Optimal Production Planning for Small-Scale Hydropower

IN

DEGREE PROJECT ELECTRICAL ENGINEERING, SECOND CYCLE, 30 CREDITS

,

STOCKHOLM SWEDEN 2018

Optimal Production Planning

for Small-Scale Hydropower

ANNA-LINNEA TOWLE

KTH ROYAL INSTITUTE OF TECHNOLOGY

OPTIMAL PRODUCTION

PLANNING FOR

SMALL-SCALE HYDROPOWER

Anna-Linnea Towle

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Abstract

As more and more renewable energy sources like wind and solar power are added to the electric grid, reliable sources of power like hydropower become more important. Hydropower is

abundant in Scandinavia, and helps to maintain a stable and reliable grid with added irregularities from wind and solar power, as well as more fluctuations in demand. Aside from the reliability aspect of hydropower, power producers want to maximize their profit from sold electricity. In Sweden, power is bid to the spot market at Nord Pool every day, and a final spot price is decided within the electricity market. There is a different electricity price each hour of the day, so it is more profitable to generate power during some hours than others.

There are many other factors that can change when it is most profitable for a hydropower plant to operate, like how much local inflow of water there is. Hydropower production is an ideal case for using optimisation models, and they are widely used throughout industry already. Though the optimisation calculations are done by a computer, there is a lot of manual work from the spot traders that goes into specifying the inputs to the model, such as local inflow, price forecasts, and perhaps most importantly, market strategy. Due to the large amount of work that needs to be done for each hydropower plant, many of the smaller power plants are not optimised at all, but are left to run on an automatic control that typically tries to maintain a constant water level. In Fortum, this is called, VNR, or vattennivåreglering (water level regulation).

The purpose of this thesis is to develop an optimisation algorithm for a small hydropower plant, using Fortum owned and operated Båthusströmmen as a test case. An optimisation model is built in Fortum’s current modelling system and is tested for 2016. In addition, a mathematical model is also built and tested using GAMS. It is found that by optimising the plant instead of running it on VNR, an increase of about 15-16% in profit could be seen for the year 2016. This is a significant improvement, and is a strong motivator to being optimising the small hydropower plants.

Since the main reason many small hydropower plants are not optimised is because it takes too much of employees time, a second phase of this thesis was conducted in conjunction with two other students, Jenny Möller and Johan Wiklund. The focus of this portion was to develop a centralized controller to automatically optimise the production schedule and communicate with the central database. This would completely remove the workload from the spot traders, as well as increase the overall profit of the plant. This thesis describes the results from both theFortum model and the GAMS model, as well as the mathematical formulation of the GAMS model. The basic structure of the automatic controller is also presented, and more can be read in the thesis by Möller and Wiklund (Möller & Wiklund, 2018).

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Sammanfattning

Tillförlitliga energikällor som vattenkraft blir allt viktigare vart eftersom elkraftsystemet utökas med fler förnybara energikällor som vindkraft och solenergi. I Norden finns det rikligt med vattenkraft, vilket bidrar till att upprätthålla ett stabilt och pålitligt elnät även med ökade oregelbundenheter från vindkraft och solkraft samt större variationer i efterfrågan. Bortsett från vattenkraftens tillförlitlighetsaspekter vill kraftproducenter maximera sin vinst från såld el. I Sverige läggs dagligen bud på effektvolym till spotmarknaden Nord Pool och ett slutgiltigt marknadspris bestäms därefter av elmarknaden. Varje timme under dygnet motsvarar ett enskilt elpris, därmed är det mer lönsamt att generera effekt under de timmar där priset är som högst. Det finns många andra faktorer som påverkar när det är mest lönsamt för ett vattenkraftverk att producera el, exempelvis hur stort det lokala inflödet av vatten är. Vattenkraftproduktion är idealt för tillämpning av optimeringsmodeller, vilka är vanligt förekommande inom verksamhetsområdet. Även om optimeringsberäkningarna utförs av en dator innebär optimeringen mycket manuellt arbete för Fortums elhandlare som specificerar indata till modellen. Exempel på indata är lokalt inflöde, prisprognoser och kanske viktigast av allt marknadsstrategi. På grund av den stora mängden arbete som fordras för varje vattenkraftverk, optimeras inte produktionen för många av de småskaliga kraftverken utan de regleras automatiskt med mål att upprätthålla en konstant vattennivå. Denna typ av reglering kallas vattennivåreglering, VNR.

Syftet med examensarbetet var att utveckla en optimeringsalgoritm för ett småskaligt vattenkraftverk, där Fortumägda vattenkraftverket Båthusströmmen används som testobjekt. En optimeringsmodell utvecklades i Fortums befintliga system och testades för 2016. Dessutom har en matematisk modell utvecklats och testades med GAMS. Det konstaterades att genom att optimera produktionen från vattenkraftverket istället för att reglera den via VNR kan en vinstökning med cirka 15-16 % för noteras år 2016. Detta är en väsentlig förbättring och är ett starkt argument för att optimera produktionen från småskaliga vattenkraftverk.

Eftersom den främsta orsaken till att många småskaliga vattenkraftverk inte optimeras är den utökade arbetsbelastningen det skulle innebära för de anställda, genomfördes en andra fas i examensarbetet i samverkan med två andra studenter, Jenny Möller och Johan Wiklund. Fokus för denna del var att utveckla en centraliserad styrenhet för att automatiskt optimera produktionsplaner och kommunicera med det befintliga centrala systemet. Detta innebär att utökad arbetsbelastningen från elhandlarna undviks, samt öka vattenkraftverkets totala vinst. Denna rapport beskriver resultaten från både Fortum-modellen och GAMS-modellen, liksom den matematiska formuleringen av GAMS-modellen. Även grundstrukturen för det självreglerande optimeringsverktyget presenteras, mer kan läsas i rapporten av Möller och Wiklund (Möller & Wiklund, 2018).

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Acknowledgements

There are many people who I wish to thank for their help with this thesis.

I want to sincerely thank my suprvisors at Fortum, Zahra Faridoon and Hans Bjerhag, as well as Erik Byström, for their continuous help and guidance throughout this project. Thank you also to my intructor, Mikael Amelin, and supervisor, Meng Song, from KTH for their valuable support and assistance. I am deeply grateful to the staff at Fortum for their help and advice from the very beginning, and their willingness to teach me as much as I could learn.

Lastly, thank you to Jenny Möller and Johan Wiklund, with whom I worked on the second portion of this thesis. It was a great collaboration, and one that I think improved the quality of both of our works.

Anna-Linnea Towle Stockholm

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T

ABLE OF

C

ONTENTS

1.2 Focus and Assumptions ... 7

1.3 Research Objectives ... 8

2 Electricity Markets ... 9

2.1 Nordic Electricity Market ... 9

2.2 Supply and Demand ... 11

2.3 Spot Market ... 13

2.4 System Spot Price ... 14

2.5 Area Spot Price ... 15

3 Hydropower System Characteristics ... 16

3.1 Hydropower Operation ... 16

3.2 Reservoir Characteristics ... 16

3.3 Structure of a Hydropower System ... 17

4 Hydropower Production planning ... 19

4.1 Planning Concepts ... 19

4.2 Steering from Mid-term Planning ... 19

4.3 Volume Coupling ... 19

4.4 Resource Cost Coupling ... 20

4.5 Short-term Planning ... 21

4.5.1 Pre-Spot Planning ... 22

4.5.2 Post-Spot Planning ... 22

4.6 Hydropower Modelling Theory ... 22

4.6.1 Linear Programming ... 22

4.6.2 Non-linear Programming ... 23

4.6.3 Dynamic Programming ... 23

4.6.4 Stochastic Programming ... 23

5 Hydropower Planning Model: Fortum System ... 25

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5.2 Optimal Planning vs VNR Control Results ... 26

