Supplementary Material for Strategic Data Center Load Shifting: Implications for Market Efficiency and Transmission Value

Supplementary Material For the Manuscript: “Strategic Data Center Load Shifting: Implications for Market Efficiency and Transmission Value” Aron Brenner, Line Roald, Saurabh Amin Proof of Proposition 1 Proposition 1. Consider a positive load shift δ > 0from Ato B. Letting pgA, pgBrespectively denote the pre-shift dispatch levels of marginal generators gAand gB, two cases arise depending on which zone’s marginal generator changes first. Case 1. (Marginal generator at Bchanges first). Suppose PgB−pgB< pgA. Let τa:= PgB−pgBbe the threshold where generator gBreaches capacity. Then, for δ≤τ1: ∆G(δ) = (∆P(δ), δ ∈[0, τ1), ∆P(δ)−κ1, δ =τ1, where the discontinuity κ1is given by κ1:= (CgB+1 −CgB)(xB+τ1)>0. Case 2. (Marginal generator at Achanges first). Suppose pgA< PgB−pgB. Let τ2:= pgAbe the threshold where generator gAis fully de-committed. Then, for δ≤τ2: ∆G(δ) = (∆P(δ), δ ∈[0, τ2), ∆P(δ) + κ2, δ =τ2, where the discontinuity κ2is given by κ2:= (CgA−CgA−1)(xA−τ2)>0. Proof. Since Aand Bare independent (with F= 0), each zone’s economic dispatch problem can be solved separately. In each zone, generators are dispatched in increasing order of marginal cost until demand is met. For a shift δ, zone Aserves xA−δand zone Bserves xB+δ. As long as the marginal generators remain the same, dispatch in each zone changes only through a one-for-one adjustment of their marginal unit. Thus, ∆G(δ)=(CgB−CgA)δ= (λB(δ)−λA(δ))δ= ∆P(δ), so system and consumer objectives are aligned for sufficiently small δ. We now examine the basis-change points at which a marginal generator switches. 1 Case 1: Generator at Bhits capacity first. At δ=τ1:= PgB−pgB, generator gBbecomes fully dispatched and gB+1 becomes marginal. Thus, λB(τ1) = CgB+1 . The change in system generation cost is given by: ∆G(τ1)=(CgB−CgA)τ1. while the flexible consumer cost changes by: ∆P(τ1) = CgA(xA−τ1) + CgB+1 (xB+τ1)−(CgAxA+CgBxB) = ∆G(τ1)+(CgB+1 −CgB)(xB+τ1). Rearranging gives the stated discontinuity: ∆G(τ1)=∆P(τ1)−κ1. Case 2: Generator at Afully de-commits first. At δ=τ2:= pgA, generator gAde-commits and gA−1becomes marginal, so λA(τ2) = CgA−1. The change in system generation cost is given by: ∆G(τ2) = (CgB−CgA)τ2, while the flexible consumer cost changes by: ∆P(τ2) = CgA−1(xA−τ2) + CgB(xB+τ2)−(CgAxA+CgBxB) = ∆G(τ2)−(CgA−CgA−1)(xA−τ2). Rearranging yields: ∆G(τ2)=∆P(τ2) + κ2. Proof of Theorem 1 Theorem 1 (Externalities at marginal basis changes).Consider the two regimes in Proposition 1. Case 1 (Marginal generator at Bchanges first). If PgB−pgB< pgA, then for all δ∈[0, τ1]with τ1=PgB−pgB, ∆G(δ)≤∆P(δ). Thus, any load shift that reduces P(δ)also reduces G(δ); misalignment cannot occur. 