The Economics of Electricity Dynamic Pricing and Demand...

The Economics of Electricity Dynamic Pricing

and Demand Response Programmes

Application to Controlling BEVs and PHEVs Charging and Storage

Christopher Andreylowast Alain Hauriedagger

April 2013

lowastORDECSYS - christopherandreyordecsyscomdaggerORDECSYS - ahaurieordecsyscom

Abstract

This document is the first deliverable of the project entitled Adaptive and TOUpricing schemes for smart technology integration funded by the Swiss Federal Officeof Energy (SFOE) Its aim is twofold (i) through an economic analysis we explorethe different possible schemes of adaptive and time-varying tariffs that could beproposed by electricity distributors in the context of a development of smart-gridtechnologies (ii) based on economic models we evaluate the potential for thesetariffs to realise load shifting andor shedding and to facilitate the integration ofbattery and plug-in hybrid electric vehicles (BEVs and PHEVs)

Please note that equations with free indices are understood to bevalid for all values the indices can take unless otherwise stated

Contents

1 Introduction and Motivation 6

2 Types of Rates and Pricing Schemes 7

21 Marginal Cost Pricing and Time-Varying Tariffs 8

22 The Electricity Price Composition in Switzerland 10

23 The Two Main Origins of Price Variation 12

24 A Survey of Time-Varying Tariffs 14

241 Time of Use 15

242 Critical Peak Pricing 15

243 Peak Time Rebates 16

244 Real-Time Pricing 16

25 Time-Varying Tariffs for Captive Consumers in Switzerland 17

251 Groupe-E - rdquo1to1 energy easyrdquo 18

252 Romande-Energie - rdquovolta doublerdquo 18

253 SIG - rdquoTarif Profile Doublerdquo 18

26 Time-Varying Tariffs for Consumers Granted with Network Access inSwitzerland 19

261 SIG - rdquoTarif 35 periodesrdquo 19

262 The FlexLast project 20

27 Smart Grids and Time-Differentiated Pricing Schemes 20

3 Customersrsquo Response to Time-Differentiated Tariffs and its Econo-metric Modelling 22

3

31 Effectiveness of Time-Varying Tariffs 22

311 Ability to Reflect the Marketrsquos Status 22

312 Technology as an Enhancer 23

313 Barriers to the Diffusion of Innovations 23

314 Rebound and Backfire Effects 24

32 Overview of Householdsrsquo Response to TOU Schemes 24

321 Empirical Evidence in Switzerland 24

322 Peak-Load Reduction Evidence 26

4 Economics of Time-Varying Rates and Load Shifting 26

41 A Global Model of TOU Pricing 27

411 Supply Side 27

412 Demand Side 29

413 Representing the Supply-Demand Equilibrium 30

414 A Numerical Illustration 30

42 Non-Cooperative Game Models 32

43 Competitive Equilibrium Models 34

431 A Global Linear Programming Formulation 34

44 Multilayer Game Models 36

5 Optimal Distributed Charging and Storage Control for a Populationof Electric Vehicles 36

51 Decentralised Control and Incentive-Based Coordinated Charging Control 37

511 A Mean-Field Game Solution 37

4

512 A Decentralised Control Scheme Based on RT Pricing 38

52 Optimal Storage and V2B Operations 40

521 Example of V2B Operations 40

522 Economic Analysis of the Use of Storage Provided by Idle PHEVsBased on Locational Marginal Pricing 41

53 Other Ancillary Services Provided by EVs 42

6 Conclusion 42

5

6

1 Introduction and Motivation

Adaptive and dynamic tariffs are designed in such a way to better inform the customerof the status of the electricity market whose volatility may increase in situations inwhich the production of electricity from renewable sources substantially penetratesIndeed the generation of electricity from solar and wind origin are intrinsically in-termittent leading to potentially large and rapid variations of the marginal price ofelectricity Due to the introduction of bidirectional communication possibilities viasmart metering systems it is expected that customers will try to optimise the timingof their different usages in order to benefit from off-peak tariffs ie periods duringwhich the price of electricity is low due to a low demand and a high level of potentialproduction from renewables

Given the time-scales on which the production of electricity from renewables mayvary adapting ones consumption would represent quite a burden Automatic solutionsintegrating decision algorithms already exist on the market Through the use of smartmeters - which are sophisticated electricity meters whose communications are bidirec-tional and in real-time connecting the decentralised production and consumption unitsto the grid - the energy distributors may influence and adapt the demand and adjustit to the production by shifting some of the deferrable loads customers did previouslydefine

This represents a dramatic shift in the electricity distribution paradigm Beforethe appearance and potentially large penetration of intermittent energy sources theproduction-side was given the role to be the one that adapts itself to the demand-side Nowadays the reverse mechanism known under the name of demand-responseis about to become prevalent However the adoption of demand-response mechanismsby consumers is not straightforward an issue Indeed in order for such strategies to beimplemented customers have to accept to give up a certain amount of control on theirconsumption behaviour in favour of their electricity distributor

In a context of high penetration of renewables another important issue concernsshort term electricity storage to alleviate the intermittent production patterns of thesetechnologies There is also an opportunity of storing the energy when available at alow cost ie when renewables are the marginal producers and to inject it back inthe electricity network at peak periods Both aspects of load shifting and possibilityof temporary storage are present in the management of a large fleet of BEVs andPHEVs1 Modern electric vehiclesrsquo batteries have large enough a capacity to servenot only as a reserve for the car itself but also as a buffer capable of absorbing theirregularities of the production patterns Adaptive dynamic pricing scheme will beamong the tools used to implement incitative schemes to control distributed chargingand storage operations But again customers will have to accept that their vehiclesrsquo

1BEV Battery electric vehicle PHEV Plug-in hybrid electric vehicle

Rue du Gothard 5 ndash Chene-Bourg ndash Switzerland Tel +41 22 940 30 20 ndash wwwordecsyscom

7

batteries are at least partially remotely managed by their electricity distributor in orderfor such mechanisms to be successfully implemented

Some surveys have already been made concerning public acceptance and responseto dynamic pricing Most of them concern time of use pricing which is the less adaptiveform of such tariffs A need exists to complement the few existing surveys concerningacceptability of dynamic pricing schemes mostly conducted in the United States ofAmerica (see eg [1]) by regional surveys where conjoint analysis (see eg [2 34]) is used to elicit the acceptability and potential effectiveness of demand-responsemechanisms

A Brief Introduction to Conjoint Analysis

Conjoint analysis is a marketing technique whose target is tomeasure preferences structure in the presence of a large numberof attributes which appeared in the early 1970s

Instead of asking consumers to evaluate on a given scale the at-tractiveness of a scenario or of a product that is characterised bya set of attributes one offers them to choose among a selectionof balanced scenarios By using simple mathematical techniquesone can extract the relative importance of the attributes Onecan then compute the attractiveness of scenarios which were notproposed to the test-panel Such a knowledge would then allowfor an adaptation of the product so as to enhance its desirability

The rest of the report is organised as follows Section 2 concerns the different typesof time-varying and adaptive tariffs that have been envisioned in utility managementwith a particular focus on the current practice in Switzerland Section 3 deals withthe modelling of customer response to dynamic tariffs using econometric analyses Sec-tion 4 is devoted to the microeconomics of time varying tariffs for electricity and itsrepresentation in planning models in Section 5 the activities of optimal distributedcharging and storage control for a population of electric vehicles are considered finallySection 6 provides a conclusion

2 Types of Rates and Pricing Schemes

This section consists of an inventory and a characterisation of the various time-varyingelectricity pricing schemes that are available to electricity distributors Such schemesare designed with the purpose of influencing the final usersrsquo consumption behaviour the


8

main target for distributors being to reduce the peak load and to ensure that productionand demand meet which is especially challenging when renewables take an importantplace in the energy panorama

Many different sorts of time-varying tariffs do exist the difference between thembeing the characteristic time-scales on which they vary and their dynamics In thefollowing we will introduce the main categories of tariffs ranging from time-of-usetariffs (TOU) to real-time pricing (RTP) The main advantage of these tariffs is thatthey allow the electricity distributors to reflect the state of the electricity generationfacilities and thus of the marginal cost of electricity on the prices paid by customers

The section begins with a brief description of the main arguments for time-varyingtariffs in the electricity sector and continues with a discussion of the main components ofthe price of electricity Finally the main categories of time-varying tariffs are consideredwith particular reference to the case of Switzerland when available

21 Marginal Cost Pricing and Time-Varying Tariffs

Before starting the survey of dynamical tariffs let us briefly discuss the economicreason for the emergence of time-varying tariffs for final consumers It is well knownthat electricity cannot be stored2 the amount of electricity generated is therefore to bestrongly correlated to the demand level This results in more electricity being generatedduring peak periods (around midday in Switzerland) and less during the night In off-peak periods only those facilities that are characterised by low operation costs areused (mostly hydro-electric and nuclear facilities in Switzerland) When the demandgrows the production adapts itself and facilities whose variable costs are higher andhigher begin to be used The cost of electricity is thereby rising This shows theimportance of the concept of marginal cost of electricity which is the cost of producingone supplementary unit of electricity (a kWh per example) On the wholesale marketwhich is competitive the electricity is naturally priced according to the marginal costof its production Figure 1 shows the fluctuations of the Swiss spot market price acrossthe day on the 5th of March 20133

However in order for consumers and the economy not to be bothered with startingconsuming only when prices are below a certain threshold electricity distributors (whoproduce and buysell electricity on the spot market) traditionally apply flat tariffs onthe retail market These tariffs allow the final consumers to plan their consumptionregardless of the state of the electricity market Static rates thus imply predictability

2Note however that electric power may be used to produce energy that can be stored in batteriesdams spinning flywheels etc and released at later times

3Retrieved from httpwwwepexspotcomenmarket-dataauctionchartauction-chart

2013-03-05CH


9

Figure 1 Electricity price evolution on the Swiss spot market on March 5th 2013

which is an advantage for a prosperous economic development to take place

However as already stated in the Introduction the situation is likely to changebecause of the following two facts

1 An important penetration of renewables suggests that the demand has to at leastpartially adapt itself to production since the latter is subject to rigid short-termcapacity constraints In order to shift some of the demand usages the electricityprice will have to act as a lever

2 In the last decades the electricity consumption hasnrsquot stopped rising Even thoughthe overall efficiency of most devices is improving the upward consumption trendis not showing any signs of slowing down as shown in Figure 24 Therefore thestress in the electricity transmission and distribution infrastructure is likely toworsen in the next years In order to reduce the need for an early infrastruc-ture upgrade and to improve the networkrsquos reliability and since the network isdimensioned in order to be able to handle peak loads time-varying prices wouldrepresent an incentive to shift some of the usages out of peak periods

In summary whereas the wholesale market prices truly reflect the state of thegeneration transport and distribution of electricity the retail market is at best onlypartially reflecting the marginal costs of electricity Economists have long been arguingthat time-differentiated pricing schemes have to be enforced in order to provide end-users with incentives convincing them to modify their consumption pattern [5] It isthen concluded that marginal pricing schemes contribute to solving the challenges thathave to be addressed by electricity distributors ie lessening the peak load by shifting

4Data retrieved from the Swiss Federal Statistical Office STAT-TAB Section 083 httpwww

pxwebbfsadminch


10

013

5013

10013

15013

20013

25013

191013 192013 193013 194013 195013 196013 197013 198013 199013 200013 201013

Elec

tric

ity c

onsu

mpt

ion

[PJ]13

Year13

Figure 2 Electricity consumption in Switzerland from 1910 to 2011 (Note that2011 has witnessed a slight decrease of the consumption in comparison with 2010 Theparticularly cold weather conditions of December 2010 are responsible for this effect)

some of the loads to off-peak periods and managing the demand so that it adapts to theproduction The second point is particularly relevant in situations in which renewablessubstantially penetrate In order to be able to qualitatively and quantitatively estimatethe effects of time-varying tariffs it is important to understand the structure of theelectricity price In the following subsection we describe the various components of theprice for a kWh of electricity in Switzerland

22 The Electricity Price Composition in Switzerland

In Switzerland as in most of the western countries the electricity market is beingliberalised Before the liberalisation consumers were obliged to buy their electricityto their local distributor meaning that the latter was charging them with a singleprice that comprised both the price of energy generation and that of transport anddistribution

As of January 1st 2009 entities consuming more than 100 MWh of electricity peryear are granted the network access5 Consequently these consumers are allowed touse their local electricity distributorrsquos network to buy electricity on the market may itbe inside or outside Switzerland and use the local infrastructure to deliver it Undersuch a scheme the structure of the traditional electricity distributor is split in two oneentity being responsible for the electricity generation import and export the other onefor the electricity delivery to the final users6 Consequently the price of a delivered kWh

5See SR 73471 Article 11 Available at httpwwwadminchchdsr73html6Electricity distributors have the legal obligation to split their activities into these two distinct


11

has several components the two most important ones being the price for generatingthe electricity and the price for its delivery

Even though only large consumers7 have access to the network and in anticipationof the complete market liberalisation that should occur in the next years in Switzerlandthe bills of all consumers do exhibit the aforementioned structure More precisely theprice of a kWh is subdivided into the following components

1 Electricity generation costThis component corresponds to the cost of producing one kWh of electricity Itmay depend on the type of generation facility on the price of the primary energythat is used on the facilityrsquos amortisation rate etc

2 Electricity transmission and distribution costsThis part of the price is related to the network use necessary to carry the elec-tricity from its generating facility to the final user It is typically further split intwo sub-components corresponding to high voltage networks (transmission im-port and export) and low voltage networks (distribution) In Switzerland thetransmission is operated by Swissgrid8 whereas local distributors are responsiblefor the final transit to the consumer ie a natural monopoly characterises thisprice component which is regulated by the ElCom9

3 Public fees and services to collectivitiesThe third component is related to the price paid to public collectivities for theuse of their assets related to the distribution of electricity For example a taxfor using public areas for the distribution of electricity is to be incorporated inthis component to the contrary of a tax corresponding to the use of water byhydroelectric facilities which is to be incorporated in the electricity generationprice10 Local authorities may also use taxes to promote renewables For examplein the Canton of Vaud 018 ctskWh are charged to encourage local initiativesin the sector of renewable energy sources (see LVLEne 73001 Art 40)

