optimal electricity demand-response contracting

Optimal Electricity Demand-Response ContractingThematic Semester on Statistics for Energy Markets

Summer School, Paris — Dourdan, June 2018

Rene Aıd Dylan Possamaı Nizar Touzi

Universite Paris-Dauphine Columbia University Ecole PolytechniqueFinance for Energy Market Research Initiative

Aıd, Possamaı & Touzi Optimal Electricity Demand-Response Contracting 1 / 55

Agenda

1 Motivations

2 Model

3 Optimal contractControled ResponsivenessUncontroled responsiveness

4 Linear case

5 Numerical studyCalibrationImplied responsiveness reduction

6 Conclusion


Motivations

MotivationsHow to cope with intermittent sources of energy in power systems?

The need for more flexibility in electric systems can be satisfied either

... by batteries or ...

... a better use of demand flexibility potential.

Possible to use distributed control of appliances (Meyn, Barooah, Busic andChen(2015), Tindemans, Trovato and Strbac (2015))

Also possible to use demand-response.


Motivations

Demand-response

Contract between a consumer and a retailer (or the TSO or a DSO or aproducer...)

The consumer pays a certain lower fare for power all the days of the year

... except on a certain number of days (or periods ot the day) decided on arandom basis by the retailer.

Example: Low Carbon London Pricing trial experiment 2012-2013

Consumer enrolled in the dynamic Time of Use tariff would pay their power:11.76 p/kWh on Normal days, 67.2 p/kWh on High price days, 3.99 p/kWhon Low price days

The number of days was limited at the inception.

The period of High or Low price could be 3, 6, 9 or 12 hours.


Motivations

Demand-response

DR contracts ishould not be confounded n the zoology of retail tariffcontracts.

TOU: Time of Use: price depends on the period of the day (peak, off-peakfor instance) fixed at the inception of the contract.

RTP: Real Time Pricing, transmission of spot prices to the consumer.

CPP: Critical Peak Pricing, a low fare price is define all the days of the yearexcept at a certain number of unspecified days at the inception when theprice will be signficantly higher but either fixed at the inception or statedependent (variable Critical Peak Pricing)

Peak-Time Rebate (PTR) works the other way around: the consumer is givenan amount of money (100$) and each he consumes energy on event days, thecorresponding amount of cash is deduced from his account.


Motivations

Remarks

Important demand-response (DR) and smart grid world wide. EU investmentin smart metering: 45 billions e to reach 200 millions smart meters.

DR programs reduce consumption level on average but at a significantinfrastructure cost and with a significant variance in consumers response.


Motivations

source: RTE Socio-economic assessment of smart grid, summary, 2015.

DSR is probably not the less costly way to cope with renewable intermittency.


Motivations

source: Faruqui & Sergici (J. Reg. Econ., 2010) Fig. 1.

DSR presents significant variance in peak-load reduction consumption.


Motivations

Low Carbon London Pricing Trial simple responsiveness analysis

Publicly avaible data of pricing trial held in London in 2012-2013.

Data can be found at https://data.london.gov.uk (raw data) or athttps://www.ukdataservice.ac.uk (treated data)

Experiment and results described in Tindemans et al. (2014) and Schofield(2015).

5,567 London households whose consumption have been measured at an halfhourly time-step on the period from February, 2011 to February, 2014.


https://data.london.gov.uk

https://www.ukdataservice.ac.uk

Motivations

For the dynamic Time-of-Use (dToU) tariff trial, the population was dividedin two groups. One group of approximately 1,117 households were offered thedToU tariff while the remaining 4,500 households were not subject to thisdynamic tariff.

The dToU was applied during the year 2013 (January, 1st to December,31st). Tariffs were sent to the households on a day-ahead basis using a HomeDisplay or a text message to the mobile phone of the customer.

Prices had three levels: High (67.20 p/kWh), Normal (11.76 p/kWh) andLow (3.99 p/kWh).

Standard tariff is made of a flat tariff of 14.228 p/kWh.


Motivations

We consider only consumers without missing data. We have 880 consumersin the control group and 250 in the dToU group.

For each consumer, we decompose the consumption C it of consumer i into a

random part X it and a seasonal part S i

t .

We are interested in the random part X it .

We compute the mean and standard deviation across the two populationduring events and non-events periods.


