Analysis of Hydropower Systems' Ability to Follow Square Wave Load Variations

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STOCKHOLM SVERIGE 2016,

Analysis of Hydropower Systems' Ability to Follow Square Wave

Load Variations

FADI HANNA

KTH

SKOLAN FÖR ELEKTRO- OCH SYSTEMTEKNIK

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Ability to Follow Square Wave Load Variations

Fadi Hanna

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Abstract

The interest for renewable energies is increasing in Sweden and wind power is a major factor for the increase. Therefore, it is expected that with increasing installed capacity from renewable resources the variations in the power system will increase drastically due to the unexpected energy production

from these types of power sources. Thus, a flexible power production is required to balance these variations which can cause serious problems in the power system such as voltage flicker and harmonics. In Sweden, the hydropower is ideal for balancing due to the high installed capacity and

the flexible generation of hydropower. Therefore it is import to study the capability of the hydropower systems at balancing these variations and this is done by studying the ability of the hydropower systems at producing with extreme variations on their power production. A short-term

mixed-integer linear programming model is utilised to study the capability of a hydropower system while facing extreme variations in its power production. The extreme variations are modelled as

square waves where the production of the hydropower system is forced to follow these square waves. The hydropower system model considers both the water delay time and the head- dependency of the power plants. Another interesting factor that is considered in the study is how the

arrangement of the power plants in a hydropower system affects the power variation capabilities.

The special aspect in this study is that the hydropower systems simulated are completely fictitious systems. This means that the crucial aspects that influence the variation in the power production of a

hydropower system can be manipulated to study the effects. The results obtained showed that the capability of a hydropower system at following extreme variation in power production is highly dependent on the reservoir water contents of the different power plants during the simulation period. Another obvious factor that impacted the capability of a hydropower system was the arrangement of the power plants included in the system. Lastly it was shown that if a hydropower system is dominated by small reservoirs, then this will mean more variation in the efficiency of the

power production.

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Sammanfattning

Intresset för förnybara energikällor ökar i Sverige och vindkraft utgör en större del av denna ökning.

Därför förväntas variationerna i det elektriska systemet öka drastiskt med en ökande effektkapacitet från förnybara energikällor. På grund av detta är det viktigt att balansera dessa variationer och för att

lyckas med detta en flexibel kraftproduktion krävs. Variationer i det elektriska systemet kan medföra seriösa spänning och frekvens problem så som flimmer och övertoner. I Sverige utgör vattenkraft en stor del av den totala installerade effekten i landet och på grund av dess flexibla kraftproduktion gör att vattenkraft är en idealisk kraftkälla för att balansera stora variationer i det elektriska systemet.

Därför är det viktigt att studera förmågan för vattenkraft systemen då dessa tvingas att producera under stora produktionsvariationer. För att studera detta har ett linjärt heltalsproblem (engelska:

MILP) använts för att simulera ett vattenkraftsystem med stora variationer i vattenkraftsystemets produktion. Dessa extrema variationer i produktionen är modellerade som fyrkantsvågor som elproduktionen från vattenkraften följer. Vattenkraftsystemet är modellerat så att den tar hänsyn till

både rinntider från ett kraftverk till ett nedströms kraftverk och till fallhöjdsberoendet för varje kraftverk. En annan intressant faktor som anses i studien är hur placeringen av dem olika kraftverken

påverkar förmågan för ett vattenkraftsystem. Den intressanta aspekten i denna studie är att dem simulerade vattenkraftsystemen är helt fiktiva system, vilket menas att dem avgörande parametrarna och faktorerna för ett vattenkraftsystem då den producerar under stora variationer

kan studeras noggrannare genom manipulation av dessa parametrar och faktorer. De erhållna resultaten visade att ett vattenkraftsystems kapabilitet att följa stora produktionsvariationer är mycket beroende av vatten innehållet i magasinerna av dem olika kraftverken. En annan faktor som

hade en tydlig påverkan på variationsförmågan av elproduktionen var hur dem olika kraftverken i systemet var placerade. Slutligen, resultaten kunde bevisa att om ett vattenkraftsystem är dominerat

av små magasiner kommer effektiviteten av vattenkraftsystemets elproduktionen att vara mer varierande.

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1. I

NTRODUCTION

The rapid increases in renewable energies have a huge impact on the power system. The impact is mainly due to the variations that are introduced with increased capacity of renewable energies such as wind power and solar power. Wind power is by far the largest renewable source that introduces variations to the power system. The reason for this is the high installed capacity of wind power and that the quality of wind power forecast is not good enough, thus large unexpected variations of wind power production can occur in the system. These variations will most certainly increase in the future due to the rapid increase in the wind power capacity, see Figure 1 [1].

Figure 1: The increase of wind power capacity and generation in Sweden 1991-2014 [1].

As seen in Figure 1, the wind power capacity is increasing in an exponential manner and this increase is expected to continue. The main reason for this is the awareness of the environmental impact and therefore the Swedish government and the EU (European Union) are setting requirements on power producers to achieve a more environmentally friendly electricity production. The Swedish

government has even as a future goal to achieve a power system consisting of 100% renewable energy sources [2]. This is of course a future goal but to consider the different renewable sources available right now, the most obvious choice would be that the non-renewable power sources will be replaced by wind power. Figure 2 represents the distribution of energy sources in Sweden [3]. As observed, the hydropower and wind power corresponds to about 50% of the total power production.

Then the main power source is nuclear power, therefore, it is expected that the nuclear power will be replaced by wind power in the upcoming years. Thus, the variations in the system will increase in future and these variations have to be balanced by another power source.

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Figure 2: Distribution of energy sources in Sweden, 2014 [3].

There are many problems that occur with variations in a power system. One of the most serious problems is the system stability; variations in the system often introduce frequency problems and voltage problems such as harmonics and voltage flicker. Another economical problem is that wind power variations can create imbalance costs for the wind farm owners. A wind farm usually signs a contract in advance that binds the wind farm to produce a specific amount of power during a specific time period. If the power production from the wind farm is much less or much more than the

contracted power, then the wind farm will face so called imbalance costs. These imbalance costs increase the market price, which in its turn effects on other players in the electricity market [4].

As observed, the system impacts are serious and therefore the need for balancing power is of great importance. The flexibility and capability of fast production increase/decrease of hydropower can provide balancing power to the electricity system [4]. The decentralized competitive electricity market and the continuous increase in the installed wind power capacity create a challenge for the planning of a short-term hydropower production in an optimal way. There is much research in this area and many introduce different approaches to achieve an optimal operation between wind power and hydropower. To consider the wind power uncertainty many have developed stochastic short- term models where the uncertainty is the wind power production.

