A Contract for Demand Response based on Probability of Call

A Contract for Demand Response based on Probability of Call Jose Vuelvas, Fredy Ruiz and Giambattista Gruosso Pontificia Universidad Javeriana - Politecnico di Milano 2018 1

2 Outline Introduction Problem Setting Contract and consumer s problem Consumer optimal behavior Numerical case study Conclusions

Net consumption per hour 3 Introduction: drawbacks of baseline estimation Traditional incentive-based DR programs require an estimated baseline against which consumer s load reduction is measured. The current methods for establishing baseline: i) averaging techniques, ii) regression approaches, etc. Incentive payment = (Reduction) x (Reward/kWh) Reduction = Baseline - Observed baseline Reduction Actual consumption Peak event Hours of day

Net consumption per hour Introduction: drawbacks of baseline estimation Traditional incentive-based DR programs require an estimated baseline against which consumer s load reduction is measured. The current methods for establishing baseline: i) averaging techniques, ii) regression approaches, etc. Incentive payment = (Reduction) x (Reward/kWh) Reduction = Baseline - Observed baseline Reduction Actual consumption Peak event Hours of day Problem: Baseline manipulation (Severin Borenstein, 2014; Vuelvas and Ruiz, 2017; Chao, 2011). 3

4 Our approach A new contract based on the probability of call is proposed.

4 Our approach A new contract based on the probability of call is proposed. Probability of call: the chance of a consumer to be selected by the aggregator to serve as DR resource at a given period.

4 Our approach A new contract based on the probability of call is proposed. Probability of call: the chance of a consumer to be selected by the aggregator to serve as DR resource at a given period. Consumer self reports his baseline (it is not estimated).

Our approach A new contract based on the probability of call is proposed. Probability of call: the chance of a consumer to be selected by the aggregator to serve as DR resource at a given period. Consumer self reports his baseline (it is not estimated). Agents bid information in terms of energy (baseline and reduction capacity). In this model, the main objective of the aggregator is to select randomly which participant consumers are called to perform DR. Generators SO DR aggregator Contracts User 1 User 2 User I 4

G(qi; bi) 5 Consumer model Utility function: G(q i (consumption) ; b i (baseline)) Customer satisfaction function. γ i 2 q2 i + [γ ib i + p]q i Parameters: p 2 2γ i + γ ib 2 i 2 + pb i γ i : marginal utility (energy preference) p: energy price b i : baseline Variable: q i : actual consumption q i b i b i + p γ i

Ui(qi; bi) Problem Setting The energy price p is given. The energy total cost is π i (q i ) = pq i The payoff function without DR is defined as U i (q i ; b i ) = G(q i ; b i ) π i (q i ) b i is the rational decision (optimal solution) of U i (q i ; b i ) q i b i Let p 2 be the rebate price. E.g. the incentive could be p 2 ( ˆb i q i ) +. The superscript ˆ means declared information. ˆq i is reported energy consumption under DR. r i is a binary variable that indicates if user i is called to participate in DR. 6

7 Contract and consumer s problem (1/2) 1. Aggregator announces p and p 2 (price and incentive)

7 Contract and consumer s problem (1/2) 1. Aggregator announces p and p 2 (price and incentive) 2. Consumers report ˆq i and ˆb i (information declaration)

7 Contract and consumer s problem (1/2) 1. Aggregator announces p and p 2 (price and incentive) 2. Consumers report ˆq i and ˆb i (information declaration) 3. Aggregator defines r i (random selection) 4. Consumers decide q i (demand response event) Questions: What is the declared information that maximizes the consumer benefit? What is the uncertainty that user faces?

8 Contract and consumer s problem (2/2) Let π r i i,3 ( ˆb i, ˆq i, q i ) be the new aggregator payment scheme: π ri i,3 ( ˆb i, ˆq i, q i ) = (non-called) r i = 0 p max( ˆb i, q i ) buy the baseline or pay the consumption r i = 1 (called) pq i p 2 ( ˆb i q i ) + + p 2 q i ˆq i energy cost incentive payment deviation penalty

Contract and consumer s problem (2/2) Let π r i i,3 ( ˆb i, ˆq i, q i ) be the new aggregator payment scheme: π ri i,3 ( ˆb i, ˆq i, q i ) = (non-called) r i = 0 p max( ˆb i, q i ) buy the baseline or pay the consumption r i = 1 (called) pq i p 2 ( ˆb i q i ) + + p 2 q i ˆq i energy cost incentive payment deviation penalty The optimization problem: [ ˆb i, ˆqi, q i ] = arg max ˆb i, ˆq i,q i {0,q max,i } J i = E(G(q i ; b i ) π r i i,3 ( ˆb i, ˆq i, q i )) The contract is settled as a two-stage procedure: Stage 1) Given the prices p and p 2, each consumer reports ˆb i and ˆq i to aggregator. Stage 2) Aggregator determines which users are called by means of the variable r i. Each agent decides his actual energy consumption q i. 8

9 Consumer optimal behavior When not called for the DR event, the optimal user response, given his previous declared information, is described by the following theorem: Theorem The optimal consumption qi,r i for the signal r i = 0 of a participant consumer in the proposed contract is: b i 0 ˆb i b i strategy A qi,r i =0 = ˆb i b i < ˆb i b i + p/γ i strategy B b i + p/γ i b i + p/γ i < ˆb i q max,i strategy C

