Spot Market Bidding Taking the Balancing Power Market Into ......The spot market, Elspot, is organized by Nord Pool Spot (NPS). This is the largest market place for purchase and sale

Spot Market Bidding Taking theBalancing Power Market Into

Account

Specialization ProjectTIOslash4500

Anders Lund EriksrudJoslashrgen Braathen

Norwegian University of Science and TechnologyDepartment of Industrial Economics and Technology Management

Supervisor Professor Stein-Erik FletenDecember 14 2012

Abstract

This paper discusses whether a hydropower producer in the Nordic region should takethe Balancing Power Market into account in the spot bidding decisions A stochasticprogramming model for the coordinated bidding problem is developed and tested for aNorwegian watercourse during the fall of 2012 Further the paper analyses in whichsituations a coordinated bidding model may outperform a sequential model The paperconcludes that a coordinated model primarily is useful in situation in which the deviationbetween the spot price and the marginal water value is small However the model needsmore testing before a final conclusion can be made The deterministic equivalent of thestochastic programming problem is implemented which resulted in very long runningtimes An appropriate solution method needs to be found

Preface

This paper is written as a specialization project for the Master of Science degree atthe Norwegian University of Science and Technology (NTNU) department of IndustrialEconomics and Technology Management within the field of Managerial Economics andOperations Research TIOslash4500 First and foremost we would like to thank our supervi-sor Professor Stein-Erik Fleten for helpful assistance throughout the semester and PhDcandidate Gro Klaeligboe at Department of Electric Power Engineering at NTNU for hourswith valuable discussions We would also like to thank Lars Thore Wibe Aarrestad andthe good folks at Powel AS Smart Generation and Pal Otto Eide and Knut-Harald Bakkeat Norsk Hydro ASA for the great collaboration

Trondheim December 14 2012

Anders Lund Eriksrud Joslashrgen Braathen

Contents

1 Introduction 1

2 The Nordic Electricity Markets and Institutional Background 221 Electricity Markets 222 The Spot Market 523 The Balancing Power Market 624 The Value of Water 825 Scheduling Hierarchy 926 Short Term Production Scheduling 11

3 A Stochastic Programming Model for the Coordinated Bidding Prob-lem 1331 Modeling the Markets 1332 Problem Formulation 16

4 Scenario Generation 2041 Spot Prices 2042 Balancing Market Volumes and Premiums 2243 Scenario Generation 26

5 Case Study 3051 Case Description 3052 Results 3253 Comparison With Sequential Model 3554 Discussion 35

6 Conclusion and Future Work 3961 Future Work 39

1 Introduction

A recent study from the European Network of Transmission System Operators for Electric-ity (ENTSO-E) concludes that the frequency quality in the Nordic region is unsatisfactoryand has declined in recent years [5] The amount of imbalances in the system is expectedto increase as a result of several new interconnectors and increasing shares of non-flexiblerenewable power generation Figure 11 illustrates that there is a distinct relationshipbetween the transmission capacity out of the Nordic region and frequency imbalancesConsequently the markets for balancing power will be growing and more important inthe years to come Sale of balancing power through the interconnectors to Europe is alsointroduced and expected to be growing which is another indication that the turnover inthe balancing markets will increase [25] The Norwegian power system consists of a largeshare of reservoir hydropower which is well suited for delivering balancing power

Software for decision support in the spot market is well developed and used by most powerproducers in Norway There is however no commercial software available for multimarketbidding ie taking all physical markets into account in the bidding decisions Currentlythe quality of the bidding across markets is largely dependent on the experience of theproduction planner Some work ([8] and [6]) has been done within the field of multimarketbidding for the Nordic markets but no study has so far considered bidding strategiesacross the spot market and the Balancing Market in a producer perspective

This paper investigates whether a producer should take the Balancing Market into accountin the spot bidding phase Section 2 gives an introduction to the Nordic power markets andthe scheduling models used by Nordic hydropower producers A stochastic programmingmodel for coordinated bidding across Elspot and the Nordic Balancing Market is developedin Section 3 Section 4 presents a methodology for generating scenarios for the uncertaindata inflow spot price Balancing Market volumes and premiums The coordinated modelis tested for a Norwegian watercourse during the fall of 2012 in Section 5 Further adiscussion of when the coordinated model is beneficial is conducted The paper concludesthat a sequential model will be sufficient only if the spot price deviates largely from thewater values and that the coordinated model may be useful in the majority of the timeThe problem needs to be studied in more detail and future work is suggested in Section6

Figure 11 Cumulative interconnector capacities out of the Nordic region (columns) com-pared to frequency deviations (line) in minutes outside the region [499 501] Hz permonth The green areas represent new capacity [25]

2 The Nordic Electricity Markets and Institutional

Background

This section gives an overview of the different power markets in the Nordic region andhydropower scheduling models The focus in this paper is on the spot market and theNordic Balancing Market which are presented in detail Moreover scheduling modelscommonly used by hydropower producers in the Nordic region are described

21 Electricity Markets

The power markets in northern Europe have been in constant development in the pastdecades The power sector has changed from being publicly regulated to a market basedindustry Nord Pool was established in 1993 as an exchange for the Norwegian marketIn 1996 the exchange was extended to include Sweden and thus became the worldrsquos firstmultinational exchange for electricity [32]

Table 21 shows an overview over the current Nordic electricity markets and tradingroutines are illustrated in Figure 21 The spot market Elspot is organized by Nord PoolSpot (NPS) This is the largest market place for purchase and sale of physical electricityin the world In 2011 73 of all power in the Nordic region was traded on NPS [24] Theremaining is traded in Bilateral contracts Section 22 describes Elspot in detail

Market Place Physical trade Financial trade

Nord Pool Spot (NPS) ElspotElbas

Transmission System Primary reserve (FNR and FDR)Operators (TSOs) Secondary reserve (FRR)

Tertiary reserve (Balancing Market)

Nasdaq OMX Commodities FuturesForwardsOptionsContracts forDifference (CfD)

Bilateral Full delivery ForwardsLoad factor contracts Options etcSpot (cap and floor) etc

Table 21 Overview of the Nordic electricity markets

Figure 21 Trading routines in the Nordic electricity markets (markets for primary andsecondary reserves are not included)

NPS also organizes the continuous intraday market Elbas It provides the opportunity fortrading power across the Nordic region Germany and Estonia Total Elbas turnover in2011 was 27 TWh The total Elbas volumes traded yearly in Norway are very small andrepresent about 01 of the Elspot turnover Elbas opens at 1400 following the closingof the Elspot auction and trading capacities available for the next day are publishedTrades in Elbas are allowed up to one hour before real-time1 which gives the participantsthe opportunity to adjust for imbalances if production and consumption schedules deviatefrom the volume committed in Elspot Prices are set based on a first-come first-servedprinciple [24]

The Transmission System Operators (TSOs) are responsible for the grid stability and thepower balance in the system Imbalances caused by deviations from the production plansand load forecasts must be leveled out in order to maintain the instant balance in thesystem at any point of time Hence the TSOs need access to balancing power normallydivided into primary secondary and tertiary reserves

The primary reserve is automatic and activated at low frequency deviations to ensureinstantaneous power balance The procurement procedures varies across the Nordic coun-tries The Norwegian TSO Statnett requires a basic delivery from producers with gen-erators larger than 10 MVA Additional needs are procured in established markets (FNRand FDR) [28] The price is set as the marginal price for each price area

If the primary reserve is unable to handle the deviations the secondary reserve will beactivated The Nordic TSOs have decided to introduce a common secondary reservesolution from 2013 called Frequency Restoration Reserves (FRR) Statnett has signedagreements with five large producers but procedures for bidding and financial settlementsare not yet set [29]

Large deviations require activation of the tertiary reserve in the Nordic Balancing MarketThis balancing does not provide instantaneous power but is activated within 15 minutesThe Nordic Balancing Market is described in detail in Section 23

The participants are also able to trade electricity derivatives in the financial marketsoperated by Nasdaq OMX Commodities [4] Producers use the financial markets for pricehedging and risk management The system price (explained in Section 22) is underlyingthe financial derivatives and there is no physical delivery in the financial contracts Thefinancial market has a time horizon up to six years ahead There is also an instrumentfor hedging against price differences between price areas called Contracts for Difference(CfD)

1Statnett has decided to change the gate closure for the Elbas market in Norway from two hoursbefore real-time to one hour before real-time from February 26 2013

Figure 22 NPS price areas October 2012 [24]

22 The Spot Market

The spot market Elspot is a day-ahead auction in which hourly power contracts aretraded daily for physical delivery Total turnover in 2011 was 2944 TWh The Elspotmarket includes Norway Sweden Finland Denmark Estonia and Lithuania [24] In orderto handle grid congestions the Nordic exchange area is divided into price areas withindividual area prices as shown in Figure 22 Absence of bottlenecks results in equalprices for all areas The system price is an artificial price calculated for the entire NordicRegion as if there were no transmission limits

Prior to 1200 noon producers and consumers submit bids for selling and buying electricityfor the next day that is the next 12-36 hours NPS then calculates the area prices foreach hour and area The prices are normally published between 1230 and 1245 Figure23 shows historical Elspot prices in the Norwegian price area NO2 Three different typesof bids can be submitted

bull Single hourly bids represents the largest share of the Elspot trading The partic-ipants specify the purchase and sales volumes for each hour and choose between aprice dependent and a price independent bid The minimum requirement for a singlehourly bid is two pricepoints at the minimum price (-e200) and the maximum price(e2000) A price dependent single hourly bid may consist of up to 62 pricepointsin addition to the upper and lower pricepoints Furthermore the bidding curve

Figure 23 Elspot prices in NO2 01012010 - 10032012 [24]

must be non-decreasing A participant that submits price dependent bids acceptthat NPS will make a linear interpolation of volumes between each adjacent pair ofsubmitted pricepoints

bull Block bids give the participants the opportunity to set an all-or-nothing conditionfor a set of at least three consecutive hours called a block A producerrsquos block bidis knocked down if the average Elspot price for the applicable hours exceeds the bidprice2 The block bid is useful for producers with high costs associated with startingand stopping production

bull Flexible hourly bids is a single hourly sales bid to which the participant sets afixed price and volume without specifying the applicable hour The bid is knockeddown at most once if the Elspot price exceeds the bid price for an hour during thebidding day Flexible bids only apply to consumers