There are many challenges while modelling the optimization problem between wind power and hydropower. The challenges are mainly due to the complex and nonlinear characteristics of the hydropower system. The complicated modelling of the hydrological constraints and the power production of hydro plants has to be accurate enough to achieve accepted results.

An optimal operation of the system is often considered when the hydropower system can balance the big variations of the wind power production without any spillage or power deficit. The water spillage of a hydropower system can occur due to many reasons, but usually a hydropower system

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spills water when the reservoirs are full, which mean no more water can be stored in the reservoirs.

This can occur if, for example the wind power production is high and the productions from the hydropower plants are low. Thus, and with consideration of the water inflow to the stations, the hydro reservoirs will be overfull and starts to spill water. The power deficit usually occurs when the discharged water from an upstream station does not reach quick enough to a downstream station, this will mean that the downstream station will not have the sufficient amount of water in its reservoir to produce the required amount of power at that time period, or the amount of water that reaches the downstream stations is not enough to produce the amount of power required.

Therefore, a power deficit in the system occurs.

As observed, the increase in renewable energy sources is obvious and with this increase the

variations in the system will occur. The main power source that will be affected by these variations is the hydropower due to its capability of flexible generation and due to the capability of water storage which means that the power generation can be produced when needed. Therefore, in this project a hydropower system will be simulated with a production profile with extreme variations. The reason for these extreme production variations is to find the capability of the hydropower system. The hydropower system introduced takes into account both the hydrological aspects of the system and the power generation of each power plant.

The unique approach in this project is that it does not consider a specific hydropower system with specific parameters; instead the idea is to come up with a fictitious hydropower system, which means many topological parameters such as the water inflow, the water delay times, the reservoir size and so on, can be changed to study different aspects of the hydropower system while the system is operating in extreme conditions such as variating its production from a minimum level to a

maximum level. Basically, the aim is to find the critical aspects that are dependent on the flexibility of production of the hydropower system.

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2. L

ITERATURE REVIEW

In this section, a literature review about the different studied cases and models will be given. The literature review will focus on the main aspects considered in this project such as: modelling of hydropower generation curve and modelling of the hydrological aspects and delay times. Other important considerations can be to review the different optimization methods used, how the dealing with the nonlinearity is done, what type of approximations are done and which aspects of the problem are neglected and how is the system affected by neglecting these aspects.

A brief review of the studied balancing cases, mainly wind power balancing, will also be given. The reason for this is to see how other papers deal with the power variation in the power system.

Therefore it is important to give an overview of the studied cases.

2.1 H

EAD

-

DEPENDENCY

One of the most important aspects to consider while modelling a hydropower system is the head- dependency. The head-dependency basically means that for different water level of reservoir

contents and discharge level, different generation efficiency can be achieved from a power plant. The problem is that the dependency from the head is a nonlinear one. This is the case because the reservoir content level changes due to the water that is discharged from the reservoir and due to the water that flows to the reservoir. The same is the case for the discharge, when the power plant is producing a lot of electricity; it is discharging a lot of water and therefore the water level where it is discharged increases. Often the water level at the reservoir is called the forebay level and the water level at the discharge is called the tailrace level.

The power output is then often calculated by multiplying the difference between the forebay level and the tailrace level with the discharged water and the generation efficiency. The forebay level can for example be modelled as a fourth degree polynomials function of the volume of stored water.

Meanwhile the tailrace level can be calculated using a fourth degree polynomials as a function of the total discharged water [5]. These polynomial functions will of course force a nonlinear dependency on the power output. In [5] different methods are used to simplify this nonlinearity. A

straightforward simplification is to neglect any polynomial function of order greater than one, which leads to linear relationship. However, the nonlinearity is still in the power output definition because the tailrace level and the forebay level still variates nonlinearly. Therefore to further simplify this, the variations are assumed to be constant and the head-dependency is modelled as the difference between the average forebay level and the average tailrace level. The average values for the forebay level and the tailrace level are calculated by considering the lower and the upper values of the reservoir contents and the total water discharge, respectively. By this approach can the variations on the net water head be neglected and the difference in the forebay and tailrace level can be constant.

The approach gives an acceptable approximation of the head-dependency but the problem with this type of approach is that the head-dependency is constant for each power plant. Which means no variations during the simulation time will occur.

In [6] another type of modelling is used that variates the generation curve depending on the reservoir levels. However, the variation in the tailrace level is neglected. Instead, two levels of

reservoir are decided for every power plant, a lower level and an upper level. These two levels results

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in three states for the head-dependency. A lower reservoir level (when the contents of reservoir is below the lower level selected for the reservoir) correspond to a lower head and therefore to a less efficient generation curve. An intermediate level, which basically means that the reservoir contents are between the lower level assigned and the upper level assigned, correspond to a better efficiency of the generation curve, and lastly if the contents of reservoir are above the upper level, the

generation curve used will have the best efficiency of the three generation curves. The generation curve here is approximated with a piecewise linear function with four equally divided segments. The slope of each segment is given as a constant.

Figure 3: Production of a power plant depending on the reservoir level and the discharge as explained in [6].

The type of modelling used in [6] requires the use of binary variables. A binary variable is a variable which only can take the value 0 or 1. Binary variables are used in [6] to obtain constraints which can find the different reservoir levels (below lower level, between lower and upper level, above upper level). The downside of binary variables is that the computational time of a MILP formulation increases compared by an LP formulation.

A similar approach to [6] is used by [7]. But here the number of reservoir contents intervals is increased. It is obvious that this will improve the approximation. However this will also increase the computational time because more binary variables will be required. Another improvement in [7] is that it tighten the linear programming relaxation of the model through a more precise estimation of the upper bound on the power production. As observed, [7] uses the same idea as [6] but improves it by using different techniques, such as new constraints and new binary variables. These

improvements improve of course the approximation of the head-dependency but also require longer computation time.

A very complicated approach is done by [8]. This complication is due to the many dependencies that the paper takes into account. The paper models the hydropower plant’s generation as a function of the water outflow (which consists of spillage and discharge), the turbine efficiency, the generation efficiency and the water head in reservoir. As observed, the modelling of the efficiency factor will be very complicated, where the turbine efficiency is modelled as a function of the net head and the

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turbine outflow, meanwhile the generation efficiency is modelled as a function of the hydro unit’s power generation. The main purpose for this type of modelling is that in [8] each hydro unit is assumed to have a certain number of turbines. Another aspect that adds to the complexity of the problem is that the spillage effects are accounted for. The authors argue that the spillage is a part of the outflow which is a part of the tailrace level. Which is true in most of the cases (some hydropower plant may have different paths for spillage and discharge, in this case, the spillage will not be a part of the tailrace), some power plants may spill a lot of water during some periods and this will of course affect the tailrace level and eventually the power output. The authors in [8] propose a four- dimensional linear model for the generation of the hydro plants. The main advantage of the approach is also the detailed representation of the spillage. However, the complexity of the model may be a problem while modelling a bigger hydropower system with many power plants. The model puts a big effort to model the spillage-effects which is a bit unnecessary with the type of problem considered in this project, where the water spillage has to be minimized.