6 Hydropower Planning Model: Theory ... 32

6.1 Nomenclature ... 33

6.2 Explanation of Flexibility of Model ... 34

6.3 Static Model ... 34

6.4 Iterative Updates Based on New Head ... 39

7 Optimal Planning Algorithm Results ... 40

8 Central Automatic Control ... 45

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L

IST OF

F

IGURES

Figure 1 Nordel electricity consumption by sector, 2008, source: (ENTSO-E, 2018) ... 11

Figure 2 Nordel electricity generation, 2008, source: (ENTSO-E, 2018) ... 12

Figure 3 Production cost curve, source: (Nord Pool, 2017) ... 13

Figure 4 Nordic power flow, source: (Statnett, 2018) ... 14

Figure 5 Area price curves in a two area market, source: (Kerola, 2006) ... 15

Figure 6 Hydropower plant operation, source: (Vattenfall, 2017) ... 16

Figure 7 Example hydropower system, source: (Hassis, 2011) ... 18

Figure 8 Water value function, source: (Hassis, 2011) ... 21

Figure 9 Båthusströmmen Location (Google Maps, 2018) ... 25

Figure 10 Marginal Water Value Curves for Båthusströmmen ... 27

Figure 11 2016 Week 1 Båthusströmmen Reservoir Level ... 28

Figure 12 2016 Week 23 Båthusströmmen Reservoir Level ... 29

Figure 13 2016 Week 40 Båthusströmmen Reservoir Level ... 30

Figure 14 Optimal Planning Algorithm ... 32

Figure 15 Piece-wise linear power curve ... 35

Figure 16 PWL example curve ... 36

Figure 17 Week 1 Reservoir level and flow rate ... 40

Figure 18 Week 1 Reservoir level and electricity price ... 41

Figure 19 Week 23 Reservoir level and flow rate ... 41

Figure 20 Week 23 Reservoir level and electricity price ... 42

Figure 21 Week 40 Reservoir level and flow rate ... 42

Figure 22 Week 40 Reservoir level and electricity price ... 43

Figure 23 Inflow and spillage for 2016 ... 43

Figure 24 Optimised and VNR discharge comparison ... 44

Figure 25 Information Flow of Central Control Unit ... 46

Figure 26 Visual Basic Main Program (Möller & Wiklund, 2018) ... 47

L

IST OF

T

ABLES

Table 2 Algorithm Nomenclature ... 33

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1 I

NTRODUCTION

1.1 B

Hydropower has been an important source of power in Sweden for centuries. Starting as early as the 12th century, water power was used in sawmills, to mill grains, and to transport logs along rivers. The industrial revolution (1871-1914) saw hydropower being used to supply electricity for the numerous factories that had sprung up across Sweden, and the electrification of the railroads motivated the building of the first large scale hydropower plants (Flood, 2015). Now hydropower makes up about 41% of Sweden’s total generation capacity of almost 40000MW (ENTSO-E, 2018). Smaller hydropower plants, under 10MW, are often run-of-the-river plants, and aren’t controlled as much as the larger plants. They are usually left to run on their own based on the inflow of the river and the height of their own reservoirs. However, about 1GW of capacity in Sweden comes from these small hydropower plants, and control and optimisation of them could have a significant influence on the total productivity of the system (ENTSO-E, 2018). As more and more variable renewable power sources like wind and solar are installed, the more need there is for stable and predictable power to balance the electricity supply. Hydropower, a renewable, fossil-fuel free power source with stable production, is the ideal solution to this growing problem.

1.2 F

A

This thesis focuses on the optimisation and control of small hydropower plants, done in

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Optimisation for hydropower dispatch is quite common, and is used for the larger plants that Fortum owns. Today, running an optimization algorithm for small plants takes too much time and effort from the employees, who could instead focus on larger, more profitable plants. To get the best of both worlds, an optimization algorithm is developed for Båthusströmmen, and a control scheme to automatically run the optimization and send the chosen schedule to the central database. The automatic control of Båthusströmmen is investigated and simulated using Microsoft Visual Basic. Communication with the central database is established, and a control scheme is developed to automatically run the optimisation algorithm and update the database with new optimal schedules. This portion of the project is done in conjunction with two undergraduate thesis students, Jenny Möller and Johan Wiklund (Möller & Wiklund, 2018).

1.3 R

O

There are two main objectives for this thesis. The first is to develop an optimal planning

algorithm that can be continuously run to plan the production at Båthusströmmen. This algorithm will be based on similar algorithms used for general hydropower modelling, and will be able to run at any time taking in updated inputs like spot price and local water inflow. Different methods for optimisation are analysed, and a final method is chosen based on accuracy, flexibility, and run time.

The second objective is to develop a control scheme to automatically control the hydropower plant based on the optimal planning. The control scheme will then be implemented and tested. This phase of the project is done in cooperation with two other thesis students.

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2 E

LECTRICITY

M

ARKETS

2.1 N

E

M

In 1991 Norway deregulated its electricity market, causing the power providers to act more competitively and be more profit driven. A deregulated market ensures that the cheaper

generation options are used first, and the more expensive ones used only when necessary. It also allows consumers to choose who they buy electricity from, promoting competition and

encouraging profitable investments (Rothwell & Gomez, 2003). In 1996, Sweden joined this deregulated market and formed Nord Pool. Today, 380 customers from 20 different countries trade in Nord Pool markets, generating around 420 TWh every year from the Nordic and Baltic countries (Nord Pool, 2017). Within a power system, there are several key players: producers, consumers, retailers, system operators, grid owners, and balance responsible players. These will be explained briefly (Söder & Amelin, 2011).

Producers and Consumers

Producers are the owners and operators of power plants and consumers are the end users of electricity. While the large economies of scale of electricity generation has resulted in very few, but large, producers, consumers operate all throughout the power system, and are much more numerous (Söder & Amelin, 2011). Consumers can be large industry or individual households, with variable and sometimes difficult to predict consumption patterns.

Retailers

It would be very complicated for each individual consumer to purchase their electricity directly from the producers, so retailers are available to act as middle men between the two. Retailers can sell electricity to consumers in many ways, including offering a simple fixed price rather than a price that varies throughout the day. This simplifies consumer sales, and places the risk of a dramatic price change on retailers. Having many retailers also increases competition, both between retailers and between producers (Söder & Amelin, 2011).

Grid Owners

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companies or municipal authorities control over an area of the same grid instead of having several competing grids. Grid owners must operate and maintain the grid with an agreed upon power quality, as well as measure the transfer of electricity from producers to consumers. Grid owners may have to buy power to make up for electric losses in the grid. To cover this cost as well as maintenance costs, they are allowed to charge grid tariffs to all users (Söder & Amelin, 2011).

System Operators

With so many players already in the market, it is best to have one player that oversees the total day to day operation. System operators, ISOs1 or TSOs2, are administrators of the power system and electricity trading. TSOs have the ultimate responsibility for maintaining the power balance of the grid at all time. This involves maintaining a constant frequency of 50Hz and also voltage and MVAr properties (Byström, 2018).

This means that system operators are usually responsible for post trading (see section Error!

Reference source not found.), and they are often also transmission grid owners (Söder &

Amelin, 2011). An electricity market, like Nord Pool, can have multiple system operators, each of which controls a certain area. The system operator for Sweden is Svenska Kraftnät.