Using market power is forbidden the sales bids must represent the marginal cost ofproduction The marginal cost of hydropower is discussed in Section 24

23 The Balancing Power Market

The common Nordic Balancing Market (BM) serves as a tool for the TSOs to ensurebalance in the power system There are three possible balancing states The first isthe situation where production equals consumption and no balancing power is needed

2If including the block bid results in lower Elspot prices such that the block bid will not be acceptedwith the new Elspot price the block bid might not be knocked down although the average price is higherthan the block bid price

Figure 24 BM-premiums (left) and BM-volumes (right) in NO2 01012010 - 10032012[24]

The second state is when consumption exceeds production Upward balancing (BMuarr) isneeded and the TSO asks a producer to increase production or a consumer to decreaseconsumption The last state is downward balancing (BMdarr) when consumption exceedsproduction The TSO asks a producer to decrease production or a consumer to increaseconsumption The difference between the Elspot price and the BM-price is defined asthe BM-premium Historical BM-premiums and BM-volumes in the Norwegian price areaNO2 are shown in Figure 24 Both the volumes and the premiums are very volatile withseveral spikes especially in BMuarr

Bids in the Balancing Market are submitted for each watercourse to the local TSO afterthe Elspot market has closed and until 45 minutes before real-time There is a preliminarydeadline for BM-bidding at 2000 the day before the respective hour Separate bids forupward and downward balancing are given and the quantity must be activated withina 15 minutes notice The bids must have a duration of one hour or more In Norwayeach bid (in NOKMWh) must be integer divisible by 5 and have a minimum quantity of25 MW The minimum bidding quantity for small producers is 10 MW [26]

The bids are rated in merit order as illustrated in Figure 25 The lowest upward balancingbids and highest downward balancing bids are knocked down and all participants receivethe same price Each participant can submit several separate bids The price setting inthe Balancing Market ought to be socio-economic efficient that is it is forbidden to abusemarket power Statnett may suspend bids that are not representing the marginal cost ofproduction and use the current Elspot price In situations with congestions or errorsStatnett has the opportunity to choose bids without following the merit order This iscalled custom balancing3

3NO spesialregulering

Hours with Hours withupward balancing downward balancing

Surplus imbalance Elspot price BMdarr priceDeficit imbalance BMuarr price Elspot price

Table 22 Imbalance prices for producers as defined by Statnett

Figure 25 Bids in merit order for the Balancing Market

Participants that does not meet their obligations in the markets must pay the imbalancecharges which is the worst price of Elspot price and the BM-price Table 22 shows theimbalance prices given by Statnett

Statnett has an option market that is used to ensure sufficient volumes in the BalancingMarket in which the participants receive an option price for committing to bid a certainvolume into the Balancing Market The option market is used when the load is highduring winter [27]

24 The Value of Water

The marginal cost of hydropower is very small because the water comes for free Howeverpricing hydropower to its marginal cost is not a good idea water is a scarce resource andmight be stored for later use Thus the alternative cost of the water should be consideredthat is the income from using the water in a subsequent time period The future incomeof a marginal unit of water is not known due to uncertainty in future prices and inflowIf the reservoir is full a marginal unit of stored water will probably be spilled hence thevalue is zero If the reservoir is almost empty the water could be stored to a period withhigh prices and the value of water is high The expected future income of a marginal unitof water is called the marginal water value From the discussion above it should be clear

Figure 26 Reservoir-future income curve

that the water value is a function of future prices future inflow and the current reservoirlevel It should also be clear that the marginal water value is decreasing with increasingreservoir levels yielding a concave4 reservoir-future income curve as shown in Figure 26

When start-up costs are not considered the total operating expenses from a hydropowerplant could be assumed constant in the long run The only relevant cost is thus themarginal water value A power plant should produce if the water value is lower than theprice

The water value could be estimated in different ways In [7] a combination of prices onfutures and forwards for electricity is used for the water value calculations If the reservoiris empty it is assumed that 50 of the inflow will be sold at the current price of a certainfuture and the remaining 50 will be sold at a certain forward price Interpolating linearlyto the point of a full reservoir with a marginal water value of zero gives an estimate of thewater value curve The use of more fundamental methodologies is however more commonin Norway The marginal water values are estimated using long term optimization modelsas explained in the next section

25 Scheduling Hierarchy

An overview of the generation hierarchy could be found in [9] Water is a scarce resourcethus the use of water tomorrow affects the future opportunity to produce In order tomake the optimal decision for tomorrow all subsequent periods should therefore be takeninto account However deciding production for all time periods to come is impossibleFurther developing detailed production plans for long planning horizons results in highcomputational times Simplifications are therefore needed in order to schedule productionfor periods far from tomorrow Figure 27 shows the hierarchy of scheduling models used

4The marginal water value is the derivative of the future income with respect to the reservoir level Anon-increasing derivative proves a concave curve

Figure 27 Scheduling hierarchy used by hydropower producers in the Nordic area

by most hydropower producers in the Nordic area Sintef Energy Research has developedthe different models described below

First long term models are run based on stochastic dynamic programming The EMPSmodel is a multi area long term model in which the total system is divided into separateareas Within each area all hydropower production is aggregated to one plant with asingle reservoir Water values for each area are calculated The marginal water value isestimated as the shadow price of the reservoir balance constraint The objective is tominimize costs or maximize social welfare when thermal generation imports and exportsetc are taken into account When the marginal water values are calculated a detailedoptimization within each area is done with a time step of one week Several heuristics areused in the area optimization which to some extent rely on user experience The EOPSmodel is a one-area model in which a producer could model its own water course witha market description with prices from the EMPS model The stochasticity in the EMPSmodel is described simulating the system with historical inflow series The generated pricescenarios from the EMPS model and the historical inflow series are the stochastic inputdata to the EOPS model

In order to estimate the marginal water values for each reservoir a seasonal model is runThe seasonal model is a multiscenario deterministic optimization model in which theinitial and the final reservoir levels are given The model is run for several initial reservoirlevels The objective is to maximize future income The marginal water values are

Figure 28 Illustration of future income curve with two reservoirs and 16 cuts

estimated as the expected shadow prices of the reservoir balance in the first period Nowa cut k is defined with a corresponding reservoir level Lkj future income Fk and marginalwater value Vkj for each reservoir j in the set of reservoirs J Thus an approximation ofthe future income v for any reservoir level lj could be given as

vj asymp Fk minus Vjk(Ljk minus lj) forallj isin J (21)

This approximation is only valid in a small region around the initial reservoir level LjkBecause the future income curve is assumed to be concave creating a set of cuts K withdifferent initial reservoir levels gives a better approximation of the future income

v le Fk minussumjisinJ

Vjk(Ljk minus lj) forallk isin K (22)

Note that this is an over-approximation of the concave future income curve The cuts areillustrated in Figure 28 with two reservoirs

The water value cuts can be used in the short term model as the value of the reservoirs inthe end of the planning horizon Short term scheduling is described in the next sectionBecause the short term model includes simplifications of the physical system a detailedsimulation is often used to evaluate the results taking non-linearities etc into account

26 Short Term Production Scheduling

In Norway the Short Term Hydropower Optimization (SHOP) model developed by SintefEnergy Research is commonly used for short term scheduling This is a deterministicmodel Below is a general description of the short term hydropower scheduling problem

The power production w equals the potential energy stored in the released water per unittime multiplied by the overall efficiency when the head is constant

w = ηgηtqγHeff (23)

where ηg is the efficiency of the generator ηt is the efficiency of the turbine q is thedischarge γ is the specific gravity of water and Heff is the effective head ie the headless the head loss The head loss is described as αq2 where α is a constant The efficienciesηg and ηt are also dependent on q thus the production-discharge curve is non-linear Thenon-linear curve can be linearized with a set of cuts F each cut f isin F has a constantmarginal production Ejf and a constant Ejf that describe a linear cut for each generatorj in the set of generators J at time t in the set of time periods T

wjt le Ejfqjt + Ejf forallf isin F j isin J t isin T (24)

Starting a generator results in increased maintenance costs [17] and should be taken intoaccount Binary variables ujt are needed for modeling whether generator5 j isin J is inoperation at time t isin T (ujt = 1) or not (ujt = 0) If Cj denotes the start-up cost forgenerator j isin J the continuous term ojt should be subtracted from the objective functionwith the following constraints

ojt ge Cj(ujt minus uj(tminus1)) forallj isin J t isin T (25)

W jujt le wjt le W jujt forallj isin J t isin T (26)

ojt ge 0 forallj isin J t isin T (27)

ujt isin 0 1 forallj isin J t isin T (28)

where W j and W j are the lower and upper limits on the production from reservoir j isin J respectively (25) forces ojt to equal Cj if there is a start-up at time t isin T whereas ojtwill be minimized to 0 if there is no start-up [3]

Further the reservoir balance is

ljt = lj(tminus1) minus qjt minus rjt + κjt +sumψisinGj

(qψ(tminusDψj) + rψ(tminusDψ)) forallj isin J t isin T (29)

where ljt denotes the reservoir level κjt the inflow and rjt the spill for reservoir j isin J attime t isin T The set of reservoirs for which discharge and spill end up in reservoir j isin Jis denoted by Gj and Dψj is the time delay from reservoir ψ to reservoir j

Each reservoir level has an upper bound Lj and a lower bound Lj

Lj le ljt le Lj forallj isin J t isin T (210)

5It is assumed that each reservoir j isin J is connected to one station with one generator

The discharge from reservoir j isin J has upper and lower bounds Qj and Qj respectively

Qjle qjt le Qj forallj isin J t isin T (211)

The objective is to maximize profits z from the spot market with spot price ρt at timet isin T less start-up costs plus the value of the end reservoir represented by v

max z =sumjisinJ

sumtisinT

ρtwjt minussumjisinJ

sumtisinT

ojt + v (212)

3 A Stochastic Programming Model for the Coordi-

nated Bidding Problem

The following section presents a stochastic model for the coordinated bidding problemFirst market modeling is discussed Then the entire mathematical model with underlyingassumptions are presented

