Optimizing profits for hydropower producers using dynamic models

Hannes Tysén, Bore Sköld

Spring 2013

Bachelor Thesis, 15 ECTS

Politices kandidatprogram 180 ECTS

Optimizing profits for hydropower

producers using dynamic models

A theoretical approach to solve the optimal use of water over

time

1

Abstract

2

Table of content

Abstract ... 1 Table of content ... 2 1. Introduction ... 3 1.1 Background ... 3

2. Method and theory ... 5

2.1 Purpose ... 5

2.2 Earlier research ... 6

2.3 Power market characteristics ... 7

2.4 Dynamic profit model ... 10

Conditions ... 12

2.5 Deriving the optimal path ... 13

The phase diagram ... 14

2.6 Interpreting the model ... 16

3. Discussion and conclusions ... 17

4. References ... 20

3

1. Introduction

1.1 Background

In the 1990s Sweden and other countries in northern Europe, namely Norway, Finland and Denmark, began a deregulation of their power markets. These deregulation from government owned monopolies, to a more open market, has led to vital changes in the fundamental way of distributing and producing electricity. To handle the exchange of electricity, the Nordic countries formed a cooperation for buying, distributing, and redistributing power within the region. This cooperation, started with the Swedish deregulation in the mid-nineties and fully integrated with the other countries in 2002 when the Danish border tariff was removed,

became Nord Pool (“Nord Pool Spot”).Data beginning from 2002 will therefore be consistent

4

a rise in the price of electricity. The effects of radial congestion and network externalities was explained further by Edward P Kahn.(Kahn 1998)

It is not only the recent deregulations that set this market apart from others but also the means of production. On the power market we have many agents using different technologies and fuel sources to produce an identical good. Power produced from wind is in no way different

from the power produced by fossil fuels like oil and coal. The majority, over 50%, of the

Nordic power comes from hydropower (NordReg 2012). These hydropower plants have many attributes that make them special, chief among these is that their fuel source is completely free, storable and only restricted by natural variables such as downpour which are reasonably predictable. Combined, these circumstances creates an interesting dynamic where the final verdict on the end result of the deregulation is still very much up for debate.

Efforts to evaluate the deregulation effects have had varying results mostly due to the fact that it is hard to isolate what changes in the market characteristics can be attributed to the deregulation reforms. One of the few apparent effects has been the sudden change in price volatility; consumers today face a more volatile price than before the regulation. New empirical research on this matter has been conducted by (Hellström, Lundgren, and Yu 2012), that came to the conclusion that “The occurrence of electricity price jumps is often loosely motivated by shocks to the electricity demand (e.g., caused by sudden large changes in temperature) or by shocks to an inelastic electricity supply (e.g., caused by production or system breakdowns)”. They tried to separate “normal” variation in price changes from so called “price jumps” using a mixed GARCH-EARJI- model. It seems that the effect demand and supply chocks have on price volatility is determined by how the market is structured. If a market is working closer to the capacity constraints, the risk for price jumps will increase. In turn, the effects that price jumps may have on the spot market in terms of the day-ahead forecasts are volatile due to exogenous effects that have to be accounted for (Haugom et al.. 2011). Even though we can see these volatilities, data shows that price will still follow a relatively predictable curve over a year, shown in Graph7.

5

level of market power (Bask, Lundgren, and Rudholm 2011). Other studies have analyzed

similar problems and ended up with the same result. However, most of these studies argue that in general these levels are not large enough to conclude that the market has monopolistic characteristics. Hjalmarsson conducted a study on this topic using a dynamic extension of the Bresnahan-Lau model and came to the conclusion that rejected the hypothesis of market power (Hjalmarsson 2000).

New environmental regulations have further complicated matters for anyone who wants to analyze the power market (Crampes and Moreaux 2001). Carbon taxes and different certificates change the shape of the aggregated supply curve in peculiar ways. These efforts to push production into the more “green” alternatives have among other things led to wind power having a negative marginal cost. In our models we will disregard the inner workings of these policies and instead assume that they are accounted for in the aggregated supply curve. With all of this, one glaring thing still remains unaccounted for if one wants to consider this market competitive. The price does not equal the marginal costs for all firms. In fact, the hydro and wind power producers meet basically constant marginal costs well below the price level for power which in turn is hard to align with the theory of perfect competition (Kotchen et al.. 2006). This is one of the reasons why we believe that market power exist and therefore we feel comfortable using monopolistic behavior when constructing our model.