A unique approach is presented by [9] where a piece-wise linearization of a two variable function is considered. Therefore, the power produced – which is formulated as a function of the reservoir contents and the discharge – by each hydropower plant can be expressed by applying the method of linearization. The method considers a piece-wise linear approximation of an arbitrary continuous function of two variables. The formulation allows an approximation of any non-linear programming problem, even non-convex ones with an objective function and constraints expressed as sums of non-linear functions of no more than two variables. Consider the following:

Let 𝑓 = 𝑓(𝑥, 𝑦) be a function defined on the rectangle (𝑥1, 𝑦₁) × (𝑥_𝑚, 𝑦_𝑛). The x and y axis can be divided in a set of intervals limited by the discrete values 𝑥𝑖 and 𝑦𝑗 (𝑖 = 1, . . , 𝑚; 𝑗 = 1, . . , 𝑛). Let suppose that 𝑓 is known for each point (𝑥_𝑖, 𝑦_𝑗) of the mesh. Then, the function 𝑓(𝑥, 𝑦) can be approximated as a linear piece-wise function using the following equations:

𝑓(𝑥, 𝑦) = ∑ ∑ 𝑓_𝑖,𝑗𝜆_𝑖,𝑗

𝑛 𝑗=1 𝑚

𝑖=1

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𝑥 = ∑ ∑ 𝑥_𝑖𝜆_𝑖,𝑗

𝑛 𝑗=1 𝑚

𝑖=1

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𝑦 = ∑ ∑ 𝑦_𝑖𝜆_𝑖,𝑗

𝑛 𝑗=1 𝑚

𝑖=1

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∑ ∑ (𝜇_𝑖,𝑗+ 𝜔_𝑖,𝑗) = 1

𝑛 𝑗=1 𝑚

𝑖=1

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∑ ∑ 𝜆_𝑖,𝑗

𝑛 𝑗=1 𝑚

𝑖=1

= 1 (5)

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𝜆_𝑖,𝑗≤ 𝜇_{𝑖,𝑗−1}+ 𝜔_{𝑖−1,𝑗}+ 𝜇_𝑖,𝑗+ 𝜔_𝑖,𝑗+1+ 𝜇_𝑖+1,𝑗+ 𝜔_𝑖,𝑗 (6)

Where 𝜆𝑖,𝑗 are positive and continuous variables, and 𝜇𝑖,𝑗 and 𝜔𝑖,𝑗 are binary variables. As seen, even this method will introduce binary variables to the problem.

As observed, all the presented examples of literature where a model of variating head-dependency is considered use some kind of binary variables. Therefore, it is nearly impossible to model the head dependencies with only linear programming. The reason for this is that some kind of constraint will always be required to activate a new level and to deactivate another one. This type of consideration will always require a binary variable. Therefore, in this project binary variables will also be used.

However, the use of binary variables has to be as limited as possible, otherwise the computational time may increase and as mentioned before, the computational time is of great importance to this project where many simulations are required to test different cases.

2.2 H

YDROLOGICAL ASPECTS

A hydropower system is highly dependent on the amount of water that flows through it. This water can come from different sources such as local inflows and water that were discharged and spilled by the upstream power plants. However, the discharged and spilled water from the different power plants reaches the next downstream power plant after some water delay time. Another important aspect to consider is that the water discharged or spilled will reach the downstream power plant in stages where each time period corresponds to a portion of the total discharged or spilled water from the upstream power plant.

As it can be observed, to represent these factors will require a very complicated formulation of the problem. As a result, many choose to approximate the water delay time by a constant or even to neglect it. Therefore, three different cases that can be found in the literature will be presented here.

The easiest representation of the hydrological constraints is to assume that the water released by upstream power plants is available for use at the downstream reservoir at instant. This basically means to totally neglecting the water delay times. This type of representation is done in papers such as [10] and [9].

A better approximation of the hydrological constraints is to assume a constant water time delay between power stations; this is the most common approach in the literature. This is done in [5] and [6] where a constant water time delay is assumed for each power station. However, both these approximations require that the water delay times between stations to be expressed in hours.

A more complicated type of representation of the hydrological constraints is done by [11] and [12]

where the water delay time is considered in both hours and minutes. Both of the aforementioned papers use a method to linearize this time delay. However, even here there still is a big assumption in the problem. The assumption is that all the water that is discharged from a power station reaches the next station after some time delay. As mentioned above, this is not the case, the water that leaves a power station reaches the next station in portions and each portion reaches after some time delay.

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Not many have dealt with this type of problem, but there are a few. Both in [13] and [14] the representation of the hydrological aspects is very well formulated. This type of representation is called “Streamflow routing constraints” and this is a term used to describe the behaviour of flood wave as it moves along a well-defined open channel. This is of course a nonlinear and nonconvex behaviour. To approximate the problem with linear behaviour the Muskingum method is used in [13]

and [14]. The method considers portions of water that reach the downstream power plant in different hours. These hours are assumed to be between a minimum allowed time and a maximum allowed time. The figure obtained from [13] describes this type of formulation:

Figure 4: Approximation of the wave propagation effect throughout a river channel [13].

Figure 4 is a good representation of the so called “streamflow routing”, where different amount of the discharged water, released by the upstream reservoir, reaches the downstream reservoir in different times in the future, with water delay times ranging between a minimum and a maximum value, often 1 hour and 24 hours.

However, it is stated in [14] that the portions of water considered are assumed constant during simulation time, which is an approximation. It is explained in [14] that the portions of water are functions of the discharged water by the upstream station. The assumption to keep the portions constant is done to keep the convexity properties of the problem and it is stated that this assumption can be a potential limitation for the procedure. Therefore, the paper presents a strategy to overcome the drawback. The strategy is simply that the different curves of the streamflow routing can be achieved for different levels of streamflow, which basically means different levels of inflow to the hydropower system. These curves are often obtained from real data; an example is shown in Fig 3 and Table I in [13]. Basically, different curves can be obtained for different values of discharge and the idea presented in [14] is to choose the curve that would be expected to occur in the period for which the system is being dispatched. These curves can be associated with season of the year or the system operating conditions, e.g. low, medium or high inflows, corresponding to the simulation required.

However, both in [13] and [14] the method seems to only consider whole hours as delay time.

Therefore a most accurate representation can be to combine the streamflow routing with the type of representation in [11] and [12], where the delay time can be expressed in hours and minutes. It was not possible to find any article that combines these two methods.