31 Modeling the Markets

There has thus far been little research on multimarket bidding for a producer in theNordic electricity market [8] presents a bidding model for a retailer taking the balancingmarket into account In [6] a model for coordination between the spot market and theElbas market for a price taking producer is developed There is done some research inmarkets with similar structure as the Nordic market [21] and [31] model the producer asa price setter in the Balancing Market by estimating of the residual demand curve Formarkets with a high share of flexible power production there is no reason to argue formarket power in the Balancing Market if there is no market power in the spot marketThe producer in this paper is modeled as a price taker in both the spot market and theBalancing Market

Market m in the set of markets M is defined such that

m isinM

1 for the spot market2 for BMuarr3 for BMdarr

The clearing of the spot market is done by interpolation of the bids as explained in Section22 Letting T B denote the set of time periods with bidding S the set of scenarios andI1 the set of bidpoints this can be expressed mathematically as

ρ1ts = P1i +P1(i+1) minus P1i

x(i+1)t minus xit(y1ts minus xit) if xit le y1ts lt x(i+1)t foralli isin I1 t isin T B s isin S

Figure 31 Interpolation of spot clearing price and volume

where ρ1ts denotes the spot clearing price at time t isin T in scenario s isin S P1i is thepredefined pricepoint at bidpoint i isin I1 xit the bid volume for bidpoint i isin I1 at timet isin T B and y1ts the volume commitment for the single hourly bids in spot market at timet isin T B in scenario s isin S (32) can also be written on the form

y1ts =ρ1ts minus P1i

P1(i+1) minus P1i

x(i+1)t +P1(i+1) minus ρ1ts

P1(i+1) minus P1i

xit if P1i le ρ1ts lt P(i+1)t foralli isin I1 t isin T B

(33)Block bids are not not modeled in this paper Further let imts be defined as

i isin Im

∣∣ Pmi le ρmts if ρmts ge Pm1

0 if ρmts lt Pm1 forallm isinM t isin T B s isin S (34)

For BMdarr the prices ρ3ts are negative because trading in BMdarr represents rdquobuying backrdquopower that is already committed in the spot market The bid prices must be non-decreasing ie Pmi le Pm(i+1) forallm isin M i isin Im For the spot market i in (33) cannow be substituted by i1ts Figure 31 shows how the interpolation is done

The definition (34) is also valid for the Balancing Market Because a bid bmits in marketm isin 2 3 for pricepoint i isin Im at time t isin T B in scenario s isin S is either knocked downor rejected the clearing volume ymts can be expressed as

ymts =imtssumi=1

bmits forallm isin 2 3 t isin T B s isin S (35)

All BM-bids that have bid prices less than the clearing price are knocked down

In order to avoid corner solutions ie overallocation to the Balancing Market there isdone an analysis to forecast the BM-volumes νmts in each market m isin 2 3 at time

t isin T B in scenario s isin S The total volume in the market represents an upper bound ofthe clearing volume of each participant ie

ymts le νmts forallm isin 2 3 t isin T B s isin S (36)

Moreover the producer might experience imbalances in some scenarios when the sum ofproduction wjts does not equal the total volume commitments ymts expressed as

y1ts + y2ts minus y3ts =sumjisinJ

wjts + ∆wminusts minus∆w+ts forallt isin T B s isin S (37)

where ∆wminusts denotes the deficit imbalance and ∆w+ts denotes the surplus imbalance at time

t isin T B in scenario s isin S The TSO requires that there is no expected imbalance whenbidding into the spot market This can be expressed assum

sisinS

πs(∆w+ts minus∆wminusts) = 0 forallt isin T B (38)

where πs denotes the probability of scenario s isin S An alternative formulation of (38)is including the BM-commitments (+y2ts minus y3ts) This results in unbiased spot biddingmeaning that the spot commitment equals the expected production for each hour How-ever constraining the BM-commitments such that the expected upward balancing equalsthe expected downward balancing will restrict the Balancing Market flexibility signifi-cantly In periods with low spot prices compared to the marginal water value a producerwill tend to bid small volumes into the spot market which gives little flexibility in BMdarrand high flexibility in BMuarr Thus on expectation the producer will probably commit tohigher BMuarr-volumes than BMdarr-volumes The situation in which the spot price exceedsthe marginal water value will result in the opposite The producer prefers to bid highvolumes into the spot market yielding higher flexibility in BMdarr than in BMuarr

Further as explained in Section 23 surplus imbalances are compensated with the lowestof the spot price and BMdarr-price and deficit imbalances are penalized with the highest ofthe spot price and the BMuarr-price In order to avoid big imbalances in both directions afactor of 12 is multiplied with the imbalance prices which gives imbalance costs σ+

ts andσminusts for time t isin T B and scenario s isin S6

σminusts 12 maxρ1ts ρ2ts σ+ts

12minρ1ts minusρ3ts forallt isin T B s isin S (39)

and sumtisinT B

sumsisinS

(σ+ts∆w

+ts minus σminusts∆wminusts

)(310)

6ρ3ts is defined as negative

is added to the objective function The additional penalty factor of 12 is used in order forthe producer to act risk averse with respect to imbalances The TSOs are expecting theparticipants to act in a way that does not jeopardize the system reliability Although (38)avoids expected imbalances it may still allow for large imbalances in both directions Thiskind of behavior will be sanctioned because the TSO rather considers the total imbalancethan the sum of surplus imbalances less the sum of deficit imbalances If a producer acts ina way that causes large imbalances in both directions over time it may risk its concessionto produce Therefore a rational producer will have a certain risk aversion when it comesto imbalances The factor 12 is set ad hoc investigating the correctness of the magnitudeof this factor is beyond the scope of this paper

The BM-volumes are also bounded by the available capacity for balancing

y2ts lesumjisinJ

W j minus y1ts forallt isin T B (311)

y3ts le y1ts forallt isin T B (312)

in order to avoid scenarios with both imbalances and BM-volume sales at the same time

32 Problem Formulation

The entire mathematical model is presented below based on the stochastic programmingframework [1] The underlying assumptions for the model are

bull The watercourse consists of two reservoirs in cascade each with a station containingone generator as illustrated in Figure 32 The discharge and spill from reservoir 1end up in reservoir 2

bull Water value curves and production-discharge curves are concave

bull The head is constant

bull The producer does not participate in any other physical market than the spot marketand the Balancing Market

bull The producer acts as a price taker in both markets

bull Custom balancing is not used

bull Block bids are not used

The list of sets data and variables is presented below Rounded capital letters (AB Cetc) denote sets greek letters denote uncertain data capital roman letters denote deter-ministic data and lower case roman letters denote variables and indices

Figure 32 An illustration of the modeled watercourse

SetsF Production curve pointsIm Bid points for market m isinMJ ReservoirsK Cuts in the water value curveM MarketsS ScenariosT Time periods (hours)T B sub T Time periods with bidding

Dataκjts Inflow to reservoir j isin J at time t isin T in scenario s isin Sνmts Volume in market m isin 2 3 at time t isin T in scenario s isin Sπs Probability of scenario s isin Sρmts Price in market m isinM at time t isin T in scenario s isin S (ρ3ts le 0)σ+ts σ

minusts Penalty of impalance at time t isin T B in scenario s isin S

Cj Start up cost reservoir j isin JD Time delay between reservoirs

Ejf Ejf Constants for the Production-Discharge CurveFk Expected future income for cut k isin KLj Lj Minimum and maximum reservoir level for reservoir j isin J respectivelyLjk Reservoir level for reservoir j isin J and cut k isin KPmi Price point i isin Im for market m isinMQj Qj Minimum and maximum discharge for reservoir j isin J respectively

Vjk Marginal Water value for reservoir j isin J for cut k isin KW jW j Minimum and maximum production for reservoir j isin J respectively

Variablesbmits Bidding volume in market m isin 2 3 for bidpoint i isin I at time t isin Tljts Reservoir level in reservoir j isin J at the end of period t isin T in scenario

s isin Sojts =1 if production from reservoir j isin J starts at time t isin T B in scenario

s isin S 0 otherwiseqjts Water discharge from reservoir j isin J at time t isin T in scenario s isin Srjts Spill from reservoir j isin J at time t isin T in scenario s isin Sujts =1 if there is production from reservoir j isin J at time t isin T in scenario s isin S

0 otherwisevs Water value of all reservoirs in scenario s isin S at the end of the planning

periodwjts Generation from reservoir j isin J at time t isin T in scenario s isin S∆w+

ts∆wminusts Generation imbalance (surplus and deficit) at time t isin T in scenario s isin S

xit Bidding volume in spot market for bidpoint i isin I at time t isin T (firststage decision)

ymts Volume commitment in market m isinM at time t isin T in scenario s isin S

max z =sumsisinS

(summisinM

sumtisinT

ρmtsymts + vs minussumjisinJ

sumtisinT

ojts +sumtisinT B

(σ+ts∆w

+ts minus σminusts∆wminusts)

)(313)

xit le x(i+1)t foralli isin I1 t isin T B (314)

y1ts =ρ1ts minus P1i1ts

P1(i1ts+1) minus P1i1ts

x(i1ts+1)t

+P1(i1ts+1) minus ρ1ts

xi1tst forallt isin T B s isin S (315)

ymts =imtssumi=1

wjts + ∆wminusts minus∆w+ts forallt isin T B s isin S (317)sum

sisinS

y2ts lesumjisinJ

W j minus y1ts forallt isin T B s isin S (319)

y3ts le y1ts forallt isin T B s isin S (320)

y1ts =sumjisinJ

wjts forallt isin T T B s isin S (321)

vs le Fk minussumjisinJ

Vjk(Ljk minus lj|T |s) forallk isin K s isin S (322)

wjts le Ejfqjts + Ejf forallj isin J f isin F t isin T s isin S (323)

ojts ge Cj(ujts minus uj(tminus1)s) forallj isin J t isin T s isin S (324)

W jujts le wjts le W jujts forallj isin J t isin T s isin S (325)

l1ts minus l1(tminus1)s + q1ts + r1ts = κ1ts forallj isin J t isin T s isin S (326)

l2ts minus l2(tminus1)s + q2ts minus r1ts + r2ts minus q1(tminusD)s = κ2ts forallj isin J t isin T s isin S (327)

0 le ymts le νmts forallm isin 2 3 t isin T s isin S (328)

Qjle qjts le Qj forallj isin J t isin T s isin S (329)

Lj le ljts le Lj forallj isin J t isin T s isin S (330)

ojts rjts ge 0 forallj isin J t isin T s isin S (331)