Balance Responsible Players

Every watt of produced power must be consumed immediately. Though companies promise to produce a certain amount of power in advance (see section 2.3), there will always be small deviations from the plan due to unpredictable things such as changes in consumption or loss of production in a plant. When such deviations occur, producers have to be compensated for generating more power or charged for not producing enough. Balance responsible players make sure that this balance is maintained, and that all energy is correctly paid for. Likewise, there are balance responsible players on the demand side accounting for any changes from planed

consumption. Electricity retailers are commonly balance responsible players acting on behalf of their customers (Söder & Amelin, 2011).

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2.2 S

D

The driving force behind electricity production and electricity cost is the balance between supply and demand. The instant a light bulb is turned on, the total production throughout the system must increase slightly to accommodate it. Electricity consumption, or demand, is usually quite independent of the price. Demand fluctuation does, however, follow several patterns, including daily and seasonal patterns. Households and industry both consume more electricity during the morning than during the middle of the night. Industry especially has a larger consumption during the week than on the weekends, as well as reduced consumption on holidays (Hassis, 2011). The consumption also changes with temperature, usually increasing during colder temperatures to supply electric heating. The breakdown of consumption in the Nord Pool area on 2008 is shown in Figure 1 (ENTSO-E, 2018). Industry makes up about half of the consumption, influencing the weekday pattern for overall consumption. Household consumption makes up over one quarter of total consumption, contributing to the daily consumption pattern.

Figure 1 Nordel electricity consumption by sector, 2008, source: (ENTSO-E, 2018)

Hydropower is the largest source of electricity in Nord Pool, accounting for 58% of production in 2008 (ENTSO-E, 2018), as shown in Figure 2. It is also the most flexible, as there are few start-up or ramping costs for the plants. In addition, it is one of few power sources that can store

28%

47% 22%

3%

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energy, in the form of reservoir water, for the future. There are many factors that affect

hydropower supply, including inflow, profit maximisation plans, reservoir levels, etc. Since the fuel for hydropower is water, it is essentially free. This means that hydropower can make a profit even at low electricity prices, and has a large influence on the spot price of the electricity market. This will be explained further in section 2.3.

Figure 2 Nordel electricity generation, 2008, source: (ENTSO-E, 2018)

Hydropower has high seasonal variations due to what is called the “spring flood”. This is when all the snow melts and fills the rivers and reservoirs during the spring. This is an annual, large event, and is prepared for by having low reservoir levels going into the spring. This means that production increases during the winter and autumn as the reservoirs are emptied.

Nuclear power is the second largest resource, making up 20% of generation in 2008. Nuclear power has high investment costs, but low variable costs, so producers want to always operate at high levels (Hassis, 2011). Since nuclear and hydropower have such low operating costs and are very predictable sources of power, they usually supply most of the load, together making up around 80% of total generation.

The “other thermal” section is mainly coal and oil condensing, which have much higher variable costs. Thermal plants have start-up and sometimes shut-down costs to operate, and the fuel itself

58% 20%

19%

3%

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is not free like with hydropower. In addition, there are costs based on CO2 emissions, driving the

variable costs higher. Therefore, production is highly dependent on the spot price, and the spot price is also highly dependent on how much thermal power is needed to meet the demand.

The electricity price is calculated based on how supply meets demand. An example of supply and demand curves is shown in Figure 3 (Nord Pool, 2017), where the demand is the dashed line and the supply curve is made up of the available generation blocks. Here it is clear that hydro power makes up a majority of the production, and keeps the price low. This price calculation will be explained further in the following sections.

Figure 3 Production cost curve, source: (Nord Pool, 2017)

2.3 S

M

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demand curve shown in Figure 3, using predicted consumptions and demand bids for the demand and the production bids for the supply.

2.4 S

S

P

Within the Nord Pool market there are different bidding areas. In Sweden there are four areas, SE1-4, each of which submits its own spot bid to the spot market.

Figure 4 Nordic power flow, source: (Statnett, 2018)

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prices. Figure 4 shows the different areas in Nord Pool and an example of power flow between areas.

2.5 A

S

P

When transmission limits are reached, individual areas need to adjust their production to match the area consumption, which results in an adjusted supply and demand curve. Figure 5 shows the price curves for a two-area market where the transmission limits between the areas A and B have been reached. Area A has much more demand than it can supply at the system price, so it must import as much as possible from area B according to transmission capacity. The area price is set to be where the difference between supply and demand is equal to the transmission capacity from area B, resulting in an area price higher than the system price. Similarly, area B now has excess power capacity. The area price is set to be where the difference between supply and demand is equal to the export to area A, resulting in an area price lower than the system price. Note that power always flows from lower price areas to higher price areas in order to generate as much power as possible at lower prices (Kerola, 2006).

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3 H

YDROPOWER

S

YSTEM

C

HARACTERISTICS

3.1 H

O

Hydropower plants use running water to generate electricity. Figure 6 shows how a basic

hydropower plant is set up. A dam holds back water in a reservoir, making the upstream, or head water level, higher than the downstream, or tail water level. This difference in height is called head. The larger the head, the larger the potential energy in the water due to gravity, and the more energy can be extracted by the turbine. The water flows through the intake rack and penstock and turns the turbine, then flows out through the draft tube. The turbine is connected to the electrical generator, which generates power and injects it into the local grid system. The amount of water flowing through the turbine, and the rate of electric power (MW) production is controlled by adjusting the guide vanes (Byström, 2018). The intake gate is fully open during operation , and only closed during emergency-stop or for maintenance reason (Bjerhag, 2018).

Figure 6 Hydropower plant operation, source: (Vattenfall, 2017)

3.2 R

C

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surrounding ecosystems, and these limits must be obeyed. Other limits include practical limits on how much the reservoir can physically handle, and also must be obeyed so as not to damage the reservoir or plant. Since these rules must be followed, these are often called hard limits. There are other tactical limits, often called soft limits or good will limits, that further constrain the water levels in order to build in a margin of safety. If an environmental limit is exceeded, there is a penalty, so producers don’t want to risk being too close to these limits in case something like a large unexpected inflow occurs.

There can also be environmental or tactical limits on flow rate and ramping rate. For example, a plant may be required to have a minimum flow rate in order for fish to continue moving

throughout the river. A producer may set their own flow rate limits to operate the generators most efficiently, or they may set ramping limits (how fast the flow rate can change) to protect different components. All of these limits can change throughout a year.

3.3 S

H

S

Hydropower plants are based on river systems and reservoirs. Unlike most electricity production types, hydropower is able to store energy by keeping water in reservoirs just upstream of the plants. This water is available, with certain limitations, to be used by the plant when it will be most profitable.

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Figure 7 Example hydropower system, source: (Hassis, 2011)

Figure 7 shows an example river system with three reservoirs R1-3, plants P1-3, spillage s1-3, and

local inflow v1-3. The discharge is indicated by q1-3. The three plants are connected, and water

from plants and reservoirs 1 and 2 go to reservoir 3. The water level in R3 is increasing all the

time by v3, q1, s1, q2, and s2. It is decreasing by s3 and q3.

An important thing to consider is that water discharged from plant 1 does not immediately arrive at reservoir 3, it takes some time to travel there through the river. This time delay is different between each plant, and the time delays of spilled water and discharged water are often very different. It is common for spilled water to take a much longer path, and thus take longer to reach the next plant. Downstream plants will thus not be affected by changes in discharge or spillage until sometime after the change occurs.

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4 H

YDROPOWER

P

RODUCTION PLANNING

4.1 P

C

As has been mentioned, hydropower has the ability to store energy in the form of reservoir waters. This allows a plant to save water for hours when the electricity price is highest. This planning is on the short-term horizon, from the day ahead to about 2-3 weeks ahead. Not only are the hours of the day considered, but also the inflow and price forecasts for the coming weeks. Short-term planning allows the river system to optimise production in its plants during the coming weeks.