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In [13], and by using Linear Programming, the computational time ranged between 13 minutes, at 4 power plants and water delay times, up to 34 minutes at 111 power plants and water delay times.

However, although [14] uses exactly the same approach and considers a real case where the Brazilian system is used with 141 hydro plants and 96 thermal plants, the computational time, for 168 hours, did not increase more than seven minutes. An exact explanation for this difference in computational time between these two papers could not be found. However, it could be noticed that there are two major differences between the models utilized in the papers. In [13], an uneven discretization of time with periods of different hours is utilised. The authors argue that this type of presentation is

reasonable to consider while simulating longer periods (e.g. several days or even weeks). The uneven discretization time means that each contribution factor for the water release by the upstream plant at previous time steps becomes more complicated and will vary according to the time index. For more detailed explanation see section III B in [13].

The described type of formulation is not used by [14]. Instead, the model in [14] concentrates on modelling the upper and lower bounds for different river-level by taking into account the amount of streamflow in the river. The modelling of river-level is not utilised in [13].

To study the effects on the computational time by applying streamflow routing, Table V in [14] is considered. Table V shows two cases of a short-term planning problem simulated with and without streamflow routing applied. In case one, which is simulated with a simulation period of 24 hours, the CPU time increased by 17 seconds by applying water delays with streamflow routing compared by only constants water delay times. If a period of 168 hours is simulated, then the CPU time difference between constant delay times and delay times with streamflow routing was 30 seconds. Keep in mind that the simulation in the aforementioned papers considers 237 power plants (141 hydro and 96 thermal).

It is obvious that the streamflow routing approach is by far the most accurate one while formulating hydrological constraints for a hydropower system. However, to apply this type of formulation to the model in this project would require somewhat realistic data from different river systems in Sweden.

This data could not be possible to obtain. To use and collect data from other papers and articles would be very time consuming and would be outside of the scope of this project.

Neglecting the water time delay will most likely cause big errors in the results. The hydropower system will then face two big problems. The first one being that the hydropower system will behave in a much quicker way because the water availability will be much faster, or even instant. This will overestimate the capability of the hydropower system. Another problem is that the system may spill much more water than what it really would, and this will be an underestimation for the system. As observed the result will then be very difficult to interpret.

Therefore, the most reasonable formulation is to choose a constant water delay time. The approach described by [11] and [12] make it possible to use a more precise delay time because it can describe the delay time in both hours and minutes. The approach does not affect the computational time in a big sense either.

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2.3 W

IND POWER BALANCING

The main problem of wind power balancing is that it affects the whole electrical system. The wind power is increasing; with increasing wind power in the system so does the variations in the system.

There are many papers that deal with balancing of the big variations of wind power. A few uses different type of wind speed prediction, other uses batteries as storage. Many uses the hydro-pump plant and often argues that this type of combination is an economic gain for both parties. However, very few consider the impacts that can occur in hydropower systems while these balance big variations. The most common approach is the hydro-pump storage as mentioned.

Here, three different approaches for balancing of wind power variation will be reviewed. These will be: Wind power predictions, balancing of hydro-pump power plants and balancing by hydropower and other methods (such as battery storage).

Wind power prediction is a very important topic. The advantage of wind power prediction is that the generation planning of, e.g. a hydropower plant, can be planned in advance. This will improve the hydropower system so it can minimize its spillage and maximize its profit and can basically be more efficient. The wind speed prediction can be divided in three different categories: a very short-term prediction – usually a few minutes to an hour; short-term prediction – hours to a few days; and lastly long-term prediction [15]. In the case of this project, a prediction of one hour is of great interest. This prediction can be used to optimize the operation of the hydropower system. However, often wind power plants and wind farms are very far from each other and the direction of every wind power turbine is different. This enhances the complexity of the problem, as discussed in [15]. Therefore, to accurately predict wind speed and direction, further research is required. As a result, wind power generation prediction cannot be presented as a reliable method, and therefore a hydropower system cannot rely entirely on it to plan its production.

Another, more successful method, is to use a so called pump-hydro plant with wind power. This type of balancing between wind power and pump-hydro plant are used by many, e.g. [16] and [17]. The main argument for using the combination between pump-hydro and wind power is that the profit increases drastically, as concluded in [17]. The idea is that during high price period – and this happen usually when wind generation is low – the hydro-pump is operating as a generator and producing electricity. On the other hand, while low price period – high wind power generation – the hydro- pump is operating as a pump, and therefore consuming electricity. The pump usually pumps water to its reservoir. As a result, the imbalance costs (penalty for overproduction and underproduction of wind generation compared by contracted power) are compensated by this type of cooperation.

However, a hydro-pump plant often requires special locations that have a combination of

geographical height and water availability. This type of locations are often founded in mountainous regions, therefore the construction of these plants is very complicated. It is even difficult to find many places with the types of requirements needed.

Some uses different types of storages to reduce the variations of wind power. In [18] the hydropower plants together with the flow batteries make it possible to balance the wind power variations. The flow batteries acts as a storage for saving electricity, if any wind power generation decrease/increase occurs, the flow battery then compensates the underproduction/overproduction of electricity.

However, in the type of case studied in [18], it was concluded that the flow batteries used in the system should have a capacity of 16.5 % of the total installed wind power. This is a very large

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percentage if a larger scale of wind power is considered. The paper does not discuss the economical or the environmental aspects of using these flow batteries either, which can be of great interest. The paper instead focuses on the technical part of the problem and concludes that with the use of the flow batteries, the imbalance costs could be fully balanced.

A similar type of approach is done by [19]. Here a battery is used to balance the fast variability of the wind generation. The battery is assumed to keep the power output constant for five minutes. It is even assumed that a very short term to short term wind prediction tool is in place. This tool predicts the average availability of wind in advance. Lastly, the paper introduces a control strategy to obtain the best balance of the system.

As it can be observed, there are a lot of different methods to balance wind power variations. Many use different methods such as wind speed predictions and all the way to using different types of storage devices to balance the variations. The capacity of installed wind power is increasing very rapidly and with it the variations in the system will increase. A system for balancing of very big variations will be needed. Big variations in the power system will be one of the biggest challenges to face in the future. Therefore, a tool to balance these big variations has to be studied. To use wind power predictions for the planning of production is usually unreliable. Pump-hydro plants is the method that shows the most promise but the hard requirements on location makes it hard to implement many of this type of power plants. The flow batteries can be a good solution but with current capability of the batteries it is hard to consider batteries for a power system of a larger scale of wind power.

As shown many papers consider the problem of high variations in the power system which means that the variations in the future are a fact. The most obvious balancing tool in Sweden to balance big variations in the power system is the hydropower due to its high installed capacity and the flexible power generation. Therefore, it is of great interest to study the capability of different hydropower systems while these extremely variate their power generation. The main reason of the study is to analyse the factors that influence the hydropower systems in such a way that it fails to variates its power output in an optimal manner.