∆wminusts∆w+ts le 0 forallt isin T B s isin S (332)

ujts isin 0 1 forallj isin J t isin T s isin S (333)

The objective function (313) maximizes expected future profits that is the sum of rev-enues from the three markets and the value of stored water less start-up costs and penal-ties of imbalances (314) forces the spot bids to be non-decreasing (315)-(320) describethe market for each scenario s isin S as presented in Section 31 (321) assigns the com-mitment in the spot market to the generators after the bidding period No imbalances areallowed after the bidding period (322) is the water value constraint for each scenariopresented in Section 25 (323)-(327) model the physical system for each scenario aspresented in Section 26 Finally (328)-(333) give the domains for the variables

In addition non-anticipativity constraints [1] are added For a given set of informationζ in the set of non-anticipativity sets Z the same decision must be taken This can bemodeled the following way for each variable (here denoted χts)

χts = χtζ forallt isin Tζ s isin Sζ ζ isin Z (334)

where χtζ is the decision done with the information set ζ isin Z containing scenarios s isin Sζand time periods t isin Tζ In order to model the stages correctly an artificial day for thespot bidding can be added see Figure 45

The first bidpoint in the spot market (at -e200) is fixed to 0 ie x1t = 0 forallt isin T B andthe last bidpoint (at e2000) is fixed to maximum production ie x|I1|t =

sumjisinJ W j forallt isin

T B according to the NPS guidelines [24]

The minimum bidding volume requirement in the Balancing Market can be accounted forby adding binary variables amits which are equal to 1 if bidpoint i isin Im is active and 0otherwise

min 25 νmts amits le bmits le νmtsamits forallm isin 2 3 i isin I t isin T B s isin S (335)

amits isin 0 1 forallm isin 2 3 i isin I t isin T B s isin S (336)

If νmts gt 25 MW each active bid bmits gt 0 will be in the region [25 MW νmts] for marketm isin 2 3 at time t isin T B in scenario s isin S However if νmts le 25 MW the bid volumemust equal the total market volume νmts or 0

4 Scenario Generation

This section presents a methodology for generating scenarios for the inflow spot priceBM-volumes and BM-premiums used in the case study in Section 5 For all parameterestimations a training period of eight weeks is used as found appropriate in [30] The setof hours in the training period is denoted T T All parameter estimations are done in thestatistical software R [22] applying the maximum likelihood method The scenarios aregenerated for the period 10042012 - 10072012 for the Norwegian price area NO2 Thedata source for historical spot prices BM-premiums and BM-volumes is [24]

41 Spot Prices

Price forecasting in the spot market is well developed and is not presented in this paperThere are two important methodologies statistical models and fundamental models Anoverview of statistical models can be found in [34] In the Nordic Market the EMPSmodel described in Section 25 is commonly used for price forecasting The future andforward markets are also relevant because they give the market value of future exchangeof electricity

The scenario generation for spot prices in this paper is based on a deterministic forecastρDt shown in Figure 41 The price has typically two peaks during the day one in themorning and one in the afternoon Night hours are expected to have significantly lowerprices than day hours Further the last two days (610-710) constitute a weekend andthe expected daily price decreases during the period

In order to describe the uncertainty in the spot price forecast an error term εts is added

Figure 41 Deterministic spot forecast for the planning horizon

The error term for the first hour is assumed to be white noise where the variance σ2ε1

isestimated from historical forecast errors ie

ρ1s = ρD1 + ε1s foralls isin S (41)

ε1s sim N (0 σ2ε1

) foralls isin S (42)

The error term in the first hour is modeled as independent of the previous error termsin order to keep the expectation of the generated scenarios equal to the deterministicforecast

The spot prices for each day (24 hours) are revealed at the same time Thus the hourlyerror is not a time series However because the error terms are highly autocorrelatedthe subsequent error terms are modeled with a seasonal autoregressive (AR) model Theseasonality comes from high correlation with the same hour the day before The modelfor the forecast error is

εat εt minus aε = Θε1ε

a(tminus1) + Θε

2εa(tminus2) + Θε

24εa(tminus24) + ωεt forallt isin T T (43)

where aε is the mean of εat Θε1 Θε

2 and Θε24 are the parameters from the AR-model and

ωεt is the residual at time t isin T T

In the scenario generation one of the residuals ωεt from the training period is drawnrandomly for each scenario s isin S The random residual ωεts is added to the deterministicforecast

εats = Θε1ε

a(tminus1)s + Θε

2εa(tminus2)s + Θε

24εa(tminus24)s forallt isin T 1 s isin S (44)

ρts = ρFt + εats + aε + ωεts forallt isin T 1 s isin S (45)

Figure 42 Correlograms for BMuarr-volume (a) BMdarr-volume (b) BMuarr-premium (c) andBMdarr-premium (d) for NO2 08182010 - 10032012

42 Balancing Market Volumes and Premiums

Price forecasting in the Nordic Balancing Market (BM) is not well studied Some workhas been done In [23] an econometric model for the BM-premiums is developed with BM-volumes and the spot price as input signals but the paper does not discuss uncertainty In[19] a Markov switching state model combined with a Seasonal Autoregressive IntegratedMoving Average (SARIMA) model for the BM-premiums is presented Further work onthe BM-premiums is done in [18] and extended in [2] in which wind power uncertainty istaken into account and non-linear time series models are used In [13] a SARIMA modelis used for the BM-state and the BM-volumes are fitted to Generalized Extreme Value(GEV) distributions The price model described below is inspired of [19] and extendedby forecasting the BM-volumes and model the dependencies between the BM-volumesand the BM-premiums

Figure 42 shows the correlograms for the BM-volumes and BM-premiums The auto-correlations are high for the first hours and there is a tendency of seasonality becausethe data is often correlated with the same hour the day before Because of the highautocorrelations for lags up to three hours time series models are used for modeling theBM-volumes and BM-premiums with orders up to 3 The seasonality is not taken intoaccount

A discrete Markov switching model [11] is used to model the state of the system in the

Figure 43 Number of hours with balancing in both directions for 2011 for NO2

price area There are three possible states st for time t isin T T defined7

2 if ν2t gt 03 if ν3t gt 01 otherwise

forallt isin T T

where νmt is the BM-volume in market m isin 2 3 at time t isin T T

In about 25 of the time there is a premium in the BM-market without any BM-volumein the price area NO2 In these hours there is export of BM-volume out of NO2 This isnot taken into account in this paper which represents a minor simplification One couldallow a limited export volume based on experience to account for this opportunity

The transition probability πij is defined as the probability of moving from state i to statej Figure 43 shows that there are more downward balancing during night hours and moreupward balancing during day hours Therefore the transition probability for each state isestimated separately for night hours (hour 1 through 7) and day hours (hours 8 through24) Because the BM-states are observable explicitly the estimations are done in thefollowing way

πTij =

∣∣∣ t isin T T | s(tminus1) = i cap st = j ∣∣∣∣∣∣ t isin T T | s(tminus1) = i

∣∣∣ foralli j isin 1 2 3T isin ND (46)

where πNij and πD

ij denote the night and day transition probabilities respectively and T N

represents the night hours in the training period and T D represents the day hours Thenotation | middot | denotes the number of entries in a set

7In the case with both upward balancing and downward balancing the state with the highest volumeis chosen Hours with both upward and downward balancing counts for about 02 of the hours in theperiod 08182010 - 10032012 hence the simplification should not cause major errors

Upward balancing counts for 222 of the hours and downward balancing 276 of thehours in the training period The estimates of π are

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

The BM-volumes have a lower bound at zero A common method when modeling non-negative stochastic variables is logarithmic transformation [16] Let νmt be the observedBM-volume in market m isin 2 3 at time t isin T T The log-transformed variable is denotedνmt and only defined for strictly positive volumes

ln(νmt) if νmt gt 0Not defined otherwise

forallm isin 2 3 t isin T T (47)

Further let aνm denote the mean (average) of νmt and

νamt νmt minus aνm forallm isin 2 3 t isin T T (48)

An Autoregressive Moving Average ARMA(p q) model is used for νamt in each marketm isin 2 3 p denotes the order of the Autoregressive terms and q denotes the orderof the Moving Average terms The model identification is done by minimizing AkaikersquosInformation Criterion (AIC) [33] For BMuarr (m = 2) p = 3 and q = 2 is chosen and p = 2and q = 1 is chosen for BMdarr (m = 3) When estimating the parameters the majority ofthe data points are missing due to no BM-volumes Thus this is a missing value problemand hours without data are not considered in the parameter estimation However hourswith balancing often come in clusters the parameter estimation is therefore possible TheBM-volume models can be written

νa2t = Θν21 ν

a2(tminus1) + Θν2

2 νa2(tminus2) + Θν2

3 νa2(tminus3) + Φν2

1 ων22(tminus1) + Φν2

2 ων22(tminus2) + ων2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

ν3t forallt isin T T

where Θν and Φν are the parameters in the models and ωνmt is the residual in marketm isin 2 3 at time t isin T T The numerical results are summarized in Table 41

Figure 44 Scatterplot for the BM-volumes against the BM-premiums in NO2 01012010- 10032012

For each time step in each scenario the BM-volume νmts is calculated as

νa2ts = Θν21 ν

a2(tminus1)s + Θν2

2 νa2(tminus2)s + Θν2

3 νa2(tminus3)s + Φν2

1 ων22(tminus1)s + Φν2

2 ων22(tminus2)s forallt isin T T s isin S

νa3ts = Θν31 ν

νmts = exp

(νamts + aνm + ωνmts +

(σνm)2

) forallm isin 2 3 t isin T T s isin S

where (σνm)2 is the variance of the residuals for market m isin 2 3 and the residuals ωνmtsare drawn randomly from the residuals in the training period The term (σνm)2

2is added

in order to generate unbiased scenarios [16] The simulation is done from the last datapoints ie the last three adjacent hours with BMuarr-volumes in the training period andthe last two adjacent hours with BMdarr-volumes

The BM-premium δmt in market m isin 2 3 at time t isin T T is defined as the differencebetween the BM-price and the spot price (ρ1t) for upward balancing and the differencebetween the spot price and the BM-price for downward balancing that is

ρ2t minus ρ1t if st = 2Not defined otherwise

forallt isin T T (414)

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

Table 41 Model parameters for the log-transformed BM-volumes and BM-premiums

Note that by construction δmt gt 0 whenever δmt is defined Thus the premium is anon-negative stochastic variable as the BM-volume A log-transformation is thereforeused also for the premiums

