The Sharing Economy for Smart Grid Chenye Wu IIIS, Tsinghua University December 20, 2016 Chenye Wu, IIIS Intro to CS December 20, 2016 0 / 25
Shared Electricity Services � The New Sharing Economy − cars, homes, services, ... − business model: exploit underutilized resources − huge growth: $ 40B in 2014 → $ 110B in 2015 � What about the grid? − what products/services can be shared? − what technology infrastructure is needed to support sharing? − what market infrastructure is needed? − is sharing good for the grid? Chenye Wu, IIIS Intro to CS December 20, 2016 1 / 25
Three Opportunities � ex 1: Sharing Unused Energy in Storage − firms face ToU prices − install storage C, excess is shared � ex 2: Sharing Distributed Generation − homes install PV − excess generation is sold to others − net metering isn’t really sharing ... price of excess is fixed by utility, not determined by market condn � ex 3: Sharing Demand Flexibility − utilities recruit flexible customers − flexibility can be modeled as a virtual battery − battery capacity is shared Chenye Wu, IIIS Intro to CS December 20, 2016 2 / 25
Challenges for Sharing in the Electricity Sector � Power tracing electricity flows according to physical laws undifferentiated good cannot claim x KWh was sold by i to firm j � Regulatory obstacles early adopters will be behind-the-meter single PCC to utility firms can do what they wish outside purvue of utility � Paying for infrastructure fair payment to distribution system owners many choices: flat connection fee, usage proportional charge, ... Chenye Wu, IIIS Intro to CS December 20, 2016 3 / 25
Sharing Electricity Storage Joint work with Dileep Kalathil, Pravin Varaiya, Kameshwar Poolla Chenye Wu, IIIS Intro to CS December 20, 2016 4 / 25
Set-up Firm 1 Aggregator Grid Firm 2 . . . Firm n − n firms, facing time-of-use pricing − Ex: industrial park, campus, housing complex − firm k invests in storage C k for arbitrage − unused stored energy is traded with other firms − AGG manages trading & power transfer − collective deficit is bought from Grid Chenye Wu, IIIS Intro to CS December 20, 2016 5 / 25
ToU Pricing and Storage power Energy Y Energy X price π h π ℓ off-peak peak − random consumption X , Y − F ( x ) = CDF of X − value of storage: firm can move some purchase from peak to off-peak Chenye Wu, IIIS Intro to CS December 20, 2016 6 / 25
Consumption Model � Energy demand for firm k is random X k in peak period, CDF F k ( · ) Y k in off peak period � Collective peak period demand � X c = X k , CDF F c ( · ) k Chenye Wu, IIIS Intro to CS December 20, 2016 7 / 25
Prices and Arbitrage capital cost of storage π s amortized per day over battery lifetime peak-period price π h off-peak price π ℓ π δ difference π h − π ℓ � Comments − today π s ≈ 20 ¢ , but falling fast − need π δ > π s to justify storage investment for arbitrage alone − rarely happens today, but many more opportunities tomorrow ... − ex: PG&E A6 tariff ... π δ ≈ 25 ¢ > π s = 20 ¢ � Arbitrage constant γ = π δ − π s γ ∈ [0 , 1] π δ Chenye Wu, IIIS Intro to CS December 20, 2016 8 / 25
Assumptions 1 Firms are price-takers for ToU tariff ... consumption is not large enough to influence π h , π ℓ 2 Demand is inelastic ... savings from using storage do not affect statistics of X k , Y k 3 Storage is lossless, inverters are perfectly efficient temporary assumption 4 All firms decide on their storage investment simultaneously temporary assumption Chenye Wu, IIIS Intro to CS December 20, 2016 9 / 25
No Sharing: Firm’s Decision � Daily cost components for firm k π s C k amortized cost for storage π h ( X k − C k ) + peak period: use storage first, buy deficit from grid π ℓ min { C k , X k } off-peak: recharge storage � Expected cost π h ( X k − C k ) + + π ℓ min { C k , X k } � � J k ( C k ) = π s C k + E 1 CDF F k ( x ) Theorem Stand alone firm γ Optimal storage investment C ∗ = arg min C k J k ( C k ) k 0 F − 1 x = k ( γ ) 0 C ∗ k Chenye Wu, IIIS Intro to CS December 20, 2016 10 / 25
Discussion � Without sharing, firms make sub-optimal investment choices: − firms may over-invest in storage! not exploiting other firms storage, if γ is large − or under-invest! not taking into account of profit opportunities, if γ is small � More precisely: − optimal storage investment for collective � c = F − 1 C ∗ ( γ ) , X k = X c ∼ F c ( · ) c k − total optimal investment for stand-alone firms � k C ∗ k − under-investment C ∗ c > � k C ∗ k c < � over-investment: C ∗ k C ∗ k Chenye Wu, IIIS Intro to CS December 20, 2016 11 / 25