4 ldquoFeed-in Remuneration at Costrdquo (KEV) taxIn order to encourage the development of energy production from renewable en-

sectors in a procedure known as unbundling see SR 7347 Article 10 Available at httpwww

adminchchdsr73html7ie more than 100 MWh per year the end-users consuming less than that amount being said to

be captive Note that a Swiss household consumes an average of around 5 MWh of electricity per year(Federal Statistical Office)

8Swissgrid is the entity operating Switzerlandrsquos transmission system with responsibility for theoperation security and expansion of the 6rsquo700 kilometre long high-voltage grid (Level 1 grid componentcorresponding to voltages in the 220 to 380kV range)

9The ElCom is Switzerlandrsquos independent regulatory authority in the electricity sector It is respon-sible for monitoring compliance with the Swiss Federal Electricity Act and the Swiss Federal EnergyAct

10See the ElCom Information issued on February 17th 2011 Available at httpwwwelcomadminchdokumentation0009100104indexhtml


12

ergy sources Switzerland has decided to guarantee producers that the electricitythey generate from renewables will be remunerated at a price that corresponds totheir production costs which may be higher than the market price the differencebeing paid through the ldquoFeed-in Remuneration at Costrdquo Tax (known in Switzer-land as the KEV Tax) which is currently fixed at the level of 035 ctskWhAnother surcharge of 010 ctskWh is currently being levied to finance waterconservation measures11

The relative importance of these components is illustrated on Figure 3 for the years2009-201212

25

0

5

10

15

20

Reacuteseau Energie Redevances RPC Total

Ct

kW

h (s

ans

TVA

)

2009

2010

2011

2012

Figure 6 Eleacutements de coucircts composant le prix total de lrsquoeacutelectriciteacute pour le profil de consommation H4

Comme le montre la figure 6 le prix total de lrsquoeacutelectriciteacute est surtout influenceacute par les prix du reacuteseau et de lrsquoeacutenergie tandis que les redevances les prestations et la redevance RPC constituent moins de dix pour cent du prix total Pour comparer les prix de lrsquoeacutelectri-citeacute il faut tenir compte du fait que les ta-rifs varient en fonction des reacutegions et que des diffeacuterences importantes sont possibles selon la quantiteacute drsquoeacutelectriciteacute consommeacutee par un client agrave un moment donneacute (profil de consommation)

Comptes annuels de reacuteseauDans sa directive 32011 agrave des fins de trans parence lrsquoElCom a fixeacute des exi gences minimales pour les comptes annuels de reacuteseau que les gestionnaires de reacuteseau de distri bution sont tenus de publier en vertu de lrsquoart 12 al 1 LApEl Les comptes annuels doivent comprendre un bilan et un compte de reacutesultat propres au reacuteseau (unbundled) preacutesenteacutes seacutepareacutement des autres domai-nes drsquoactiviteacute des montants chiffreacutes et les chiffres de lrsquoanneacutee preacuteceacutedente En outre le

Figure 3 Composition of the Swiss electricity prices in ctskWh VAT excluded TheRPC component corresponds to the ldquoFeed-in Remuneration at Costrdquo tax

23 The Two Main Origins of Price Variation

In dynamic pricing schemes both the electricity generation price and the transmissionand distribution costs may vary Periods of high demand would consequently be chargedat higher a price Two mechanisms are responsible for this phenomenon

1 Marginal cost of production

11For both these two taxes see httpswissgridchswissgriddehomeexpertstopicsgrid_

usagehtml12Illustration valid for the H4 Profile corresponding to households with a yearly consumption of 45

MWh of electricity (Source ElCom Tatigkeitsbericht 2011)


13

As the demand rises the set of facilities to be switched on evolves In periods oflow demand the only facilities running are those which are leading to cheap pricesand those that cannot be easily be stopped eg nuclear power plants When thedemand goes up gas and coal fired power plants are progressively turned onleading to higher prices The order in which the power plants are switched on isrelated to their marginal cost of production ie the cost for producing one moreunit of energy and to the price on the market The only exceptions to this ruleare those facilities that need to run close to full-time for technical reasons

The corresponding amount charged to consumers is in most cases proportional tothe quantity of delivered electricity (measured eg in kWh)

2 Transmission and distribution costs

On the other hand as the demand rises more electricity is to be conveyed bythe transmission and distribution networks Since the network capacity is finitethe price for electricity transit will tend to rise during periods of high demand(corresponding to large power flows)

The amount charged to consumers resulting from the transmission and distri-bution costs may have two components the first one being proportional to thequantity of delivered energy (in eg kWh) while the second is proportional to themaximal power13 (in eg kW) used in a given period of time The transmissionand distribution costs not only depend on the consumption profile but also on thetopography and on the density of residential development Note however thatin the following tariffs the transmission and distribution costs that are chargedto final consumers are independent of time congestion and distance The onlytime-varying tariffs are the ones of cross-border flows the capacity of the latterbeing allocated by an auction process14

We thus conclude that in order to reflect the situation of both the generationfacilities and of the transmission and distribution networks time-varying tariffs areappealing since they allow sending the right price signal to end-users [5] Indeed itis for example known that Californiarsquos 2000 electricity crisis was caused by rates thatwere not reflecting the real situation of the electricity market (the wholesale price wentup from around 30$ per MWh to 400$ per MWh without any consequence on the retailprices paid by the final users see eg [6]) leading to blackouts

Time-varying tariffs also allow a reward for the efforts done in reducing the energyconsumption during peak-periods Indeed since the price is higher during high demandperiods not consuming a unit of energy can result in potentially high savings

13The maximal power is generally defined as follows First the day is split in 15 minutes blocksthen the average power for each of the blocks is computed The maximum power is then defined as themaximum of these averaged powers

14Since April 2011 the Auction Office CASCEU performs these auctions see httpwwwcasceu

enResource-centerCWE--CSE-and-Switzerland


14

Many economists have observed that the principle of time-varying tariffs wouldsolve the problems induced by a varying production may that variation be due tospeculation as was the case in California or to intrinsic volatility as is the case for bothphotovoltaic panels and wind turbines production see eg [5] and references thereinThe theoretical appeal of such a solution is thus firmly established In order to assessthe viability of such a strategy one has to measure the behavioural changes when oneis confronted to time-varying tariffs More precisely one has to measure the relativechange in demand when a user is facing a change in price This precisely meets thedefinition of the price-elasticity of the demand denoted as η

η =∆DD∆pp

=∆ log(DD0)

∆ log(pp0)ie D prop pη (1)

Since the demand is not only sensitive to the price in the same period but also tothe price during other periods (the knowledge of a lower price in the future may tendto lessen the present demand) the corresponding elasticities must be introduced toaccount for this phenomenon One can then write in a self-explanatory notation

D(t) prop [p(t)]η1 [p(ti 6= t)]ηpi [D(ti 6= t)]η

Di (2)

where the ηs are the elasticities Note that in most of the models we will consider theelasticities will be assumed to be constant The ηs are then determined by fitting thedemand Qualitatively we can already anticipate that η1 will in most cases be negativesince the usual response to rising prices is a diminution of the demand On the otherhand the ηpi will generally tend to be positive since if the past andor future priceincreases the consumption will tend to be shifted to the current period of time

Several studies have been carried out in order to determine the effectiveness oftime-varying tariffs and to evaluate the price-elasticity Before exposing the results inSection s3 the existing types of time-varying tariffs are introduced in the next section

24 A Survey of Time-Varying Tariffs

Over the years a number of different strategies have been devised in order for theelectricity prices to reflect the state of electricity generation and network congestionSome of the pricing schemes are characterised by pre-determined adjustments othersby special hours or days when the electricity is charged at a different price and otherstruly reflect the changes in the marginal costs A more refined description of the mostwidely used tariffs is to be found in the following


15

241 Time of Use

The Time of Use (TOU) tariffs are the simplest and the most extensively used time-varying tariffs The days are typically split in 2 to 5 periods each being characterisedby a static price The splitting and prices are pre-determined and typically adaptedon a monthly basis The distributor fixes the electricity prices by using consumptionestimates The aim of TOU tariffs is to partially reflect the situation of the electricitymarket but has the advantage of being non-dynamical thereby allowing householdsand industries to schedule their activities as a function of the TOU blocks

242 Critical Peak Pricing

In Critical Peak Pricing (CPP) a normal tariff which is generally belonging to theTOU family is valid for most of the days of the year However a small number of daysper year are subject to a price change These occurrences correspond to periods ofvery high demand (peak loads) during which the generating utilities could not providesufficient a quantity of electricity if keeping the prices flat The electricity prices duringthe peak prices may be pre-determined or in some cases reflect the real-time status ofthe market These two situations are referred as CPP-F (fixed) and CPP-V (variable)

As far as the authors know CPP tariffs are not available to Swiss customers Oneof the most well known examples of such tariffs is to be found in France The ldquoTempordquotariff consists of three TOU tariffs In order of increasing tariffs these are the ldquobluerdquoone that is valid for 300 days per year the ldquowhiterdquo one for 43 days and the ldquoredrdquo onefor the remaining 22 days see Figure 4 Every day the colour of the following day iscommunicated by SMS and email A set of rules prevents the distributor from usingtoo many highly priced days in a row The tariffs range is quite wide the peak periodof a ldquoredrdquo day being charged around 7 times more than an off-peak period of a ldquobluerdquoday15

A variant of CPP is called interruptible rates The rates are following the TOUscheme most of the time but when facing critical events and potential shortages thefacility may call its customers upon to cease their consumption In the case the cus-tomers cannot stop their processes the electricity is still delivered but at rates that areincreased up to several dozens times see eg [7]

15See httpparticuliersedfcomgestion-de-mon-contratoptions-tempo-et-ejp

option-tempodetails-de-l-option-52429html


16

Figure 4 EDFrsquos Tempo tariff

243 Peak Time Rebates

The Peak Time Rebates (PTR) scheme is very similar to CPP in that only a reducednumber of periods characterised by either a high demand or a low production areimpacted by exceptional tariffs In PTR the incentive not to consume during peaktime is not realised by increasing the corresponding prices but in a non-punitive wayby rewarding consumers if they agree to reduce their loads (compared to their baselineconsumption) by either shedding them or shifting them outside of the critical peakperiods Note that no penalty is inflicted if consumers use electricity during thesespecial periods the usual tariff being applied Figure 5 illustrates the effect of offeringrebates by comparing the effective consumption to the averaged baseline consumptionNote that outside the peak period the consumption of the group being offered withinterruptible rates is higher than the baseline consumption This is an illustration ofthe rebound effect which will be discussed in 314

244 Real-Time Pricing

In Real-Time Pricing (RTP) schemes electricity tariffs are reflecting the electricitymarket situation Prices are not pre-determined and are typically subject to hourlychanges This pricing scheme being extremely difficult to handle (intensive exchangeof data new billing procedures to define etc) it is often proposed in conjunctionwith other contracts For example a fixed quantity of energy (the so-called customerbaseline load) may be sold at TOU rates the amounts exceeding the threshold being theonly ones charged according to the RTP scheme Such a hedging strategy reduces thecustomerrsquos exposition to price volatility and offers utilities a greater revenue stabilityFinancial risk management products are also offered by some utilities in order forcustomers to customise their exposure to price differences eg by entering forwardcontracts Examples of voluntary RTP schemes in the United States of America andtheir effects on load shifting may be found eg in [1]


17

Figure 5 The blue line corresponds to the control group which is charged withflat tariffs The magenta line corresponds to consumers being offered with PTR butnot responding to the incentives Finally the black line corresponds to the consumersresponding to the PTR incentives [8] The period during which rebates were offered isthe one ranging from 11am to 6pm

25 Time-Varying Tariffs for Captive Consumers in Switzerland

As anticipated in the beginning of this section the first step of the liberalisation ofthe Swiss electricity market was to allow end-users consuming more than 100 MWh ofelectricity per year to freely choose their electricity producer16 they are said to have anetwork access The remainder of the end-users are said to be captive since for themthe electricity market takes the form of a monopoly The tariffs available for captiveconsumers are regulated according to SR 7347 Article 6 and SR 73471 Article 417 Theliberalisation process will eventually allow every end-user to benefit from the networkaccess

In addition to the traditional flat rates Switzerlandrsquos main distributors proposeTOU schemes These tariffs are available both for captive consumers and for consumersthat benefit from the network access In Switzerland the only non-flat tariffs offeredto captive consumers known to the authors are of the TOU kind Examples of suchtariffs are given in the following18

16The ElCom estimates that around 20 of the energy consumed by end-users provided with thenetwork access was bought on the market in 2012 The first figures for 2013 indicate that this ratioshould attain around 26 (Sources ElCom Tatigkeitsbericht 2012 and private communications withthe ElCom)

17Available at httpwwwadminchchdsr73html18Retrieved from httpwwwstrompreiselcomadminch


18

251 Groupe-E - rdquo1to1 energy easyrdquo

In the ldquo1to1 energy easyrdquo the day is split in three periods the one ranging from 7amto 21pm being the peak period during which the electricity is around 15 times moreexpensive than in the off-peak period (1938 vs 1254 ctskWh) For clients that areconsuming more energy but are still captive the electricity price is lower but is tobe supplemented with a power component of a few CHF per kW In this case ie inthe ldquo1to1 energy easy powerrdquo pricing scheme the kW is billed at CHF 680 while theon-peak and off-peak tariffs are respectively reduced to 1508 and 874 ctskWh Notethat the power is defined as the maximal averaged power on 15 minutes blocks duringthe month at hand

Tarifs drsquoeacutelectriciteacute 2013 pour les clients consommant moins de 100 MWhan Le prix de votre eacutelectriciteacute est composeacute de trois parties lrsquoeacutenergie et la distribution (propres agrave Groupe E) et les prestations de Swissgrid (transport national)

Pour les particuliers et les petites entreprises artisanales Composition

Tarif unitaire (0-24h)

