p was in use during earlier relevant time steps. Hotstart(p,t) is a binary variable that has value 1 if plant p is in use at time t and was in use during earlier relevant time steps, but was not in use at time t – 1. This variable has value 0 if plant p is not in use at time t or if plant p is in use at time t and plant p was also in use at time t – 1.

The constraints force the variable Start(p,t) to have value 1 if plant p is in use at time t, but was not in use at time t – 1 and value 0 if plant p is not in use at time t.

The last two variables are Relevanthotstart(p,t) and Relevantcoldstart(p,t). The first one is a binary variable used to determine if a hot start is allowed and the second one is a variable, which is used to determine if a cold start is allowed.

The next four constraints determine if there is a start if plant p is used at time t.

0 ≤ Start(p,t) ≤ 1 ∀ p, t,

Start(p,t) ≥ Inuseatanylevel(p,t) – Inuseatanylevel(p,t – 1) ∀ p, t, Start(p,t) ≤ 2 - Inuseatanylevel(p,t) – Inuseatanylevel(p,t – 1) ∀ p, t and

Start(p,t) ≤ Inuseatanylevel(p,t) + Inuseatanylevel(p,t – 1) ∀ p, t. At t=1 the variable Inuseatanylevel(p,0) will be replaced with the independent variable HistInuseatanylevel(p,0).

The following constraints determine the values of Relevanthotstart(p,t) and Relevantcoldstart(p,t) for each plant p and each time t.

Relevanthotstart(p,t) ≥ ∑tt | Relevanthotstartperiod(p,t,tt) Inuseatanylevel(p,tt)

/ ∑tt | Relevanthotstartperiod(p,t,tt) Relevanthotstartperiod(p,t,tt) ∀ p, t,

Relevanthotstart(p,t) ≤ ∑tt | Relevanthotstartperiod(p,t,tt) Inuseatanylevel(p,tt) ∀ p, t,

0 ≤ Relevantcoldstart (p,t) ≤ 1 ∀ p, t,

Relevantcoldstart(p,t) ≥ ∑tt | Relevantcoldstartperiod(p,t,tt) Inuseatanylevel(p,tt)

/ ∑tt | Relevantcoldstartperiod(p,t,tt) Relevantcoldstartperiod(p,t,tt) ∀ p, t,

Relevantcoldstart(p,t) ≤ ∑tt | Relevantcoldstartperiod(p,t,tt) Inuseatanylevel(p,tt) ∀ p, t

and

Relevantcoldstart(p,t) – Relevanthotstart(p,t) ≥ 0 ∀ p, t.

In these constraints "tt | Relevanthotstartperiod(p,t,tt)" means look only at those tt for which Relevanthotstartperiod(p,t,tt) has value 1. For Relevantcoldstartperiod a similar definition applies. When t<1 the variable Inuseatanylevel(p,t) will be replaced with the independent variable HistInuseatanylevel(p,ht).

The last five constraints determine the values of the variables Start(p,t), Hotstart(p,t) and Coldstart(p,t). It is possible that according to these constraints a start is not allowed, while according to the first three constraints with Inuseatanylevel(p,t) a start is necessary. If this happens the constraints will force Inuseatanylevel(p,t) or Inuseatanylevel(p,t – 1) to take a different value. The constraints are

Coldstart(p,t) + Hotstart(p,t) – Start(p,t) = 0 ∀ p, t,

Start(p,t) ≤ 1 + Relevanthotstart(p,t) – Relevantcoldstart(p,t) ∀ p, t,

Hotstart(p,t) ≥ Relevanthotstart(p,t) / 2 + Start(p,t) – 1 ∀ p, t,

Hotstart(p,t) ≤ Relevanthotstart(p,t) + Relevantcoldstart(p,t) ∀ p, t and

Coldstart(p,t) ≤ 1 – Relevantcoldstart(p,t) / 2 ∀ p, t.

5.3.5. Operation & maintenance

The only variable in this category is VarOMcosts(p,t), which is the variable operation & maintenance costs for plant p at time t. This variable is in € and has as definition

VarOMcosts(p,t) := PlantvarOM(p) * Dispatchlevel(p,t) ∀ p, t.

5.4. The objective

The objective is minimizing the total variable costs, Totalvarcosts, in €. The definition of Totalvarcosts is

Totalvarcosts := ∑t ∑p[Fuelcosts(p,t) + Rampingcosts(p,t)

+ Startupcosts(p,t) + VarOMcosts(p,t)].