2. Method and theory

2.1 Purpose

6

the year will evolve (Byström 2005). To simplify things we will assume that only the output from hydropower will vary over time in the supply curve, this might not be entirely true but the assumption is not as farfetched as it first appears. Among the fuel sources, water is the only one that is truly at nature’s mercy as the net inflow of new water varies greatly over a year. Spring floods, frozen water and dry periods all occur over time with regular intervals. Defining the price function can be difficult due to the varying nature of the market, there are a lot of things at play that is affecting the market price and we cannot take all aspects into account. Based on this, and the reasoning about market power discussed earlier, we will assume a monopolistic behavior in the price function when creating the model.

Hydro- and wind power are two quite unique production forms as both input variables are free to the producers. Wind power has one major drawback in that it is impossible to store, no wind, means no power. Water on the other hand has none of these issues as it is easily storable in reservoirs without any major upkeep costs where the only issue is the rate of evaporation and the capacity of the reservoir.

2.2 Earlier research

Optimal control is commonly used for analyzing the use of resources over time. Since the usage of power is ever increasing while the natural resources available are limited, any type of optimization will require the usage of dynamic models. This has become an important area of research in order to satisfy the wide range of requirements one might face dealing with limited resources. Applying optimal control to solve problems is not only used in economic research but in a wide range of scientific disciplines, of which many deal with similar issues when working with these matters. For instance, the rate of deforestation is not just interesting for the economist trying to maximize a logging company’s profit, but also for an ecologist trying to maintain a natural balance.

7

On the subject of managing water release, different approaches can be taken. Kyung, Kim, Jung and Eom studied a river system with several dams in South Korea in order to stabilize power outputs to offset seasonal effects on power supply. Using optimal control methods they successfully constructed a model to solve this issue. By simulating data, using this method, they came to the conclusion that an increased output could be achieved (Kyung et al.. 2010). However, this paper was written by engineers and thusly most economic theory was absent. Regardless the basic premise remains the same and many of its components can be altered to fit into a more economic-oriented model.

As mentioned earlier, Rudholm and Lundgren along with Hjalmarsson had a different approach when studying the power market. Using different varieties of the Bresnahan-Lau model they sought to check if any agents exercised market power. Hjalmarsson, using a dynamic augmentation of the Breshnahan-Lau model could conclude that the spot market was completely competitive and that it showed no signs of market power (Hjalmarsson 2000). Both of these studies regard the market more or less as a single entity, which is by no means a faulty assumption. However, we would argue that there is merit in using a different approach that tries to take into account the very unique properties of hydropower production and its potential influence on the price of power.

2.3 Power market characteristics

Normally we would assume that on a market characterized by perfect competition, price would be determined by the market equilibrium and that each agent on the market will act accordingly. However, if we assume some sort of quasi-competitive market instead where each agent is subject to the market equilibrium except the hydropower producers, who have some degree of control over the supply curve, then these producers will be able to affect the price by regulation of their output. This assumption is possible due to the characteristics of the power market where the price is determined by the last power source used, i.e. the market will consume every unit of power supplied from a cheaper source before moving on to a more expensive one. Nord Pool explains that the price formation process is economically effective for society because of this argument (NordPool 2013). They state two arguments for balanced price:

8

ii) The price that the consumer group is willing to pay for the final kWh required to satisfy demand.

The reference price, i.e. the spot markets equilibrium price for futures and forwards markets, are for Nord Pools’ system of vital importance concerning their use of hydropower. It is necessary for efficiency that the reference price reflects the water level in the reservoirs at different times to obtain optimal use of hydropower in different periods of time. Since price levels are determined by supply and demand on the spot market, hydropower generation and its water levels are of great importance to our study (Hjalmarsson 2000). The static supply-demand graph for the power market is illustrated in Graph 1.

Graph 1, Power supply & demand, Energy Markets Expectorate

9

deregulated market might give an indication of as to why the market is somewhat chaotic. We argue that in a few years we might see a more stable oscillation.

Even though price peaks are more common in winter, price jumps follow an opposite seasonality and occur more often during the summer months. They might not be as high, and have such a great impact as the once we get during winter, but they occur more frequently. Hellström, Lundgren, Yu explains this as being normal since we actually expect price jumps in winter, and therefore are better prepared contrary to the summer (Hellström, Lundgren, and Yu 2012). The difference is that price jumps that occur during the darkest and coldest months have a greater impact since the power production is peaking in winter. The higher demand of power puts pressure on production triggering supply to increase to a level where import is needed to sustain balance in supply-demand. Import of power mainly consists of more expensive sources of power, such as oil, gas, coal etc. The same price shocks would theoretically occur when we experience a negative supply shock; a dam breaks, a nuclear reactor has to shut down etc. This would be consistent with our hypothesis that there might exist incentives for a firm to lower output of hydropower in order to increase price and therefore gain more profit. Combined, all of these structural characteristics of the power market concludes that no long-run equilibrium price can be achieved. The price should however stabilize around a certain pattern that is illustated in Graph 2. This helps to understand the complex nature of this market and the mechanics that determine the price of electricity. It appear to be a market that reacts quickly to changes in supply and demand which we argue can make it susceptible to abuse from firms with the type of control over their output that the hydropower producers have.