Prix de lrsquoeacutelectriciteacute

1to1 energy easy light

dirgssiwS E epuorG latoT

tropsnarT noitubirtsiD eigrenE lanoitaN

Montant de base (CHFan) 7560 - 7560 -

Tarif unitaire (ctkWh) 1986 1015 883 088

Pour les installations interruptibles telles que chauffe-eau-eacutelectriques en compleacutement drsquoune installation principale au tarif 1to1 energy easy light easy ou easy power Composition



1to1 energy break light





Pour les particuliers et les petites entreprises Composition

ffeacuterencieacute en haut et bas tarif


1to1 energy easy




Haut tarif (ctkWh) 1938 1010 840 088

Bas tarif (ctkWh) 1254 660 506 088

Pour les proprieacutetaires drsquoappareils et drsquoinstallations fixes interruptibles tels que les pompes agrave chaleur ou chauffages eacutelectriques Composition



1to1 energy easy comfort




Haut tarif (ctkWh) 1874 1030 756 088

Bas tarif (ctkWh) 1116 650 378 088

Heures

Lundi agrave dimanche

10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif

Pour les petites et moyennes entreprises qui soutirent une puissance eacuteleveacutee du reacuteseau Composition


1to1 energy easy power





Puissance mensuelle (CHFkWmois) 680 - 680 -

Haut tarif (ctkWh) 1508 1090 330 088

Bas tarif (ctkWh) 874 660 126 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif

La puissance mensuelle correspond agrave la puissance maximale mesureacutee sur un quart drsquoheure du mois de facturation

- Les prix mentionneacutes sont hors TVA

- En fonction du canton et de la commune du site de consommation des taxes redevances et eacutemoluments communauxcantonaux ou feacutedeacuteraux (dont la taxe d encouragement pour les eacutenergies renouvelables 045 ctkWh) normalementmajoreacutes de la TVA selon les exigences leacutegales sont perccedilus par Groupe E pour le compte des autoriteacutes compeacutetentes via lesfactures aux consommateurs finaux Les montants ou taux en vigueur sont disponibles sur le site Internet wwwgroupe-ech

Figure 6 Groupe-E - rdquo1to1 energy easyrdquo

252 Romande-Energie - rdquovolta doublerdquo

Romande Energiersquos ldquovolta doublerdquo tariff is very similar to the previous scheme we havediscussed The prices are pre-determined at 2189 ctskWh during the peak period and1334 ctskWh during the rest of the day and during weekends

Lundi agrave vendredi

Samedi et dimanche0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Heures Pleines Heures Creuses

Figure 7 Romande-Energie - rdquovolta doublerdquo

253 SIG - rdquoTarif Profile Doublerdquo

In SIGrsquos ldquoTarif Profile Doublerdquo the same kind of tariffs apply Depending on theproportion of renewables the current incorporates the peak periods are billed from2210 ctskWh to 2782 ctskWh while the off-peak ones range from 2046 ctskWh to2782 ctskWh


19

Figure 8 SIG - rdquoTarif Profile Doublerdquo

26 Time-Varying Tariffs for Consumers Granted with Network Ac-cess in Switzerland

In Switzerland end-users who annually consume more that 100 MWh of electricityare granted the right to access the network Thereby they are allowed to buy theirelectricity on the market in or outside Switzerland19 As a second step of the Swisselectricity market liberalisation process all consumers will eventually have the right toaccess the market and will not be bound anymore to buy their electricity to their localdistributor The procedure which is described in SR 7347 Article 3420 is plannedto take place five years after the law has been enforced and may be challenged byreferendum

In the following we give two examples of possibilities that are available for customersthat have access to the network even though the prices for large companies remainlargely unknown for understandable reasons An in-depth evaluation of the effect oftime-differentiated tariffs for large consumers may be found in [9]

261 SIG - rdquoTarif 35 periodesrdquo

In SIGrsquos tariffs for large consumers ndash also available for captive clients consuming be-tween 30 and 100 MWh of electricity per year ndash the days are subdivided in 3 or 5periods as indicated by Figure 9 Each period is characterised by its own energy (egin kWh) and power (eg in kW) prices

Figure 9 SIG - rdquoTarif 35 periodesrdquo

19Customers who buy their electricity abroad have to purchase the corresponding cross-border ca-pacity for which auctions are organised

20Available at httpwwwadminchchdsr73html


20

From this example one can see that tariffs designed for large consumers are notvery different from those captive consumers are charged with However very largeconsumers are known to regroup in order to emit calls for bids and thereby to beoffered competitive contracts together with negotiation possibilities21

262 The FlexLast project

The FlexLast project22 illustrates the innovative mechanisms available to large elec-tricity consumers Originating from a collaboration among Migros BKW Swissgridand IBM the FlexLast project aims at using a giant 200rsquo000-m2-sized refrigerator as abuffer to cope with the intermittent character of the electricity generated from renew-able sources and that is injected in the grid The warehouse which consumes 6 GWh ofelectricity per year is equipped with an algorithm taking into account the grid statusthe status (activity temperature etc) of the refrigerator the production of renewablesetc and controlling the cooling strategy of the building

The lesson emerging from this example is that it is indeed possible to modify onersquosconsumption pattern so as to dynamically absorb the variability that is inherent to theelectricity production from wind and solar technologies Since such a strategy stronglyrelies on real-time data communication technologies and automation are crucial ingre-dients which are the subject of the next subsection

27 Smart Grids and Time-Differentiated Pricing Schemes

Under the Smart Grid paradigm the electricity distribution network is supplementedwith bidirectional real-time communication capacities The latter aptitude may thenbe exploited to transmit close to real-time price signals As anticipated in the afore-mentioned FlexLast project the problems emerging due to the variability of both sun-and wind-powered electricity generation may find their solutions in a real-time partialadaptation of the consumption to the production In order to exploit the load shiftingpotential of households and industry market incentives should be properly designed soas to stimulate a sizeable response from end-users (demand-response mechanisms)

Implementing such a vision requires the installation of decentralised energy man-agement systems that dynamically adapt the consumption behaviour of the home orbusiness by exchanging data with the grid eg short-term prices storage capacitiescomfort requirements occupancy pattern etc Both the demand- and the production-side would benefit from dynamical pricing

21See eg httpwwwswisselectricitycom22See httpwwwbkw-fmbchbkwfmbfrhomeklimafreundliche_stromproduktion

tatbeweiseenergieeffizienzsmartgrid-pilotprojekthtml


21

1 Demand-sideSince Smart Grid schemes are dynamic the demand level is automatically adaptedso as to optimise the usage storing and selling of electricity while maintainingcomfort above a pre-defined threshold Robust strategies incorporating algo-rithms that can face the uncertainties regarding future prices and conditions aredeployed in such situations The consumption is thereby occurring during periodscharacterised by low prices resulting in the satisfaction of the end-usersrsquo needstogether with a lowering of their bills

2 Production-sideOn the production side Smart Grid schemes not only allow for an optimisationof the way energy from renewables is used but also represent a solution to theproblem of stability that occurs as soon as renewables substantially penetrateThe Smart Grid concept indeed provides the framework (at both the physicaland algorithmic levels) enabling the energy emerging from renewable sources tooffer regulation services despite their intrinsic variability Issues regarding thecombined effect of multiple users using different parametrisations may be solvedthanks to the IT capacities brought by Smart Grids

For both the demand and production sides the question of the control of a certainnumber of domestic appliances by the local distributor is raised The following twooptions are available

1 Automatic controlA certain subset of the processes would be dynamically controlled by a localcomputer according to pre-defined criteria among which the price of electricity

2 Remote controlThe remainder of the devices of which certain loads may be deferred would becontrolled by the distributor according to pre-defined criteria The distributorwould then be in charge of delivering and storing the electricity whenever suitsthe grid best

Remotely controlled devices (including local storage capacities) would open the pos-sibility for grid operators to dynamically manage the grid status by adjusting thedemand A critical issue for the success of the Smart Grid concept is thus the degreeof acceptance by end-users This is the subject of Section 3


22

3 Customersrsquo Response to Time-Differentiated Tariffs andits Econometric Modelling

This section is devoted to the main factors that influence the end-usersrsquo acceptance oftime-differentiated tariffs and more generally of the Smart Grid paradigm Experi-mental studies evaluating the price elasticity in the Swiss context and the peak-loadreduction at the international level are then presented

31 Effectiveness of Time-Varying Tariffs

311 Ability to Reflect the Marketrsquos Status

The efficiency of time-varying pricing schemes may be evaluated from various perspec-tives From the distributorrsquos point of view one of the criteria that applies is the abilityto reflect the market situation in order for its consumers to dynamically adapt their be-haviours to the present context Among the pricing schemes that have been presented(TOU CPP-F CPP-V PTR RTP) RTP pricing schemes are the ones that are themost likely to generate a change in the end-usersrsquo response whose effects on the electric-ity market are beneficial according to [10] Indeed since the rate of tariff adaptationin RTP schemes generally is occurring once every hour the image the customers canbuild of the electricity market is quite accurate23

On the other hand TOUrsquos ability to capture the state of the electricity market islimited since the time-blocks and the corresponding prices are generally kept fixed forseveral months in a row As an illustration the author of [10] evaluates that a TOUscheme would only have reflected around 15 to 20 of the price variation that hasoccurred in the wholesale market during the California crisis of 2000-2001

Since TOU is only partially representing the real-time situation of the electricitymarket it is often proposed in conjunction with load charges ie not only is the energycharged but the peak power too Load charges represent a convenient way of chargingthe end-users with a tariff that partially reflects their power usage pattern Howeversince load charges are only related to the peak usage they do not prevent end-users toconcentrate their usage around the peak period

23Moreover RTP prices are typically announced with a delay of less than a day they are therebycapable of reflecting unplanned events


23

312 Technology as an Enhancer

Even if energy behaviours may have an impact on energy savings (see eg [11] fora review) one of the most important factors that influence the effectiveness of time-varying tariffs measured eg by the reduction in the peak-load is the one of enablingtechnologies Indeed on average the end-usersrsquo enthusiasm to dynamically adapt theirbehaviour to price incentives drops off very quickly even more in the electricity marketdue to the low level of prices that are nowadays faced In order to witness a sustainableeffect enabling technologies are to be provided These technologies may take formsas simple as smart thermostats provided with algorithms that take into account theaverage prices during the day the weather conditions and forecast the pre-definedcomfort zone etc The effect of such devices have shown to be more important andmore durable see eg [12 13 14] and [15] in the case of substantial penetration ofenergy generation from renewable sources

313 Barriers to the Diffusion of Innovations

Even though some innovations would have a positive impact ie they are rationally de-sirable they are not perforce adopted Several obstacles to the adoption of innovationsmay be identified

Too small benefits considering the implementation efforts

Lack of knowledge about the existence of alternatives (bounded rationality)

Emotional bias andor attraction towards a particular brand

Analysis restricted to short-term consequences (see eg [16 17])

Group dynamics (peer support or pressure)

A comprehensive review of the major obstacles faced by innovations and of success-ful adoption strategies in the energy sector may be found in [18] The electricity sectoris certainly concerned by all of the obstacles in the above list but the emotional onewhich is most probably negligible

An estimate of the adoption potential can be found by measuring the price past andfuture demand elasticities see equation (2) The next subsection consists in a reviewof the works performed in the Swiss context which measure the price elasticities in theSwiss residential sector


24

314 Rebound and Backfire Effects

In the above efficiency issues have been approached under the angle of peak-loadreduction and of the tariffrsquos ability to reflect the market dynamics Even though theperformance of dynamic pricing schemes in particular RTP have been assessed in thesedomains it remains largely unclear whether dynamic pricing schemes tend to lessenor not the overall consumption Unless distributorsrsquo revenues are decoupled from thequantity of electricity sold (see eg [19]) their interests are certainly more to be foundin shifting loads rather than in shedding them Figure 5 is a representative example ofload shifting It was moreover noticed in [13] that the dynamical tariffs introduced inCalifornia did not result in any consumption reduction

There was essentially no change in total energy use across the entire yearbased on average SPP24 prices That is the reduction in energy use duringhigh-price periods was almost exactly offset by increases in energy use duringoff-peak periods

As is very often the case when introducing new technologies an increased efficiencyin one domain tends to favourably impact the amount of time and resources that canbe devoted to other activities Progress realised in the electricity sector may then becompensated by other highly carbon-emitting activities The owner of an electric carmay for example feel more comfortable with the idea of flying more often thereby par-tially annihilating his efforts when viewed from an holistic perspective For a discussionof this effect known as the rebound effect and of situations in which the overall effectis negative (backfire effect) see eg [20]

32 Overview of Householdsrsquo Response to TOU Schemes

321 Empirical Evidence in Switzerland

Estimating the elasticity of the Swiss householdsrsquo electricity demand is a complex taskTo tackle this challenge the author of [21] has collected data from several Swiss elec-tricity distributors proposing two-period TOU tariffs Assuming that the demand takesthe form

Doni (t) prop

[poni (t)

]ηonon middot[poffi (t)

]ηoffon

Doffi (t) prop

[poni (t)

]ηonoff middot[poffi (t)

]ηoffoff

i isin cities t isin years (3)

24SPP California Statewide Pricing Program see [14]


25

where the proportionality factors take care of various other influences such as the levelof income the householdsrsquo size the number of heating and cooling days and whererdquoonrdquo and rdquooffrdquo respectively stand for on- and off-peak the author found elasticities ofthe following magnitude

ηonon ηoff

on ηonoff ηoff

off

-085 110 047 -092

Note that the elasticities defined in (3) are relating the demand and the price evaluatedat the same time t they are therefore known as short-term elasticities If one were toconsider the effect of the prices at time t minus 1 on the demand at time t one wouldintroduce long-term elasticities In [21] the author finds long-term elasticities thatare generically larger in absolute value than the aforementioned short-term ones Thiseffect is interpreted as being due the fact that if end-users are given more time to adaptto a change of price not only their behaviour will change but also the kind of devicesthey own By adapting their devices end-users change their consumption habits byeither being able to consume less in the case of an increasing price or by shifting moreof their loads more effectively leading to larger elasticities