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3. M

ATHEMATICAL MODEL

In this section, a description of the model required for analysing the capabilities of different hydropower systems while these are facing very large production fluctuations will be given. The purpose of the model is to find the amount of electrical power that is lost and also the water spillage that the hydropower system is forced to spill during power generation with huge variations. The model uses a discrete time-step of one hour. As a result, the model can be used to study the ability of the hydropower systems to change their generation from hour to hour. To obtain a better

analysation of the different cases, a longer time period is required. Otherwise the true capability of the hydropower model can be overrated, e.g. a power plant with a small reservoir will have no problem to produce a lot of energy during a short period of time. However, during longer periods of low power production can the power plant’s reservoir be forced to store more water and therefore will this reservoir be vulnerable for spillage if this reservoir has not the capacity of storing all this water. As a results of this, it is of great interest to analyse longer period of time (e.g. one week).

The model that is presented below is a model originally used in [12] and describes several

hydropower plants and each hydropower plant have about four or five variables that are optimised for every new time period. With a time period of 168 hours (one week) the optimisation problem will be very large with many variables.

The main improvement in this report compared to [12] is that the reservoir head-dependency of each hydro power plant is accounted for. This means that the described type of Linear Programming (LP) in [12] was not possible anymore. Therefore a more complicated Mixed Integer Linear

Programming (MILP) was used in this report. The reason for using MILP instead of LP is mainly due to the binary variables used for modelling of the head-dependency. Binary variables are basically variables that only can have the value 0 or 1. The drawback of using binary variables is that this will require more computational time and data capacity. The reason is that there is no coherent area of allowed solutions anymore; instead the allowed solutions are more like scattered points.

3.1 O

BJECTIVE OF THE MODEL

As mentioned above, the objective of the model is to minimize the spillage of the hydropower system, which means that a power plant spilled the water instead of producing it or storing it for other periods. The objective of the model is also to minimize the power that is not served by the hydropower system. By the unserved power it is meant that the hydropower system could not follow the extreme variations and therefore some of the power that should have been produced by the hydropower system, was not produced. Therefore, the objective function can be defined as follows:

𝑚𝑖𝑛 ∑ [∑ 𝑆_𝑖,𝑡

𝑖∈𝐼

+ 𝑈_𝑡]

𝑡∈𝑇

(7)

Equation (7) considers the spillage 𝑆𝑖,𝑡 in power plant i and during simulation period t and it also considers the unserved power 𝑈𝑡 during period t. The summation considers all power plants 𝐼 during the whole planning period 𝑇. However, it should be mentioned that the original model was

formulated in a different way.

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The original model was defined to test real wind power variations which mean that variating wind power generation data was obtained to study the effects of these variations on different hydropower systems. However, it was found that these variations were not very interesting due to the fact that the different hydropower systems could balance the variations easily. Therefore, it was decided to change the real wind power variations by huge fluctuations on the power output of the hydropower system, and then to study the impact of implementing these huge variations in the hydropower output. The old formulation of the problem is presented below.

The objective of the old formulation was to minimize the loss in electrical power by the hydropower system during high wind power variations. The power loss is defined as the power that is lost due to the water spillage from the hydropower stations. Another type of loss is the loss of wind power production. This basically means that the electricity produced by wind power is not used to supply the load. Therefore the optimisation problem will try to minimise the unused wind power too. Lastly, the unserved power has to be minimized. Unserved power means that at a specific hour the demand could not be fully supplied. Therefore, the equation defining the objective function can be

formulated as follows:

𝑚𝑖𝑛 ∑ [∑ 𝑃𝑆_𝑖,𝑡

𝑖∈𝐼

+ 𝑈_𝑡+ 𝑋_𝑡]

𝑡∈𝑇

(8)

Where 𝑃𝑆𝑖,𝑡 describes the power that would be produced by the spilled water in power plant i during hour t. 𝑈_𝑡 is defined as the unserved power during hour t, and lastly 𝑋_𝑡 is the unused wind power during hour t. The summation includes all power plants 𝐼 during the whole planning period 𝑇.

A comparison between (7) and (8) reveals two differences, the unused wind 𝑋𝑡 and the different definition of the loss from spillage. Basically in (7) the spillage is minimized as an amount of water meanwhile in (8) it is defined as the loss in energy that would be produced due to the water spillage.

𝑋_𝑡 Considers the wind power generation that is not used during a specific hour, this means that a hydropower plant overproduced power and therefore the wind power generation was not needed to supply any load.

However, both (7) and (8) still represents the same problem. This is the case because a “problem” in a hydropower system is defined as soon as any of the above variables occur in the system. Which means the value of the different variables presented is not important, what is important is that if any of the problems described above occurs in the system or not. The unused wind power in (8) basically means that a hydropower plant overproduced power. This means that the hydropower plant

discharged water that was not needed to be discharged, or basically discharged water that was not necessarily to supply the load. This is the exact same thing for spillage. A power plant spills water and therefore the power plant spilled water that was not used to supply the load. Therefore, if a problem occurs in one of the formulations above, that means it will certainly occur in the other formulation but maybe the problem occurs in another form, for example spillage expressed as unused wind power.

The same thing is valid for the definition of spillage, if spillage occurs in definition (7), then it is expressed as an amount of water. Meanwhile in definition (8) it is expressed as an amount of power.

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As a result, both definitions will show a problem in the system which indicates that the hydropower system failed at following the variations in the system.

3.2 H

YDROPOWER MODELLING

The most important part of the modelling is the hydropower modelling. To obtain as real power output variation capability as possible, the model of the hydropower system has to be as close to reality as possible. There are two main challenges while formulating a hydropower model:

The first challenge is to model the hydrological aspects of the hydropower system, which means that the water inflow as well as the water outflow of every power station in the hydropower system has to be described. The problem is further complicated when the water delay time from a station to another downstream one is accounted for. This delay time has usually nonlinear characteristics and therefore need to be linearized.

The second challenge is the modelling of the hydropower generation curve. A hydropower

generation is dependent on the so called head of the hydropower station. This means that the hydro power plant generates electric power by utilizing the difference in potential energy between the upper and the lower water level. The upper water level is defined as the level at the water intake and the lower water level is defined as the level of the tailrace, see Figure 5.

Figure 5: Simple description of a hydro power plant with a reservoir [20].

The difference in these two water levels is called the head of the hydropower station. The efficiency of a hydro power plant is depending on the head and the discharge through the turbine [20]. This dependency causes nonlinear relation between the power generation, the water discharge and the head. Therefore, to obtain a good approximation of hydropower generation, this nonlinearity has to be approximated by a linear function. One of the most important improvements in this report compared to [12] is that the head-dependence is modelled which means that the hydropower generation is now dependent on the water levels of the system.