Let δmt ln(δmt) forallm isin 2 3 t isin T T and aδm be the mean of δmt Further let δamt δmt minus aδm forallm isin 2 3 t isin T T A Moving Average model of order 2 (MA(2)) is foundto generate unbiased scenarios for δamt with the BM-volume as external input Figure 44shows that there is a correlation between the BM-premium and the BM-volume Themodel is also tested with the spot price as an external input however the correlation isnot significant

δamt = Φδm1 ωδm(tminus1) + Φδm

2 ωδm(tminus2) + Γmln(νmt) + ωδmt forallm isin 2 3 t isin T T s isin S (416)

where Φδm and Γm are the parameters in the models and ωδmt denotes the residuals inmarket m isin 2 3 at time t isin T T The numerical results are summarized in Table 41

For each time step in each scenario the BM-premium δmts is calculated as

δamts = Φδm1 ωδm(tminus1)s + Φδm

2 ωδm(tminus2)s + Γmln(νmts) forallm isin 2 3 t isin T B (417)

δmts = exp

(δamts + aδm + ωδmts +

(σδm)2

) forallm isin 2 3 t isin T B (418)

where (σδm)2 is the variance of the residuals in market m isin 2 3 and the residuals ωδmtsare drawn randomly from the residuals in the training period

The simulations are done from the last two adjacent hours in the training period withvolumes in market market m isin 2 3

The procedure below generates 3000 individual scenarios and is implemented in MatlabTable 42 shows the descriptive statistics for the input data The characteristics of the

Mean Standard Min Maxdeviation

ρD1t [eMWh] 2886 298 2308 3597ν2t [eMWh] 11859 9478 6 502ν3t [eMWh] 11497 10942 2 719δ2t [eMWh] 541 293 021 1512δ3t [eMWh] 56 292 001 1581κ1t [m3s] 752 220 402 1411κ2t [m3s] 905 479 476 2890

Table 42 Descriptive statistics BM-data is for hours with balancing volumes

BMuarr and BMdarr data are quite similar The ranges and the standard deviations for the BM-volumes are very high indicating the high uncertainty associated with these variablesThe volatilities in the BM-premiums are somewhat smaller than for the BM-volumesstill noteworthy high The table also shows that the uncertainties in the spot market andinflows are significantly smaller compared to the Balancing Market data

For each scenario s isin 1 3000 doPick one inflow scenario randomlyCalculate the spot price for hour t = 1 ρ11s using (41)-(42)For each hour t isin T 1 do

Calculate the spot price ρ1ts using (44)-(45)End doFor each hour t isin T B do

Determine the BM-state sts randomly using the probabilities found in (46)For each market m isin 2 3 do

Calculate the BM-volume using (411)-(413)Calculate the BM-premium using (417)-(418)If sts = m do

Set νmts equal to the calculated BM-volumeSet δmts equal to the calculated BM-premium

Else doSet νmts = 0 and δmts = 0

End ifEnd do

End doEnd do

Figure 45 Illustration of scenario fan before scenario reduction (upper) and scenariotree after scenario reduction (lower) The figure shows when the decisions for the biddingperiod are done

Figure 46 Distributions of BM-volumes (left) and BM-premiums (right) for historicaldata (upper) and scenario generated data (lower) the historical data is for the period08182010 - 10032012

Figure 47 Generated spot price scenarios (a) and inflow scenarios for reservoir 1 (b) andreservoir 2 (c)

This procedure generates a scenario fan as illustrated in the upper tree of Figure 45In order to model the uncertainty correctly one should reveal the uncertain parametersgradually The methodology presented in [12] is used for scenario reduction and forgenerating a scenario tree A reduction percentage of 02 is used which resulted in 801scenarios A higher reduction percentage yielded fewer scenarios but tended to removetoo many hours with balancing The means and variances of the generated scenarioswere checked to be approximately equal to the observed means and variances Figure46 shows the distributions of BM-volumes and BM-premiums for the historical data andin the generated scenarios The kurtosis for the BMdarr-premium is slightly higher for thescenarios Still the scenarios for all stochastic data represent the historical distributionssufficiently

Figure 47a shows the generated spot price scenarios The variance is increasing from thesecond stage (first day) to the last stage (day 2 through 4) 50 scenarios for the inflow areinput to the scenario generation Figures 47b-c show the generated inflow scenarios κjtsfor reservoir j isin J at time t isin T in scenario s isin S The expected inflow is decreasingduring the planning horizon The variance is not substantially increasing

Because this paper is mainly concerned with the uncertainty in the Balancing Marketthree stages are modeled The first stage contains one period where the decision xit is donefor each hour in the bidding period The next stage contains an artificial day (24 hours)for modeling the spot market correctly It consists of spot clearing prices for the nextday and is used because the spot prices are revealed the day before operation whereasthe BM-volumes BM-premiums and inflows are revealed in real-time The decisionsy1ts forallt isin T B s isin S are done in the second stage The rest of the time periods arecontained in the last stage In the three stage scenario tree all information about BM-volumes BM-premiums inflows and spot prices for subsequent days are revealed at thesame time This is a simplification However the scope of this paper is to model how aproducer should bid into the spot market the first day without knowing the BM-pricesand volumes This is modeled correctly in the scenario tree Figure 45 illustrates howthe scenario tree is constructed The spot volumes for the artificial day corresponds tothe commitment in the first day ie y1(tminus24)s is the commitment at time t isin T B

5 Case Study

The case study presented in this section discusses whether a hydropower producer shouldaccount for the Balancing Market in the spot bidding phase The performance from thecoordinated model developed in Section 3 is disussed A sequential model that suggestsspot bids without considering the Balancing Market then it bids into the BalancingMarket with fixed spot commitments is proposed as a benchmark to the coordinatedmodel Finally an analysis of when the coordinated model may outperform the sequentialmodel is done The modeled watercourse is located in the Norwegian price area NO2 andoperated by Norsk Hydro ASA The study is done for the fall of 2012 with high initialreservoir levels

51 Case Description

There is no evidence of systematic use of market power in the Nordic spot market [10]NO2 is the largest price area in Norway corresponding to 29 of the total production in2011 consisting mainly of reservoir hydropower [24] The producer is therefore assumed toact as a price taker in the spot market The number of suppliers in the Balancing Market isthe same as in the spot market the fundamental assumptions about market power shouldhence be the same in the two markets despite the low liquidity in the Balancing MarketThe producer is therefore assumed to act as a price taker in the Balancing Market

The case study consists of two reservoirs in cascade each connected to a power stationas shown in Figure 51 The upper reservoir is large (178 middot106 m3) whereas the lowerreservoir is small (16 middot106 m3) with little flexibility The risk of spill is significant for thelower reservoir The upper reservoir is an aggregation of three reservoirs as shown in the

Figure 51 Illustration of watercourse in reality (left) and simplified (right)

figure The marginal water values for reservoir 1 are calculated as the volume weightedaverage of the individual water values The differences in the marginal water values arehowever small Further the two stations is a part of a larger watercourse but none ofthe discharges are affected by the rest of the watercourse There are no restrictions onthe discharges and no time delay between the reservoirs

The head is assumed to be constant which is a minor simplification in the short termscheduling process Reservoir 1 can in reality be kept at approximately constant head bycontrolling the upper two reservoirs Head changes for reservoir 2 are small Thus theconstant head assumption will not cause major errors

The production from each generator is modeled with three linear cuts according to (24)There is one generator (Figure 52a) in the upper station with a maximum production of45 MW The lower station contains two generators (Figure 52b-c) with a total productioncapacity of 150 MW In order to model start-up costs for each generator in the lowerstation the following heuristic is used The second generator is assumed to operate onlyif the first generator is operating The production from the first generator is loadedas if it is operating alone (with no head loss from running the second generator) Theproduction from the second generator is calculated as the additional production whenthe first generator is operating at the best point level Lastly the maximum productionfrom both generators is calculated Because of the increasing head loss when loading thesecond generator its efficiency is substantially lower compared to the efficiency of the firstgenerator The second generator will therefore never operate when the first generator isnot operating

The model is run with a planning horizon of four days from October 4 2012 through

Figure 52 Production-Discharge curves for generator 1 (a) generator 2 (b) and genera-tor 3 (c) with maximum production (Max) best-point production (BP) and minimumproduction (Min)

Large model Simplified model

Constraints (presolved) 1 226 229 495 717Variables (presolved) 971 511 404 403Running time (s) 748 818 399 792

Table 51 Statistics from running model the large model and the simplified model

October 7 2012 Seven water value cuts are implemented valid for the end of October7 The period has high reservoir levels and one should hence anticipate high productionrates The currency used in the model is Euro (e) 24 bidpoints from 0 to e110 with aspacing of e5 are used in addition to one bidpoint at e1999 The bidding constraint (316)is relaxed such that the BM-production decisions are done in the last stage Moreoverthe model is not implemented with the minimum volume restriction in the BalancingMarket (335)-(336) in order to limit the number of binary variables

52 Results

The model is run with 801 scenarios generated from the procedure described in Section43 The deterministic equivalent of the three-stage model presented in Section 32 isimplemented in Xpress MP [20] on an Intel Core i7 340 GHz processor with 16 GB RAMOptimality was proven in the first node of the Branch-and-Bound tree Table 51 showsthe statistics from solving the model The running time of approximately eight days is waytoo long for the model to be implemented for decision support an appropriate solutionmethod needs to be implemented see Section 61 The idea is rather to explore whetherthe Balancing Market should be taken into account in the spot bidding phase

Figure 53 Bidding curve for the spot market

The spot bidding curve is shown in Figure 53 Due to the high reservoir levels the modeltends to produce at maximum when the price reaches a certain level Therefore fewbidpoints are active There are however a few hours with more active bidpoints Thisis caused by different marginal water values in the reservoirs and the breakpoints in theproduction-discharge curves

Figure 54 shows how the reservoir levels develop in the solution Reservoir 1 is signif-icantly reduced in the period The reservoir level in reservoir 2 is expected to increaseThis is a small reservoir The solution tends to allocate water from the upper reservoir (1)to the lower reservoir (2) because the difference in water value is lower than the expectedprices Many of the scenarios result in maximum reservoir levels for reservoir 2 Themodel does not account for the risk of spill after the planning period and sees only themarginal water values This is a weakness of the coupling between the scheduling modelsand a producer will not allow for full reservoirs if the expected inflows after the planningperiod are high