Example: Two Firms − X 1 , X 2 ∼ U [0 , 1], independent k = F − 1 − individual investments: C ∗ k ( γ ) = γ c = F − 1 − collective investment: C ∗ ( γ ) where X c = X 1 + X 2 c √ 2 γ � if γ ∈ [0 , 0 . 5] 2 − √ 2 − 2 γ C ∗ c = if γ ∈ [0 . 5 , 1] Storage capacity C ∗ c C ∗ 1 + C ∗ 2 γ Chenye Wu, IIIS Intro to CS December 20, 2016 12 / 25
Sharing Storage � Firm k has surplus energy in storage ( C k − X k ) + − can be sold to other firms who might have a deficit − willing to sell at acquisition price π ℓ � Supply and demand k ( C k − X k ) + − collective surplus: S = � k ( X k − C k ) + − collective deficit: D = � � Spot market for sharing storage − if S > D firms with surplus compete energy trades at the price floor π ℓ − if S < D firms with deficit must buy some energy from grid energy trades at price ceiling π h Chenye Wu, IIIS Intro to CS December 20, 2016 13 / 25
Spot Market � Market clearing price � π l if S > D π eq = π h if S < D � Random, depends on daily market condns price price demand demand schedule schedule equil supply price equil schedule π h π h price π ℓ π ℓ energy energy D S D S Chenye Wu, IIIS Intro to CS December 20, 2016 14 / 25
Firm’s Decisions Under Sharing � Expected cost for firm k J k ( C k | C − k ) = π s C k + π l C k + E [ π eq ( X k − C k ) + − π eq ( C k − X k ) + ] � Storage Sharing Game − players: n firms, decisions: storage investments C k − optimal investment C ∗ k depends on the investment of other firms − non-convex game � Expected cost for collection of firms � k J k − simplifies to: J c ( C c ) = π s C c + E [ π h ( X c − C c ) + + π ℓ min { C c , X c } ] − like a single firm without sharing � Social Planner’s Problem c = F − 1 min C c J c ( C c ) solution: C ∗ ( γ ) c Chenye Wu, IIIS Intro to CS December 20, 2016 15 / 25
Firm’s Decisions Under Sharing Theorem Assume the existence of Nash equilibrium. (a) Storage Sharing Game admits unique Nash Equilibrium (b) Optimal storage investments: � C ∗ k = E [ X k | X c = C c ] , where C c = C ∗ k , F c ( C c ) = γ k (c) Nash equilibrium supports the social welfare (d) Equilibrium is coalitional stable – no subset of firms will defect (e) Nash equilibrium is in the core of the corresponding cooperative game Chenye Wu, IIIS Intro to CS December 20, 2016 16 / 25
Sufficient Condn for the Existence of Equilibrium Theorem Assume technical alignment condn: E [ X k | X c = β ] is non-decreasing in β , then the Storage Sharing Game admits unique Nash Equilibrium. Natural interpretation of the technical alignment condn: (a) The expected demand X k of firm k increases if the total demand X c increases. (b) This is not unreasonable. (c) For example, if X k and X c are jointly Gaussian, then this condition holds if they are positively correlated. Chenye Wu, IIIS Intro to CS December 20, 2016 17 / 25
Example: Violating the Technical Condn − Let W be a random variable uniformly distributed on [0 , 10]. − Define the peak period consumption for two firms by X 1 = W sin 2 ( W ); X 2 = W cos 2 ( W ) . − X c = X 1 + X 2 = W . − E [ X 1 | X 1 + X 2 = β ] = β sin 2 ( β ). Non-increasing in β . 3 10 8 2 ) J ( C 1 , C ∗ 2 6 X 2 4 1 2 0 0 2 4 6 8 10 0 2 4 6 8 C 1 X 1 Chenye Wu, IIIS Intro to CS December 20, 2016 18 / 25
Competitive Equilibrium? − Our problem formulation above does not assume a perfect competition model. − Indeed, under perfect competition, the celebrated welfare theorems assure existence of a unique Nash equilibrium. − In our analysis, we allow firm k to take into account the influence its investment decision C k has on the statistics of the clearing price π eq . Λ 1 C ∗ ≈ m + c − 1 T m ) E [ X ] = m , cov( X ) = Λ = ⇒ 1 T Λ 1 ( C ∗ − This is a Cournot model of competition, under which Nash equilibria do not necessarily exist. Chenye Wu, IIIS Intro to CS December 20, 2016 19 / 25
Lossy Storage � More realistic storage model − charging efficiency η i ≈ 0 . 95 − discharging efficiency η o ≈ 0 . 95 − daily leakage ǫ (holding cost) � Storage parameters modify arbitrage constant Theorem Optimal investment of collective is a = 1 where γ = π h η o η i − π ℓ − η i π s · F − 1 C ∗ a ( γ ) , π h η o η i − π ℓ (1 − ǫ ) η o Chenye Wu, IIIS Intro to CS December 20, 2016 20 / 25
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