5.5. Model size

This model contains 258,217 variables from which 139,776 are binary. The total number of constraints is 301,904. There are 457,896 independent variables and 2 parameters used in this model.

6. The solver

I build this model in the programming language AIMMS. The letters AIMMS stand for Advanced Interactive Mathematical Modelling Software. One of the advantages of using an algebraic modelling language like AIMMS over using spreadsheets or manual coding is, that you only have to focus on specifying your application. There is no need for worrying about the implementation of your model. AIMMS will translate the algebraic presentation into a matrix representation, which is passed to a solver [7]. Through AIMMS solvers like CPLEX and CONOPT are accessible for solving all kinds of optimisation models.

The solver I used for solving this model is CPLEX 7.0 Mixed Integer Optimiser. CPLEX uses branch and bound techniques when solving the model. The default setting is to let CPLEX choose the techniques that should be used [8]. Although it is possible to choose which techniques I want C-PLEX to use, the solver time hardly improves when I do so.

The techniques that are used in solving this model are GUB (Generalized Upper Bound) cover cuts, Cover cuts, Flow cuts and Gomory fractional cuts. A Gomory cut is defined as a linear constraint which has the property that it is strictly stronger than its parent. A Gomory cut does not exclude any feasible integer solution of the LP problem under consideration [9]. For a long time Gomory cuts were seen as useless from a practical point of view, but with the current software available, these are some of the most powerful cuts available. This is shown by Bixby, Fenelon, Gu, Rothberg and Wunderling [10].

The time that is needed to solve 1 run of the model, that is to solve one week, is approximately 35 minutes. The memory that by CPLEX 7.0 is used during while running is 28 MB.

7. The results

For presentation purposes I defined 6 plant categories. These are:

o Must run (all plants that must run at a minimum level all the time, fuelled with gas or blast-furnace gas and including renewable capacity)

o Uranium (the nuclear plant)

o Import auction (auctioned import capacity) o Coal (all plants that are fuelled with coal) o Import contracts (contracted import capacity)

o Gas (all plants that are fuelled with gas, but don’t have a must run status)

7.2. Expected results

To give a presentation of the results I expect I looked at 3 scenarios.

By definition the must run plants are used, these plants can be used on the minimum level at all times. The uranium-fired plant will stay at maximum all the time, since this is the cheapest fuel. Although just based on fuel costs and variable operation and maintenance costs the next categories would be contracted import capacity and auctioned import capacity, I expect that the coal fired plants will stay on a minimum level all the time, because of the start and stop costs for this category.

In the first scenario I looked at the average volume in each time block. The period that has the lowest demand is the night. In addition to the minimum level of must run plants and the maximum level of nuclear power, the maximum level of coal and part of the contracted import capacity is used during this period. For the Sunday peak, the rest of the contracted import and part of the auctioned import will be added to this. The Saturday peak, the Shoulders and the evenings show an increase in volume for must run plants, or the cheapest gas fired plants. Finally in the Superpeak more volume comes from the gas fired plants.

0 1,000 2,000 3,000 4,000 5,000 6,000 0 12 0 12 0 12 0 12 0 12 0 12 0 12 Hours Monday - Sunday MW 0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 16,000 MW D e m a n d Must run Uranium Coal Import auction Import contracts Gas Demand

Figure 9. Expected results: average volume.

The second scenario that I looked at was the minimum volume in each time block. Again the period that has the lowest demand is the night. Part of the coal fired plants is used in addition to the must run plants and the nuclear plant. For the other periods the coal-fired plants run at maximum, import contracts run at maximum and a part of the auctioned import will be used. The third scenario was the maximum volume in each time block. In this scenario the period with the lowest volume is Sunday peak. Here coal plants are used at maximum and all import volume is used. The must run plants stay at minimum level and gas-fired plants are not used. For all other periods, the must run plants dispatch at a higher level and / or gas-fired plants will be used.

7.3. Realised results

I will compare the realised results of the three scenarios outlined above. Contrary to expectations, the must run plants are used at a higher level than minimum at all times. This can be explained by the relatively low costs of the blast-furnace gas plants. Another explanation can be that the efficiency improves when plants are dispatched at a higher level which can make it more attractive for a must run plant to be dispatched at a higher level compared to another plant to be dispatched at a low level. Further the uranium-fired plant is used at maximum level at all times.