3.2.1 MODELLING OF HYDROLOGICAL CONSTRAINTS

As mentioned above, the hydrological aspects of the hydropower model are defined. The

hydrological constraints describe the contents of water in the reservoir for each hour. There is of course water that flow in to a reservoir and there is water that flow out from a reservoir. The water that flow in to a reservoir is often the natural inflow and the water that was discharge from the

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upstream power stations. The natural inflow is often described as the water from rainfall and smelting ice. A simple representation of the reservoir contents at a specific hour can be as follows:

𝑅𝑒𝑠𝑒𝑟𝑣𝑜𝑖𝑟 𝑐𝑜𝑛𝑡𝑒𝑛𝑡𝑠 𝑎𝑡 𝑡ℎ𝑖𝑠 ℎ𝑜𝑢𝑟

= 𝑟𝑒𝑠𝑒𝑟𝑣𝑜𝑖𝑟 𝑐𝑜𝑛𝑡𝑒𝑛𝑡𝑠 𝑎𝑡 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 ℎ𝑜𝑢𝑟

+ 𝑤𝑎𝑡𝑒𝑟 𝑓𝑙𝑜𝑤 𝑇𝑂 𝑟𝑒𝑠𝑒𝑟𝑣𝑜𝑖𝑟 − 𝑤𝑎𝑡𝑒𝑟 𝑓𝑙𝑜𝑤 𝐹𝑅𝑂𝑀 𝑟𝑒𝑠𝑒𝑟𝑣𝑜𝑖𝑟

(9)

As it can be observed, the reservoir content at a specific hour is dependent on the contents at the previous hour. However, for the first hour, the definition of contents at the previous hour is not straight forward. Therefore, a parameter that defines the contents in the first hour has to be obtained, and it is defined as:

𝑀𝑠𝑡𝑎𝑟𝑡_𝑖 = 𝑃𝑒𝑟𝑐_{𝑠𝑡𝑎𝑟𝑡}∙ 𝑀_𝑖 (10)

Equation (10) defines a percentage of the maximal contents of the specified reservoir, 𝑀_𝑖 is a parameters which defines the maximal contents of reservoir i and 𝑃𝑒𝑟𝑐𝑠𝑡𝑎𝑟𝑡 is the percentage.

The most complicated formulation in (9) is the water flow to reservoir. As mentioned above this water inflow considers the natural inflow and the water discharged from the upstream power stations. This can be defined as follows:

𝑤𝑎𝑡𝑒𝑟 𝑓𝑙𝑜𝑤 𝑇𝑂 𝑟𝑒𝑠𝑒𝑟𝑣𝑜𝑖𝑟 = 𝑉_𝑖,𝑡+ ∑ 𝑄_{𝑘,𝑡−𝜏}_𝑘

𝑘∈𝑢𝑝𝑄

+ ∑ 𝑆_{𝑘,𝑡−𝜏}_𝑘

𝑘∈𝑢𝑝𝑆

(11)

Equation (11) considers the natural inflow (𝑉𝑖,𝑡) to a reservoir and the discharged (𝑄𝑘,𝑡−𝜏_𝑘) and spilled (𝑆_{𝑘,𝑡−𝜏}_𝑘) water to a reservoir from all the upstream power stations (upQ and upS is the number of upstream reservoirs that discharge and spill water to reservoir i). It can be observed that there is a time delay τ in the equation, the reason for this is that the water that is discharged from a specific power station will reach the appointed downstream reservoir after some time τ.

The delay time (τ) of the water from an upstream reservoir to a downstream one is a very

complicated function which can give the term 𝑄𝑘,𝑡−𝜏_𝑘 nonlinear properties. Therefore, to be able to obtain a linear function, the time delay in the discharge and spillage representation has to be linearized.

As mentioned before, this model uses a discrete time step in hours. Therefore, the time delay has to be given in hours (t-τ); otherwise the model may face problems. The delay times of the discharged water from a power station to a downstream one is often given in hours and minutes. Therefore to compensate for this, the discharged water to a reservoir with time delay dependency can be approximated with the following equation [12]:

𝑄_{𝑖,𝑡−𝜏}_𝑖=𝑚_𝑖

60𝑄_{𝑖,𝑡−ℎ}_𝑖₋₁+60 − 𝑚_𝑖

60 𝑄_{𝑖,𝑡−ℎ}_𝑖 (12)

(23)

The delay time in (12) is divided in hours and minutes, e.g. if a delay time of 90 minutes is obtained, then ℎ𝑖 will equal to 1 (60 minutes) and 𝑚𝑖 will equal 30 (remaining minutes). The same type of representation is valid for the spillage to a reservoir.

Lastly, the water that flow out from a power plant has to be considered. This water can be defined as the water that is discharged from the power station and the water that is spilled from the power station at a specific hour. This definition can be formulated as:

𝑤𝑎𝑡𝑒𝑟 𝑓𝑙𝑜𝑤 𝐹𝑅𝑂𝑀 𝑟𝑒𝑠𝑒𝑟𝑣𝑜𝑖𝑟 = 𝑄_𝑖,𝑡+ 𝑆_𝑖,𝑡 (13)

This completes the hydrological constraints, all the factors, such as: local inflow, spilled water, discharge water and old reservoir contents, are now considered and the hydrological constraints can be modelled.

LIMITATION OF PARAMETERS

There are some limitations on different parameters such as reservoir contents and water discharge.

These limitations are often introduced to avoid drying out lakes and even to avoid flooding. These limitations are often decided by environmental courts and the reason for this is to prevent destruction or altering of the environment around the power plants.

Therefore some parameters have to be limited and these are:

𝑀_𝑖≤ 𝑀_𝑖,𝑡≤ 𝑀_𝑖 (14)

𝑄_𝑖 ≤ 𝑄_𝑖,𝑡≤ 𝑄_𝑖 (15)

𝑀𝑒𝑛𝑑_𝑖 = 𝑃𝑒𝑟𝑐_𝑖· 𝑀_𝑖 (16)

𝑀𝑒𝑛𝑑_𝑖 ≤ 𝑀_{𝑖,𝑡=𝑛} (17)

Equation (14) describes the limit of the contents of reservoir at each hour and (15) describes the limit of the minimal and maximal allowed discharge at each hour and power plant. This limit is decided to avoid drought or flood of rivers. Lastly, (16) and (17) describes the limitation on reservoir contents at last hour. This is a very important consideration. The simulated period is often limited to 1-7 days in short-term planning, and if no limitation on last contents of reservoir is considered, then there is a risk that the hydropower system will use all of its saved water just to satisfy the requirements on the simulated period. This is of course a big fault because in reality there is no last period and the hydropower system needs always water in its reservoirs.

