Firming Renewable Power with Demand Response: An End-to-end Aggregator Business Model Clay Campaigne joint work with Shmuel Oren November 5, 2015 1 / 16
Motivation Problem: Expansion of renewables increases uncertainty and intermittency Uncertainty and intermittency must be managed Currently, system operator buys more reserves, costs are socialized Policy Solution: Make renewable producers pay imbalance prices Incentivize production of more valuable and controllable power Incentivize abatement of pollution and GHGs, not intermittency Business Model: Owner of stochastic power resource signs contracts, as an Aggregator, with Demand Response participants to firm up its net production level Aggregator co-optimizes a day-ahead offer quantity together with a DR dispatch policy 2 / 16
The Aggregator s Problem Random variables: p : Day Ahead (DA) price a, b : Real time (RT) overproduction and shortfall prices s : RT VER realization, wind θ τ : Ex post valuation for each increment type τ Control policy variables q : ω DA q(ω DA ) 0 : DA offer quantity DR : ω RT DR(ω RT ) 0 : DR dispatch quantity T is determined by DR, using contract theory, explained below max J EA(q, DR, T) q,dr,t overproduction = max E [ { }} { p,a,b,s p q + a( DR + s q) + b( q,dr,t Day-ahead revenue shortfall { }} { q DR s) +] T payment to DR 3 / 16
Information Structure / Order of Events overproduction max E[ { }} { p q + a( DR + s q) + b( q,dr,t Day-ahead revenue shortfall { }} { q DR s) +] T payment to DR 0 Ex ante (EA) stage: aggregator knows probability distributions, signs contracts with DR participants 1 Day ahead (DA) stage: aggregator learns p, possibly a and b. Aggregator chooses q: how much to commit day ahead. 2 Real time (RT) stage: imbalance prices a and b, and wind supply s, are realized. Aggregator decides who to curtail. 3 Ex post (EP) stage: increments realize their valuations and make consumption decisions 4 / 16
DR Population and Curtailment Policy DR population comprises increments of consumption capacity Each increment indexed by private information τ, realizes ex post valuation for energy θ $/kwh First-order stochastic dominance: τ F(θ τ, ω RT) < 0 Higher types have higher distributions over valuations Merit order curtailment: cutoff type policy ˆτ(ω RT ) Type τ is curtailed in event ω RT if τ ˆτ(ω RT ) 5 / 16
DR production DR(ˆτ(ω RT ); ω EP ) ˆτ(ωRT ) τ 1{τ : θ(τ, ω EP ) [R, H]}g(τ) dτ, E [ ] ˆτ(ωRT ) DR ωrt = Pr{θ(τ, ω EP ) [R, H]} ω RT }g(τ) dτ ˆτ(ωRT ) τ τ y(τ, ω RT ) g(τ) dτ } {{ } expected yield 6 / 16
DR Utility Model and Determination of Payment, T Contracting process assigns a curtailment status to each type: k(τ, ω RT ) 1{τ ˆτ(ω RT )} Net ex post option value if curtailed: L(θ) (θ H) + (θ R) + 0 Expected net option value if curtailed: z(τ, ω RT ) Θ L(θ)f(θ τ, ω RT) dθ 0 Direct mechanism increment reports its type as τ, receives curtailment function k( τ, ) and payment t( τ) reporting value U( τ τ) Ω RT z(τ, ω RT ) k( τ, ω RT ) dp(ω RT ) net expected utility U( τ τ) + t( τ) Revelation principle: any contracting equilibrium k(τ, ), t(τ) τ can be implemented as a direct revelation mechanism Incentive compatibility (IC): τ arg max τ {U( τ τ) + t( τ)} This entails linking constraints across types, but our assumptions (FOSD and merit order) guarantee them Individual rationality (IR): u(τ) U(τ τ) + t(τ) 0 7 / 16
Payment required to implement curtailment policy (IC) and (IR) imply: increment is paid the reservation utility of the highest curtailed type in every scenario it s curtailed t(τ) = z(ˆτ(ω RT ), ω RT )k(τ, ω RT ) dp(ω RT ) Ω RT T τ t(τ)g(τ) dτ τ Write T as linear functional against g(τ) dτ dp(ω RT ): T = ψ(τ, ω RT )g(τ) dτ dp(ω RT ) ψ(τ, ω RT ) = z(τ, ω RT ) + G(τ) g(τ) τ z(τ, ω RT): virtual payment to τ This attributes the cost of marginally curtailing type τ to τ itself (no reference to ˆτ). Useful for first order condition on ˆτ(ω RT ). :: whiteboard :: 8 / 16
Solving end-to-end problem: Example 1 Example 1: Idiosyncratic valuation shocks: θ(τ, ω EP ) = m(τ, ω RT ) + ɛ Law of Large Numbers: Expected DR a.s. = Realized DR c(τ) ψ(τ)/y(τ) = cost of DR per unit yield 9 / 16
Solving end-to-end problem: Example 1 Example 1 First order condition for ˆτ (ω RT ): J RT ( b q = p q + a ˆτ τ ( ˆτ τ ) + y(τ)g(τ) dτ + s q ) + ˆτ y(τ)g(τ) dτ s ψ(τ)g(τ) dτ ; τ First order condition for ˆτ(ω RT ): c(ˆτ ) = b if DR(ˆτ ) + s < q c(ˆτ ) = a if DR(ˆτ ) + s > q ˆτ = DR 1 (q s) otherwise 10 / 16
Solving end-to-end problem: Example 1 First order condition for q : p = E[a DR(ˆτ ) + s > q ] Pr{DR(ˆτ ) + s > q } + E[b DR(ˆτ ) + s < q ] Pr{DR(ˆτ ) + s < q } + E[c(ˆτ ) DR(ˆτ ) + s = q] Pr{DR(ˆτ ) + s = q }. 11 / 16
Example 2: parameterized uniform distributions g(τ) = d dτ G(τ) = N1 τ [τ,τ] s Uniform[0, s] θ τ (ω RTD ) τ R = τ H = τ, ω RTD a and b known day ahead z(τ) = τ τ z(τ) = 1 G(τ)/g(τ) = τ ψ(τ) = 2τ y(τ) = 1 c(τ) = 2τ J RT = p q + a ( s + νˆτ DR(ˆτ) q ) + b ( q s νˆτ ) + ˆτ 2 T 12 / 16
Solution to Example 2 13 / 16
Graphs Figure: Supply curves 14 / 16
The contract menu in Example 2 Assume market statistics to obtain curtailment probabilities p Uniform[10, 100] a = (1 δ)p b = (1 + δ)p δ Uniform[0.1, 0.9] Figure: Menu as function of Type Figure: Payment as function of curtailment probability 15 / 16
Conclusions We have embedded a contracting problem into a newsvendor style problem The optimal ex ante contract is equivalent to waiting until ω RT realizes, and making a uniform offer to all increments, equal to reservation price of marginally curtailed increment follows from FOSD and merit order assumptions: no increment would want to switch service plans with another after the realization of ω RT Since ψ(τ) > z(τ), the aggregator has an incentive to under-purchase DR, in order to suppress the price Standard monopsony result In future work we will solve more general instances approximately, using simulation / Monte Carlo Questions? 16 / 16