In the first scenario - average volume in each time block - during the night all fuels are used in access of a minimum required level. There are a few reasons for this. At first, if gas plants that need to be used in the morning have a long start up period, they already start during the night. Second, part of the auctioned import is cheaper than the contracted import; therefore this will be

used instead of total contracted import capacity. Further the most expensive coal plants will run at a lower level to make way for cheaper fuels.

During the weekend the must run plants go to a higher level, like coal and auctioned import. Contracted import is at maximum level during the weekend. The Shoulders show in addition to this some usage of gas-fired plants, again because of power needed in the Superpeak. Finally Superpeak shows an increase in must run volume and gas-fired volume. Both coal-fired plants and imported auction capacity almost reach their maximum.

0 1,000 2,000 3,000 4,000 5,000 6,000 0 12 0 12 0 12 0 12 0 12 0 12 21 11 Hours Monday - Sunday MW 0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 16,000 MW D e m a n d Must run Uranium Coal Import auction Import contracts Gas Demand

Figure 10. Realised results: average volume.

The second scenario, i.e. the minimum level, shows that during some hours coal-fired plants will shut down, while contracted import is still used at a low level. The auctioned import is in this scenario always at approximately half of the possible volume. The shoulders show a slight increase in import volume and some coal-fired plants are not in use. During the rest of the hours contracted import goes to maximum and the coal-fired plants are in use, but not at maximum. The third scenario, i.e. the maximum level in each time block, shows a maximum usage of all import and coal-fired plants. Further must run plants go to a higher level and during the weekdays and nights gas-fired plants are used.

8. Sensitivity analysis & scenario analysis

Although sensitivity analysis on linear programming models is quite easy using shadow prices, it is more difficult to perform sensitivity analysis on mixed integer programming models. The fact that the integers in this model are binary makes it even more complex. For example while the value of an integer change from 1000 to 999 does not always have a huge effect on the outcome of a model, the effect of changing from 0 to 1 means that the outcome will be completely different.

In linear programming models the right hand side of the dual shows how sensitive the solution of the primal model is to perturbations of the right-hand side of the constraints. This kind of duality is a special case of inference duality. Inference duality can be used as a general approach for sensitivity analysis.

Dawande and Hooker [10] presented a method for sensitivity analysis for mixed integer linear programming based on the idea of inference duality. In their paper they show that any perturbation that satisfies a certain system of linear inequalities will reduce the optimal value by no more than a pre-specified amount. The sensitivity analysis which is based on inference duality consists of two parts: dual analysis and primal analysis. The dual analysis determines how much the problem can be perturbed, while keeping the value of the objective function within a certain range. The primal analysis shows an upper bound on the value of the objective function, when the problem is perturbed by a certain amount.

To obtain the dual solution inference methods are used to generate constraints at every node that is violated by the branching cuts. This dual solution can be seen as proof of the optimality of the primal solution.

Unfortunately this model is too big for performing a sensitivity analysis as just described. Another problem is that most parameters in this model are fixed, because they are based on the technical aspects of the plants. The results of a sensitivity analysis wouldn’t therefore be very useful. Instead of a sensitivity analysis, I chose to do a scenario analysis. I looked at what changes in input data would logically have an influence on the results of the model.

8.2. Scenario analysis

I have analysed 4 scenarios. The first scenario that I have choosen is to remove the decentralized production capacity. I selected this scenario since it is not unlikely that if the electricity demand of the owners of this capacity grows, that less capacity becomes available for general usage.

In the second scenario I lowered the gas prices. This scenario shows the effect of the fuel prices on the capacity that is used. This is relevant because in the last years we have seen that there is a high volatility in fuel prices.

The third scenario shows the effect of the growing demand in combination with the current group of plants. I chose this scenario since in practice we see a steady growth in the demand in the Netherlands.

In the last scenario I removed the nuclear plant. Although this scenario might seem unrealistic, there are discussions on a political level about when the nuclear plant has to be closed.

8.2.1. The results without decentralised production capacity

Most of the decentralised capacity has a must run status. Part of the volume that this capacity produces is used for own supply of the owners of the capacity. The rest is used for supplying normal demand. Because of this, the demand in this scenario is lower, but not by as much as total supply.