10

2.4 Dynamic profit model

To create our model we will use optimal control theory and Pontryagin’s maximum principle and with it illustrate how to derive what the optimum path is (Chiang 1992). Our base model will be a simple profit function: , where in our model is the price of electricity, which is the same on the market no matter what power resource we include. This price will have stochastic attributes in the supply and demand due to the randomness of their fluctuations. We will however disregard this to avoid complications and instead use a monopolistic price function. As stated earlier we will deviate from the usual course where we have agents as price takers and instead assume that hydropower producers can affect the price by regulating their output.

The model contains a number of variables that are all generalized to fit the model. is a price function for the aggregated market, where is an output function for just the hydro power and is a constant representing all other sources excluding hydropower. To create a similar problem for a single producer all that is needed is to use for one firm and then create a sum of all to use as the new . Obviously the other sources can’t be described with a single constant but we use this to simplify matters for easier computing. Regarding , it is simply a function where is multiplied by a constant to represent the rate at which water passing through the plant is turned into power. is the water release and is the technology constant. That means that explains at which rate each unit of water is turned into units of power. The cost function will remain constant here, again for easier computation but partly because it is not at all far from the truth, hydro power plants have almost no varying costs and the total cost will remain almost the same regardless of how many units of power are produced (Kahn 1998). The price-function depends on both the amount of output we produce as hydropower producers and on the rest of the power being produced by all other sources (wind, nuclear, oil, gas etc.) The price-function

will have the characteristics of a monopoly price function: . Here

is varying over time, which is true but to simplify we will treat it as constant.

As for the restriction function explains the water level in the dam at any given time

is the natural rate of evaporation and is a constant for the external changes to dam water

11

again quite a simplification of the truth when we claim that is constant since this variable

will take on a stochastic characteristic. We do this to avoid complicating things needlessly. The profit maximization model is stated in equation (1.1):

∫[ ( ) ]

_(1.1)

(1.2)

̇

(1.3)

T

he change of water is explained by the formula in equation (1.3) which includes inflow,

evaporation and release. These are general statements which has to be determined if one wants to evaluate a specific hydropower plant or group of producers (Kyung et al.. 2010). The variables regarding natural occurrences such as downpour has, much like the price, a stochastic nature. We will disregard this for the same reason as for the price.

To be able to maximize profit in our dynamic model (1.1), we will construct the current value Hamiltonian for this problem with the restriction being the change in water level over time. The Hamiltonian model follows:

( )

(1.4)

{ | ( ) }

(1.5) will be chosen from a vector stated in equation (1.5). The lower limit will be restricted by the minimum amount of release that is allowed while the upper limit is restricted by the remaining water or the maximum amount of release possible at any given time period.

The level of water in the reservoirs has their own restrictions which is why we state

12

reservoir maximum capacity. Secondly, the reservoirs must keep a minimum level of water at all times due to regulations demanding that the hydropower plants spare supply of water can be used as a buffer if needed as well as for environmental reasons. The environmental reasons

are regulated by law in most countries. Hence we have where is the minimum

and is the maximum water level. Other papers on hydropower optimization uses similar

restrictions for the same reasons such as Zheng, Fu and Wei (Zheng, Fu, and Wei 2013).

{ | }

(1.6)

The last two restrictions mentioned in (1.5) and (1.6) will also be disregarded in our simplified modeled for the same reasons we disregard the stochastic variables. While it is interesting these restrictions complicates matters immensely without necessarily bringing much worth to the table.

Conditions i.

ii.

̇

iii.

̇

iv.

(2.1)

(2.1)

13

_{[ ( )]}

_(2.2)

2.5 Deriving the optimal path

In order to find if there is an optimal path, we want to create a phase diagram. To do so we

must first derive suitable expressions for

̇,

the time differentiated state variable, as well as

for ̇

,

the time differentiated control variable. Below follows the optimization by using partial

derivatives.

The first-order condition to maximize follows from equation (2.4)

(3.1)

Followed by the second-order condition that proves that we do indeed have a maximum.

(3.2)

Using the costate-equation and substituting in (2.5) we derive an expression for ̇ depending on as shown in (3.5).