3.2.2 HYDROPOWER GENERATION MODEL

As mentioned above the hydropower generation is one of the most complicated aspects while modelling a hydropower system. In this project a more complicated model for hydropower generation is used. This model is dependent on the reservoir contents and uses three different generation curves depending on the contents of the reservoir. However, a brief presentation of the

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linear model that were used in [12] will first be given to ease the presentation of the problem. This linear model represents a power generation which is only dependent on the water discharge.

Another improved linear model will also be presented. This model is still only dependent on the water discharge but here the shape of the generation curve is modelled by taking more specification of the power plant into account, such as the forebay level. This type of model is originally presented in [5].

MODELLING OF HYDROPOWER GENERATION BY LP

It is mentioned before that the hydropower generation, the discharge and the head have a nonlinear relation. The nonlinearity must somehow be approximated by a linear, often piecewise linear, function. This is necessary to be able to include the hydropower generation curve in the LP model.

However, the impact of the head is often considered as small and is often neglected in LP modelling [12, 20]. The problem is then to find a linear relation between the power generation and the discharge. The problem is that the hydropower generation does not increase linearly with increased discharge, and each discharge level can have a different effect on the generation. The quota between the hydropower generation and the discharge is called the production equivalent and it is measured in MWh/HE and defined as:

𝛾(𝑄) =𝐻(𝑄)

𝑄 (18)

The hydropower generation is defined as 𝐻(𝑄) and the discharge as 𝑄. Another important notion to obtain is the marginal production equivalent which is basically a measure of how much the power generation will increase for a small change of the discharge, it is also measured in MWh/HE and it is defined as:

𝜇 =𝑑𝐻(𝑄)

𝑑𝑄 (19)

The efficiency of the hydropower plants is often given as the current production equivalent divided by the maximum production equivalent. The efficiency of the system is measured in percent and can be described as:

𝜂 =𝛾(𝑄) 𝛾𝑚𝑎𝑥

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This relative efficiency is often maximised for some specific discharge levels. Therefore, to obtain a piecewise linear function, these maximal efficiency points are used as breakpoints between the linear segments. Another important formulation is to use the whole previous production segment before the next segment is utilised. Therefore, the production equivalent for the next segment should always be decreasing compared to the previous segment. As a result, an optimal solution will always use the whole lower segment before starting at a new segment However, if there are limits on the maximum allowed power outputs for the different power plants then this will not be the case.

To ensure that the previous segment is fully utilised before the next one starts, binary variables have to be included, and this is not possible in LP.

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According to [12] it is assumed that the highest efficiency is achieved at 75% of the maximal discharge. Therefore the discharge is divided into two segments with the following formulation:

𝑄_𝑖,1= 0,75 ∙ 𝑄_𝑖 (21)

𝑄_𝑖,2= 0,25 ∙ 𝑄_𝑖 (22)

Where 𝑄_𝑖,1 and 𝑄_𝑖,2 represent the maximal discharge for segment 1 and segment 2, respectively. As mentioned above, the second segment should have a lower marginal production equivalent than the first one and therefore the marginal production equivalent, which is basically the slope, is assumed to be 5% lower for the second segment compared to the first one:

𝜇_𝑖,2= 0,95 ∙ 𝜇_𝑖,1 (23)

The maximal production of the hydro power plant, which is the maximal capacity, is then given as:

𝐻_𝑖= 𝑄_𝑖,1∙ 𝜇_𝑖,1+ 𝑄_𝑖,2∙ 𝜇_𝑖,2 (24)

From (24) the slope for segment one can then be obtained. By using (21) and (22) it can be simplified to:

𝜇𝑖,1= 𝐻_𝑖

0,9875 ∙ 𝑄_𝑖 (25)

The slope for the second segment can then be obtained by using (23). Lastly, the power generation from each station and at each hour can be defined as:

𝐻_𝑖,𝑡 = ∑ 𝜇_𝑖,𝑗∙ 𝑄_{𝑖,𝑗,𝑡}

𝑗

(26)

Where 𝑄𝑖,𝑗,𝑡 is the discharge at power plant i, segment j and hour t. The new formulation of the discharge will require new constraints such as 𝑄_{𝑖,𝑗,𝑡}≤ 𝑄_𝑖,𝑗 which basically means that the discharge at segment j at any moment cannot be greater than the maximal discharge at segment j.

With this formulation, the LP model for a hydropower plant is done.

IMPROVED LP MODEL

As mentioned in INTRODUCTION, an accurate representation of the head-dependency is not possible with LP methods. However, to represent a generation curve closer to the parameters chosen for the power plant, an improved LP model is presented. The idea that will be presented here is originally from [5] where the head-height is represented as the difference between the average forebay level and the average tailrace level. These averages are calculated by observing the lower and the upper values of the reservoir levels and the discharge levels. As a result, the power generation curve will be

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a constant but will correspond to a more realistic operation due to more parameters are included in the shape of the curve. The nonlinear dependencies can also be avoided by assigning constant forebay and tailrace levels. The procedure from the nonlinear formulation to the linear formulation of the problem will be described in detail below.

The forebay level and the tailrace level can be described by considering fourth degree polynomials as a function of the water levels at the reservoir and the discharge, respectively. The following

formulate the forebay level and the tailrace level [5]:

𝐹𝐵_𝑖,𝑡= 𝑎_𝑖,0+ 𝑎_𝑖,1∙ 𝑀_𝑖,𝑡+ 𝑎_𝑖,2∙ 𝑀_𝑖,𝑡² + 𝑎_𝑖,3∙ 𝑀_𝑖,𝑡³ + 𝑎_𝑖,4∙ 𝑀_𝑖,𝑡⁴ (27)

𝑇𝑅_𝑖,𝑡 = 𝑏_𝑖,0+ 𝑏_𝑖,1∙ 𝑄_𝑖,𝑡+ 𝑏_𝑖,2∙ 𝑄_𝑖,𝑡² + 𝑏_𝑖,3∙ 𝑄_𝑖,𝑡³ + 𝑏_𝑖,4∙ 𝑄_𝑖,𝑡⁴ (28)

Equation (27) describes the forebay level as a function of the reservoir contents (𝑀𝑖,𝑡). As observed, the reservoir contents will increase and decrease nonlinearly and therefore the nonlinear

characteristics are also obtained for the forebay level. Equation (28) describes the tailrace level and it is a function of the total discharged water (𝑄𝑖,𝑡), once again the discharged water increases and decreases nonlinearly and therefore the nonlinear behaviour of the tailrace level. The head-height of a power plant is then simply described as the difference between the forebay level and the tailrace level and can be formulated as follows:

𝐻𝐻_𝑖,𝑡= 𝐹𝐵_𝑖,𝑡− 𝑇𝑅_𝑖,𝑡 (29)

As mentioned, the nonlinearity of the formulation above will cause problems in the optimization model and therefore this nonlinearity has to be formulated by a linear approach. A straightforward solution is to decrease the degree of the polynomials to one, this will simplify (27) and (28) to:

𝐹𝐵_𝑖,𝑡= 𝑎_𝑖,0+ 𝑎_𝑖,1∙ 𝑀_𝑖,𝑡 (30)

𝑇𝑅_𝑖,𝑡 = 𝑏_𝑖,0+ 𝑏_𝑖,1∙ 𝑄_𝑖,𝑡 (31)

Even in (30) and (31) the nonlinearity is still there due to the variables 𝑀𝑖,𝑡 and 𝑄𝑖,𝑡. Therefore, to avoid the nonlinearity, the forebay level and the tailrace level are assumed instead to be constants and are defined here at the maximal values of the reservoir contents and the discharged water respectively.