Motivations

Std Tariff ToU Tariff p-valuePopulation Population

X Non-Event (W) 3.8 3.2 0.149

σ(X ) Non–Event (W) 34.7 26.8 5.1 10−10

X Event (W) 13.9 3.5 1.3 10−15

σ(X ) Event (W) 87.2 66.2 0.128

Statistical tests confirm the observation that the deviations X it of the control

group and the dToU group have the same mean and different (but close) varianceduring non-event daysOn event days, they exhibit different means (there is a reduction effect) but withthe same variances.


Motivations

Responsiveness

Achieving a precise target of consumption reduction is an important concernin the power system literature.

See Kwac and Rajagopal (IEEE Int. Conf. on Big Data, 2013) andKuppannagari et al. (ICCS 2016, Proc. Comp. Scien. vol. 80, 2016).

Use statistical models of DR. Fit DR on observed past consumption for eachcustomer.


Motivations

Our modelprovides a framework to take into account the rationality of variance on theresponsiveness of consumer in DSR contract

prodives a tool to increase the responsiveness of consumers and thus, DSRprograms.

is based on optimal contract theory and in particular, on Principal-Agentmoral hazard situations.


Motivations

Principal-Agent framework

Examples

Insurer / insured

Land owner / farmer

Stockholder / manager

Game theory situation

with a leader (the Principal) and a follower (the Agent)

where the Principal tries to find the contract to incentivise the Agent to dohis best effort so that the Principal’s utility is maximised.

Information asymetry between actors.

Two families of situations: adverse selection and moral-hazard


Motivations

Adverse selectionThe Principal is facing a population of agents with unknown types.

The Principal observe the distribution of the type in the population butcannot identify the type of a given agent.

She tries to figure out the best menus of contracts that would maximise herutility while incentivizing the agents to choose the contract that correspondsto his type (prevent agents from lying on their type).

Moral hazardThe Principal is facing one agent.

The Principal observe the result of the action of the agent but not the effortof the agent.

She tries to figure out the best that would maximise her utility whileincentivizing the agents to do his best effort so that it maximises thePrincipal’s utility.


Motivations

RemarkIn the economic literature, DSR contracting has been identified as an adverseselection problem (see Fahrioglu and Alvarado (IEEE Trans. Power Systems,200) and Crampes and Leautier (J. Regul. Econ. 2015)).

We frame demand-response contracting in a moral-hazard context.

In particular, we frame our model in the Principal-Agent in continuous-timeframework.


Motivations

2016 Nobel Prize winners

Figure: Prof. Bengt Holmstrom (MIT) and Prof. Oliver Hart (Harvard University)


Motivations

Some references

A (very) few references

Laffont and Martimort, The Theory of Incentives, Princeton University Press,2002.

Holmstrom and Milgrom, Econometrica, 1987.

Sannikov Rev. Econ. Stud., 2008.

Cvitanic & Zang, Contrat Theory in Continuous-time models, Springer, 2013.

Cvitanic, Possamaı & Touzi, Dynamic programming approach toPrincipal-Agent problems, arxiv, 2015.


Model

Model


Model

Interaction between one producer (the Principal, she) and one consumer (theAgent, he).

We consider the deviation Xt of consumer’s from his deterministicconsumption.

Increase (resp. decreasing) consumption with deviation Xt provides a value(resp. a loss) to the consumer f (Xt).

Consumer can reduce her level and variation of consumption at cost c(a, b)where a is the effort on the level and b the effort on the variation.

Consumer’s agrees to enter in the contract under participation constraint: hisutility should not be below his reservation utility R.


Model

Efforts and costs are differentiated per usage (tv, oven, heating, cooling,refrigerator, lights...)

Producer provides power to consumer’s at energy cost (or avoided cost)g(Xt) and variation cost proportional to 〈X 〉.The objective of the producer is to minimize his total cost of generation(energy and variation) plus the payment ξ to the consumer.

The objective of the consumer is to maximise the utility of consumptionminus effort cost plus payment from the producer.

Producer observes the deviation X and its quadratic variation 〈X 〉 but notthe efforts of the consumer (case of hidden actions or moral hazard).


Model

The consumer (The Agent)

Dynamics of the deviation from baseline consumption

X a,bt = X0 +

∫ t

0

(−N∑i=1

ai (s))ds +

∫ t

0

N∑i=1

σi√bi (s)dW i

s

Cost function for efforts ν := (a, b):

c(a, b) :=1

2

N∑i=1

ai2

µi︸︷︷︸c1(a)

+1

2

N∑i=1

σi (bi−ηi − 1)

λiηi︸︷︷︸c2(b)

, 0 ≤ ai , 0 < bi ≤ 1.