Let us now return to the short-term elasticities As anticipated in subsection 23 theown-price elasticities are negative and cross-prices elasticities positive The rationalebehind this is simply that as the price rises during a peak period end-users tend toconsume less during that period (ηon

on lt 0) and shift their consumption towards off-peakperiods (ηon

off gt 0) The same happens when the off-peak price rises the demand duringthe off-peak period diminishes (ηoff

off lt 0) while the on-peak periodrsquos demand augments(ηoff

on gt 0) One can thus conclude that on- and off-peak electricity are substitutes

As can be seen from the numerical values reported in the above table the substitu-tion is however not symmetric the sensitivity to a off-peak price rise is greater than theone to an on-peak price rise (ηoff

on gt ηonoff) as illustrated by Figure 10 On the left-hand

side the rise in price lessens the price difference among the two periods consumerswould thus tend to shift some loads towards the on-peak period since it would improvetheir comfort On the other hand even though an on-peak price rise augments thedifference among the two periods any load shifting would result in less comfort Agood estimate of the amount of deferrable loads together with comfort issues are thusabsolutely crucial to understand the demand response to a price variation

Note that the estimation of [21] were realised by using present-day data ie they donot include the effects automation could bring in the case of a Smart Grid deployment


26

ONOFF ONOFF

Price

ocrarron on

ocrarr

p

p

gt

Figure 10 Cross-demand adaptation to a price increase (∆p) The size of the arrowindicates the sensitivity of the demand response to ∆p ie that ηoff

on gt ηonoff

322 Peak-Load Reduction Evidence

In order to assess the efficiency of the peak-load reduction due to time-varying tariffs[22] gathered data from 15 experiments ranging from simple TOU schemes to RTP oneswith enabling technology mostly in the United States of America The conclusion fromthis study is that the effect may be substantial and even more if enabling technologiesare present see Figure 11

demand responses vary from modest to substantial largely depending onthe data used in the experiments and the availability of enabling technologiesAcross the range of experiments studied time-of-use rates induce a dropin peak demand that ranges between three to six percent and critical-peakpricing tariffs lead to a drop in peak demand of 13 to 20 percent Whenaccompanied with enabling technologies the latter set of tariffs lead to adrop in peak demand in the 27 to 44 percent range25

4 Economics of Time-Varying Rates and Load Shifting

This section contains a review of the microeconomics that lie behind the design oftime-varying rates In a first subsection the modelling approach proposed in [23] ispresented It consists of a dynamic partial (supply-demand) equilibrium model usedto determine the optimal dynamic pricing when the demand responds to price change

25Quote from [22]rsquos conclusion


27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech

Figure 11 Relative reduction of peak load observed in the 15 experiments studied in[22]

with some delay In a second subsection the more general concept of optimisation withequilibrium constraint is considered It seems to the authors that this framework is thebest adapted to study the design of time-varying or adaptive pricing for electricity orto include these schemes in planning models

41 A Global Model of TOU Pricing

In this section we present the model elaborated by Celebi and Fuller in [23] Thesupply- and demand-side of the model are successively reported After the merge ofthese two parts an illustration in the Swiss context is performed

411 Supply Side

Let us first consider the supply-side It contains m facilities (indexed by i) n demandor TOU blocks (indexed by j) the model describes the evolution during T months(indexed by t) The discount factor is r

The model is characterised by the following parameters that may depend on themonth t


28

Number of hours in demand block j Hj(t)

Cost per produced MWh by facility i ci(t)

Capacity in MW of facility i Ki(t)

Energy demand in MWh during the hour h of block j djh(t)

Decision Variables The decision variables consist of the energy flowing from facilityi during the hour h of block j zijh(t)

Supply model If the demand was known and fixed the supplier would solve thefollowing linear program

minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)

subject to the following constraints

msum

i=1

zijh(t) ge djh(t) (5)

zijh(t) le Ki(t) (6)

zijh(t) ge 0 (7)

The quantity between brackets in (4) represents the daily cost during month tEquations (5) to (7) respectively correspond to the requirements that the demand ismet (5) the facilities are running below their maximal capacities (6) and that theenergy flow is positive (7) The dual variables corresponding to equations (5) and (6)respectively are pjh(t) and uijh(t) which are greater or equal to zero The Lagrangiancan thus be written as

L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)

Note that we can interpret pjh(t) as the cost of increasing the demand djh(t) by oneunit (a MWh in the case at hand) while uijh(t) may be interpreted as the cost ofdecreasing the capacity Ki(t) by one unit (a MW in the case at hand)


29

If the demand was known to the distributor he would solve the above optimisationproblem find the optimal solution zijh(t) (using eg the simplex algorithm) and thenproduce zijh(t) MWh during the hour h of block j from facility i and this all days ofmonth t

Let us finally define the δjh(t) parameter as

δjh(t) equiv djh(t)

dj(t)where dj(t) equiv

Hj(t)sum

h=1

djh(t) (9)

for a later use It measures the proportion of the demand in block j that takes placeduring hour h

412 Demand Side

In the above supply model the demand djh(t) is considered as known and fixed How-ever this assumption should be relaxed and a model of the demand elaborated Sincethe response of consumers to changing prices is distributed over time (see eg [24]) ageometric distributed lag (GDL) demand law is used by [23] For a single commoditymodel with constant elasticities b and e it takes the form

d(t) = a(t)p(t)bd(tminus 1)e (10)

which in its log form is given by

ln[d(t)] = ln[a(t)] + b ln[p(t)] + e ln[d(tminus 1)] (11)

Equation (11) can be extended to the multi-commodity case the commodities beingthe electricity demand in the different blocks of the day

ln[Dj(t)] = Aj(t) +nsum

k=1

Bjk ln[Pk(t)] + Ejj ln[Dj(tminus 1)] (12)

where

A(t) Vector representing non-price effects of period tD(t) Vector of electricity demand for each block in period tP (t) Vector of electricity price for each block in period tB Square matrix of own- and cross-price elasticitiesE Square diagonal matrix of lag elasticities

The outcome of (12) is the knowledge of the demand for each TOU block dependingon the electricity price of the block the past demand the elasticities and on non-priceeffects


30

413 Representing the Supply-Demand Equilibrium

We now merge the two parts of the problem the supply (cf 411) and demand (cf412) sides In order to do so dj(t) and Dj(t) are first identified The problem isthen represented by a mixed complementarity problem (MCP) composed of the com-plementarity conditions for the optimal supply problem derived from the Lagrangian(8)

zijh(t) ge 0 perp rtci(t)minus pjh(t) + uijh(t) ge 0 (13)

pjh(t) ge 0 perpmsum

i=1

zijh(t)minus djh(t) ge 0 (14)

uijh(t) ge 0 perp Ki(t)minus zijh(t) ge 0 (15)

and of the demand side where the price pj(t) for each of the TOU blocks is derivedfrom the dual variables associated with the discounted marginal cost

ln[dj(t)] = aj(t) +nsum

k=1

bjk ln[pk(t)] + ejj ln[dj(tminus 1)] (16)

djh(t) = δjh(t) dj(t) (17)

pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)

414 A Numerical Illustration

In order to illustrate the aforementioned model let us apply it to a region roughlythe size of Switzerland on a three-months period The implementation code has beenwritten in GAMS26 and solved using the PATH solver27The generation facilities are taken to be

Type Installed capacity [MW] Costs [CHFMWh] Availability []

Hydro 11400 10 70Nuclear 560 76 90Renewables

(non hydro)3800 150 0 - 15

Imports 10000 200 100

26General Algebraic Modelling System see httpwwwgamscom27PATH solves Mixed Complementarity Problems see httppagescswiscedu~ferrispath

html


31

where the costs have been extracted from the ETSAP database28 and where the lastcolumn indicated the averaged availability of the corresponding electricity source duringa typical day The availability for renewables is taken to be varying across the day withsolar activity peaking at around 2pm and wind activity around 6pm Note that astraightforward procedure would enable the model to take exports into accountThe days are subdivided in two periods (ie demand blocks) which respectively are

Name Period Consumption []

On-Peak 7 am to 9 pm 72Off-Peak 9 pm to 7 am 28

The demand for electricity is taken to be around 240 PJyear (ie 667 TWhyear)Since the numerical example is only designed to be illustrative the elasticities used aredirectly taken from [21]

bjk =

(minus08 0805 minus07

)and ejj = 05 forallj isin 1 2 (19)

ResultsThe following table summarises the prices in CHF per MWh computed for the on-peakand off-peak blocks by solving the model designed in [23] which has been refined totake into account the variability of electric production emerging from renewable energysources

On-Peak Off-Peak

Month 1 391 206Month 2 461 324Month 3 478 336

Table 1 Prices pj(t) [CHFMWh]

From the last table one can observe that even though the hydro capacities are sufficientto produce all the electricity for the off-peak period

Off-peak hydro production 07 middot 114 GW middot 10 hday middot 30 daymonth

= 2394 GWhmonth

Off-peak demand 028 middot 675 TWhyear middot 112 yearmonth

= 1575 GWhmonth

(20)

the corresponding prices are higher than the operationsrsquo costs per MWh (10CHFMWhfor hydro) the reason being that thanks to the fact that on-peak and off-peak electricity

28ETSAP is the International Energy Agencyrsquos Energy Technology Systems Analysis ProgramDatabase available at httpwwwiea-etsaporgwebSupplyasp


32

are substitutes as may be seen from (19) an important part of the consumption isshifted towards the off-peak period The demands are given below in TWh per month

On-Peak Off-Peak


Table 2 Demand dj(t) [TWhmonth]

The off-peak price thus takes into account the fact that the hydro capacities are satu-rated and that any increase in the demand would either have to be met by using moreexpensive facilities or by rising the price so as to lower the demand (using the negativityof the b matrixrsquos diagonal elements)

42 Non-Cooperative Game Models

In [25] the relationship between a retailer practising real-time pricing and a finite setof customers optimising the timing of their electricity consumption is modelled as anon-cooperative game which admits under some general conditions a unique and sta-ble Nash equilibrium The model is summarised below in a slightly more generalformulation than the one used in [25]

Retailer The distributor has access to its own production equipment and also to thewholesale market Depending of the total demand D(t) in a time slot t of the day themarginal cost of production is given by γ(D(t)) which is the price that will be charged

Consumers Each consumer i has a minimum daily requirement of electricity βij for

each type of service j Let xij(t) the demand by consumer i to satisfy service j at timeslot t The following constraints must thus be satisfied

sum

t

xij(t) ge βij (21)

together with

xij(t) ge xij [min](t) (22)

xij(t) le xij [max](t) (23)

where xij [min](t) and xij [max](t) are given bounds The dual variables corresponding

to the constraints (21) to (23) respectively are ηij microij(t) and νij(t) which are greater or


33

equal to zero Let us define the following sets x = (x(t))tisinT where x(t) = (xi(t))iisinI with xi(t) = (xij(t))jisinJ

Payoff to consumer The total demand in time slot t is given by

D(t) =sum

i

sum

j

xij(t) (24)

This determines the marginal cost γ(D(t)) and hence the tariff payed at time slot t byeach customer The aim of the i-th customer is thus to minimise

ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)

under the constraints (21) to (23) Note that the interdependence among customerscomes from the price determination equation see (24) The Lagrangian for the i-thconsumer can thus be written as

Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)

(xij [min](t)minus xij(t)

)

+sum

tj

νij(t)

(xij(t)minus xij [max](t)

)

(26)

The first order conditions for a Nash equilibrium are given by

0 =partLipartxij(t)

=partψi(x)

partxij(t)minus ηij minus microij(t) + νij(t) (27)

which can be written as(γ(D(t)) + γprime(D(t))

(sum

k

xik(t)

))minus ηij minus microij(t) + νij(t) = 0 (28)

with the following complementarity conditions

ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)

microij(t) ge 0 and microij(t)(xij(t)minus xij [min](t)

)= 0 (30)

νij(t) ge 0 and νij(t)(xij(t)minus xij [max](t)

)= 0 (31)

Applying classical theorems (see eg [26]) we can find easily conditions which assurethat an equilibrium exists and that it is unique if the γ(D(t)) function is strictly convexand increasing


34

43 Competitive Equilibrium Models

The model of the previous subsection may be adapted to situation in which thereare a consequent number of players To do so let us assume that each customeri is replicated n times with demand parameters βijn and bounds xij [min](t)n and

xij [max](t)n This describes a game where the number of players increases while theinfluence of each player diminishes The first order conditions for a Nash equilibrium(28) is now given by

(γ(D(t)) + γprime(D(t))

sumk x

ik(t)

n

)minus ηij + microij(t)minus νij(t) = 0 (32)

When nrarrinfin the conditions of a competitive equilibrium are met Note that for eachtype of player i and type of service j the following constraint must hold

sum

t

xij(t)

nminusβijnge 0 (33)

which is the same as (21) The same reasoning applies for the other constraints andas a consequence the KKT multipliers are the same as before

In the large n limit the term γprime(D(t))sumk x

ik(t)n tends to 0 the condition (32) thus

becomesγ(D(t))minus ηij minus microij(t) + νij(t) = 0 (34)

Each consumer is now a price taker His decisions have no influence on the price Thequantities xij(t) are then determined by using (34) together with the complementarityconditions (29)-(31)

In [25] these games are shown to be similar to a class of games defined on transportor communication networks In [27] the convergence of Nash equilibria to a trafficequilibrium called Wardrop equilibrium was proven along very similar lines to thoseexposed in this section

431 A Global Linear Programming Formulation

The above model only considers the demand side In order to represent the supply sidethe description developed in 411 is particularly well adapted It contains m facilities(indexed by κ) n demand or TOU blocks (indexed by θ) The model is characterisedby the following parameters29

29In this model no monthly variations nor discount factor are considered but these are straightfor-ward to implement