As observed, to be able to calculate the forebay level and the tailrace level the coefficients have to be obtained. These coefficients are obtained by studying real cases. In [21], a test case of the Brazilian hydropower system is studied and the calculation of the forebay level and tailrace level is done in the same approach as (27) and (28). The coefficients are obtained for 32 power plants for both the tailrace level and the forebay level. Therefore, in this project, these coefficients will also be used. However, in [21] the coefficients are specified for specific stations, which takes into accounts the design of the reservoirs and the discharge specifications of the power stations. As described earlier, the study in this report focuses on fictitious hydropower systems. Therefore, specific forebay

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coefficients will be used for all power plants and the same is applied for the tailrace coefficients. This will mean that depending on the reservoir size and the maximal discharge of the power plant will the power plants achieve different head-heights. This will also mean that if two power plants have the same reservoir size and maximal allowed discharge, then these power plants will have the same head-height. Therefore, the final presentation of the head-height will be as follows:

𝐹𝐵_𝑖= 𝑎₀+ 𝑎₁∙ 𝑀_𝑖+ 𝑎₂∙ 𝑀_𝑖²+ 𝑎₃∙ 𝑀_𝑖³+ 𝑎₄∙ 𝑀_𝑖⁴ (32)

𝑇𝑅_𝑖 = 𝑏₀+ 𝑏₁∙ 𝑄_𝑖+ 𝑏₂∙ 𝑄_𝑖²+ 𝑏₃∙ 𝑄_𝑖³+ 𝑏₄∙ 𝑄_𝑖⁴ (33)

𝐻𝐻𝑖= 𝐹𝐵𝑖− 𝑇𝑅𝑖 (34)

Where the coefficients 𝑎𝑛 and 𝑏𝑚 are values taken from Table A.II and Table A.III presented in [21]. It is important to mention that the forebay level is calculated by considering Hm³ (cubic hectometres, which equals 10⁶ cubic metres) instead of Hour Equivalent as reservoir contents. In this project, the reservoir content is usually measured in HE (hour equivalent), one HE correspond to the water flow of 1 m³/s during 1 hour (which means 3600 m³). Therefore, to obtain the reservoir contents in Hm³ instead of HE, a multiplication with 3600 m³ and then a division by 10⁶ m³ is required. The maximal discharge in (33) is modelled as m³/s which is the same unit as the one used in this project. Notice that if a discharge of 200 m³/s is applied, that is exactly the same thing as a discharge of 200 HE/h, because 200 m³/s during 3600 s equals 720 000 m³, which equals to 200 HE.

By applying the coefficient values obtained from [21] to equation (32) and (33) the following figures can be obtained:

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Figure 6: Forebay level as a function of reservoir contents for four different (four different power plants) forebay coefficients obtained from [21].

Figure 7: Tailrace level as a function of water discharge for the same power plants used as Figure 6.

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As seen in Figure 6 and Figure 7, four power plants are studied. Each power plant has its special specifications (coefficients) and therefore the difference. The height of the forebay and the tailrace level is given in metres over the sea level. In the simulation it is decided that power plant 4 (see figures above) will be used because the difference in the forebay and tailrace level is highest for this power plant. This will mean that the dependency and variations of the head-height will be greater for different values of the maximal discharge and the reservoir size.

To make the power generation of each hydropower plant to be dependent on the head-height of the power plant a new formulation of the power generation has to be done. As seen in (26) the power production is formulated by multiplying the discharge by the marginal production equivalent. This means that a power plant will reach its maximal production only when the power plant is discharging water at its maximal discharge. Therefore, a multiplication with the head-height will overestimate the production capability of the power plant and the power plant will start to produce at the maximal capacity at much lower discharge values. Of course the units of the equation would be wrong too. Therefore, the power generation will instead be modelled as follows [5]:

𝑃_𝑖,𝑡= 𝐾 ∙ ∑ 𝑄_{𝑖,𝑗,𝑡}∙ 𝜂_𝑗∙ 𝐻_𝑖

𝑗

(35)

There are two major differences between (35) and (26). The first thing is that 𝜂_𝑗, which represent the efficiency of the power plant at segment j, is modelled instead of 𝜇 which was called the marginal production equivalent in (26). As presented in [6] and [21], a constant efficiency at each segment is applied for the power plants. Instead in this project, the value of the efficiency 𝜂 is calculated as an average of the values of efficiencies presented in [21]. Notice that the efficiency values presented in [21] are calculated with inclusion of the constant K represented here, see below.

The other difference between (35) and (26) is that a constant K is now included to the definition. The constant represents the water density and the acceleration of gravity and has a value of 0.00981, which basically is calculated by multiplying the water density (1000 kg/m³) with the acceleration of gravity (9.81 m/s²) and then divided by 10⁶ to obtain the power output in MW. The reason for the inclusion of the water density and the gravity is to take into account the water drop which is given as head-height in metres. By analysing the different units in (35) it can be obtained that the power will be given in watt which has the SI units ^{𝑘𝑔∙𝑚}_𝑠₃ ².

The formulation above takes into account the head-height of the power plants and depending on the maximal discharge and the maximal reservoir contents different head-height can be achieved.

However, the hydropower generation is still constant during simulation, which means that the power generation of a hydropower plant remains only as a function of discharge. But, an improvement to the first formulation is that the power generation of the power plant can now be more accurate and the shape of the generation curve is now more dependent on the specifications (parameters) of the power plant. If a real hydropower system is simulated, then this approach would represent a closer case of the reality because it takes into account more parameters of each power plant in the system.

MODELLING OF HYDROPOWER GENERATION BY MILP

As observed from the previous section, the head-height is assumed to be constant and the

generation is assumed to be piecewise linear with two segments. This section will show another type

Analysis of Hydropower Systems' Ability to Follow Square Wave Load Variations