(3.3)

̇

14

̇

(3.5)

In (3.6) we differentiate from equation (2.5) over time in order to get a

time-differentiated expression for our control variable yielding (3.7).

[ ( )]

(

)

̇

(3.6)

̇

̇

(3.7)

In the next stage we substitute ̇ which gives the following expression:

̇

(3.8)

The phase diagram

Using these expressions we can construct a phase diagram

15

̇

(3.12)

(3.13)

Graph 3, the phase diagram

With these results we can create a phase diagram (Graph 3) in the

-

space where the two

16

2.6 Interpreting the model

The price functions negative relation with is defined by differentiating the profit function

with respect to , and can best be described graphically. Due to the characteristics of the

aggregated supply and demand functions, any attempts to derive a function for how the price at equilibrium shifts over time would be complicated unless assumptions are made since the function is not continuously differentiable. The assumptions we have made regarding the monopolistic price function solves some of these issues without deviating too much from reality. We want the supply of hydropower to be excluded from the other power sources to be able to evaluate how a shift in hydropower supply affects the market while keeping other sources unchanged.

Using Graph 4 as intuition we can quite clearly see that shifts in the supply of hydropower will have the negative relation between price and hydropower we previously concluded. The step-like character of the function means that we might have price plateaus, for simplicity we will however assume that the change in price is continuous. One way to visualize this would be a deliberately induced supply chock on the market, where all other agents treat the lowered supply as such.

Graph 4, cut in power supply

We should not forget how the upper and lower limits on and , as defined by (1.5) and

17

explain the limits put on each variable and and when included into the Hamiltonian these would be inserted as extensions on the model along with their respective multiplier

functions and

.

As we optimize our expression we will assume that these Lagrange

multipliers are equal to zero (Kamien and Schwartz 1991). Worth noting here is that forcing these two to equal zero effectivly means that the water levels are irrelevant, an aspect that has to be taken into account. Looking at our phase diagram (Graph 3) it shows us that any point beyond the equilibrium will most likely break the restrictions. The streamlines all move away

from the stable point towards unsustainable levels of , it either goes toward infinity or to

zero both breaking the ristrictions.The same analysis for the -variable is not as obvious

however logically if move toward its extremes x would also end up breaking the

restrictions. In practice this would mean that when the water release rate reaches its extremes, over time the reservoir will either overflow, or empty.

Our final expressions (3.9), (3.10) indicate that there is an optimal price that is fixed and independent of our state- and control-variable. There is a given that will optimize our profit independent of and to achieve a stable equilibrium we need a level of that can sustain this

given . This point is when the inflow of water equals the combined outflow of our given

and the evaporation rate.

3. Discussion and conclusions

18

wants to evaluate this situation a more rigorous model would be needed or a completely different approach should be taken.

One way to develop our model would be to extend the theoretical part of the model, beyond the above mentioned, to include more restrictions and other effects that most certainly occurs on the market. While developing it should lead to a model that better explains the inner workings of hydropower one has to be careful not to end up with unusable results. As Conrad & Clark argues, more extensive models quickly become very hard to interpret as the mathematics required tends to turn into a complicated mess of undefined results. Including the stochastic nature of some of the variables would nevertheless seem reasonable as both the environmental variables as well as the variables affecting the price have these properties. The reason we excluded these characteristics is simply because we wanted to create a general problem why adding stochasticity would complicate matters immensely. Without the stochasticity the model is simply unfinished.

Another way to develop it would be to create a specific problem for a single, or a group of power plants. Deriving all the variables would make it possible to simulate these effects with real data. However determining certain variables such as the effects on the price could prove difficult because of the volatile nature of the market price mentioned by Jens Lundgren. In creating this problem with real world data, the inclusion of stochasticity in the variables mentioned above should be somewhat easier than in a general model since fewer values end up unknown when we have data to test for. In general it is easier to extend the model when using a specific set of data to determine variables for the same reason. However with these added variables comes the ever present uncertainty inherit in any statistical estimation.

19

20

4. References

Abel, Andrew B. 1982. “Andrew B. ABEL*” 9: 353–73.

Bask, Mikael, Jens Lundgren, and Niklas Rudholm. 2011. “Market Power in the Expanding Nordic Power Market.” Applied Economics 43 (9): 1035–43.

doi:10.1080/00036840802600269.

Byström, Hans N.E. 2005. “Extreme Value Theory and Extremely Large Electricity Price Changes.” International Review of Economics & Finance 14 (1): 41–55.

doi:10.1016/S1059-0560(03)00032-7.

Chiang, Alpha C. 1992. Elements of Dynamic Optimization. Search. Vol. 47. McGraw-Hill. Conrad, Jon M., and Collin W. Clark. 1987. Natural Resource Economics.