35

Number of hours in demand block θ Hθ

Cost per produced MWh by facility κ cκ

Capacity in MW of facility κ Kκ

Let zκθ be the energy flowing from facility κ during the time slot θ If the timingof demands of type j for consumer type i were under direct control of the retailer itwould solve the following linear programme

minimisezκθ xij(θ)

nsum

θ=1

msum

κ=1

cκzκθ (35)

under the following constraints

msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)

minuszκθ ge minusKκHθ (37)

zκθ ge 0 (38)sum

θ

xij(θ) ge βij (39)

xij(θ) ge xij [min](θ) (40)

xij(θ) le xij [max](θ) (41)

These constraints correspond to the demand having to be met (36) the capacity havingto be kept below its threshold (37) and the energy flows having to be positive (38)The dual variables for equations (36) (39) (40) and (41) are respectively given by πθηij micro

ij(θ) and νij(θ)

By applying the optimality condition for a linear programme and if the variablesxij(θ) are in the optimal basis meaning that their reduced costs must be zero we getthe following

νij(θ)minus microij(θ)minus ηij + πθ = 0 (42)

This equation is to be compared with (34) The dual variable πθ corresponds to themarginal production cost to satisfy demand Therefore we conclude that (42) and(34) are identical The other duality conditions would lead to similar complementarityconditions Henceforth the solution of the linear programme gives the optimal responseof consumers to a marginal cost pricing

Indeed the same result remains valid if one models the retailer over a long periodwith investment activities as is done in the ETEM-SG model for example (see eg[28]) In this case the price of electricity will be given by the long term marginal costwhich includes the investment cost


36

44 Multilayer Game Models

The gamesrsquo structure considered in the previous subsections is occurring when differentelectricity customers are facing a supplier with a marginal cost pricing scheme In [29]and [30] the game structure considered is more encompassing as it involves three layersof players the customers (many) the retailers (a few competing to attract customers)and the producer (wholesale market actors) We will not develop this analysis anyfurther because it does not correspond to the electricity market structure in SwitzerlandThe total realisation of the Swiss electricity market liberalisation would however requiresuch models to be studied

5 Optimal Distributed Charging and Storage Control fora Population of Electric Vehicles

In this section the focus is placed on the use of dynamic TOU pricing or RTP schemesto foster the integration of BEVs or PHEVs in a Smart Grid environment

As noted in [31] the electrification of transportation is often seen as one of thesolutions to challenges such as global warming sustainability and geopolitical concernson the availability of oil From the perspective of power systems an introduction ofplug-in electric vehicles presents many challenges but also opportunities to the operationand planning of power systems If vehicles charging operations are considered regularloads without flexibility uncontrolled charging can lead to problems at different networklevels endangering secure operation of installed assets However with direct or indirectcontrol approaches the charging of vehicles can be managed in a desirable way egshifted to low-load hours Moreover BEVs and PHEVs can be used as distributedstorage resources to contribute to ancillary services for the system such as frequencyregulation and peak-shaving power or help integrate fluctuating renewable resources

In most of the proposed distributed control approaches a so-called aggregator isin charge of directly or indirectly controlling the charging of vehicles and can serve asan interface with other entities such as the transmission system operator or the energyservice providers Communication thus plays a key role since as in most of the controlschemes a significant amount of information needs to be transmitted between vehiclesand utility management This fits well in the paradigm of Smart Grids in which anadvanced use of communication technologies and metering infrastructure increasedcontrollability and load flexibility and a larger share of fluctuating and distributedresources are foreseen


37

51 Decentralised Control and Incentive-Based Coordinated Charg-ing Control

Each BEV or PHEV is entering the market as an independent actor in the interactionbetween the utility providing the charging energy and the users deciding when to plugin their vehicles Decentralised control schemes based on game theoretic conceptsmust thus be envisioned

511 A Mean-Field Game Solution

In [32 33] the decentralised control of a large population of BEVs or PHEVs (calledPEVs hereafter) is considered In [33] an adaptive real-time tariff is proposed as a wayto achieve the optimal timing of recharging electric vehicles Basically the situation isvery similar to the ones addressed in sections 42 and 43 Adopting the notations of[33] the days are split time steps t isin 0 1 T The relative state of charge of then-th PEV at time t is denoted by xnt The charging dynamics can thus be written as

xnt+1 = xnt +αn

βnunt (43)

where utn is the charging rate of the n-th PEV at time t (in MWhtimestep) αn is thecharging efficiency for the n-th PEV βn is the battery size of the n-th PEV (in MWh)and xn0 is a given initial state of charge for the n-th PEV The aim of the chargingstrategy is to get all PEVs charged at the end of the period

xnT = 1 (44)

Let Dt be the aggregate non-PEV base demand at time t We assume that the retailprice is based on the marginal cost pricing scheme The price will thus be an increasingfunction of the ratio of total demand to total electric generation capacity denoted C

p

(Dt +

sumNn=1 u

nt

C

) (45)

The daily cost for the n-th PEV is thus given by

Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)

which is the same equation as (25) and where dt = DtN c = CN and

avg(ut) =

sumNn=1 u

nt

N (47)


38

In this model each PEV is a player in a dynamic non-cooperative game One shouldthus tries to characterise a Nash equilibrium As usual for control and open-loop Nashequilibrium problems (see eg [34]) the following Hamiltonian is introduced for PEVn

Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)

At a Nash equilibrium for each PEV n the following first order optimality conditionsholds

0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)

The adjoint equation of which being given by

λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)

with transversality conditionλnt+1 = ζn (51)

where ζn is the Lagrange multiplier associated with the constraint (44) for the n-thPEV

In this game each player reacts to the average decision taken by the set of allplayers When the number of players becomes large the decision of one player has asmall impact on the average of all decisions and the optimality condition simplifies Inthe limit where N rarrinfin while c and dt remain finite which is representative of a largefleet of PEVs the conditions which determines the optimal charge unt is thus given by

0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)

At the limit each player (ie each PEV n) plays against the average of all otherplayers This is a situation very similar to a competitive equilibrium as already visitedin section 43 The control community calls such a dynamic game structure a Mean-Field Game see [35 36 37] The obtained solution is also called a Mean-Field NashEquilibrium

512 A Decentralised Control Scheme Based on RT Pricing

Let us modify slightly the cost function of the agent controlling the n-th PEV

Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt + δ(unt minus avg(ut))

2

(53)


39

where δ is a given non-negative constant common to all agents The presence of δ ispenalising the strategies that depart from the average strategy In [33] it is shown howa decentralised control is implemented through a charging negotiation procedure whichtakes place at some time prior to the actual charging interval The introduction of δmay be necessary for the negotiation process to converge The negotiation procedureis the following quoting [33]

Step 1 The retailer broadcasts the prediction of non-PEV base demand dtto all the PEV agents

Step 2 Each of the PEVs proposes a charging control that minimises itscharging cost with respect to a common a-priori aggregate PEVdemand broadcast by the utility

Step 3 The retailer collects all the individual optimal charging strategiesproposed in step 2 and updates the aggregate PEV demand corre-sponding to the proposed charging strategies This updated aggre-gate PEV demand is rebroadcast to all of the PEVs

Step 4 Repeat steps 2 and 3 until the optimal strategies proposed by theagents no longer change

When the actual charging start time is reached each PEV will implement theoptimal strategy obtained from steps 1-4 In the negotiation procedure each of thePEVs independently updates its own optimal feedback charging strategy with respectto the one-dimensional average value of all PEV agent strategies At convergence if itoccurs the collection of proposed individual charging strategies is a Mean-Field NashEquilibrium see [33] Figure 12 illustrates the valley-filling effect due to PEVs

1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10

August 15 minus 16 2007

No

rmal

ized

po

wer

(k

W)

Fig 5 Simulation results with = 0007 Although the tracking parameterdoes not satisfy the conditions of Theorem 32 convergence still occurs

1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)

Fig 6 Simulation results with = 0003 At this point the trackingparameter is small enough that the negotiation process does not converge

PEVs charge for less time than others As a consequencetotal demand ramps down at the beginning of the charginginterval and ramps up at the end

V CONCLUSIONS

In this paper decentralized charging control of largepopulations of PEVs is formulated as a class of finite-horizondynamic games The decentralized approach works by solv-ing a relatively simple local problem and iterates quickly toa global Nash equilibrium This strategy does not requiresignificant central computing resources or communicationsinfrastructure

The paper establishes under certain mild conditions ex-istence uniqueness and social optimality of the Nash equi-librium attained through decentralized control A negotiationprocedure is proposed that converges to a charging strategythat is nearly optimal In fact for a homogeneous PEV pop-ulation the charging strategy degenerates to a purely socialoptimal lsquovalley-fillingrsquo strategy The results are illustratedwith various numerical examples

1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)

best charging strategy of PEV 2


nonminusPEV base demand

average charging strategy

Fig 7 Converged Nash equilibrium for a heterogeneous population ofPEVs with = 0015

APPENDIX

The proof of Theorem 33 proceeds by considering with-out loss of generality adjacent time instants t and s = t+1Local charging controls (un

t unt+1) that are optimal with

respect to u and xnt can be decomposed as un

t = bn an

and unt+1 = bn + an respectively It is possible to show

that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2

with Sbn an bn an bnRelationship (11a) can be established by contradiction

If (11a) were not true then it can be shown that an lt12 (ut+1 ut) implying that un

t+1 unt lt ut+1 ut for all

n and hence that

avg(ut+1) avg(ut) lt ut+1 ut

where u (un 1 n lt $

) This however conflicts with

the fact that un n lt $ is a Nash equilibrium with respectto u see Theorem 21 Hence a contradiction

Relationship (11b) is also proved by contradiction andfollows a similar argument as the proof of (11a) In this casethough it is determined that

avg(ut+1) avg(ut) gt ut+1 ut

which conflicts with un n lt $ being a Nash equilibriumwith respect to u

Proof by contradiction is also used to establish (11c)Assume there are adjacent times t t + 1 [t0ts] such that

dt+1 + ut+1 = dt + ut + B (14)

where B gt 0 without loss of generality Then there will alsoexist an n and C amp B such that

dt+1 + unt+1 = dt + un

t + C

The theorem states that uns gt 0 for all n and all s [t0ts]

so there always exists a sufficiently small gt 0 such that

1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



Figure 12 Illustration of the valley-filling due to PEVs the solid pink line correspondsto Dt On the left negotiation strategies do not converge due to a too small value of δ


40

52 Optimal Storage and V2B Operations

521 Example of V2B Operations

As argued in [38 39] there are also new opportunities offered by correlating the move-ment of cars in a city and the movement of the power load BEVsPHEVs may beused to feed power back to home or office buildings this is known as Vehicle-to-Building(V2B) operation Such operation can be used either for demand response (DR) duringpeak load or for outage mangement during faults of the distribution system

Simple economics of DR In [39] the authors consider a garage for a large commer-cial building where 80 vehicles arrive at 800 am and stay until 1700 The garage isequipped with a bidirectional charger and controller It can then play the role of anaggregator and charge the car batteries when the building demand is below peak loadand discharge the batteries to partially supply the building to reduce the peak demand

One assumes that there are two charge rates that vary during the day r1e and r2

erespectively corresponding to mid-peak and on-peak periods and a monthly demandcharge rp which applies to peak power demand The monthly revenue from DR is thusgiven by

G = Eec(r2e minus r1

e)times ϑ+ rp(Pmax minus PDRmax) (54)

where Eec is the energy shifted from on-peak to mid-peak time ϑ the number of daysin month Pmax the maximum on-peak demand (kW) and PDR

max the new maximumon-peak demand after the DR (kW)

As an example let us consider 80 cars each having a 15 kWh battery and usethe electricity tariffs exposed in 251 r1

e = 009 CHFkWh r2e = 015 CHFkWh and

rp = 680 CHF per maximum on-peak kW The cars are fully charged at night whenthe off-peak rate is the lowest In the morning the garage is charging the batteries atmid-peak rate and the energy is shifted to the afternoon thus reducing both the energyand the peak demand bill A safety 6kWh energy level is maintained in each batteryto allow for the return trip of the driving cycle

In this example 720 kWh of energy is shifted (80 times 9 kWh) and the maximumon-peak demand reduces by roughly 225 kW Hence according to (54) the monthlyrevenue is 2394 CHF for a month of 20 weekdays Notice that the major part of therevenue comes from the reduction of the peak-demand charge (1530 CHF)


41

522 Economic Analysis of the Use of Storage Provided by Idle PHEVsBased on Locational Marginal Pricing

Profitability of temporary storage activities based on current zonalmarginal pricesIn [40] it is claimed that the potential benefits to users will not be sufficientto provide incentives for using BEV or PHEVs for temporary energy storageA very detailed analysis is based on the assumption that the vehicle owner ischarged under a real time pricing (RTP) scheme which is obtained by adding atransmission and distribution cost of 007 $kWh to the hourly nodal price thatpermit the computation of locational marginal price (LMP)30 These prices reflectthe congestion occurring on the transmission network and reflect the real economicvalue of each kWh stored at different times of the day by vehicles connected tothe different hubs of the transmission grid In [40] the authors use LMP datapublished by three American cities (Boston Rochester and Philadelphia) andsimulate the optimal use of storage capacity of PHEVs by the owners Theyobserve a very low yearly profit (typically in the range $10-50) which could notbe a credible incentive to participate in this trade

This analysis reported in [40] seems to contradict the analysis of [39] summarisedabove in 521 A closer look shows that in both cases the trading of energy is notprofitable In the example of 521 it was mostly the component of tariff based onthe maximum on-peak demand which was generating profit from demand shiftingIn conclusion it seems that the zonal prices used in [40] where not reflecting thesame economic value as the one characterising the reduction of the maximumon-peak demand as in the commercial tariff mimicked in 521

Profitability of temporary storage activities based on prospective nodalmarginal pricesThe same analysis in a prospective context with nodal prices reflecting conges-tions due to increased use of renewable energies like solar and wind to produceelectricity could lead to different conclusions The computation of nodal pricesis shown to be obtained from the dual solution of the optimal dispatch problemunder constraints of capacity of the transmission network This development isbased on the papers by Ruiz et al [42 43 44] and Stiel [45] and is exposed inthe following

It is possible to define a linear power flow model do describe the distribution ofpower in the different lines of a transmission network ie a linear relation