The differences with the original results are that where coal-fired plants or auctioned import were contributing a small volume, coal-fired plants will produce more or more volume is imported through auctions in these hours. In all other hours the gas-fired plants will produce more.

-3,000 -2,500 -2,000 -1,500 -1,000 -500 0 500 1,000 1,500 2,000 0 12 0 12 0 12 0 12 0 12 0 12 0 12 Hours Monday - Sunday MW Demand Must run Uranium Coal Import auction Import contracts Gas

Figure 11. Results of the model without decentralized capacity.

8.2.2. The results with low gas prices

The gas prices in this run are about one third of the gas prices in the original model. This implies that about 1810 MW of gas-fired plants and 1760 MW of must run plants are cheaper to use than coal-fired plants. Also a big part of the auctioned import capacity is more expensive than these gas-fired and must run plants in this scenario. This results in a shift from auctioned import capacity and coal-fired capacity towards gas-fired capacity and must run capacity.

-3,000 -2,000 -1,000 0 1,000 2,000 3,000 0 12 0 12 0 12 0 12 0 12 0 12 0 12 Hours Monday - Sunday MW Must run Uranium Coal Import auction Import contracts Gas Demand

Figure 12. Results of the model with low gas prices.

8.2.3. The results with high demand

The demand in this run is 20% higher than in the original model. During the nights the extra volume is produced with coal fired plants and with a small part of the must run plants. These are the cheapest must run plants. They are fired with blast furnace gas or with a combination of gas and blast furnace gas.

During the weekend days the auctioned import volume goes to maximum. Furthermore during all days, both gas fired plants and must run plants go to a higher level.

-500 0 500 1,000 1,500 2,000 2,500 0 12 0 12 0 12 0 12 0 12 0 12 0 12 Hours Monday - Sunday MW -750 0 750 1,500 2,250 3,000 3,750 4,500 MW D e m a n d Must run Uranium Coal Import auction Import contracts Gas Demand

Figure 13. Results of the model with high demand.

8.2.4. The results without the nuclear plant

Without the nuclear plant more volume is produced with coal and / or auctioned import. At times where there is no auctioned import available, the rest of the volume is produced by the must run plants. -1,500 -1,000 -500 0 500 1,000 1,500 0 12 0 12 0 12 0 12 0 12 0 12 21 11 Hours Monday - Sunday MW Must run Uranium Coal Import auction Import contracts Gas Demand

Figure 14. Results of the model without the nuclear plant.

9. Conclusions and recommendations

This model gives an idea of the expected hourly costs and therefore of the expected APX prices. My analysis shows that the availability of plants and the fuel costs can have a lot of influence on which plant is marginal and on the expected hourly prices.

The definition of marginal costs in this model is not including the start- and stop-costs. These costs are taken into account for determining if a plant is running, but not for determining the marginal plant. A request from the future users of this model is to include these costs in the marginal costs. This means that the total start-costs for a certain plant need to be spread out over the period that this plant is running.

Another thing that might be worthwhile to add is transport costs. This is only relevant when coal is used as a fuel. It depends on the coal prices compared with the other fuel prices wether this will have impact on the results of the model.

References

[1] ECN (2001), Energie Markt Trends 2001, ECN-P—01-009, pp. 8-9. [2] ECN (2001), Energie Markt Trends 2001, ECN-P—01-009, pp. 28.

[3] Energieonderzoek Centrum Nederland, Beleidsstudies; Energie in Nederland

(www.energie.nl); Energie in Cijfers.

[4] ECN (2001), Energie Markt Trends 2001, ECN-P—01-009, pp. 31. [5] ECN (2001), Energie Markt Trends 2001, ECN-P—01-009, pp. 32. [6] Tennet: www.tennet.org. [7] NEOS Guide: AIMMS http://www-fp.mcs.anl.gov/otc/Guide/SoftwareGuide/Blurbs/aimms.html. [8] NEOS Guide: CPLEX http://www-fp.mcs.anl.gov/otc/Guide/SoftwareGuide/Blurbs/cplex.html. [9] http://www.ms.unimelb.edu.au/~moshe/620-362/gomory/index.html.

[10] R.E. Bixby, M. Fenelon, Z. Gu, E. Rothberg, R. Wunderling, MIP : Theory and practice – Closing the gap.

[11] M.W. Dawande and J.N. Hooker (1996, revised 1998), Inference-Based Sensitivity Analysis for Mixed Integer/Linear Programming.