Consumer’s criterion:

JA(ξ, ν) := Eν[UA

(ξ +

∫ T

0

(f (X νs )− c(νs)) ds

)],

with UA(x) = −e−r x .


Model

The producer (The Principal)

JP(ξ,Pν) := EPν[U

(− ξ −

∫ T

0

g(Xs)ds − h

2〈X 〉T

)]

g generation cost function, convexe centered at zero

h direct unitary cost of volatility

U(x) = −e−p x .

The producer’s problem is:

VP := supξ∈Ξ

supPν∈P?(ξ)

JP(ξ,Pν).

with the participation constraint: the consumer enters in the contract only if hisexpected utility is above R := R0e

−rπ where

R0 := supPν∈P

JA(0,Pν) = EPν[UA

(∫ T

0

(f (Xs)− c(νs)) ds)]

,

is the utility he gets without contract and π is a premium.


Model

Quadratic variation for dummies

0 2 4 6 8 10 12 140

2

4

6

8

10

12

Total consumption X = Total consumption X

〈X 〉 = 12 + 12 + ...+ 12 = 12 〈X 〉 = 122 + 112 + 101..+ 12 = 650


Model

Interpretations of volatility cost and control

Incentives to smoothen consumption by moving around scheduled usages

Incentives to provide a regular response to incentives.


Model

RemarksThe producer does not observe the efforts a and b on usages. She onlyobserves the consumption X and also its quadratic variation 〈X 〉.The problem is not Markovian. The contract is written on the observation ofthe whole path of the consumption on [0,T ].

The consumer has never an interest in making an effort to reduceconsumption without contract.

Because of risk-aversion, the consumer has an interest in making an effort toreduce volatility even without contract


Model

Consumer’s reservation utility

The consumer’s reservation utility is given by R0 = −e−ru(0,X0), where u is theunique viscosity solution of the HJB equation

−∂tu = f + H(ux , uxx − ru2x ), with u(T , .) = 0,

where H is the consumer’s Hamiltonian

H(z , γ) := sup(a,b)

{−a · 1z +

1

2|σ(b)|2γ − c(a, b)

}, 1 := (1, · · · , 1),

which optimizers are

a(z) := µz−, bj(γ) := 1 ∧ (λjγ−)− 1

1+ηj .

If in addition u is smooth, then the optimal efforts of the consumer are

a0 := 0, b0j := 1 ∧

(λj ( uxx − ru2

x )−)− 1

1+ηj

.


Optimal contract

Optimal contract


Optimal contract Controled Responsiveness

Optimal contract for controled responsiveness

Cvitanic, Possamaı & Touzi (2015) proves that the optimal contract is of theform

Y Y0,Z ,Γ := Y0+

∫ t

0

ZsdXs+1

2

∫ t

0

(Γs+rZs2)d〈X 〉s−

∫ t

0

(H(Zs , Γs)+f (Xs)

)ds.

Y0 is going to be the certainty equivalent of reservation utility of theconsumer.

Payment (Zt ≤ 0) if consumption decreases (dX ≤ 0)

Payment (Γt ≤ 0) if volatility decreases

Compensation for induced volatility cost rZ 2s

Minus the natural benefits the consumer earns when making efforts (Zt , Γt),i.e. H(Zs , Γs) + f (Xs)



Producer optimisation problem reduction

Any contract of the form above provides a utility E[UA(Y0)

].

When the Agent is being proposed the contract above, his optimal responseat each instant t is precisely given by a(Zt) and b(Γt).

For sake of simplicity consider the case r = 0 and plug the contractξ = Y Y0,Z ,Γ

T in the Agent’s optimsation problem.

supν=(a,b)

Eν[Y0 +

∫ t

0

ZsdXs +1

2

∫ t

0

Γsd〈X 〉s −∫ t

0

(H(Zs , Γs) + f (X ν

s ))ds

+

∫ T

0

f (X νs )− c(νs)ds

],

whose Hamiltonian is precisely:

H(Z , Γ) = supa,b

{− a · 1Z +

1

2Γ2‖σ(b)|2 − c(a, b)

}



Producer optimisation problem reduction

Thus, the optimization problem of the producer reduces to

V sb = supZ ,Γ

E[U(− LZ ,ΓT

)], with LZ ,Γt := Y Z ,Γ

t +

∫ t

0

g(XZ ,Γs

)ds +

h

2d〈XZ ,Γ〉s ,

and L0 = Y0 = −r−1 log(−R).