Pf = Ψ(PG minus PL) (55)

where Pf is the vector of power flows on each line of the network and PG minus PLis the vector net power injection (generation power PG minus load PL) at each

30See [41] for a description of the construction of zonal marginal prices from hourly nodal pricesobtained from the optimal power dispatch on the European transmission grid


42

bus (node) of the network The transmission sensitivity matrix Ψ also known asthe injection shift factor matrix gives the variations in flows due to changes inthe nodal injections The shift factor matrix is a function of the characteristicsof the transmission elements and of the state of the transmission switches Fora given point in time the system operator dispatches the committed units soas to minimise the total costs of operations Assume that the generation costsare piecewise linear and denote the vector of nodal generation annualised costsin CHFMWh by cG The economic dispatch solved is a linearised lossless DCoptimal power flow (OPF) problem formulated as a linear programme

minimisePG

cTG middot PG (56)

under the following set of constraints

1T (PG minus PL) = 0harr λ (57)

Pf min le Ψ (PG minus PL) le Pf max harr micromin micromax (58)

PGmin le PG le PGmax harr γmin γmax (59)

where 1 stands for a vector whose components are all equal to 1 The constraint(57) ensures the total load-generation balance (58) enforces the flow limits ontransmission elements and flowgates where lower limits usually represent the limitin the opposite flow direction and (59) models the lower and upper generationlimits The dual variables are indicated next to the corresponding constraints

In [44] the nodal marginal prices is then derived

π = minus(λ1 + ΨT (micromax minus micromin)) (60)

So if we introduce the dispatch problem description in a linear programmingmodel like ETEM-SG [28] the implicit nodal price given by (60) will serve toguide the potential use of BEVs or PHEVs for temporary storage and for charging

53 Other Ancillary Services Provided by EVs

In Refs [46] [47] and [48] it is shown how electrical vehicles can be used to provideancillary system services like frequency and voltage stabilisation and load followingpower injection These activities may have significative economic value however arelative small number of BEVs seems to be needed to provide those services as notedin Ref [40]

6 Conclusion

This report has focused on the economic rationale of dynamic adaptive tariffs that willnecessarily be introduced when a Smart Grid concept is implemented in the context


43

of major renewable energy development and important penetration of plug-in electricvehicles After having reviewed and reported in these pages some of the most recentinternational contributions to the economic analysis of smart-grids and their pricingschemes we arrive at the following observations

Variable tariffs also called Time-of-use pricing are already in use in severalcountries including Switzerland A few econometric analyses have permittedthe assessment of their relatively small effect on demand shifting and peak-loadreduction

Much higher price elasticities are expected when a Smart Grid scheme is imple-mented and households are provided with devices (home computers) permittingan optimal timing of flexible loads based on real time prices

The issue of public acceptability of two-way communication of energy and pricedata and their use in regulating daily electricity usages is posed and some surveyanalyses will have to be done to get a better grasp at this delicate questionConjoint analysis seems to be the technique that could be used to exploit theresult of surveys concerning acceptability of new technologies

For planning purpose one has to represent demand-response in the prospectivesupply-demand equilibrium models The Celebi and Fuller model exposed insubsection 41 is particularly convenient if we can model the dynamic adjustmentof the electricity demand law using hereditary and price elasticity effects

Another approach exposed in 42 consists in representing the optimisation prob-lem concerning the optimal timing of the usages which is solved by the con-sumer when a new tariff is announced The situation has the structure of anon-cooperative game for which a unique Nash equilibrium exists When thenumber of customers increases each customer having less influence on the pricedetermination the Nash-equilibrium tends to become a market equilibrium Itis possible to show that the optimal demand-response can be represented in anenergy model structured as a mathematical (or linear) programming model bysimply representing the constraints of the optimal timing problem in the globalenergy model This approach will be implemented in the ETEM-SG model thathas been recently developed to model energy planning decisions in a context ofSmart Grids deployment in the Arc-Lemanique region and the Canton of Bern[49]

The possible massive penetration of electric vehicles poses a challenge to the elec-tric grids Real-time pricing schemes should be used to ensure that the rechargeperiod for electrical vehicles remains in the off-peak period so that this newload will have a ldquovalley-fillingrdquo effect on the regional power system load curveHere again the interactions between the utility and the vehicle owners is a non-cooperative game where each player reacts to the average action of the other


44

players a structure called ldquomean field gamerdquo which corresponds grossly to acompetitive economic equilibrium

Plug-in electric cars will also offer temporary storage services With the currentelectricity tariffs this activity could prove to be profitable in a V2B system whenthe business organisation has an important part of the tariff which is determinedby the peak-demand load If the tariff seen by the vehicle owner is only the zonalmarginal price of electricity the current values for these prices do not seem toprovide a sufficient potential profit to the owner entering this type of temporarystorage transaction However the future zonal marginal prices when a very largepenetration of renewable energy technologies has taken place could be quitedifferent from what they are today and the temporary storage business coulddevelop under these new circumstances

In conclusion we are comforted in our conviction that dynamic electricity pricing willdevelop simultaneously with the diffusion of smart metering systems and with thepenetration of renewable energy generation technologies and the diffusion of electricvehicles with plug-in batteries Smart Grids will permit a large implementation of tariffsbased on real-time-pricing and home computers will facilitate the implementation oflocal optimisation of the timing of demand battery charging loads and temporarystorage The economic analysis of these pricing schemes has shown that they leadto some form of competitive market equilibrium which can be easily represented inenergy models like ETEM-SG These models should then be able to anticipate thepotential impact of Smart Grids on the future supply-demand equilibrium for electricityin different regions of the country However a big unknown still exists it concerns theacceptance by the end-users of smart-grid protocols involving day-to-day consumptionusages and management of the vehicle battery This will be explored in a survey duringthe second phase of this research project


45

References

[1] G Barbose C Goldman and B Neenan A Survey of Utility Experience withReal Time Pricing Ernest Orlando Lawrence Berkeley National Laboratory 2004

[2] P E Green and Y Wood New way to measure consumersrsquo judgments HarvardBusiness Review 1975

[3] P E Green and V Srinivasan Conjoint Analysis in Consumer Research Issuesand Outlook Journal of Consumer Research 5(2) 1978

[4] P E Green and V Srinivasan Conjoint Analysis in Marketing New Developmentswith Implications for Research and Practice Journal of Marketing 1990

[5] S Borenstein The Long-Run Efficiency of Real-Time Electricity Pricing TheEnergy Journal 26(3) 2005

[6] C Weare The California Electricity Crisis Causes and Policy Options PublicPolicy Institute of California 2003

[7] S Borenstein Time-Varying Retail Electricity Prices Theory and Practice inElectricity Deregulation Griffin and Puller University of Chicago Press 2009

[8] S D Braithwait D G Hansen and D A Armstrong 2011 Impact Evaluation ofSan Diego Gas amp Electricrsquos Peak-Time Rebate Pilot Program Technical reportCALMAC Study ID SDG0255 2011

[9] C Goldman N Hopper R Bharvirkar B Neenan and P Cappers EstimatingDemand Response Market Potential among Large Commercial and Industrial Cus-tomers A Scoping Study Ernest Orlando Lawrence Berkeley National Laboratory2007

[10] S Borenstein M Jaske and A Rosenfeld Dynamic Pricing Advanced Meteringand Demand Response in Electricity Markets Center for the Study of EnergyMarkets UC Berkeley 2002

[11] M A R Lopes C H Antunes and N Martins Energy behaviours as promotersof energy efficiency A 21st century review Renewable and Sustainable EnergyReviews 16 2012

[12] A Faruqui and S George The Value of Dynamic Pricing in Mass Markets TheElectricity Journal 2002

[13] A Faruqui and S George Quantifying Customer Response to Dynamic PricingThe Electricity Journal 18(4) 2005

[14] Charles Rivers Associates Impact Evaluation of the California Statewide PricingPilot 2005


46

[15] G Deconinck and B Decroix Smart metering tariff schemes combined with dis-tributed energy resources In Proceedings International conference on critical in-frastructures pages 1ndash8 CRIS 2009 March 27-April 30 2009

[16] K A Hassett and G E Metcalf Energy conservation investment Do consumersdiscount the future correctly Energy Policy 21(6) 1993

[17] A Alberini S Banfi and C Ramseier Energy Efficiency Investments in theHome Swiss Homeowners and Expectations about Future Energy Prices IAEEThe Energy Journal 34(1) 2013

[18] Department of Energy amp Climate Change UK What Works in Changing Energy-Using Behaviours in the Home 2012

[19] P G Lesh Rate Impacts and Key Design Elements of Gas and Electric UtilityDecoupling A Comprehensive Review The Electricity Journal 22(8) 2009

[20] A Druckman M Chitnis S Sorrell and T Jackson Missing carbon reductionsExploring rebound and backfire effects in UK households Energy Policy 39 2011

[21] M Filippini Short- and long-run time-of-use price elasticities in Swiss residentialelectricity demand Energy Policy 39(19) 2011

[22] A Faruqui and S Sergici Household response to dynamic pricing of electricity - asurvey of 15 experiments Journal of Regulatory Economics 38(2)193ndash225 2010

[23] E Celebi and JD Fuller A model for efficient consumer pricing schemes inelectricity markets IEEE Trans Power Syst 22(1)60ndash67 Feb 2007

[24] Y Wu and J D Fuller Introduction of geometric distributed lag demand intoenergy-process models Energy J 20(7)647ndash656 1995

[25] TAgarwal and SCui Noncooperative games for autonomous consumer load bal-ancing over smart grid In Game Theory for Networks Third International ICSTConference GameNets 2012 Vancouver BC Canada May 24-26 2012 RevisedSelected Papers volume 105 of Lecture Notes of the Institute for Computer Sci-ences Social Informatics and Telecommunications Engineering pages 163ndash175Springer Berlin Heidelberg 2012

[26] JB Rosen Existence and uniqueness of equilibrium points for concave n-persongames Econometrica 33(3)520ndash534 1965

[27] A Haurie and P Marcotte On the relationship between Nash-Cournot andWardrop equilibria Networks 15(3)295ndash308 1985

[28] F Babonneau A Haurie G J Tarel and J Thenie Assessing the future ofrenewable and smart grid technologies in regional energy systems Swiss Journalof Economics and Statistics 148(2)229ndash273 2012


47

[29] E Pettersen AB Philpott and SW Wallace An electricity market game be-tween consumers retailers and network operators Decision Support Systems 40(3-4)427ndash438 2005

[30] A Pandharipande S Oruc and SW Cunningham An electricity market incen-tive game based on time-of-use tariff Presented at Fourth Workshop on GameTheory in Energy Resources and Environment GERAD-HEC-Montreal Nov 29-30 2012 httpwwwgeradcacolloques4Workshop-GTEREpresentationsOrucpdf

[31] M D Galus M G Vaya T Krause and G Andersson The role of electricvehicles in smart grids WIREs Energy Environ doi 101002wene56 October2012

[32] M D Galus R A Waraich and G Andersson Predictive distributed hierar-chical charging control of PHEVs in the distribution system of a large urban areaincorporating a multi agent transportation simulation In 17th Power SystemsComputation Conference Stockholm Sweden August 22-26 2011

[33] Z Ma D Callaway and I Hiskens Decentralized charging control of large pop-ulations of plug-in electric vehicles Application of the Nash certainty equivalenceprinciple IEEE Transactions on Control Systems Technology 2167ndash78 January2013

[34] A Haurie J B Krawczyk and G Zaccour Games and Dynamic Games WorldScientific 2012

[35] M Huang P Caines and R Malhamrsquoe Large-population cost coupled LQGproblem with nonuniform agents IEEE Trans Automatic Control 52(9)1560ndash1571 2008

[36] J-M Lasry and PL Lions Jeux lsquoa champs moyens i - le cas stationnaire CRAcad Sci Paris Ser I 343619ndash625 2006

[37] J-M Lasry and PL Lions Mean field games Japan J Math 2(1)229ndash2602007

[38] C Pang P Dutta S Kim M Kezunovic and I Damnjanovic PHEVs as dy-namically configurable dispersed energy storage for V2B uses in the smart grid InIEEE editor Power Generation Transmission Distribution and Energy IEEE7th Mediterranean Conference and Exhibition on Power Generation Transmis-sion Distribution and Energy Conversion (MedPower 2010) pages 1ndash6 2010

[39] C Pang P Dutta and M Kezunovic EVsPHEVs as dispersed energy storagefor V2B uses in the smart grid IEEE Transactions on Smart Grid 3(1)473ndash482March 2012


48

[40] SB Petersom JF Whitacre and J Apt The economics of using plug-in hybridelectric vehicle battery packs for grid storage Journal of Power Sources 1952377ndash2384 2010

[41] B Burstedde From nodal to zonal pricing A bottom-up approach to the second-best In IEEE middot 9th International Conference on the European Energy Market(EEM) Florence Italy 10-12 May 2012

[42] P A Ruiz J M Foster A Rudkevich and M C Caramanis Tractable trans-mission topology control using sensitivity analysis IEEE Transactions on PowerSystems to appear

[43] P A Ruiz A Rudkevich MC Caramanis E Goldis E Ntakou and R PhilbrickReduced mip formulation for transmission topology control In USA Universityof Illinois at Urbana-Champaign IL editor 50th Annual Allerton Conference onCommunication Control and Computing Monticello 2012

[44] PA Ruiz JM Foster A Rudkevich and M Caramanis On fast transmissiontopology control heuristics In Proc 2011 IEEE Power and Energy Society GeneralMeeting Detroit MI IEEE July 2011

[45] ADJ Stiel Modelling liberalised power markets Masterrsquos thesis ETH ZurichCentre for Energy Policy and Economics September 2011

[46] D S Callaway Tapping the energy storage potential in electric loads to deliverload following and regulation with application to wind energy Energy Conversionand Management 501389ndash1400 2009

[47] D S Callaway and I A Hiskens Achieving controllability of electric loads Pro-ceedings of the IEEE 99(1)184ndash199 January 2011