Yearly inflow and price changes are important as well. Every year there is a spring flood, and the reservoirs must be relatively empty going into spring to accommodate this. Mid-term planning takes place between the end of the short-term planning until about 1-2 years out. When the mid-term plans are optimised, a start value for the reservoirs is found. This value is the end point for the short-term planning, and provides steering for the short-term planning.

Similarly, long-term planning provides steering for where the mid-term planning should end up at the end of its planning horizon. Long-term planning considers the time from the end of the mid-term planning to about 3-5 years out. The coupling between these different planning periods allows the models to be very specific in the short-term while still following an overall plan for the coming years. This thesis focuses on the short-term planning. More about the actual models will be discussed in the following sections.

4.2 S

M

-

P

As mentioned above, the mid-term planning period begins at the end of the short-term planning period, and determines the end point for the short-term planning. There are two main ways to provide this steering, or coupling, for the short-term optimisation: volume coupling and resource cost coupling (Hassis, 2011).

4.3 V

C

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reservoirs depends highly on the flow of upstream reservoirs, not just their volume. This is especially true because of the time delay between reservoirs. It is difficult to achieve consistent coupling between models without taking this time delay into account in the final hours (Hassis, 2011). Volume coupling also does not consider the marginal value of water, which changes how much the water is worth and may change the optimal plan.

4.4 R

C

C

Water stored in a reservoir has the potential to create power in the future. When there is a lot of water stored in a reservoir, adding a few more cubic meters does not increase the value of the water significantly, so we say the marginal value of the water is low. When a reservoir is

approaching its lower limit, a few extra cubic meters of water makes much more of a difference, so that water has a higher marginal value.

Resource cost coupling takes the marginal water value more into account. The mid-term planning instead passes an end marginal cost value for each reservoir to the short-term planning. Figure 8 shows an example of a water value function. Each reservoir in a river system has its own water value function. The future expected income increases with increased reservoir content, but the slope decreases, meaning additional water is less valuable as the total reservoir content increases. Note that the derivative of the water value function, or its slope, is the marginal water value function. Points A, B, and C show the slope, or the marginal water value, at three different points. At point A, the reservoir content is low, so adding more water increases the future income

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Figure 8 Water value function, source: (Hassis, 2011)

By requiring the water value function to be at a certain point at the end of the short-term planning period, the models can take price dependency better into account. Also, this improves the

flexibility of the model to adapt the resources to the inflow conditions and reduce the amount of start/stop cycles (Hassis, 2011).

Since different reservoirs have different volumes and flow rates, it is beneficial to use reservoir-specific water values (Hassis, 2011). This means assuming that marginal water values in a river system are independent of each other for each reservoir. A major drawback of this is that short-term optimisation is very sensitive to the relative difference between marginal water values and short-term spot price.

4.5 S

-

P

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4.5.1 Pre-Spot Planning

Every morning the short-term model is used to plan the production for the next day. The inflow and spot price forecasts, as well as the steering from the mid-term planning, are input into the model, and the bid for the following day is formed. This optimisation considers several different price scenarios and even different inflow scenarios, and the results are used to create a piece-wise bid curve.

4.5.2 Post-Spot Planning

Once the actual spot prices have been determined and released at 12:42 CET, the short-term model is run again. This time, the model has another constraint to generate exactly as much power as was sold on the spot market per hour. It does not have to be generated by the same plants as in the original optimisation in the pre-spot planning, so the plans can change in order to be the most beneficial. The results of this optimisation are the production plans for the following day, and are used by the dispatch center to actually run the plants and realize the production that was promised. .

4.6 H

M

T

There are many different methods for modelling hydropower systems, and the basic layouts have been discussed thoroughly in literature, for example in (Söder & Amelin, 2011). The following sections will introduce several mathematical methods for modelling.

4.6.1 Linear Programming

Linear programming is the simplest modelling method. The objective function and all constraints are linear. It can efficiently solve large-scale problems, and will converge to global optimums (Labadie, 2004). The most common linear programming method, simplex, is presented in (S. Nash, 1996). When non-linear functions are required, separable programming is used to replace them with piece-wise linear curves. Mixed Integer Linear Programming, MILP, allows the use of both continuous and integer variables like binaries (Hassis, 2011). MILP is often used to

represent hydropower efficiency curves or unit commitment (deciding to have a unit, or a

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4.6.2 Non-linear Programming

Non-linear programming is able to more accurately represent real-life systems like the efficiency curves of hydropower plants. The most robust methods are successive linear programming (SLP), successive quadratic programming (SQP), and the method of multipliers (MOM) (Labadie, 2004). All equations must be differentiable, which can sometimes be problematic and require creative solutions. SLP is the most efficient non-linear programming method (Hiew, 1987), but it does not always converge to a global optimum (Bazaraa, 1993). With SLP, the objective function is often quite flat near the optimum, and this point can change very slightly each iteration,

preventing the model from converging. A penalty can be applied to prevent the optimal value from changing too much between iterations, forcing it to converge (Belsnes, Wolfgang, Follestad, & Aasgård, 2015).

4.6.3 Dynamic Programming

Discrete dynamic programming breaks large problems down into sub-problems that can be solved sequentially each time period. It is often used when there are dynamic features like system control variables (discharges and spillages) and state variables (reservoir levels) (Hassis, 2011), and is often used when the state of the system depends on the previous system state and the time of occurrence. Dynamic programming is a flexible method and works with both convex and non-convex problems. However, as the number of state-variables increases, the calculation time grows exponentially as every discrete dimension is tested. This is known as the “curse of dimensionality” (Labadie, 2004). A different method, differential dynamic programming, was developed by (Jacobson & Mayne, 1970) to solve the dimensionality problems using analytical instead of discretized methods. Differential dynamic programming was applied to the Mad River System in northern California, and it was determined that the same problem would have taken 16 times longer in an LP formulation (Jones, Willis, & Finney, 1986).

4.6.4 Stochastic Programming

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problems. Stochastic programming allows uncertainties to be accounted for and recommends the best decisions to make based on these uncertainties. In hydropower planning, the main

uncertainties come from the inflow and price forecasts.

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5 H

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5.1 B

Båthusströmmen is a small3 3.3 MW hydropower plant in Dalälven connected to the Hösthån reservoir and is owned and operated by Fortum. It is the furthest upstream plant in its branch of Dalälven, meaning there are no other plants upstream of Båthusströmmen. The next closest downstream plant is Trängslet, one of the largest hydropower plants and reservoirs in Sweden.

Figure 9 Båthusströmmen Location (Google Maps, 2018)

Since Trängslet is so much larger than Båthusströmmen, 300 MW compared to 3.3 MW,

anything that Båthusströmmen does will not have any significant effect on Trängslet. This means that Båthusströmmen can be modelled as an independent hydropower plant with no hydrological coupling constraints (Bjerhag, 2018). This is an ideal case study since the impact of a single plant can be analysed, and any results can be directly linked to changes in that specific plant instead of some unknown factor throughout the river system.

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Båthusströmmen, as well as many other small hydropower plants, is run on VNR

(vattennivåreglering), or water level regulation (Faridoon, 2018). There is a simple controller located in the plant that measures the water level of the reservoir and controls the amount of discharge and spillage in order to maintain a constant water level. This type of control is used in many small plants to simplify the operation. The central dispatch table is in charge of the

operation of every Fortum hydropower plant in Sweden, and there are many factors that have to be monitored. Though an optimal schedule is created for each plant every day (see section 4.5), the operators still must monitor the plants and adjust the schedule to account for any unexpected changes, such as a larger inflow than expected. In order to be sure that the larger plants like Trängslet are operating in the best way possible, the smaller, more “insignificant” plants are put on automatic VNR control. This allows the operators to focus on the larger, more profitable plants.