The state variable L represents the loss of the Principal under the optimalresponse of the Agent, and is defined by the dynamics

dLZ ,Γt =1

2

[2(g − f )(XZ ,Γ

t ) + µ(Z−t )2 + c2(Γt) +(rZ 2

t + h)|σ(Γt)|2

]dt

+ Zt σ(Γt) · dWt .

The dynamics of X is given by

dXZ ,Γt = −a(Zt)dt + σ(Γt) · dWt .



RemarkThe contract can be written formally:

dY =(c(a(Zt), b(Γt)) +

1

2rZ 2

t |σ(Γt)|2 − f (Xt))dt + Zt σ(Γt) · dWt .

The consumer’s rate of payment is made of the payment for his effort whenreceiving incentives (Zt , Γt), plus the risk-premium induced by hisrisk-aversion for volatility, minus the value he receives from the consumption.



Controled responsiveness optimal contract

The optimal payment rates zsb and γsb are given by:

γsb := −(q(vx , vxx , zsb) ∨ 1

λ

),

zsb ∈(vx ,

pr+p vx

), when vx ≤ 0, and zsb = p

r+p vx when vx ≥ 0,where v is the unique viscosity solution of the PDE

−∂tv = (f − g) +1

2µ v2

x −1

2infz∈R

{F0

(q(vx , vxx , z)

)+ µ(z− + vx)2

},

v(T , x) = 0,

with F0(q) = q|σ(−q)|2 + c2(−q), and q(vx , vxx , z) := h− vxx + rz2 + p(z − vx)2.Further, the value function of the producer is VP = −e−p(v(0,X0)−L0) withL0 = −r−1 log(−R).



0 5 10 15 20 25 30q

0

100

200

300

400

500

600

700

800F0(q)

Figure: The function F0 for 2 usages λ1 = 1/2, λ2 = 1/5, σ1 = 2, σ2 = 5.


Optimal contract Uncontroled responsiveness

Uncontroled responsiveness

In the case where the consumer only controls the average rate of consumption andreceived incentives on it, the preceeding result holds with:

γsb := −(q(wx ,wxx , zsb) ∨ 1

λ

),

zsb = Λwx , when wx ≤ 0, and zsb = pr+pwx when wx ≥ 0, with Λ :=

1+p |σ|2

µ

1+(r+p) |σ|2

µ

and where v is the unique viscosity solution of the PDE

−∂tw = (f − g) +1

2µw2

x −1

2infz∈R{q(wx ,wxx , z)|σ|2 + µ(z− + wx)2

},

w(T , x) = 0.


Optimal contract Uncontroled responsiveness

Remarks

Assume that (f − g)(x) = δx .

Then, we guess that v(t, x) = A(t)x + B(t) with

−A′(t) = δ,

−B ′(t) =1

2µA2(t)− 1

2infz∈R

{F0

(h + rz2 + p(z − A(t))2

)+ µ(z− + A(t))2

},

A(T ) = B(T ) = 0.

Thus, we have A(t) = δ(T − t) and the sign of vx is given by the sign δ.


Linear case

Linear Case


Linear case

Consumer’s reservation utility in the linear case f (x) = κ x

Then, the reservation utility of the consumer is

R0 = − exp(− r(κX0T + E (T )

)),

where E (T ) := − 12

∫ T

0F0(−γ0

s

)ds, γ0

s := −rκ2(T − s)2.

The consumer’s optimal effort is

a0 = 0, and b0j (t) := 1 ∧

(λj rκ

2(T − t)2)− 1

1+ηj ,

thus inducing the dynamics

dX 0t = σ0 · dWt ,

with σ0 := σ(γ0t

).


Linear case

Consequence

If for all usages, rκ2T 2 < 1λj

, the consumer has no interest in making any effort to

reduce the volatility of his consumption.


Linear case

Optimal contract when energy has more value for the consumer δ ≥ 0

If δ ≥ 0, the optimal payments rate are

z?t =p

r + pδ(T − t), γ?t = −

[(h + ρ δ2(T − t)2

)∨ 1

λ

],

1

ρ:=

1

r+

1

p.