[48] R Hermans M Almassalkhi and IA Hiskens Incentive-based coordinatedcharging control of plug-in electric vehicles at the distribution-transformer levelIn AACC editor Proceedings of the American Control Conference MontrealCanada June pages 264ndash269 June 27-29 2012

[49] ORDECSYS RITES Reseaux intelligents de transporttransmission delrsquoelectricite en suisse Technical report Office federal de lrsquoenergie ConfederationHelvetique 2013


Introduction and Motivation

Types of Rates and Pricing Schemes

Marginal Cost Pricing and Time-Varying Tariffs

The Electricity Price Composition in Switzerland

The Two Main Origins of Price Variation

A Survey of Time-Varying Tariffs

Time of Use

Critical Peak Pricing

Peak Time Rebates

Real-Time Pricing

Time-Varying Tariffs for Captive Consumers in Switzerland

Groupe-E - 1to1 energy easy

Romande-Energie - volta double

SIG - Tarif Profile Double

Time-Varying Tariffs for Consumers Granted with Network Access in Switzerland

SIG - Tarif 35 peacuteriodes

The FlexLast project

Smart Grids and Time-Differentiated Pricing Schemes

Customers Response to Time-Differentiated Tariffs and its Econometric Modelling

Effectiveness of Time-Varying Tariffs

Ability to Reflect the Markets Status

Technology as an Enhancer

Barriers to the Diffusion of Innovations

Rebound and Backfire Effects

Overview of Households Response to TOU Schemes

Empirical Evidence in Switzerland

Peak-Load Reduction Evidence

Economics of Time-Varying Rates and Load Shifting

A Global Model of TOU Pricing

Supply Side

Demand Side

Representing the Supply-Demand Equilibrium

A Numerical Illustration

Non-Cooperative Game Models

Competitive Equilibrium Models

A Global Linear Programming Formulation

Multilayer Game Models

Optimal Distributed Charging and Storage Control for a Population of Electric Vehicles

Decentralised Control and Incentive-Based Coordinated Charging Control

A Mean-Field Game Solution

A Decentralised Control Scheme Based on RT Pricing

Optimal Storage and V2B Operations

Example of V2B Operations

Economic Analysis of the Use of Storage Provided by Idle PHEVs Based on Locational Marginal Pricing

Other Ancillary Services Provided by EVs

Conclusion

Abstract

This document is the first deliverable of the project entitled Adaptive and TOUpricing schemes for smart technology integration funded by the Swiss Federal Officeof Energy (SFOE) Its aim is twofold (i) through an economic analysis we explorethe different possible schemes of adaptive and time-varying tariffs that could beproposed by electricity distributors in the context of a development of smart-gridtechnologies (ii) based on economic models we evaluate the potential for thesetariffs to realise load shifting andor shedding and to facilitate the integration ofbattery and plug-in hybrid electric vehicles (BEVs and PHEVs)

Please note that equations with free indices are understood to bevalid for all values the indices can take unless otherwise stated

Contents







241 Time of Use 15













3











411 Supply Side 27

412 Demand Side 29










4






6 Conclusion 42

5

6








7










8









2013-03-05CH


9








pxwebbfsadminch


10

013

5013

10013

15013

20013

25013

191013 192013 193013 194013 195013 196013 197013 198013 199013 200013 201013

Elec

tric

ity c

onsu

mpt

ion

[PJ]13

Year13








11














12



25

0

5

10

15

20


Ct

kW

h (s

ans

TVA

)

2009

2010

2011

2012












13












14


η =∆DD∆pp

=∆ log(DD0)




Di (2)






15

241 Time of Use









16







17








18























1to1 energy easy




Haut tarif (ctkWh) 1938 1010 840 088

Bas tarif (ctkWh) 1254 660 506 088








Haut tarif (ctkWh) 1874 1030 756 088

Bas tarif (ctkWh) 1116 650 378 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









Haut tarif (ctkWh) 1508 1090 330 088

Bas tarif (ctkWh) 874 660 126 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24






19











20











21








22










23













24








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26

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27

41

Figure 1

0

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20

30

40

50

60O

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er-2

PSE

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PUO

lym

pic

PX

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w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

Contents







241 Time of Use 15













3











411 Supply Side 27

412 Demand Side 29










4






6 Conclusion 42

5

6








7










8









2013-03-05CH


9








pxwebbfsadminch


10

013

5013

10013

15013

20013

25013

191013 192013 193013 194013 195013 196013 197013 198013 199013 200013 201013

Elec

tric

ity c

onsu

mpt

ion

[PJ]13

Year13








11














12



25

0

5

10

15

20


Ct

kW

h (s

ans

TVA

)

2009

2010

2011

2012












13












14


η =∆DD∆pp

=∆ log(DD0)




Di (2)






15

241 Time of Use









16







17








18























1to1 energy easy




Haut tarif (ctkWh) 1938 1010 840 088

Bas tarif (ctkWh) 1254 660 506 088








Haut tarif (ctkWh) 1874 1030 756 088

Bas tarif (ctkWh) 1116 650 378 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









Haut tarif (ctkWh) 1508 1090 330 088

Bas tarif (ctkWh) 874 660 126 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24






19











20











21








22










23













24








Doni (t) prop

[poni (t)


]ηoffon

Doffi (t) prop

[poni (t)


]ηoffoff




25


ηonon ηoff

on ηonoff ηoff

off

-085 110 047 -092












26

ONOFF ONOFF

Price

ocrarron on

ocrarr

p

p

gt


on gt ηonoff








27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion











411 Supply Side 27

412 Demand Side 29










4






6 Conclusion 42

5

6








7










8









2013-03-05CH


9








pxwebbfsadminch


10

013

5013

10013

15013

20013

25013

191013 192013 193013 194013 195013 196013 197013 198013 199013 200013 201013

Elec

tric

ity c

onsu

mpt

ion

[PJ]13

Year13








11














12



25

0

5

10

15

20


Ct

kW

h (s

ans

TVA

)

2009

2010

2011

2012












13












14


η =∆DD∆pp

=∆ log(DD0)




Di (2)






15

241 Time of Use









16







17








18























1to1 energy easy




Haut tarif (ctkWh) 1938 1010 840 088

Bas tarif (ctkWh) 1254 660 506 088








Haut tarif (ctkWh) 1874 1030 756 088

Bas tarif (ctkWh) 1116 650 378 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









Haut tarif (ctkWh) 1508 1090 330 088

Bas tarif (ctkWh) 874 660 126 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24






19











20











21








22










23













24








Doni (t) prop

[poni (t)


]ηoffon

Doffi (t) prop

[poni (t)


]ηoffoff




25


ηonon ηoff

on ηonoff ηoff

off

-085 110 047 -092












26

ONOFF ONOFF

Price

ocrarron on

ocrarr

p

p

gt


on gt ηonoff








27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion






6 Conclusion 42

5

6








7










8









2013-03-05CH


9








pxwebbfsadminch


10

013

5013

10013

15013

20013

25013

191013 192013 193013 194013 195013 196013 197013 198013 199013 200013 201013

Elec

tric

ity c

onsu

mpt

ion

[PJ]13

Year13








11














12



25

0

5

10

15

20


Ct

kW

h (s

ans

TVA

)

2009

2010

2011

2012












13












14


η =∆DD∆pp

=∆ log(DD0)




Di (2)






15

241 Time of Use









16







17








18























1to1 energy easy




Haut tarif (ctkWh) 1938 1010 840 088

Bas tarif (ctkWh) 1254 660 506 088








Haut tarif (ctkWh) 1874 1030 756 088

Bas tarif (ctkWh) 1116 650 378 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









Haut tarif (ctkWh) 1508 1090 330 088

Bas tarif (ctkWh) 874 660 126 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24






19











20











21








22










23













24








Doni (t) prop

[poni (t)


]ηoffon

Doffi (t) prop

[poni (t)


]ηoffoff




25


ηonon ηoff

on ηonoff ηoff

off

-085 110 047 -092












26

ONOFF ONOFF

Price

ocrarron on

ocrarr

p

p

gt


on gt ηonoff








27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

6








7










8









2013-03-05CH


9








pxwebbfsadminch


10

013

5013

10013

15013

20013

25013

191013 192013 193013 194013 195013 196013 197013 198013 199013 200013 201013

Elec

tric

ity c

onsu

mpt

ion

[PJ]13

Year13








11














12



25

0

5

10

15

20


Ct

kW

h (s

ans

TVA

)

2009

2010

2011

2012












13












14


η =∆DD∆pp

=∆ log(DD0)




Di (2)






15

241 Time of Use









16







17








18























1to1 energy easy




Haut tarif (ctkWh) 1938 1010 840 088

Bas tarif (ctkWh) 1254 660 506 088








Haut tarif (ctkWh) 1874 1030 756 088

Bas tarif (ctkWh) 1116 650 378 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









Haut tarif (ctkWh) 1508 1090 330 088

Bas tarif (ctkWh) 874 660 126 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24






19











20











21








22










23













24








Doni (t) prop

[poni (t)


]ηoffon

Doffi (t) prop

[poni (t)


]ηoffoff




25


ηonon ηoff

on ηonoff ηoff

off

-085 110 047 -092












26

ONOFF ONOFF

Price

ocrarron on

ocrarr

p

p

gt


on gt ηonoff








27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

7










8









2013-03-05CH


9








pxwebbfsadminch


10

013

5013

10013

15013

20013

25013

191013 192013 193013 194013 195013 196013 197013 198013 199013 200013 201013

Elec

tric

ity c

onsu

mpt

ion

[PJ]13

Year13








11














12



25

0

5

10

15

20


Ct

kW

h (s

ans

TVA

)

2009

2010

2011

2012












13












14


η =∆DD∆pp

=∆ log(DD0)




Di (2)






15

241 Time of Use









16







17








18























1to1 energy easy




Haut tarif (ctkWh) 1938 1010 840 088

Bas tarif (ctkWh) 1254 660 506 088








Haut tarif (ctkWh) 1874 1030 756 088

Bas tarif (ctkWh) 1116 650 378 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









Haut tarif (ctkWh) 1508 1090 330 088

Bas tarif (ctkWh) 874 660 126 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24






19











20











21








22










23













24








Doni (t) prop

[poni (t)


]ηoffon

Doffi (t) prop

[poni (t)


]ηoffoff




25


ηonon ηoff

on ηonoff ηoff

off

-085 110 047 -092












26

ONOFF ONOFF

Price

ocrarron on

ocrarr

p

p

gt


on gt ηonoff








27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

8









2013-03-05CH


9








pxwebbfsadminch


10

013

5013

10013

15013

20013

25013

191013 192013 193013 194013 195013 196013 197013 198013 199013 200013 201013

Elec

tric

ity c

onsu

mpt

ion

[PJ]13

Year13








11














12



25

0

5

10

15

20


Ct

kW

h (s

ans

TVA

)

2009

2010

2011

2012












13












14


η =∆DD∆pp

=∆ log(DD0)




Di (2)






15

241 Time of Use









16







17








18























1to1 energy easy




Haut tarif (ctkWh) 1938 1010 840 088

Bas tarif (ctkWh) 1254 660 506 088








Haut tarif (ctkWh) 1874 1030 756 088

Bas tarif (ctkWh) 1116 650 378 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









Haut tarif (ctkWh) 1508 1090 330 088

Bas tarif (ctkWh) 874 660 126 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24






19











20











21








22










23













24








Doni (t) prop

[poni (t)


]ηoffon

Doffi (t) prop

[poni (t)


]ηoffoff




25


ηonon ηoff

on ηonoff ηoff

off

-085 110 047 -092












26

ONOFF ONOFF

Price

ocrarron on

ocrarr

p

p

gt


on gt ηonoff








27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

9








pxwebbfsadminch


10

013

5013

10013

15013

20013

25013

191013 192013 193013 194013 195013 196013 197013 198013 199013 200013 201013

Elec

tric

ity c

onsu

mpt

ion

[PJ]13

Year13








11














12



25

0

5

10

15

20


Ct

kW

h (s

ans

TVA

)

2009

2010

2011

2012












13












14


η =∆DD∆pp

=∆ log(DD0)




Di (2)






15

241 Time of Use









16







17








18























1to1 energy easy




Haut tarif (ctkWh) 1938 1010 840 088

Bas tarif (ctkWh) 1254 660 506 088








Haut tarif (ctkWh) 1874 1030 756 088

Bas tarif (ctkWh) 1116 650 378 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









Haut tarif (ctkWh) 1508 1090 330 088

Bas tarif (ctkWh) 874 660 126 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24






19











20











21








22










23













24








Doni (t) prop

[poni (t)


]ηoffon

Doffi (t) prop

[poni (t)


]ηoffoff




25


ηonon ηoff

on ηonoff ηoff

off

-085 110 047 -092












26

ONOFF ONOFF

Price

ocrarron on

ocrarr

p

p

gt


on gt ηonoff








27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

10

013

5013

10013

15013

20013

25013

191013 192013 193013 194013 195013 196013 197013 198013 199013 200013 201013

Elec

tric

ity c

onsu

mpt

ion

[PJ]13

Year13








11














12



25

0

5

10

15

20


Ct

kW

h (s

ans

TVA

)

2009

2010

2011

2012












13












14


η =∆DD∆pp

=∆ log(DD0)




Di (2)






15

241 Time of Use









16







17








18























1to1 energy easy




Haut tarif (ctkWh) 1938 1010 840 088

Bas tarif (ctkWh) 1254 660 506 088








Haut tarif (ctkWh) 1874 1030 756 088

Bas tarif (ctkWh) 1116 650 378 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









Haut tarif (ctkWh) 1508 1090 330 088

Bas tarif (ctkWh) 874 660 126 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24






19











20











21








22










23













24








Doni (t) prop

[poni (t)


]ηoffon

Doffi (t) prop

[poni (t)


]ηoffoff




25


ηonon ηoff

on ηonoff ηoff

off

-085 110 047 -092












26

ONOFF ONOFF

Price

ocrarron on

ocrarr

p

p

gt


on gt ηonoff








27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

11














12



25

0

5

10

15

20


Ct

kW

h (s

ans

TVA

)