There are imbalances8 for about 41 of the hours and there is no spill in any scenarioBecause the minimum volume requirement in the Balancing Market is relaxed the resultsgive many hours with very small volumes in the Balancing Market

The solution gives BMdarr-commitments (y3ts ge 5 MW) in 84 of the hours with avail-able BMdarr-volumes corresponding to 232 of all bidding hours in every scenario Thisindicates that BMdarr is used almost always when possible and is reasonable because theexpected spot prices lie just above the marginal water values and bidding high volumesinto the spot market gives high flexibility in BMdarr

8Only imbalances over 2 MW are counted

Figure 54 Results for the reservoir levels in the different scenarios for reservoir 1 (a) andreservoir 2 (b)

There are BMuarr-commitments (y2ts gt 5 MW) in about 5 of the hours with availableBMuarr-volumes corresponding to about 11 of the all hours Bidding high volumes intothe spot market reduces the flexibility in BMuarr Due to high reservoir levels the model isnot willing to risk lower total production in order achieve high flexibility in BMuarr

The objective value and the revenues in the different markets are presented in Table52 The results show that the objective function value is quite stable The standarddeviation in the objective function is below 2 of the spot revenues in the planninghorizon Furthermore the spot volume commitment is very high in almost all scenariosfor the bidding period which is reasonable both because the spot prices decrease duringthe period and because the BMdarr opportunity drives the spot volumes up

In order to reduce the running time the model is also run in a simplified version Thenumber of bidpoints is reduced to seven each day after the bidding period has beendivided into six periods containing four hours and start-up costs are only implementedfor the bidding day The running time and the numbers of variables and constraints aresignificantly reduced as shown in Table 51 Still the running time is way too long tobe used for operational decision support The spot bidding curve for the first day isalmost identical to the result from the original model There is a small bias to bid smallervolumes into the spot market the first day but this counts for only 02 MW on averageThus the bid curve from the simplified model gives more or less the same results as theoriginal model

The value of stochastic solution [1] is not investigated in this paper because using expectedvalues for the Balancing Market will not make sense If the BM-volume is known at the

Coordinatedmodel

Mean Stdev

Planninghorizon

Objective value[e] 7 929 195 6 840Spot revenue[e] 392 545 9 973Spot volume[MWh] 13 526 336Reservoir level j = 1[106 m3] 167752 0506Reservoir level j = 2[106 m3] 1527 0518

Biddingperiod

Spot revenue[e] 139 573 1 240Spot volume[MWh] 4 652 39BMuarr-revenue[e] 66 706BMuarr-volume[MWh] 219 2471BMdarr-revenue[e] -2 619 3 681BMdarr-volume[MWh] 12540 18629Reservoir level j = 1[106 m3] 170026 0127Reservoir level j = 2[106 m3] 0305 0115

Table 52 Results for the coordinated model

time of spot bidding the BM-volume will always be assigned before the spot volumebecause the price in the Balancing Market by construction is higher than the spot priceThe value of the stochastic solution for the spot bidding problem is studied in [7]

53 Comparison With Sequential Model

In order to evaluate whether the Balancing Market should be taken into account in thespot bidding phase the model can also be run in sequence That is the stochastic modelis first run without the Balancing Market opportunity (fixing bmt = 0 forallm isin 2 3 t isinT B) which gives spot bids and spot clearing prices and volumes Then the BalancingMarket model can be run which is the same model with fixed spot commitments y1ts

for the first day (t isin T B) Because the last model starts in the third and last stagethis is a deterministic model for each scenario It is left to future research to comparethe preformance of the sequential model with the results from the coordinated modelHowever a discussion of when the coordinated model will be beneficial is conductedbelow

54 Discussion

The coordinated model developed in this paper represents an extension of the availabledecision support tools commercially available today This section discusses in which situ-ations such an extension might be helpful for the bidding decisions and when a sequentialmodel is sufficient

Figure 55 Illustration of possible states for BMdarr

Considering an hour in state 3 (downward balancing) the profit per unit production isthe difference between the spot price ρ1 and the marginal water value V if BMdarr is notused and the BM-premium δ3 if BMdarr is used The model seeks to maximize profitschoosing between no bidding (zero profit) only spot bidding or spot bidding and BMdarrbidding There are four possible states for the BM-premium illustrated in Figure 559

(a) ρ1 V It is not profitable to bid into any of the two marketsThe producer will not risk not being commited toBMdarr-volumes

(b) ρ1 lt V It is profitable to bid into both markets if one expectshigh volumes in BMdarr and not profitable to bid intoany market if one expects low volumes in BMdarr

(c) ρ1 gt V and δ3 gt (ρ1 minus V ) It is profitable to bid into the spot market and oneshould bid into BMdarr whenever possible

(d) ρ1 gt V and δ3 lt (ρ1 minus V ) It is profitable to bid into spot market and not intoBMdarr because it yields smaller profits than thepure spot profit

The high spot volumes and BMdarr-volumes indicate that the system in the case study isoften in state (b) or (c) If the system is often in state (b) taking the Balancing Marketinto account will tend to increase the bidding volume in the spot market

9Note that the water value is dependent on the production level which complicates the analysis butis taken care of in the optimization model

Figure 56 Illustration of possible states for BMuarr

For an hour in state 2 (upward balancing) the profit per unit production equals the spotprice less the marginal water value if the spot market is used and BMuarr is not used IfBMuarr is used the profit is the BMuarr-price less the marginal water value A similar analysisas done above can be carried out for BMuarr The four possible states are described belowand illustrated in Figure 56

(e) ρ1 lt V and (ρ1 + δ2) lt V It is not profitable to bid into any of the two markets

(f) ρ1 lt V and (ρ1 + δ2) ge V It is not profitable to bid into the spot market and itis profitable to bid into BMuarr

(g) ρ1 gt V and (ρ1 + δ2) gt V It is profitable to bid into both markets If one anticipateshigh volumes in BMuarr one might be willing to reducethe spot volume to achieve higher volumes in BMuarr

(h) ρ1 V and (ρ1 + δ2) V It is very profitable to bid into the spot market and theproducer will not risk reducing its total commitments inorder to obtain flexibility in BMuarr

The high spot volumes and the fact that BMuarr is rarely used indicate that the systemin the case study is often in state (b) or (c) for downward balancing and (g) for upwardbalancing Thus the spot price is slightly above the marginal water value and themodel choses to achieve flexibility in BMdarr rather than in BMuarr A rational decision wouldtherefore be to bid higher volumes into the spot market in a coordinated model comparedto a sequential model

Figure 57 Scatterplots for the spot price against the BMuarr-premium (a) and the BMdarr-premium (b) in NO2 08182010 - 10032012

There are complicating factors to the analyses above First the marginal water valuesare dependent on the production level both because of the non-constant efficiencies andbecause the marginal water value is dependent on the total use of water in the planninghorizon Furthermore although the balancing premiums had no significant correlationswith the spot price in the training period there are correlations in the long run Figure57 shows scatterplots of the BM-premiums against the spot price The correlations withthe spot price are 014 for the BMuarr-premium and 018 for the BMdarr-premium The figureshows that there is no trivial dependency between the BMuarr-premium and the spot pricebut when the spot price exceeds a certain level (about e35MWh) the expected premiumincreases The dependency between the spot price and the BMdarr-premium is somewhatlinear in the long run The fact that the correlation for BMuarr is smaller may be explainedby the option market for BMuarr-volumes that is active when the expected spot prices arehigh (during winter) The options ensure sufficient BMuarr-volumes at any point of timeyielding no incentive to bid BMuarr-prices that are substantially higher than the marginalwater value for committed producers When the spot price is high the producers will notbid volumes in BMdarr for premiums that are smaller than the difference between the spotprice and the marginal water value This difference is expected to increase with increasingspot prices Thus the state (d) is in general unlikely if the marginal water values for allproducers in a price area are strongly correlated Therefore it is likely that the amountof time the coordinated model is not useful is further limited

Summing up if the spot price is very low ((a) and (e)) or very high ((d) and (h)) comparedto the marginal water value the Balancing Market will not be used and needs not beaccounted for in the spot bidding phase If the spot price is near the marginal water valueit may be profitable to include the BM-flexibility in the spot bidding decisions Moreover

data from the Nordic market indicates that state (h) is unlikely

6 Conclusion and Future Work

The coordinated bidding model developed in this paper may prove to outperform a se-quential model unless there is a big difference between the spot price and the marginalwater value compared to the expected premium in the Balancing Market Market dataindicates that the coordinated model will be beneficial in the majority of the time theproducer is considering bidding volumes into the spot market

61 Future Work

The case presented in this paper examines a bidding period of four days in October 2012with high reservoir levels and prices slightly above the marginal water values To gainmore knowledge about when it is profitable to account for the Balancing Market in thespot bidding phase the model should be tested for more cases It is especially inter-esting to model different seasons with different characteristics in the markets Modelingmore complicated watercourses could give further knowledge about the problem Further-more the performance of the coordinated model should be compared to results from thesequential model to gain further insight about when the developed model is favorable

The long running times indicate that an appropriate solution method needs to be inves-tigated The L-Shaped Method [1] or Linear Decision Rules [15] can be tested Bothrequire continuous problem formulations start-up costs and minimum balancing volumebids can hence not be modeled correctly with these methods Other solution methodscan be found in [14]

Producers also consider if they should participate in other physical power markets likeElbas the balancing option market and the primary and secondary reserve markets Astudy that includes all physical markets is left to future research

The volatility in the spot price varies substantially over the day This varying volatility isnot taken into account in the scenario generation in this paper Neither is seasonality Theuse of more advanced time series models can account for these effects and thus describethe uncertainty better

References

[1] JR Birge and F Louveaux Introduction to Stochastic Programming Springer2011

[2] MO Brolin and L Soder ldquoModeling Swedish real-time balancing power prices usingnonlinear time series modelsrdquo In Probabilistic Methods Applied to Power Systems(PMAPS) 2010 pp 358ndash363

[3] M Carrion and JM Arroyo ldquoA computationally efficient mixed-integer linear for-mulation for the thermal unit commitment problemrdquo In Power Systems IEEETransactions on 21 (3) (2006) pp 1371 ndash1378