When on VNR control, plants like Båthusströmmen are not following an optimal dispatch

schedule and thus are not making as much profit as they could. The following sections show what the operation of Båthusströmmen in 2016 would have looked like if an optimal dispatch schedule were utilized instead of VNR control. The optimal schedule was made for January 1 to December 5 of 2016 using price forecasts made by Fortum and realized inflow values. The price forecasts for the rest of December were not available. It is worth noting that 2016 was a leap year. The optimisation was done as if a real spot trader were doing them, so a plan was created every day for the following day, but considering up to 4 weeks ahead. The price forecasts are updated every day, and the plans are assumed to have been followed exactly, with no unforeseen changes in, for example, local inflow.

5.2 O

P

VNR

C

R

A simple model was built in Fortum modelling system to model only Båthusströmmen and its reservoir, Hösthån. The model uses the same constraints and efficiency curves as the current model that Fortum uses for the other plants in Dalälven, but a marginal water value curve had to be generated for Hösthån. Recall from section 4.4 that the water value curve can be used to give the expected future value of the remaining water in a reservoir. The marginal water curve is simply the derivative of a water value curve, and is used to estimate how much the water value would increase by increasing the reservoir height. This curve is used in the model to create an optimal schedule. The available curve was designed to simulate a VNR operation of

Båthusströmmen, and needed to be adapted to a more realistic curve. The new marginal water value curve was created based on the existing curve as well as marginal water value curves of similar plants. Two different curves were tested, and the difference in results was deemed to be insignificant enough that a more accurate curve did not have to be generated, and the curves used are just for calculation purposes.

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limit the reservoir levels further (Faridoon, 2018). The original marginal water value curve and two newly generated curves (V2 and V3) are shown below. The new curves represent the reservoir having a 2 meter range, giving it a total usable capacity of 1Mm3. These curves were generated based on the existing water value curves for similar sized Fortum plants, as well as looking at water value curves for larger plants. The exact formulation of these curves is not the focus of this thesis, and is a good candidate for further research.

Figure 10 Marginal Water Value Curves for Båthusströmmen

In total, 4 cases were simulated for Båthusströmmen:

1. VNR: Using the current model in the Fortum system with a 20cm range for reservoir height and original marginal water value curve

2. V2: Using the model of Båthusströmmen with V2 of the marginal water value curve and a 2m range for the reservoir level

3. V3: Using the model of Båthusströmmen with V3 of the marginal water value curve and a 2m range for the reservoir level

4. Minimum Discharge (Min Q): Using the model of Båthusströmmen with V3 of the marginal water value curve and 2m range for the reservoir level. There is also a lower limit for the discharge to avoid many start/stops of the generator. Q here represents the water discharged through the turbine.

Case 4 was added after the results from cases 2-3 showed that the generator should start more than once a day. This adds a lot of wear and tear on the generator and turbine, and would require a lot of extra maintenance. In addition, the maintenance cost is very high for Båthusströmmen because it takes a worker about 2 hours to travel from Trängslet to perform the maintenance, plus 2 hours to return to Trängslet. Not only does the maintenance require a lot of time just in travel,

0 10000 20000 30000 40000 50000 60000 70000 0 0,2 0,4 0,6 0,8 1 1,2 Ma rgin al Wa ter Va lu e (E U R/Mm 3) Reservoir Filling (Mm3)

Marginal Water Value

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but it takes a worker away from Trängslet, a much more significant plant. Thus, case 4 was simulated to prevent the generator from stopping unnecessarily.

Figure 11Figure 13 show the reservoir level for weeks 1, 20, and 40 respectively and for each of the 4 cases. The realized reservoir level is also shown for comparison. The reservoir level is measured in meters above sea level (masl). Case 1, the VNR case, forces the reservoir level to vary only within a 20cm range. In reality, the VNR control kept the level nearly constant throughout the year, only varying by a few centimeters. Case 2 and 3 utilize the full 2 meters of reservoir level range, often causing the generator to turn off when the lower limit is reached. In case 4, with a minimum discharge limit, the reservoir level does not dip as low because the optimisation algorithm plans ahead to have enough water to satisfy the minimum discharge constraint.

Figure 11 2016 Week 1 Båthusströmmen Reservoir Level

During June, the spring flood is still occurring, meaning that there is a lot of water running through all of the rivers from ice and snow melt. This is why the reservoir level remains near the upper limit during the spring and summertime.

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Figure 13 2016 Week 40 Båthusströmmen Reservoir Level

Table 1 Båthusströmmen Simulation Comparison

% Increase Profit Total Production (MWh) % Increase Production Total start-ups Realized 11192 40 Case 1: VNR -1.79 13215 18.08 615 Case 2: V2 2.89 11236 0.39 406 Case 3: V3 2.61 11224 0.29 421 Case 4: Min Q 15.05 9574 -14.46 0

Table 1 shows the increase in profit and production for the 4 cases as compared to the realized operation. The profit here is calculated as the power sold at the predicted electricity price minus the start-up cost of the generator. Every day there were new electricity predictions for the

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optimisation period, and the first day of predictions from each set was saved as the final predicted price. The predicted electricity cost is used in the profit calculations, including the realized

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6 H

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HEORY

When using the model at Fortum, the exact model formulation is not known. In addition, the model cannot update the efficiency curve in between iterations, which would improve the accuracy of the optimisation. The following chapter describes an optimisation model developed to resemble, and improve, the Fortum model. First the theory is presented, then a case study is presented in Chapter 7. This fictitious plant is comparable to Båthusströmmen and is also not hydrologically coupled to other plants.

In addition to the reservoir upper level and minimum discharge constraints, a variable minimum reservoir level constraint has been added. During the winter, the surface of the reservoirs freeze. If the surface ice is cracked, which happens if the reservoir level varies too much, small cracks can appear in the ice. These cracks allow cold air to flow through them into the running water below, causing ice to form in the water and clog the intake grates of the power plant. This

phenomena is called frazil ice. To prevent this, the range for the reservoir level is restricted to just 20 centimeters during the winter time. This will limit the power production, but is a practical limitation to prevent costly maintenance.

The second main difference between this model and the Fortum model is that the

power-discharge (PQ) curve is updated each iteration based on the reservoir level. When the reservoir is completely full, there is more potential energy stored in the water because it has a larger head. This means that the power that will result from the generator is greater than if the reservoir was at its minimum level. In (Belsnes, Wolfgang, Follestad, & Aasgård, 2015), an iterative method for updating the efficiency curve based on a new head is demonstrated. The following algorithm implements this method on the PQ curve instead of the efficiency curve directly, since this is what is practically available in many hydropower plants. A black box diagram of the described algorithm is shown below.

Figure 14 Optimal Planning Algorithm

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6.1 N

The sets, indices, parameters, and variables are described below. Here, a unit is a generator-turbine pair. PWL stands for piece-wise linear, used to represent nonlinear curves.