The dynamics of the consumption deviation is

dX ?t = σ?t · dWt ,

with σ?t := σ(γ?t ). And the optimal contract is

ξ? = L0 +

∫ T

0

(1

2c2(γ?t )− κXt +

1

2

rp2δ2

(p + r)2(T − t)2|σ?t |2

)dt +

∫ T

0

z?t σ?t · dWt .


Linear case

RemarksIf δ = 0 and h = 0, the producer induces no effort from the consumer andthus, the volatility under optimal contract is |σ|2 ≥ |σ0

t |2.


Linear case

Certainty equivalent gain

When δ ≥ 0, the certainty equivalent gain from the contract for the producer is:

GP = −π +1

2

∫ T

0

F0(−γ0s )ds − 1

2

∫ T

0

c2(γ?s )ds +h

2

(∫ T

0

(|σ0

s |2 − |σ?s |2)ds

)+

p

2

∫ T

0

((κ− δ)2|σ0

s |2 −r

r + pδ2|σ?s |2

)(T − s)2ds

)︸︷︷︸

Indirect volatility cost compromise

.


Linear case

0 0.2 0.4 0.6 0.8 1t

1

1.5

2

2.5

3

3.5

4

4.5

5<?

1

<?2

<01

<02

0 0.2 0.4 0.6 0.8 1t

0

5

10

15

20

25

30hX?ihX0ihXi

Figure: (Left) Volatilities of two usages without contract (red) and with optimal contract(blue). (Right) Quadratic variation when no efforts are done (black) without contract(red) and with optimal contract (blue).

µ = (1, 5), σ = (2.0, 5.0), λ = (1/2, 1/5), η = (1, 1), r = 1, π = 0, p = 2,h = 4.5, κ = 5 δ = 3


Linear case

0

5100

p

15

5 10

h

20

R T 0j<

tj2dt

10

25

1515

30

-1500150

-1000

-500

v(0

;0)

0

105

500

hp

510015

Figure: (Left) Total volatility of consumption deviation under optimal contract as afunction of the direct volatility cost h and the risk-aversion parameter p of the consumercompared to the total volatility without contract (flat surface). (Right) Certaintyequivalent of the producer with contract and without contract as a function of the directvolatility cost h and the risk-aversion parameter p.


Linear case

Optimal contract with δ ≤ 0

If δ ≤ 0 and h + rδ2T 2 ≤ 1λ, the optimal payments rate are

γ?t = − 1

λ, z?t = Λδ(T − t),with Λ :=

1 + p |σ|2

µ

1 + (r + p) |σ|2

µ

The dynamics of the consumption deviation is

dX ?t = µz?t dt + σ · dWt .

And the optimal contract is

ξ? = L0 +1

2

∫ t

0

(µ+ r |σ|2

)(z?s )2ds −

∫ t

0

κXsds +

∫ T

0

z?s σ · dWs


Linear case

Certainty equivalent gain

When δ ≤ 0 and h + rδ2T 2 ≤ 1λ

, the certainty equivalent gain from the contractfor the producer is:

GP =− π + κTX0 +1

2

∫ T

0

F0(−γ0t )dt +

h

2

∫ T

0

(|σ0t |2 − |σ|2)dt

+

∫ T

0

δµ(T − t)z?t dt −1

2

∫ T

0

(µ+ r |σ|2)(z?t )2dt

+p

2

∫ T

0

(κ2|σ0t |2 − (1− Λ)2δ2|σ|2(z?t )2)(T − t)2dt.

Remark

The positive term δµ(T − t)z?t is the rate of revenue from the energyreduction while 1

2 (µ+ r |σ|2)(z?t )2 is the rate of cost.

This cost is made of two terms: the direct cost of effort made by theconsumer to reduce consumption (µ(z?t )2) and the indirect cost of volatilityinduced by this reduction on the mean consumption (r |σ|2(z?t )2).


Numerical study

Numerical study


Numerical study Calibration

ParametersExtensive use of the Low Carbon London data.

Cost function parameters: δ given by the difference between High price andNormal price |δ| = 67.2− 11.76 = 55.44 p/kWh. The value of energy for theconsumer taken to be κ = 11.76.

Risk aversion of the producer p. Literature on day-ahead risk premia (Bessembinder and Lemon (2002), Longstaff and Wang (2004), Benth et al.(2008) and Viehmann (2011)). Find consistent and convergent estimates forp ≈ 0.6 per e.