[4] Nasdaq OMX Commodities Website Dec 2012 url httpwwwnasdaqomxcomcommodities

[5] ENTSO-E Deterministic Frequency Deviations Root Causes and Proposals forPotential Solutions 2012 url https www entsoe eu index php id = 42

ampno_cache=1amptx_ttnews[tt_news]=173

[6] E Faria and S-E Fleten ldquoDay-ahead market bidding for a Nordic hydropowerproducer taking the Elbas market into accountrdquo In Computational ManagementScience 8 (1) (2011) pp 75ndash101

[7] S-E Fleten and TK Kristoffersen ldquoStochastic programming for optimizing bid-ding strategies of a Nordic hydropower producerrdquo In European Journal of Opera-tional Research (2007)

[8] S-E Fleten and E Pettersen ldquoConstructing Bidding Curves for a Price-TakingRetailer in the Norwegian Electricity Marketrdquo In IEEE Transactions on PowerSystems 20 (2) (2005) pp 701ndash708

[9] OB Fosso A Gjelsvik A Haugstad B Mo and I Wangensteen ldquoGeneration Schedul-ing in a Deregulated System The Norwegian Caserdquo In IEEE Transaction on PowerSystems 14 (1) (1999) pp 75ndash81

[10] S-O Fridolfsson and T Tangeras ldquoMarket power in the Nordic electricity whole-sale market A survey of the empirical evidencerdquo In Energy Policy 37 (9) (2009)pp 3681ndash3692

[11] JD Hamilton Time Series Analysis Princeton University Press 1994

[12] H Heitsch and W Romisch ldquoScenario tree modeling for multistage stochastic pro-gramsrdquo In Math Program 118 (2) (2009) pp 371ndash406

[13] S Jaehnert H Farahmand and GL Doorman ldquoModelling of Prices Using the Vol-ume in the Norwegian Regulating Power Marketrdquo In IEEE Bucharest PowerTech2009

[14] P Kall and SW Wallace Stochastic programming Wiley-Interscience series in sys-tems and optimization Wiley 1994

[15] D Kuhn W Wiesemann and A Georghiou ldquoPrimal and Dual Linear DecisionRules in Stochastic and Robust Optimizationrdquo In Mathematical Programming 130(1) (2011) pp 177ndash209

[16] MC Newman ldquoRegression Analysis of Log-Transformed Data Statistical Bias andits Correctionrdquo In Environmental Toxicology and Chemistry 12 (1993) pp 1129ndash1133

[17] O Nilsson and D Sjelvgren ldquoHydro Unit Start-up Costs and Their Impact on theShort Term Scheduling Strategies of Swedish Power Producersrdquo In IEEE Transac-tions on Power Systems 12 (1997) pp 38ndash44

[18] M Olsson ldquoOn Optimal Hydropower Bidding in Systems With Wind Powerrdquo PhDthesis KTH Electrical Engineering 2009

[19] M Olsson and L Soder ldquoModeling Real-Time Balancing Power Market PricesUsing Combined SARIMA and Markov Processesrdquo In IEEE Transactions on PowerSystems 23 (2) (2008) pp 443ndash450

[20] Dash Optimization Xpress MP Essentials 2002

[21] MA Plazas AJ Conejo and FJ Prieto ldquoMultimarket optimal bidding for apower producerrdquo In IEEE Transactions on Power Systems 20 (4) (2005) pp 2041ndash2050

[22] R Core Team R A Language and Environment for Statistical Computing R Foun-dation for Statistical Computing Vienna Austria 2012 url httpwwwR-projectorg

[23] K Skytte ldquoThe Regulating Power Market on the Nordic Power Exchange NordPool An Econometric Analysisrdquo In Energy Economics 21 (4) (1999) pp 295ndash308

[24] Nord Pool Spot Website Sept 2012 url httpwwwnpspotcom

[25] Statnett Systemdrifts- og markedsutviklingsplan 2012 url httpwwwstatn

ettnoDocumentsKraftsystemetSystemansvaretStatnett_SMUP_2405_lnk_

Lowpdf

[26] Statnett Vilkar for anmelding handtering av bud og prissetting i regulerkraftmarkedet2009 url httpwwwstatnettnonoKraftsystemetMarkedsinformasjon

Regulerkraft-RKMVilkar

[27] Statnett Vilkar for tilbud aksept og bruk av regulerkraftopsjoner i produksjon forbruk 2012 url httpwwwstatnettnonoKraftsystemetMarkedsinformasjonRegulerkraftopsjoner-RKOMVilkar-for-RKOM

[28] Statnett Vilkar for tilbud aksept rapportering og avregning i Marked for primaeligr-reserver 2011 url httpwwwstatnettnonoKraftsystemetMarkedsinformasjonFrekvensstyrte-reserverVilkar

[29] Statnett Website Nov 2012 url httpwwwstatnettno

[30] K Stenshorne ldquoEvaluating Probabilistic Forecasts of Electric Prices - A Case Studyof the Nord Pool Marketrdquo MA thesis NTNU 2011

[31] A Ugedo E Lobato A Franco L Rouco J Fernandez-Caro and J Chofre ldquoStrate-gic bidding in sequential electricity marketsrdquo In IEE Proceedings- GenerationTransmission and Distribution 153 (2006) pp 431 ndash442

[32] I Wangensteen Power System Economics - the Nordic Electricity Markets TapirAcademic Press 2006

[33] WS Wei Time Series Analysis - Univariate and Multivariate Methods SecondEdition Pearsi Education Inc 2006

[34] R Weron Modelling and Forecasting Electricity Loads and Prices - A StatisticalApproach John Wiley amp Sons Ltd 2006

Introduction

The Nordic Electricity Markets and Institutional Background

Electricity Markets

The Spot Market

The Balancing Power Market

The Value of Water

Scheduling Hierarchy

Short Term Production Scheduling

A Stochastic Programming Model for the Coordinated Bidding Problem

Modeling the Markets

Problem Formulation

Scenario Generation

Spot Prices

Balancing Market Volumes and Premiums

Scenario Generation

Case Study

Case Description

Results

Comparison With Sequential Model

Discussion

Conclusion and Future Work

Future Work

Abstract

This paper discusses whether a hydropower producer in the Nordic region should takethe Balancing Power Market into account in the spot bidding decisions A stochasticprogramming model for the coordinated bidding problem is developed and tested for aNorwegian watercourse during the fall of 2012 Further the paper analyses in whichsituations a coordinated bidding model may outperform a sequential model The paperconcludes that a coordinated model primarily is useful in situation in which the deviationbetween the spot price and the marginal water value is small However the model needsmore testing before a final conclusion can be made The deterministic equivalent of thestochastic programming problem is implemented which resulted in very long runningtimes An appropriate solution method needs to be found

Preface

Contents

1 Introduction 1

1 Introduction

Background

22 The Spot Market

max z =sumjisinJ

sumtisinT

ojt + v (212)

m isinM

P1(i+1) minus P1i

i isin Im

ymts =imtssumi=1

sisinS

and sumtisinT B

sumsisinS

(σ+ts∆w

)(310)

y2ts lesumjisinJ

max z =sumsisinS

(summisinM

sumtisinT

ojts +sumtisinT B

(σ+ts∆w

)(313)

x(i1ts+1)t

ymts =imtssumi=1

sisinS

y2ts lesumjisinJ

y1ts =sumjisinJ

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

Preface

Contents

1 Introduction 1

1 Introduction

Background

22 The Spot Market

max z =sumjisinJ

sumtisinT

ojt + v (212)

m isinM

P1(i+1) minus P1i

i isin Im

ymts =imtssumi=1

sisinS

and sumtisinT B

sumsisinS

(σ+ts∆w

)(310)

y2ts lesumjisinJ

max z =sumsisinS

(summisinM

sumtisinT

ojts +sumtisinT B

(σ+ts∆w

)(313)

x(i1ts+1)t

ymts =imtssumi=1

sisinS

y2ts lesumjisinJ

y1ts =sumjisinJ

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

Preface

Contents

1 Introduction 1

1 Introduction

Background

22 The Spot Market

max z =sumjisinJ

sumtisinT

ojt + v (212)

m isinM

P1(i+1) minus P1i

i isin Im

ymts =imtssumi=1

sisinS

and sumtisinT B

sumsisinS

(σ+ts∆w

)(310)

y2ts lesumjisinJ

max z =sumsisinS

(summisinM

sumtisinT

ojts +sumtisinT B

(σ+ts∆w

)(313)

x(i1ts+1)t

ymts =imtssumi=1

sisinS

y2ts lesumjisinJ

y1ts =sumjisinJ

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

Contents

1 Introduction 1

1 Introduction

Background

22 The Spot Market

max z =sumjisinJ

sumtisinT

ojt + v (212)

m isinM

P1(i+1) minus P1i

i isin Im

ymts =imtssumi=1

sisinS

and sumtisinT B

sumsisinS

(σ+ts∆w

)(310)

y2ts lesumjisinJ

max z =sumsisinS

(summisinM

sumtisinT

ojts +sumtisinT B

(σ+ts∆w

)(313)

x(i1ts+1)t

ymts =imtssumi=1

sisinS

y2ts lesumjisinJ

y1ts =sumjisinJ

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

Contents

1 Introduction 1

1 Introduction

Background

22 The Spot Market

max z =sumjisinJ

sumtisinT

ojt + v (212)

m isinM

P1(i+1) minus P1i

i isin Im

ymts =imtssumi=1

sisinS

and sumtisinT B

sumsisinS

(σ+ts∆w

)(310)

y2ts lesumjisinJ

max z =sumsisinS

(summisinM

sumtisinT

ojts +sumtisinT B

(σ+ts∆w

)(313)

x(i1ts+1)t

ymts =imtssumi=1

sisinS

y2ts lesumjisinJ

y1ts =sumjisinJ

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

1 Introduction

Background

22 The Spot Market

max z =sumjisinJ

sumtisinT

ojt + v (212)

m isinM

P1(i+1) minus P1i

i isin Im

ymts =imtssumi=1

sisinS

and sumtisinT B

sumsisinS

(σ+ts∆w

)(310)

y2ts lesumjisinJ

max z =sumsisinS

(summisinM

sumtisinT

ojts +sumtisinT B

(σ+ts∆w

)(313)