Table 2 Algorithm Nomenclature Sets and Indices

Set T, index t Time

Set I, index i Unit, including generator and turbine Set J, index j Spillage gates

Set A, index a Linear segments for PWL PQ curve

Set N, index n Indices for binary values in PQ PWL curve Set W, index w Linear segment for PWL water value curve

Set H, index h Indices for binary values in water value PWL curve Variables

pi,t Power produced by unit i at time t (MW)

Qi,t Discharge through unit i at time t (m3/s)

Sj,t Spillage through gate j at time t (m3/s)

Mt Reservoir level at time t (masl, meters above sea level)

qi,a,t Amount of discharge for segment a of power curve (m3/s)

yi,n,t Binary variables for PQ PWL curve

dt Binary variable for discharge lower limit

zj,t Binary variable for spillage lower limit

WVt Water value of the reservoir (€)

mw,t Width of segment, reservoir level (meters)

kh,t Binary variable for water value PWL curve

ui,t Unit commitment binary for generator i

𝑠_𝑖,𝑡+ Start-up binary for generator i 𝑠_𝑖,𝑡− Stop binary for generator i Parameters

Pt Spot price (predicted or real) at time t (€/MW)

Vt Inflow to reservoir at time t (m3/s)

Δxi,a Initial power per segment for unit at defined head (MW)

Δqi,a Width of segment, discharge (m3/s)

M0 Initial reservoir height (masl)

𝑄 Minimum discharge (m3_/s)

𝑄̅ Maximum discharge (m3_/s)

𝑆 Minimum spillage (m3_/s)

𝑆̅ Maximum spillage (m3_/s)

𝑀_𝑡 Minimum reservoir level (masl)

𝑀̅ Maximum reservoir level (masl)

𝑝 Minimum power (MW)

𝑝̅ Maximum power (MW)

TE Time equivalent, converts water flow to reservoir height, unit is hour equivalents/m 𝑊𝑉_𝑤𝑠𝑙𝑜𝑝𝑒 Slope of segment w of water value curve

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Ci Start-up cost for generator I (€)

ui,0 Initial state of generator, on or off

6.2 E

F

M

For this simple test case, only one plant with one generator (i=1) and one spillway (j=1) is used. However, the algorithm is written in such a way that a more complex system could be used as well. If more plants are added to the model, the hydrological coupling between them must be added as well. This is highly dependent on the river system, especially the connection and travel time between plants. Only one plant was used for this thesis both as a proof of concept and also to be able to clearly see the benefit of using optimal planning. By using just one plant, any changes in production are trackable, and are not dependent on many factors caused by the coupling with other plants.

The model is implemented in GAMS, a commercial optimisation program.

6.3 S

M

A basic hydropower model is presented below where all parameters remain the same between time steps except the electricity price and local inflow.

Objective Function

The goal of the model is to maximise the profit from the sales to the spot market. Often in hydropower optimisation the future value of water is also included in the objective function, as it is here. The water value curve is found by integrating the marginal water value curve in Figure 10. Version 3 of the curve is used in this simulation. Mid-term steering, as described in section 4.2, is not utilized here. That function will be explored later on in this thesis. Thus, the objective function is

𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 ∑_{𝑡∈𝑇,𝑖∈𝐼} (𝑃_𝑡𝑝_𝑖,𝑡 − 𝐶_𝑖𝑠_𝑖,𝑡+_{) + 𝑊𝑉}

𝑒𝑛𝑑 ( 1 )

Hydrological Constraint

There are no plants connected up- or downstream, so there are no hydrological couplings to consider. The plant is therefore only affected by the local inflow, discharge, and spillage, as well as the local hour equivalent of the reservoir. This hour equivalent, TE, is based on the size and shape of the reservoir, and converts m3/s to reservoir height in meters.

𝑀_𝑡= 𝑀_𝑡−1+𝑉𝑡− ∑𝑖∈𝐼𝑄𝑖,𝑡− ∑𝑗∈𝐽𝑆𝑗,𝑡

𝑇𝐸

∀𝑡 ∈ 𝑇 ( 2 )

Piece-wise Linear Power-Discharge Curve

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Figure 15 Piece-wise linear power curve

The above graph shows variables for one unit (i), which is omitted from the subscripts. The following constraints describe the piece-wise linear curve (Almassalkhi & Towle, 2016).

𝑄_𝑖,𝑡 = ∑ 𝑞_{𝑖,𝑎,𝑡} 𝑎∈𝐴 ∀𝑡 ∈ 𝑇 ( 3 ) 𝑝_𝑖,𝑡 = ∑ 𝑞_{𝑖,𝑎,𝑡}∆𝑥𝑖,𝑎 ∆𝑞𝑖,𝑎 𝑎∈𝐴 ∀𝑡 ∈ 𝑇, 𝑖 ∈ 𝐼 ( 4 ) 𝑦𝑖,1,𝑡∆𝑞𝑖,1 ≤ 𝑞𝑖,1,𝑡 ≤ ∆𝑞𝑖,1 ∀𝑡 ∈ 𝑇, 𝑖 ∈ 𝐼 ( 5 ) 𝑦𝑖,2𝑎−2,𝑡∆𝑞𝑖,𝑎 ≤ 𝑞𝑖,𝑎,𝑡 ≤ 𝑦𝑖,2𝑎−1,𝑡∆𝑞𝑖,𝑎 𝑓𝑜𝑟 𝑎 ∈ 𝐴 − {1, 𝑒𝑛𝑑}, ∀𝑡 ∈ 𝑇, 𝑖 ∈ 𝐼 ( 6 ) 0 ≤ 𝑞_{𝑖,𝑎,𝑡} ≤ 𝑦_{𝑖,2𝑎−2,𝑡}∆𝑞_𝑖,𝑎 𝑓𝑜𝑟 𝑎 ∈ 𝐴{𝑒𝑛𝑑}, ∀𝑡 ∈ 𝑇, 𝑖 ∈ 𝐼 ( 7 ) 𝑦𝑖,2,𝑡 ≤ 𝑦𝑖,1,𝑡 ∀𝑡 ∈ 𝑇, 𝑖 ∈ 𝐼 ( 8 ) 𝑦𝑖,𝑛+2,𝑡 ≤ 𝑦𝑖,𝑛,𝑡 𝑓𝑜𝑟 𝑛 ∈ 1,2,4,6 … 𝑒𝑛𝑑 − 2, ∀𝑡 ∈ 𝑇, 𝑖 ∈ 𝐼 ( 9 ) 𝑦_{𝑖,𝑛+3,𝑡} ≤ 𝑦_{𝑖,𝑛,𝑡} 𝑓𝑜𝑟 𝑛 ∈ 2,4,6 … 𝑒𝑛𝑑 − 4, ∀𝑡 ∈ 𝑇, 𝑖 ∈ 𝐼 ( 10 )

Let’s break this down. Equations ( 3 )-( 4 ) show how the total power and discharge are

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of qi,a,t to the size of each segment ∆𝑞𝑖,𝑡. The use of the binary variable yi,n,tin equations ( 5 )-( 10

) makes sure that the curve is utilized in order, meaning that qi,3,t can only be non-zero if qi,1,t and

qi,2,t are at their maximum values. For example, Figure 16 shows the same curve as above but

with values filled in for ∆𝑞_𝑖,𝑎 and ∆𝑥_𝑖,𝑎 .