10 Consumer optimal behavior When called for the DR event, the optimal user response, given his previous declared information, is described by the following theorem: Theorem The optimal consumption qi,r i for the signal r i = 1 of a participant consumer in the proposed contract is: qi,r i =1 = b i b i ˆb i q max,i, b i ˆq i ˆb i q max,i strategy U ˆq i b i p 2 γ i ˆb i q max,i, b i 2p 2 γ i ˆq i ˆb i b i strategy V ˆq i α ˆb i b i p 2 γ i, b i 2p 2 γ i ˆq i b i p 2 γ i (b i p 2 γ i ) + b i 2p 2 γ i ˆb i α, b i 2p 2 γ i ˆq i ˆb i b i p 2 γ i (b i p 2 γ i ) + 0 ˆb i b i 3p 2 2γ i, 0 ˆq i ˆb i b i 2p 2 γ i (b i 2p 2 γ i ) + b i 3p 2 2γ i ˆb i q max,i, 0 ˆq i b i 2p 2 γ i with α = γ i b 2 i 2p 2 γ i b i ˆq i p 2 b i + γ i ˆq i 2 2p 2 + 2 ˆq i + p 2 2γ i. strategy W strategy Y strategy Z strategy X

11 Consumer optimal behavior Knowing the optimal consumer responses to the random signal r i, the best strategy in the first stage is to report the baseline and reduction level of the following theorem: Theorem Given qi,r i from Theorems 1 and 2, then the optimal reports ˆb i and ˆqi are: { pri p 2 ˆb i = γ i (1 p ri ) + b i 0 p ri p p 2 +p p q max,i p 2 +p p r i 1 ( ˆq i = b i p ) 2 γ i +

12 Consumer optimal behavior A user decides to participate in the program if his profit (net benefit) is greater, or at least equal, to what he gets when not participating. From the previous theorems, the expected profit of the consumer is: Collorary The optimal expected profit J is: bi 2γ i 2 + pr i p2 2 2γ i (1 p ri ) 0 p ri p p 2 +p Ji = p 2 2γ i + b i p pq max,i + b2 i γ i 2 p2 p ri 2γ i + p2 2 pr i 2γ i p b i p ri b i p 2 p ri + pq max,i p ri + p 2 q max,i p ri p 2 +p p r i 1

13 Contract properties Individually rational (voluntary participation): a user that participates in this approach obtains a profit at least as good as he does not signing the DR contract. Incentive compatibility on the reported energy consumption under DR: a consumer informs the truthful consumption under DR according to his preferences. Asymptotic incentive compatibility on the reported baseline: as the probability of call tends to zero, the consumer s optimal strategy is to declare ˆb i = b i.

14 Numerical case study Consumer s decision problem [ ˆb J i = E(G(q i ; b i ) π ri i,3 ( ˆb i, ˆqi, qi ] = arg max i, ˆq i, q i )) actual consumption declared reduction capacity reported baseline What are the consumer s optimal decisions under this DR contract? Simulation info: The retail price is p = 0.26 $/kwh. True baseline is b i = 8 kwh. The incentive/penalty price is p 2 = 0.3 $/kwh. The marginal utility is γ i = 0.05 $/kwh 2. The maximum allowable consumption is q max,i = 16 kwh. A Monte Carlo simulation is performed with 1000 realizations of r i for each value of probability.

Results: optimal consumer s choice (p ri of call) is the probability Consumer's optimal choices (kwh) 16 14 12 10 8 6 ˆb i q i,ri=0 ˆq i q i,ri=1 4 2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 p ri Aggregator should call for DR to users with a probability of call between zero and the threshold one to limit gaming opportunities (asymptotic truthfulness for baseline). An agent declares what he is willing to reduce according to his true preferences ˆq i = q i,r i =1 irrespective of the probability of call. 15

Results: Percentage of gaming limitation on the reported baseline (p ri is the probability of call) 2 1.9 1.8 1.7 1.6 ˆbi/bi 1.5 1.4 1.3 1.2 1.1 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 p ri For instance, an aggregator calls a group of agents with a probability of call equals to p ri = 0.1 then rational consumers have incentives to overreport the baseline by an 11%. 16

17 Results: Optimal expected profit of a consumer (p ri probability of call). is the 5 4.5 DR No DR Optimal expected profit ($) 4 3.5 3 2.5 2 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 p ri The user s benefit when he participates in this contract is at least as good as when he does not join in the incentive-based DR program (voluntary participation).

Results: Threshold probability of call (p ri of call) is the probability 1 0.9 Threshold probability of call 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0 0.5 1 1.5 2 2.5 3 p 2 /p If p = p 2, the critical point is 0.5 of likelihood. The threshold probability is designed by the aggregator through the selection of prices. 18

19 Conclusions DR contract was proposed which induces voluntary participation and truthfulness based on the probability of call. The main goal of the aggregator is to call a subset of users that meets the probability criterion. A contract for incentive-based DR based on probability of call enables to limit the gaming opportunities. Advantages: no computation by aggregator, information exchange in terms of energy, easy to understand by agents, implementable.

20 References I Chao, H., 2011. Demand response in wholesale electricity markets: the choice of customer baseline. Journal of Regulatory Economics 39 (1), 68 88. Severin Borenstein, 2014. Peak-Time Rebates: Money for Nothing? URL http://www.greentechmedia.com/articles/read/ Peak-Time-Rebates-Money-for-Nothing Vuelvas, J., Ruiz, F., 2017. Rational consumer decisions in a peak time rebate program. Electric Power Systems Research 143, 533 543.

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