2009

2010

2011

2012












13












14


η =∆DD∆pp

=∆ log(DD0)




Di (2)






15

241 Time of Use









16







17








18























1to1 energy easy




Haut tarif (ctkWh) 1938 1010 840 088

Bas tarif (ctkWh) 1254 660 506 088








Haut tarif (ctkWh) 1874 1030 756 088

Bas tarif (ctkWh) 1116 650 378 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









Haut tarif (ctkWh) 1508 1090 330 088

Bas tarif (ctkWh) 874 660 126 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24






19











20











21








22










23













24








Doni (t) prop

[poni (t)


]ηoffon

Doffi (t) prop

[poni (t)


]ηoffoff




25


ηonon ηoff

on ηonoff ηoff

off

-085 110 047 -092












26

ONOFF ONOFF

Price

ocrarron on

ocrarr

p

p

gt


on gt ηonoff








27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

12



25

0

5

10

15

20


Ct

kW

h (s

ans

TVA

)

2009

2010

2011

2012












13












14


η =∆DD∆pp

=∆ log(DD0)




Di (2)






15

241 Time of Use









16







17








18























1to1 energy easy




Haut tarif (ctkWh) 1938 1010 840 088

Bas tarif (ctkWh) 1254 660 506 088








Haut tarif (ctkWh) 1874 1030 756 088

Bas tarif (ctkWh) 1116 650 378 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









Haut tarif (ctkWh) 1508 1090 330 088

Bas tarif (ctkWh) 874 660 126 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24






19











20











21








22










23













24








Doni (t) prop

[poni (t)


]ηoffon

Doffi (t) prop

[poni (t)


]ηoffoff




25


ηonon ηoff

on ηonoff ηoff

off

-085 110 047 -092












26

ONOFF ONOFF

Price

ocrarron on

ocrarr

p

p

gt


on gt ηonoff








27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

13












14


η =∆DD∆pp

=∆ log(DD0)




Di (2)






15

241 Time of Use









16







17








18























1to1 energy easy




Haut tarif (ctkWh) 1938 1010 840 088

Bas tarif (ctkWh) 1254 660 506 088








Haut tarif (ctkWh) 1874 1030 756 088

Bas tarif (ctkWh) 1116 650 378 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









Haut tarif (ctkWh) 1508 1090 330 088

Bas tarif (ctkWh) 874 660 126 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24






19











20











21








22










23













24








Doni (t) prop

[poni (t)


]ηoffon

Doffi (t) prop

[poni (t)


]ηoffoff




25


ηonon ηoff

on ηonoff ηoff

off

-085 110 047 -092












26

ONOFF ONOFF

Price

ocrarron on

ocrarr

p

p

gt


on gt ηonoff








27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

14


η =∆DD∆pp

=∆ log(DD0)




Di (2)






15

241 Time of Use









16







17








18























1to1 energy easy




Haut tarif (ctkWh) 1938 1010 840 088

Bas tarif (ctkWh) 1254 660 506 088








Haut tarif (ctkWh) 1874 1030 756 088

Bas tarif (ctkWh) 1116 650 378 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









Haut tarif (ctkWh) 1508 1090 330 088

Bas tarif (ctkWh) 874 660 126 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24






19











20











21








22










23













24








Doni (t) prop

[poni (t)


]ηoffon

Doffi (t) prop

[poni (t)


]ηoffoff




25


ηonon ηoff

on ηonoff ηoff

off

-085 110 047 -092












26

ONOFF ONOFF

Price

ocrarron on

ocrarr

p

p

gt


on gt ηonoff








27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

15

241 Time of Use









16







17








18























1to1 energy easy




Haut tarif (ctkWh) 1938 1010 840 088

Bas tarif (ctkWh) 1254 660 506 088








Haut tarif (ctkWh) 1874 1030 756 088

Bas tarif (ctkWh) 1116 650 378 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









Haut tarif (ctkWh) 1508 1090 330 088

Bas tarif (ctkWh) 874 660 126 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24






19











20











21








22










23













24








Doni (t) prop

[poni (t)


]ηoffon

Doffi (t) prop

[poni (t)


]ηoffoff




25


ηonon ηoff

on ηonoff ηoff

off

-085 110 047 -092












26

ONOFF ONOFF

Price

ocrarron on

ocrarr

p

p

gt


on gt ηonoff








27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

16







17








18























1to1 energy easy




Haut tarif (ctkWh) 1938 1010 840 088

Bas tarif (ctkWh) 1254 660 506 088








Haut tarif (ctkWh) 1874 1030 756 088

Bas tarif (ctkWh) 1116 650 378 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









Haut tarif (ctkWh) 1508 1090 330 088

Bas tarif (ctkWh) 874 660 126 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24






19











20











21








22










23













24








Doni (t) prop

[poni (t)


]ηoffon

Doffi (t) prop

[poni (t)


]ηoffoff




25


ηonon ηoff

on ηonoff ηoff

off

-085 110 047 -092












26

ONOFF ONOFF

Price

ocrarron on

ocrarr

p

p

gt


on gt ηonoff








27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

17








18























1to1 energy easy




Haut tarif (ctkWh) 1938 1010 840 088

Bas tarif (ctkWh) 1254 660 506 088








Haut tarif (ctkWh) 1874 1030 756 088

Bas tarif (ctkWh) 1116 650 378 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









Haut tarif (ctkWh) 1508 1090 330 088

Bas tarif (ctkWh) 874 660 126 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24






19











20











21








22










23













24








Doni (t) prop

[poni (t)


]ηoffon

Doffi (t) prop

[poni (t)


]ηoffoff




25


ηonon ηoff

on ηonoff ηoff

off

-085 110 047 -092












26

ONOFF ONOFF

Price

ocrarron on

ocrarr

p

p

gt


on gt ηonoff








27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

18























1to1 energy easy




Haut tarif (ctkWh) 1938 1010 840 088

Bas tarif (ctkWh) 1254 660 506 088








Haut tarif (ctkWh) 1874 1030 756 088

Bas tarif (ctkWh) 1116 650 378 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









Haut tarif (ctkWh) 1508 1090 330 088

Bas tarif (ctkWh) 874 660 126 088

Heures


10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Haut tarif

Bas tarif









0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24






19











20











21








22










23













24








Doni (t) prop

[poni (t)


]ηoffon

Doffi (t) prop

[poni (t)


]ηoffoff




25


ηonon ηoff

on ηonoff ηoff

off

-085 110 047 -092












26

ONOFF ONOFF

Price

ocrarron on

ocrarr

p

p

gt


on gt ηonoff








27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

19











20











21








22










23













24








Doni (t) prop

[poni (t)


]ηoffon

Doffi (t) prop

[poni (t)


]ηoffoff




25


ηonon ηoff

on ηonoff ηoff

off

-085 110 047 -092












26

ONOFF ONOFF

Price

ocrarron on

ocrarr

p

p

gt


on gt ηonoff








27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

20











21








22










23













24








Doni (t) prop

[poni (t)


]ηoffon

Doffi (t) prop

[poni (t)


]ηoffoff




25


ηonon ηoff

on ηonoff ηoff

off

-085 110 047 -092












26

ONOFF ONOFF

Price

ocrarron on

ocrarr

p

p

gt


on gt ηonoff








27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

21








22










23













24








Doni (t) prop

[poni (t)


]ηoffon

Doffi (t) prop

[poni (t)


]ηoffoff




25


ηonon ηoff

on ηonoff ηoff

off

-085 110 047 -092












26

ONOFF ONOFF

Price

ocrarron on

ocrarr

p

p

gt


on gt ηonoff








27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

22










23













24








Doni (t) prop

[poni (t)


]ηoffon

Doffi (t) prop

[poni (t)


]ηoffoff




25


ηonon ηoff

on ηonoff ηoff

off

-085 110 047 -092












26

ONOFF ONOFF

Price

ocrarron on

ocrarr

p

p

gt


on gt ηonoff








27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

23













24








Doni (t) prop

[poni (t)


]ηoffon

Doffi (t) prop

[poni (t)


]ηoffoff




25


ηonon ηoff

on ηonoff ηoff

off

-085 110 047 -092












26

ONOFF ONOFF

Price

ocrarron on

ocrarr

p

p

gt


on gt ηonoff








27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

24








Doni (t) prop

[poni (t)


]ηoffon

Doffi (t) prop

[poni (t)


]ηoffoff




25


ηonon ηoff

on ηonoff ηoff

off

-085 110 047 -092












26

ONOFF ONOFF

Price

ocrarron on

ocrarr

p

p

gt


on gt ηonoff








27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

25


ηonon ηoff

on ηonoff ηoff

off

-085 110 047 -092












26

ONOFF ONOFF

Price

ocrarron on

ocrarr

p

p

gt


on gt ηonoff








27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

26

ONOFF ONOFF

Price

ocrarron on

ocrarr

p

p

gt


on gt ηonoff








27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

27

41

Figure 1

0

10

20

30

40

50

60O

ntar

io- 1

Ont

ario

- 2 SPP

PSE

ampG

w C

AC

PSEamp

G w

o C

AC

PSE

X-T

OU

w C

AC

X-T

OU

wo

CA

C

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-1

PSE

ampG

Ana

heim

Ont

ario

- 1O

ntar

io- 2

Ont

ario

- 1O

ntar

io- 2 SPP

Aus

tral

iaA

me r

en- 0

4A

mer

en- 0

5PS

Eamp

G w

CA

CPS

EampG

wo

CA

CX

-CPP

w C

AC

X-C

PP w

o C

AC

X-C

TO

U w

CA

CX

-CT

OU

wo

CA

CId

aho

SPP-

ASP

P- C

Am

eren

- 04

Am

e ren

- 05

AD

RS-

04

AD

RS-

05

Gul

f Pow

er-2

PSE

ampG

G

PUO

lym

pic

PX

-AC

- CPP

w C

AC

X-A

C- C

TOU

w C

AC

X-P

CT-

CT

OU

w C

AC

ESP

P

Oly

mpi

c P

Pricing Pilot

R

educ

tion

in P

eak

Loa

d

TOUTOU

w Tech

PTR CPPCPP w

TechRTP

RTPw

Tech





411 Supply Side




28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

28







minimisezijh(t)

Tsum

t=1

msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

(4)


msum

i=1



zijh(t) ge 0 (7)


L =Tsum

t=1

[msum

i=1

nsum

j=1

Hj(t)sum

h=1

rtci(t)zijh(t)

+

nsum

j=1

Hj(t)sum

h=1

pjh(t)

(minussum

i

zijh(t) + djh(t)

)

+

msum

i=1

nsum

j=1

Hj(t)sum

h=1

uijh(t)

(zijh(t)minusKi(t)

)]

(8)



29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

29





Hj(t)sum

h=1

djh(t) (9)


412 Demand Side







k=1


where




30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

30





i=1





k=1



pj(t) =1

rt

Hj(t)sum

h=1

δjh(t) pjh(t) (18)





(non hydro)3800 150 0 - 15

Imports 10000 200 100


html


31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

31





bjk =




On-Peak Off-Peak





= 2394 GWhmonth


= 1575 GWhmonth

(20)




32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

32


On-Peak Off-Peak









sum

t

xij(t) ge βij (21)

together with






33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

33



D(t) =sum

i

sum

j

xij(t) (24)


ψi(x) =sum

t

γ(D(t))

sum

j

xij(t)

(25)


Li = ψi(x) +sum

j

ηij

(βij minus

sum

t

xij(t)

)

+sum

tj

microij(t)


)

+sum

tj

νij(t)


)

(26)


0 =partLipartxij(t)

=partψi(x)



(sum

k

xik(t)



ηij ge 0 and ηij

(sum

t

xij(t)minus βij)

= 0 (29)


)= 0 (30)


)= 0 (31)



34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

34





sumk x

ik(t)

n



sum

t

xij(t)












35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

35






nsum

θ=1

msum

κ=1

cκzκθ (35)


msum

κ=1

zκθ minussum

ij

xij(θ) ge 0 (36)


zκθ ge 0 (38)sum

θ





ij(θ) and νij(θ)






36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

36








37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

37





xnt+1 = xnt +αn

βnunt (43)


xnT = 1 (44)


p

(Dt +

sumNn=1 u

nt

C

) (45)


Jn(u) =Tminus1sum

t=0

p

(Dt +

sumNm=1 u

mt

C

)unt =

Tminus1sum

t=0

p

(dt + avg(ut)

c

)unt (46)


avg(ut) =

sumNn=1 u

nt

N (47)


38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

38


Hn(λnt+1 xnt u

nt ) = p

(dt + avg(ut)

c

)unt + λnt+1

(xnt +

αn

βnunt

) (48)


0 =part

partuntHn(λnt+1 x

nt u

nt )

= p

(dt + avg(ut)

c

)+ pprime

(dt + avg(ut)

c

)untcN

+ λnt+1

αn

βn (49)


λnt =part

partxntHn(λnt+1 x

nt u

nt )

= λnt+1 (50)




0 = p

(dt + avg(ut)

c

)+ ζn

αn

βn (52)




Jn(u) =Tminus1sum

t=0

p

(dt + avg(ut)

c


2

(53)


39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

39







1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


No

rmal

ized

po

wer

(k

W)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)


1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)



V CONCLUSIONS



1200 1600 2000 000 400 800 12005

55

6

65

7

75

8

85

9

95

10


Norm

aliz

ed p

ow

er (

kW

)






APPENDIX




t = bn an


that

an = arginfanSbn

an 1

2(ut+1 ut)

+1

4

$p(dt+1 + ut+1) p(dt + ut)

amp2




n and hence that









dt+1 + ut+1 = dt + ut + B (14)



t + C





40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

40
















41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

41










42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

42


minimisePG

cTG middot PG (56)











6 Conclusion



43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

43









44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

44





45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

45

References
















46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

46
















47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

47













48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

48


















Time of Use


Peak Time Rebates

Real-Time Pricing




















Supply Side

Demand Side















Conclusion

The Economics of Electricity Dynamic Pricing and Demand...

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Transcript of The Economics of Electricity Dynamic Pricing and Demand...