Figure 16 PWL example curve

Suppose we want to calculate the shown point, where the discharge is 7.5 m3_{/s and the power is}

15MW. This means that qi,1,t=3, qi,2,t=3, and qi,3,t=1.5. Equations ( 5 ) and ( 6 ) become:

𝑦𝑖,1,𝑡∗ 3 ≤ 𝑞𝑖,1,𝑡 ≤ 3

𝑦𝑖,2,𝑡∗ 3 ≤ 𝑞𝑖,2,𝑡 ≤ 𝑦𝑖,3,𝑡∗ 3

𝑦_𝑖,4,𝑡∗ 3 ≤ 𝑞_𝑖,3,𝑡 ≤ 𝑦_𝑖,5,𝑡∗ 3 And equation ( 7 ) becomes

0 ≤ 𝑞𝑖,4,𝑡 ≤ 𝑦𝑖,6,𝑡∗ 3

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3 ≤ 𝑞_𝑖,1,𝑡 ≤ 3 3 ≤ 𝑞_𝑖,2,𝑡 ≤ 3

The next segment variable, qi,3,t can be any value in its allowed range, so 𝑦𝑖,4,𝑡 must be 0, and

𝑦𝑖,5,𝑡 must be 1, resulting in

0 ≤ 𝑞𝑖,3,𝑡 ≤ 3

The last segment must be empty, since the desired point is to the left of the entire segment. To set 𝑞_𝑖,4,𝑡 to 0, 𝑦_𝑖,6,𝑡 must also be 0, making

0 ≤ 𝑞_𝑖,4,𝑡 ≤ 0

These binary choices force a segment q to either be full, partially full, or empty. To ensure that the segments are “filled up” in order from left to right, equations ( 8 )-( 10 ) are used. From this example, equation ( 8 ) becomes

𝑦𝑖,2,𝑡(1) ≤ 𝑦𝑖,1,𝑡(1)

The values are shown in parenthesis, and the equation holds true. Equation ( 9 ) becomes 𝑦_𝑖,3,𝑡(1) ≤ 𝑦_𝑖,1,𝑡(1)

𝑦_𝑖,4,𝑡(0) ≤ 𝑦_𝑖,2,𝑡(1) 𝑦𝑖,6,𝑡(0) ≤ 𝑦𝑖,4,𝑡(0)

The above equations are all true. Lastly, equation ( 10 ) becomes 𝑦_𝑖,5,𝑡(1) ≤ 𝑦_𝑖,2,𝑡(1)

Once again, this constraint holds true. That the segments must fill up in order is called an adjacency constraint in (Almassalkhi & Towle, 2016). Since the power in each linear segment, xi,a,t is based on the discharge from each linear segment, qi,a,t, those too will fill up in order and

the total power will be the correct value, 15, from this example.

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Note that the discharge has a minimum level, as was determined to be best from the simulation in Chapter 5. The reservoir lower limit can also be different for different time steps. This is to prevent frazil ice from forming, as explained previously.

Piece-wise Linear Water Value Curve

Similarly, the water value curve must also be represented as piece-wise linear. The marginal water value curve in Figure 10 has reservoir filling volume on the x-axis, and so will the water value curve that is derived from it. To more easily implement the water value curve, the x-axis is converted from volume(Mm3) to reservoir height (masl). The reservoir, within the available reservoir levels, is assumed to be rectangular, so the height increases linearly with the volume.

WVt = ∑ mw,tWVwslope w∈W ∀𝑡 ∈ 𝑇 ( 15 ) M_t = ∑ m_w,t w∈W ∀𝑡 ∈ 𝑇 ( 16 ) 𝑘𝑖,1,𝑡∆𝑚1 ≤ 𝑚1,𝑡≤ ∆𝑚1 ∀𝑡 ∈ 𝑇, 𝑖 ∈ 𝐼 ( 17 ) 𝑘_{𝑖,2𝑤−2,𝑡}∆𝑚_𝑤 ≤ 𝑚_𝑤,𝑡 ≤ 𝑘_{𝑖,2𝑤−1,𝑡}∆𝑚_𝑤 𝑓𝑜𝑟 𝑤 ∈ 𝑊 − {1, 𝑒𝑛𝑑}, ∀𝑡 ∈ 𝑇, 𝑖 ∈ 𝐼 ( 18 ) 0 ≤ 𝑚_𝑤,𝑡 ≤ 𝑘_{𝑖,2𝑤−2,𝑡}∆𝑚_𝑤 𝑓𝑜𝑟 𝑤 ∈ 𝑊{𝑒𝑛𝑑}, ∀𝑡 ∈ 𝑇, 𝑖 ∈ 𝐼 ( 19 ) 𝑘𝑖,2,𝑡 ≤ 𝑘𝑖,1,𝑡 ∀𝑡 ∈ 𝑇, 𝑖 ∈ 𝐼 ( 20 ) 𝑘_{𝑖,ℎ+2,𝑡} ≤ 𝑘_{𝑖,ℎ,𝑡} 𝑓𝑜𝑟 ℎ ∈ 1,2,4,6 … 𝑒𝑛𝑑 − 2, ∀𝑡 ∈ 𝑇, 𝑖 ∈ 𝐼 ( 21 ) 𝑘_{𝑖,ℎ+3,𝑡} ≤ 𝑘_{𝑖,ℎ,𝑡} 𝑓𝑜𝑟 ℎ ∈ 2,4,6 … 𝑒𝑛𝑑 − 4, ∀𝑡 ∈ 𝑇, 𝑖 ∈ 𝐼 ( 22 ) Start-up Costs

Lastly, the generator starts need to be kept track of and the start cost subtracted from the profit. With a minimum discharge constraint the generator should not have to turn off, but the start cost should be included just in case a constraint must be violated in order to reach a solution. For this, the time when the generator is on (unit commitment) must be kept track of. If a minimum discharge constraint larger than zero is set, the unit will never turn off, so these equations can be omitted.

𝑢_𝑖,𝑡− 𝑢_{𝑖,𝑡−1} = 𝑠_𝑖,𝑡+ _{− 𝑠}

𝑖,𝑡− 𝑓𝑜𝑟 𝑡 = 2 … 𝑒𝑛𝑑, ∀𝑖 ∈ 𝐼 ( 23 )

𝑢_𝑖,𝑡 − 𝑢_𝑖,0 = 𝑠_𝑖,𝑡+ _{− 𝑠}

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6.4 I

U

B

N

H

The above static model does not change between iterations. In reality, the efficiency of the turbine/generator is greater when there is greater head since there is more potential energy stored in the water. This change in efficiency can be represented by updating the power curve each time step (Belsnes, Wolfgang, Follestad, & Aasgård, 2015). The entire power curve is scaled based on the most efficient point. This update is shown below. Note that the index a=best refers to the point on the curve with the best efficiency, and a=rest refers to the remaining points. Note also that instead of using the segment variables, 𝑥_{𝑖,𝑎,𝑡}, the points on the graph, or the y-coordinates, are updated. The segments are then reformed from the new points. The points here are

represented as 𝑥_{𝑖,𝑎,𝑡}𝑝 .

𝑥_{𝑖,best,𝑡}𝑝 = 𝑥_{𝑖,min 𝑒𝑓𝑓,𝑡−1}𝑝 + (𝑥_{𝑖,best,𝑡−1}𝑝 − 𝑥_{𝑖,min 𝑒𝑓𝑓,𝑡−1}𝑝 )𝑀𝑡−1 𝑀̅ ∀𝑡 ∈ 𝑇, 𝑖 ∈ 𝐼 ( 25 ) 𝑥_{𝑖,𝑟𝑒𝑠𝑡,𝑡}𝑝 = 𝑥_{𝑖,𝑟𝑒𝑠𝑡,𝑡−1}𝑝 𝑥𝑖,𝑏𝑒𝑠𝑡,𝑡 𝑝 𝑥_{𝑖,𝑏𝑒𝑠𝑡,𝑡−1}𝑝 ∀𝑡 ∈ 𝑇, 𝑖 ∈ 𝐼 ( 26 )

This update is done outside of the above model. For example, the optimisation is done every day for the following day and considering the next several weeks. The optimisation will be done using the original PQ curve, then the optimal schedule for the first hour is kept. The PQ curve is updated as shown above, and the program iterates and runs the optimisation again starting from the second hour. This continues until the full 24 hours are planned. It was decided that iterating through each hour of the planning period (up to 4 weeks) was not necessary, and that just

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7 O

PTIMAL

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A

LGORITHM

R

ESULTS

Once again, 2016 was chosen as the simulation year. For the electricity price predictions, the realized spot price was used with an introduced random error of up to 4%. The inflow was generated based on real measurements and with a random error of up to 6%. The results for the optimal planning algorithm are shown below. The parameters used are shown below.