Risk aversion of the consumers in the LCL pricing trial. Consumers whereoffered 150 pounds to enroll in the dToU to accept a risk of 23 pounds asmeasured by the standard deviation of annual electricity bill of the controlgroup. It lead to a constant absolute risk aversion of 0.57 per pounds.

Per unit cost of volatility h. Extensive literature on the induced externalitycost of intermittency on power systems (see Hirth (2015) for a review). Herewe take conservative values based on estimates of raping costs for gas powerplants provided by Oxera (2003) (Table 3.2 p. 8) and Van den Bergh andDelarue (2015) Table IV. We take h = 40 e/MW2.h.



What about the µi?

When the optimal contract is limited to reduction of the averageconsumption, the absolute value of the average consumption deviation isgiven by:

1

T

∣∣∣∣E[ ∫ T

0

Xtdt

]∣∣∣∣ =1

3Λµ|δ|T 2.

Using LCL data and the mean deviation reduction estimation given by thedifference between the control group consumption during event 13.9 and thedToU consumption 3.5, one has a mean consumption deviation of 10 W.

Thus, taking T to be one hour and using the estimates of p and r , one findsthat µ = 6.0 10−4.



What about the λi?

No experiment reports a framework with incentives to provide regularresponses.

Note that there is no interest to incite the consumer to reduce its volatility ofconsumption if the costs 1/λi are larger than h.

Mental experiment: what would be the highest cost the consumer would bearto reduce volatility if he was to respond to an incentive?

Answer: using the value of h, we numerically compute λ = 0.587.


Numerical study Implied responsiveness reduction

Benefit from contracting

Controled Uncontroledresponsiveness responsiveness

Consumer’s reservation utility -9.0 10−4 -9.0 10−4

Payment for the mean 0.29 0.27Volatility risk premium 9.5 10−3 9.5 10−3

Payment for the volatility 4.6 10−3 0−f (X ) 0.12 0.12Total payment 0.43 0.40

Cost without contract -0.03 -0.03Cost with contract 0.33 0.29Gain from the contract 0.36 0.32

Volatility reduction 17% 0

Table: Volatility reduction, payments and benefits from the contract in pennies.


Conclusion

Conclusion & Perspectives

ConclusionOptimal contract theory in continuous-time provides a framework to accountfor responsiveness control of demand-response contract.

Calibration with large scale DR experiment indicates a significant benefitfrom responsiveness increase.

More theoretically, trading flexibility would allow power system to bear morerisk.

Perspectives

Limited liability (no negative payments)

Extension to a group of consumers

Identification of consumers types (adverse selection)

Making a pricing trial experiement with a responsiveness incentive mechanism


Conclusion

References

C. Crampes and T.-O. Leautier. Demand response in adjustment markets forelectricity. Journal of Regulatory Economics, 48(2):169–193, 2015.

J. Cvitanic, D. Possamaı, and N. Touzi. Dynamic programming approach toprincipal–agent problems. arXiv preprint arXiv:1510.07111, 2015.

M. Fahrioglu, F. L. Alvarado. Designing Incentive Compatible Contracts forEffective Demand Management. IEEE Transactions On Power Systems, Vol.15, No. 4, November 2000.

Holmstrom, B., Milgrom, P. Aggregation and linearity in the provision ofintertemporal incentives, Econometrica, 55(2):303–328. 1987.

J. Kwac and R. Rajagopal. Targeting customers for demand response basedon big data. arXiv preprint arXiv:1409.4119, 2014.


Conclusion

References

S. Meyn, P. Barooah, A. Busic, Y. Chen, J. Ehren. Ancillary Service to theGrid Using Intelligent Deferrable Loads. IEEE Trans. Automat. Control. 60(11): 2847 - 2862. 2015

Y. Sannikov. A continuous-time version of the principal–Agent problem, TheReview of Financial Studies, 75(3):957–984. 2008.

Schofield, J., Carmichael, R., Woolf, M., Bilton, M., Ozaki, R., Strbac, G.Residential consumer attitudes to time–varying pricing, report A2 for the LowCarbon London LCNF project, Imperial College London. 2014.

S. Tindemans, P. Djapic, J. Schofield, T. Ustinova, and G. Strbac. Resilienceperformance of smart distribution networks. Technical report D4 for the ”LowCarbon London” LCNF project, Imperial College London, 2014.

S. Tindemans, V. Trovato, G Strbac. Decentralized Control of ThermostaticLoads for Flexible Demand Response. IEEE Transactions on Control SystemsTechnology, 61, 2015.


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