2 Case 2 (Marginal generator at Achanges first). If pgA< PgB−pgB, then a load shift that reduces P(δ)increases G(δ)if and only if: (i) CgB> CgA, (ii) (CgA−CgA−1)(xA−τ2)>(CgB−CgA)τ2, where τ2=pgA. Proof. Case 1: Generator at Bhits capacity first. From Proposition 1, we have ∆G(δ) = (∆P(δ), δ ∈[0, τ1), ∆P(δ)−κ1, δ =τ1, with κ1>0. Hence ∆G(δ)≤∆P(δ)∀δ∈[0, τ1], with strict inequality at δ=τ1. In particular, for the basis-change shift δ=τ1, ∆P(τ1)<0 =⇒∆G(τ1)<0. Thus, any load shift that reduces the flexible consumer cost also reduces system cost; there is no misalignment at δ=τ1. Case 2: Generator at Afully decommits first. From Proposition 1, for δ < τ2, ∆G(δ)=∆P(δ), and at the basis-change shift δ=τ2, ∆G(τ2)=(CgB−CgA)τ2, ∆P(τ2)=(CgB−CgA)τ2−(CgA−CgA−1)(xA−τ2). A flexible consumer is incentivized to choose δ=τ2if and only if this shift reduces its procurement cost: ∆P(τ2)<0⇐⇒ (CgA−CgA−1)(xA−τ2)>(CgB−CgA)τ2. Likewise, shifting increases system generation cost if and only if ∆G(τ2)>0⇐⇒ CgB> CgA. Combining the two, strategic misalignment at the basis-change shift occurs exactly when shifting by δ=τ2is privately beneficial yet socially costly: ∆P(τ2)<0 and ∆G(τ2)>0⇐⇒ (CgB> CgA, (CgA−CgA−1)(xA−τ2)>(CgB−CgA)τ2. This is precisely the set of conditions stated in the theorem. 3 Proof of Theorem 2 Theorem 2 (Zero bilevel marginal value of transmission).Fix initial transmission capacity F0≥0 and let δ0:= δ∗(F0)be the consumer-optimal load shift. Suppose the following conditions hold: (A1) Marginal generation costs satisfy CgA< CgA+1 < CgBand the transmission line is congested from Ato B: λA(δ0, F0) = CgA< CgB=λB(δ0, F0), f =F0. (A2) Strategic positioning is profitable: (CgA+1 −CgA) (xA−δ0)>(CgB−CgA+1 )δ0, so that gAis being dispatched at level: pgA=PgA. (A3) The marginal generator at Bhas downward headroom: pgB>0. Define the threshold transmission capacity: Γ := CgA+1 −CgA CgB−CgAxA−δ0 and the upper limit: F1:= F0+ min {Γ, α −δ0} Then for all F∈[F0, F1): (i) The transmission constraint remains binding with positive shadow price: µ+(δ∗(F), F) = CgB−CgA>0. (ii) The marginal value of transmission with respect to the system objective is zero under load shifting: −d dF G(δ∗(F), F)=0. (iii) The marginal value of transmission with respect to the flexible consumer objective is negative under load shifting: −d dF P(δ∗(F), F) = CgA−CgB<0. 4 Proof. Let us first express the stationarity, dual feasibility, and complementary slackness conditions for the economic dispatch problem: ∂ ∂pg =Cg−λi−η− g+η+ g= 0, g ∈ Gi, i ∈ {A, B} ∂ ∂f =λA−λB+µ+−µ−= 0 µ+(f−F) = 0, µ−(−f−F) = 0 η− g(−pg)=0, η+ g(pg−Pg) = 0, g ∈ GA∪ GB µ+, µ−,η+,η−>0. Together with primal feasibility, these provide sufficient conditions for optimality of the economic dispatch problem. By assumption, the optimality conditions for the economic dispatch problem with δ0, F0are satisfied by: pgA=PgA=⇒λA=CgA, η+ gA= 0, η− gA≥0 (2a) pgB>0 =⇒λB=CgB, η+ gB= 0, η− gB≥0 (2b) f=F0=⇒µ+=λB−λA, µ−= 0,(2c) pg=Pg=⇒η− g= 0, η+ g=CgA−Cg, g ∈ GA, g < gA(2d) pg= 0 =⇒η− g=Cg−CgA, η+ g= 0, g ∈ GA, g > gA(2e) pg=Pg=⇒η− g= 0, η+ g=CgB−Cg, g ∈ GB, g < gB(2f) pg= 0 =⇒η− g=Cg−CgB, η+ g= 0, g ∈ GB, g > gB(2g) Let p′ gand f′denote the optimal dispatch decision for generator gand transmission flow respectively as functions of the load shift δand transmission capacity F. Additionally, let ε > 0 be an infinitesimal increase in transmission capacity and, fixing δ0, consider a solution given by: p′ gA+1 (δ0, F0+ε) = ε=⇒λA=CgA+1 , η+ gA+1 = 0, η− gA+1 = 0 p′ gB(δ0, F0+ε) = pgB−ε=⇒λB=CgB, η+ gB= 0, η− gB= 0 f=F0+ε=⇒µ+=λB−λA, µ−= 0, pg=Pg=⇒η− g= 0, η+ g=CgA−Cg, g ∈ GA, g < gA+1 pg= 0 =⇒η− g=Cg−CgA, η+ g= 0, g ∈ GA, g > gA+1 pg=Pg=⇒η− g= 0, η+ g=CgB−Cg, g ∈ GB, g < gB pg= 0 =⇒η− g=Cg−CgB, η+ g= 0, g ∈ GB, g > gB Note that this solution satisfies primal feasibility as well as the stationarity, dual feasibility, and complementary slackness conditions given in (1). Consequently, this solution is optimal for the economic dispatch problem with (δ0, F0+ε). Increasing Fby εallows the system operator to reduce expensive dispatch at Bby εand increase dispatch at Aby ε. Since gAis already at capacity, the next-cheapest unit gA+1 enters the margin and the flexible consumer’s cost increases by: P(δ0, F0+ε)−P(δ0, F0)=(CgA+1 −CgA) (xA−δ0). Because gA+1 raises the price at A, the flexible consumer strictly prefers to restore gAas the marginal generator. This is achieved by increasing the load shift by exactly the same amount: δ∗(F0+ε) = δ0+ε. 5 Doing so reverts the solution to the pre-expansion dispatch for all variables (2) except the flow: f(δ∗(F0+ε), F0+ε) = F0+ε. Because the redispatch is exactly reversed, the price difference remains unchanged: µ+(δ∗(F), F) = λB−λA=CgB−CgA>0. Consequently, the shadow price of the transmission capacity constraint remains equal to the price difference: µ+(δ∗(F0+ε), F0+ε)λB−λA=CgB−CgA. while the marginal value of transmission to the system is zero: −d dF G(δ∗(F), F) = lim ε→0+−G(δ∗(F0+ε), F0+ε)−G(δ0, F0) ε= 0 and the marginal value of transmission to the consumer is negative: −d dF P(δ∗(F), F) = lim ε→0+−P(δ∗(F0+ε), F0+ε)−P(δ0, F0) ε =−(CgB−CgA)ε ε=CgA−CgB<0. The consumer can continue offsetting transmission expansions so long as: 1. It does not violate the flexibility limit: δ0+ (F−F0)≤α. 2. Strategic positioning at the upper capacity limit of gAremains profitable: (CgA+1 −CgA) (xA−(δ0+ (F−F0))) >(CgB−CgA+1 ) (δ0+ (F−F0)) (CgA+1 −CgA) (xA−δ0)−(CgA+1 −CgA) (F−F0)>(CgB−CgA+1 )δ0+ (CgB−CgA+1 ) (F−F0) (CgA+1 −CgA) (xA−δ0)−(CgB−CgA+1 )δ0>(CgB−CgA) (F−F0) CgA+1 xA−CgAxA+CgAδ0−CgBδ0>(CgB−CgA) (F−F0) CgA+1 −CgA CgB−CgAxA−δ0>(F−F0). Together these give F < F1, completing the proof. 6