Table 3 Test Case Parameters

M0 495.4 masl

𝑄 3 m3/s 𝑄̅ 24 m3/s 𝑆 0 m3_/s

𝑆̅ 252 m3/s

𝑀_𝑡 495.0 masl (December 1-February 29) , 493.5 masl (March 1-November 30) 𝑀̅ 495.5 masl 𝑝 0 MW 𝑝̅ 3.3 MW TE 350 hour equivalents/m C 151 € u0 1, on

The following graphs show the reservoir level, discharge, spillage, and electricity price for week 1, 23, and 40 of 2016.

Figure 17 Week 1 Reservoir level and flow rate

0 5 10 15 20 25 30 495,44 495,45 495,46 495,47 495,48 495,49 495,5 495,51 01-04 01-05 01-06 01-07 01-08 01-09 01-10 01-11 Flow Rat e (m^ 3/s ) Re se rv o ir Lev el (m asl )

January 4-10, 2016

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Figure 18 Week 1 Reservoir level and electricity price

During the winter, the reservoir level is more tightly restricted to stay above 495.0 masl. There is no spillage needed during week 1. There is a clear correlation between the electricity price and the reservoir level in Figure 18 where the drops in reservoir level occur when there are spikes in electricity price.

Figure 19 Week 23 Reservoir level and flow rate

0 20 40 60 80 100 120 495,44 495,45 495,46 495,47 495,48 495,49 495,5 495,51 01-04 01-05 01-06 01-07 01-08 01-09 01-10 01-11 Ele ctricity P rice ( € /MW) Re se rv o ir Lev el (m asl )

January 4-10, 2016

M (masl) Price (Eur/MW)

0 5 10 15 20 25 30 495,48 495,485 495,49 495,495 495,5 495,505 06-06 06-07 06-08 06-09 06-10 06-11 06-12 06-13 Fl ow Rate (m ^3/ s) Re se rv o ir Lev el (m asl )

June 6-12, 2016

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Figure 20 Week 23 Reservoir level and electricity price

During the summer the reservoir level is allowed to decrease until 493.5 masl, but the spring flood is in full effect. At this time there is a large amount of inflow, so the reservoir level does not get too low. During the fall, shown in Figure 21 and Figure 22, the reservoir level changes daily based on the price, but still does not go below approximately 495.47 masl, refilling during the night when the electricity price is low.

Figure 21 Week 40 Reservoir level and flow rate

0 5 10 15 20 25 30 35 40 45 50 495,48 495,485 495,49 495,495 495,5 495,505 06-06 06-07 06-08 06-09 06-10 06-11 06-12 06-13 Ele ctricity P rice ( € /MW) Re se rv o ir Lev el (m asl )

June 6-12, 2016

M (masl) Price (Eur/MW)

0 2 4 6 8 10 12 14 16 18 20 495,465 495,47 495,475 495,48 495,485 495,49 495,495 495,5 10-03 10-04 10-05 10-06 10-07 10-08 10-09 10-10 Flow Rat e (m^ 3/s ) Re se rv o ir Lev el (m asl )

October 3-9, 2016

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Figure 22 Week 40 Reservoir level and electricity price

During most of the year there is no need to spill any water. However, during the spring flood the inflow is well over the upper discharge limit, so some water must be spilled.

Figure 23 Inflow and spillage for 2016

0 5 10 15 20 25 30 35 40 45 495,465 495,47 495,475 495,48 495,485 495,49 495,495 495,5 10-03 10-04 10-05 10-06 10-07 10-08 10-09 10-10 Ele ctricity P rice ( € /MW) Re se rv o ir Lev el (m asl )

October 3-9, 2016

M (masl) Price (Eur/MW)

0 50 100 150 200 250 300 01-01 02-20 04-10 05-30 07-19 09-07 10-27 12-16 Flow ra te (m 3/s )

Inflow and Spillage

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Even when the reservoir level is allowed to go to the lower limit of 495.0 masl, it remains relatively high. This is most likely due to the fact that mid-term steering was not used in this model, so the end water value was always a part of the objective function. This causes the model to want to maintain a high reservoir level at the end of the planning period (short term) to have a high end water value, and high objective function. To see the effect of the optimised planning, a simple VNR simulation was done where the discharge exactly equalled the inflow. When the inflow was larger than the allowed discharged, the remaining inflow was spilled. Using this ‘VNR’ discharge schedule, the benefit of using an optimising algorithm can be seen. For example, week 1 discharge from the VNR and optimised versions are shown below. While the VNR discharge remains quite constant, the optimised discharge varies throughout each day, following the price.

Figure 24 Optimised and VNR discharge comparison

The optimal discharge plan varies much more on an hourly basis than the VNR discharge in order to optimise the profit. There is a 16% increase in profit during 2016 by using the optimisation algorithm. This is comparable to the 15% increase from case 4 in section 5.2, so the discussed algorithm is similar to that implemented in the Fortum model.

0 5 10 15 20 25 30

04-jan 05-jan 06-jan 07-jan 08-jan 09-jan 10-jan 11-jan

Dis ch ar ge (m 3/s )

VNR vs Optimised Discharge January 4-10, 2016

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8 C

ENTRAL

A

UTOMATIC

C

ONTROL

8.1 D

The algorithm described above is capable of planning the operation of a hydropower plant for varying lengths of time. With this in mind, it can be used for the mid-term planning as well as short-term planning. As described in section 4.2, steering from mid-term planning can ensure that future factors are considered. The above algorithm can be used for mid-term planning, which then passes the steering, in the form of a reservoir water value, to a short-term planning model. This short-term planning model will have two main differences: the objective function will no longer consider future value of stored water, and there will be an additional steering constraint. Equation ( 1 ) becomes

𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 ∑𝑡∈𝑇,𝑖∈𝐼 (𝑃𝑡𝑝𝑖,𝑡− 𝐶𝑖𝑠𝑖,𝑡+) ( 27 )

The steering constraint will simply be

𝑊𝑉_𝑒𝑛𝑑 = 𝑊𝑉_{𝑠𝑡𝑒𝑒𝑟𝑖𝑛𝑔} ( 28 )

where WVsteering is the water value at the end of the short-term optimisation as determined in the

mid-term optimisation. With the mid-term and short-term planning algorithms, a hydropower plant can be optimised in real time.

The following section describes the implementation of the planning algorithms with Fortum’s central control. This work was done in collaboration with Jenny Möller and Johan Wiklund for their Undergraduate thesis, titled “Optimisation tool for automated planning and control of production in small-scale hydropower plants ” (Möller & Wiklund, 2018). For more details on the results of the implementation, please refer to their thesis. Note that this work is written in Swedish.

8.2 C

F

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Figure 25 Information Flow of Central Control Unit

At Fortum there are already tools built in Excel to communicate with the central database, so the central controller was written in Microsoft Visual Basic to take advantage of these tools. The central controller reads the input values it needs, executes the algorithm, then writes the optimal schedules back to the database, which are then automatically sent to the local controller at the plant. In practice, the central controller will be a program located on a central server.

8.3 F

P

Initially the algorithm was rewritten to run in Visual Basic using the Microsoft Solver Foundation, but the model was too large to be solved. Finally, communication was set up between the Visual Basic project and a model built in GAMS. Since the algorithm was already tested in Chapter 6, the main focus here was establishing communication between the database, the Visual Basic project, and the GAMS code, as well as utilizing both the mid-term and short-term algorithms. The Visual Basic project acts as the central controller, so it is capable of running the algorithms and communicating with the database in real time. The basic structure of the code is shown in Figure 26. For a detailed axplanation, refer to (Möller & Wiklund, 2018).

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