x(i1ts+1)t

ymts =imtssumi=1

sisinS

y2ts lesumjisinJ

y1ts =sumjisinJ

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

1 Introduction

Background

22 The Spot Market

max z =sumjisinJ

sumtisinT

ojt + v (212)

m isinM

P1(i+1) minus P1i

i isin Im

ymts =imtssumi=1

sisinS

and sumtisinT B

sumsisinS

(σ+ts∆w

)(310)

y2ts lesumjisinJ

max z =sumsisinS

(summisinM

sumtisinT

ojts +sumtisinT B

(σ+ts∆w

)(313)

x(i1ts+1)t

ymts =imtssumi=1

sisinS

y2ts lesumjisinJ

y1ts =sumjisinJ

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

Background

22 The Spot Market

max z =sumjisinJ

sumtisinT

ojt + v (212)

m isinM

P1(i+1) minus P1i

i isin Im

ymts =imtssumi=1

sisinS

and sumtisinT B

sumsisinS

(σ+ts∆w

)(310)

y2ts lesumjisinJ

max z =sumsisinS

(summisinM

sumtisinT

ojts +sumtisinT B

(σ+ts∆w

)(313)

x(i1ts+1)t

ymts =imtssumi=1

sisinS

y2ts lesumjisinJ

y1ts =sumjisinJ

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

22 The Spot Market

max z =sumjisinJ

sumtisinT

ojt + v (212)

m isinM

P1(i+1) minus P1i

i isin Im

ymts =imtssumi=1

sisinS

and sumtisinT B

sumsisinS

(σ+ts∆w

)(310)

y2ts lesumjisinJ

max z =sumsisinS

(summisinM

sumtisinT

ojts +sumtisinT B

(σ+ts∆w

)(313)

x(i1ts+1)t

ymts =imtssumi=1

sisinS

y2ts lesumjisinJ

y1ts =sumjisinJ

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

22 The Spot Market

max z =sumjisinJ

sumtisinT

ojt + v (212)

m isinM

P1(i+1) minus P1i

i isin Im

ymts =imtssumi=1

sisinS

and sumtisinT B

sumsisinS

(σ+ts∆w

)(310)

y2ts lesumjisinJ

max z =sumsisinS

(summisinM

sumtisinT

ojts +sumtisinT B

(σ+ts∆w

)(313)

x(i1ts+1)t

ymts =imtssumi=1

sisinS

y2ts lesumjisinJ

y1ts =sumjisinJ

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

22 The Spot Market

max z =sumjisinJ

sumtisinT

ojt + v (212)

m isinM

P1(i+1) minus P1i

i isin Im

ymts =imtssumi=1

sisinS

and sumtisinT B

sumsisinS

(σ+ts∆w

)(310)

y2ts lesumjisinJ

max z =sumsisinS

(summisinM

sumtisinT

ojts +sumtisinT B

(σ+ts∆w

)(313)

x(i1ts+1)t

ymts =imtssumi=1

sisinS

y2ts lesumjisinJ

y1ts =sumjisinJ

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

max z =sumjisinJ

sumtisinT

ojt + v (212)

m isinM

P1(i+1) minus P1i

i isin Im

ymts =imtssumi=1

sisinS

and sumtisinT B

sumsisinS

(σ+ts∆w

)(310)

y2ts lesumjisinJ

max z =sumsisinS

(summisinM

sumtisinT

ojts +sumtisinT B

(σ+ts∆w

)(313)

x(i1ts+1)t

ymts =imtssumi=1

sisinS

y2ts lesumjisinJ

y1ts =sumjisinJ

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

max z =sumjisinJ

sumtisinT

ojt + v (212)

m isinM

P1(i+1) minus P1i

i isin Im

ymts =imtssumi=1

sisinS

and sumtisinT B

sumsisinS

(σ+ts∆w

)(310)

y2ts lesumjisinJ

max z =sumsisinS

(summisinM

sumtisinT

ojts +sumtisinT B

(σ+ts∆w

)(313)

x(i1ts+1)t

ymts =imtssumi=1

sisinS

y2ts lesumjisinJ

y1ts =sumjisinJ

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

max z =sumjisinJ

sumtisinT

ojt + v (212)

m isinM

P1(i+1) minus P1i

i isin Im

ymts =imtssumi=1

sisinS

and sumtisinT B

sumsisinS

(σ+ts∆w

)(310)

y2ts lesumjisinJ

max z =sumsisinS

(summisinM

sumtisinT

ojts +sumtisinT B

(σ+ts∆w

)(313)

x(i1ts+1)t

ymts =imtssumi=1

sisinS

y2ts lesumjisinJ

y1ts =sumjisinJ

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

max z =sumjisinJ

sumtisinT

ojt + v (212)

m isinM

P1(i+1) minus P1i

i isin Im

ymts =imtssumi=1

sisinS

and sumtisinT B

sumsisinS

(σ+ts∆w

)(310)

y2ts lesumjisinJ

max z =sumsisinS

(summisinM

sumtisinT

ojts +sumtisinT B

(σ+ts∆w

)(313)

x(i1ts+1)t

ymts =imtssumi=1

sisinS

y2ts lesumjisinJ

y1ts =sumjisinJ

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

max z =sumjisinJ

sumtisinT

ojt + v (212)

m isinM

P1(i+1) minus P1i

i isin Im

ymts =imtssumi=1

sisinS

and sumtisinT B

sumsisinS

(σ+ts∆w

)(310)

y2ts lesumjisinJ

max z =sumsisinS

(summisinM

sumtisinT

ojts +sumtisinT B

(σ+ts∆w

)(313)

x(i1ts+1)t

ymts =imtssumi=1

sisinS

y2ts lesumjisinJ

y1ts =sumjisinJ

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

max z =sumjisinJ

sumtisinT

ojt + v (212)

m isinM

P1(i+1) minus P1i

i isin Im

ymts =imtssumi=1

sisinS

and sumtisinT B

sumsisinS

(σ+ts∆w

)(310)

y2ts lesumjisinJ

max z =sumsisinS

(summisinM

sumtisinT

ojts +sumtisinT B

(σ+ts∆w

)(313)

x(i1ts+1)t

ymts =imtssumi=1

sisinS

y2ts lesumjisinJ

y1ts =sumjisinJ

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

max z =sumjisinJ

sumtisinT

ojt + v (212)

m isinM

P1(i+1) minus P1i

i isin Im

ymts =imtssumi=1

sisinS

and sumtisinT B

sumsisinS

(σ+ts∆w

)(310)

y2ts lesumjisinJ

max z =sumsisinS

(summisinM

sumtisinT

ojts +sumtisinT B

(σ+ts∆w

)(313)

x(i1ts+1)t

ymts =imtssumi=1

sisinS

y2ts lesumjisinJ

y1ts =sumjisinJ

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

max z =sumjisinJ

sumtisinT

ojt + v (212)

m isinM

P1(i+1) minus P1i

i isin Im

ymts =imtssumi=1

sisinS

and sumtisinT B

sumsisinS

(σ+ts∆w

)(310)

y2ts lesumjisinJ

max z =sumsisinS

(summisinM

sumtisinT

ojts +sumtisinT B

(σ+ts∆w

)(313)

x(i1ts+1)t

ymts =imtssumi=1

sisinS

y2ts lesumjisinJ

y1ts =sumjisinJ

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

P1(i+1) minus P1i

i isin Im

ymts =imtssumi=1

sisinS

and sumtisinT B

sumsisinS

(σ+ts∆w

)(310)

y2ts lesumjisinJ

max z =sumsisinS

(summisinM

sumtisinT

ojts +sumtisinT B

(σ+ts∆w

)(313)

x(i1ts+1)t

ymts =imtssumi=1

sisinS

y2ts lesumjisinJ

y1ts =sumjisinJ

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

sisinS

and sumtisinT B

sumsisinS

(σ+ts∆w

)(310)

y2ts lesumjisinJ

max z =sumsisinS

(summisinM

sumtisinT

ojts +sumtisinT B

(σ+ts∆w

)(313)

x(i1ts+1)t

ymts =imtssumi=1

sisinS

y2ts lesumjisinJ

y1ts =sumjisinJ

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

y2ts lesumjisinJ

max z =sumsisinS

(summisinM

sumtisinT

ojts +sumtisinT B

(σ+ts∆w

)(313)

x(i1ts+1)t

ymts =imtssumi=1

sisinS

y2ts lesumjisinJ

y1ts =sumjisinJ

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

max z =sumsisinS

(summisinM

sumtisinT

ojts +sumtisinT B

(σ+ts∆w

)(313)

x(i1ts+1)t

ymts =imtssumi=1

sisinS

y2ts lesumjisinJ

y1ts =sumjisinJ

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

max z =sumsisinS

(summisinM

sumtisinT

ojts +sumtisinT B

(σ+ts∆w

)(313)

x(i1ts+1)t

ymts =imtssumi=1

sisinS

y2ts lesumjisinJ

y1ts =sumjisinJ

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

y2ts lesumjisinJ

y1ts =sumjisinJ

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

41 Spot Prices

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

a(tminus1) + Θε

εats = Θε1ε

a(tminus1)s + Θε

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

forallt isin T T

πTij =

where πNij and πD

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

077 009 014026 070 004022 002 076

084 008 008018 080 002015 001 084

νa2t = Θν21 ν

a2(tminus1) + Θν2

2t forallt isin T T

νa3t = Θν31 ν

a3(tminus1) + Θν3

1 ων3(tminus1)ω

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

νa2ts = Θν21 ν

νa3ts = Θν31 ν

νmts = exp

(σνm)2

2is added

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

a Θ1 Θ2 Θ3 Φ1 Φ2 Γ σ2

νa2t 440 167 -165 077 -100 100 NA 042νa3t 434 -010 047 NA 076 NA NA 056

δa2t 167 NA NA NA 106 058 002 010

δa3t 185 NA NA NA 075 035 001 009

δmts = exp

(σδm)2

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

End ifEnd do

End doEnd do

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

5 Case Study

51 Case Description

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

52 Results

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

Coordinatedmodel

Mean Stdev

Planninghorizon

Biddingperiod

54 Discussion

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

61 Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

References

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

Lowpdf

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work

Introduction

Electricity Markets

The Spot Market


The Value of Water





Problem Formulation

Scenario Generation

Spot Prices


Scenario Generation

Case Study

Case Description

Results


Discussion


Future Work