Crampes, Claude, and Michel Moreaux. 2001. “Water Resource and Power Generation.”

International Journal of Industrial Organization 19 (6): 975–97.

Fortum, Rakel. 2010. “RAKEL Fortum 1.” (2014-03-25)

Haugom, Erik, Sjur Westgaard, Per Bjarte Solibakke, and Gudbrand Lien. 2011. “Realized Volatility and the Influence of Market Measures on Predictability: Analysis of Nord Pool Forward Electricity Data.” Energy Economics 33 (6). Elsevier B.V. 1206–15. doi:10.1016/j.eneco.2011.01.013.

Hayashi, Fumio. 2013. “Tobin's Marginal Q and Average Q : A Neoclassical Interpretation Author ( S ): Fumio Hayashi Published by : The Econometric Society Stable URL : http://www.jstor.org/stable/1912538 .” 50 (1): 213–24.

Hellström, Jörgen, Jens Lundgren, and Haishan Yu. 2012. “Why Do Electricity Prices Jump? Empirical Evidence from the Nordic Electricity Market.” Energy Economics 34 (6). Elsevier B.V. 1774–81. doi:10.1016/j.eneco.2012.07.006.

Hjalmarsson, Erik. 2000. “Nord Pool: A Power Market without Market Power.” Rapport Nr.:

Working Papers in Economics, 1–39.

Kahn, Edward P. 1998. “Numerical Techniques for Analyzing Market Power in Electricity.”

The Electricity Journal 11 (6): 34–43. doi:10.1016/S1040-6190(98)00057-8.

Kamien, Morton I., and Nancy L. Schwartz. 1991. “Dynamic Optimization.”

Kotchen, Matthew J., Michael R. Moore, Frank Lupi, and Edward S. Rutherford. 2006. “Environmental Constraints on Hydropower: An Ex Post Benefit-Cost Analysis of Dam Relicensing in Michigan.” Land Economics 82 (3): 384–403.

Kyung, Yeosun, Joo-woong Kim, Sung-boo Jung, and Ki-hwan Eom. 2010. “Optimal Control Method for a Hydroelectric Power Development in Multi Level Dams” 3 (3): 13–24. Lundgren, Jens. 2012. “Market Liberalization and Market Integration : Essays on the Nordic Electricity Market.” http://www.diva-portal.org/smash/record.jsf?pid=diva2:570731. “Nord Pool Spot.” http://www.nordpoolspot.com/About-us/History/.(2014-03-25)

NordPool. 2013. “Nord Pool Spot.” http://www.nordpoolspot.com/How-does-it-work/Day-ahead-market-Elspot-/Price-formation-in-Nord-Pool-Spot/.(2013-12-10)

NordReg. 2012. “Nordic Market Report 2012.”

Zheng, Ying, Xudong Fu, and Jiahua Wei. 2013. “Evaluation of Power Generation Efficiency of Cascade Hydropower Plants: A Case Study.” Energies 6 (2): 1165–77.

21

5. Appendix

Graph 5, Total electricity supply Sweden 2008-2012, SCB

Graph 6, Total electricity consumed Sweden 2008-2012, SCB

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20 08M0 1 20 08M0 4 20 08M0 7 20 08M1 0 20 09M0 1 20 09M0 4 20 09M0 7 20 09M1 0 20 10M0 1 20 10 M0 4 20 10M0 7 20 10M1 0 20 11M0 1 20 11M0 4 20 11M0 7 20 11M1 0 20 12M0 1 20 12 M0 4 20 12M0 7 20 12M1 0

total supply of electricity

hydro power, net

wind power

nuclear power, net

conventional hydro power, net

resistance, industrial

resistance,combined heat and power (CHP)

conventional hydro power, condensation production conventional CHP, gas turbines and other production

import

produktionsslag:

vattenkraft (inkl pumpkraft), netto:

Förbrukning för pumpkraft fråndragen

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20 08 M0 1 20 08M0 4 20 08M0 7 20 08M1 0 20 09M0 1 20 09M0 4 20 09M0 7 20 09M1 0 20 10M0 1 20 10M0 4 20 10M0 7 20 10M1 0 20 11M0 1 20 11 M0 4 20 11M0 7 20 11M1 0 20 12M0 1 20 12M0 4 20 12M0 7 20 12M1 0

total electricity consumed export

domestic consumption mining and manufacturing electricity, gas, heat, hydro power train, trams, busses

housing, services etc. housing, temp.-compensated losses

grid losses

22

Graph 7, Price of electricity on Nord Pool Spot 2008-2012, Nord Pool Spot