Duration and deadline differentiated demand: a model of flexible demand

Transcription

1 Duration and deadline differentiated demand: a model of flexible demand A. Nayyar, M. Negrete-Pincetić, K. Poolla, W. Chen, Y.Mo, L. Qiu, P. Varaiya May, / 21

2 Outline Duration-differentiated (DD) service Aggregate demand and supply Social optimum Market implementation Different deadlines 2 / 21

3 Why flexible demand? Today power sources are controlled to match variable demand Renewable sources (wind, solar) are variable and uncontrollable Better to match variability by shaping flexible demand than scheduling reserve generation to reduce emissions and reserve capacity, Issues: Identify different kinds of flexible tasks (EV, HVAC, preemptible, ) Design incentives so consumers offer their task flexibility Match flexible demand to variable supply 3 / 21

4 Identify flexible demand Differentiate electricity service to support different kinds of flexibility Priority pricing [Chao and Wilson, 1987]: urgent tasks pay more Interruptible electricity [Tan and Varaiya, 1992]: more reliable service costs more Demand response [FERC, 2005]: pays for demand reduction Deadline-differentiated [Bitar and Low, 2012]: deferrable tasks pay less Other proposals: TOU, RTP Duration-differentiated service: preemptible tasks of different duration 4 / 21

5 Duration-differentiated service DD service (l, h): provides l kw for any h out of T time slots; so same service can be produced in many ways flexible: 5 / 21

6 Aggregate demand K tasks; consumers x [0, 1]. Task i of x needs l i (x) kw for h i (x) slots, i.e. service (l i (x), h i (x)); l i (x) 0, h i (x) {0,, T }. So x needs i 1{h i(x) t}l i (x) kw for t slots. Aggregate demand is d t = 1 0 K 1{h i (x) t}l i (x)dx kw for t slots i=1 Note d 1 d T. s d s is the demand duration curve. So t t s=1 d s is concave. 6 / 21

7 Interpretation of continuum of consumers Fraction of the population that carries out task k = 1,, K is C k = 1 0 1{l k (x)h k (x) > 0}dx Average energy spent on this task is 1 0 l k(x)h k (x)dx C k 7 / 21

8 Aggregate supply Given renewable power supply q t kw in slot t = 1,, T. Reorder q to get p so that p 1 p 2 p T. s p s is the generation duration curve. t t s=1 p s is also concave. 8 / 21

9 Shaping demand to match supply Say that p is adequate for d if demand for t slots supply: d i i=t p i, t = 1,, T i=t An allocation A i (x, t) {0, 1} serves task i of consumer x if and only if A i (x, t) = 1. It is feasible if for all x, i, t 1 0 A i (x, t) = h i (x) t=1 K l i (x)a i (x, t)dx p t i=1 Theorem There is a feasible allocation iff p is adequate for d. 9 / 21

10 LLDF allocation Suppose q(t) is supply in slot t. Longest leftover duration first (LLDF) allocation A i (x, t) (i) In slot 1 use q(1) to serve i, x tasks with longest duration h i (x) (ii) In slot t calculate leftover duration (LD) h i (x) t 1 s=1 A i(x, s). Use q(t) to serve (i, x) tasks with longest LD. Theorem LLDF is feasible iff p is adequate for d. 10 / 21

11 Benefits If supply is inadequate we must choose an allocation that maximizes (benefits - cost). Since cost = 0, we model benefits. Benefit to consumer x from tasks (l(x), h(x)) = ((l 1 (x), h 1 (x)),, (l K (x), h K (x))) is U(x, l(x), h(x)) U 0, continuous, bounded, and U(x, 0, 0) = 0. U need not be convex. The social benefit of feasible allocation x (l(x), h(x)) is W = 1 0 U(x, l(x), h(x))dx Optimal allocation maximizes W subject to feasibility. 11 / 21

12 Optimal allocation The optimal allocation is solution of optimal control problem: max W = l(x),h(x) s.t. d i = 1 0 d i i=t 1 0 U(x, l(x), h(x))dx p i, all t i=t K 1{h k (x) t}l k (x)dx k=1 l k (x) 0, h k (x) {1,, T } Theorem An optimal allocation exists. Requires knowing everyone s utility function and the renewable power supply. We need a decentralized approach. 12 / 21

13 Market for DD services The DD services market consists of Prices: π h per kw for duration h = 1,, T ; so price of service (l, h) is π h l. Consumers: x maximizes net benefit by purchasing (l (x), h (x)) arg max l,h U(x, l, h) K π h h=1 k=1 Supplier: uses supply (p 1,..., p T ) to produce services L = {(L h, h), h = 1,, T } to maximize revenue L arg max π h L h L s.t. i=t h=i h L h p i, t = 1, T i=t 1{h k = h}l k 13 / 21

14 Equilibrium {π, (l (x), h (x)), L } is an equilibrium if (l (x), h (x)) maximizes benefits for x L maximizes revenue of supplier market clears, i.e. demand for duration h less than its supply, 1 0 K 1{h k (x) = h}l k (x)dx L h. k=1 Theorem There exists an equilibrium. Every equilibrium maximizes social benefit. 14 / 21

15 Equilibrium Characteristics (1) π h1 +h 2 π h1 + π h2. (2) Price discovery Supply drives service contracts h L h offered inelastically to consumers, resulting in prices {π h }. Adjust supplies {L h } to increase revenues. This can lead to monopoly pricing, depending on supply competition. 15 / 21

16 Comparison with RTP (1) There are real-time prices r t per kw in slot t such that π h is the least cost for 1 kw for h slots, i.e, π 1 = min{r t }, π 2 = min{r s + r t, s < t}, etc So RTP is more general, as expected. (2) But RTP discovery is more difficult. Suppose per kw prices r 1,, r T are announced at time 0. A consumer wants services (l 1, h 1 ),, (l K, h K ). The demand for task k will be concentrated on cheapest h k slots (t 1,, t hk ) arg min{r t1 + + r thk t 1 < < t hk } This is a highly discontinuous demand function. That is it is difficult to extract inter-temporal demand flexibility with RTP. 16 / 21

17 Two deadlines Tasks 1,, K 1 must be completed before deadline T 1, tasks K 1 + 1,, K must be completed before T. Available power in slot t is p t. Reorder time so p 1 p T 1, p T 1 +1 p T. Allocation A i (x, t) {0, 1} is feasible if for all x, i, t 1 0 A i (x, t) = h i (x), t=1 A i (x, t) = 0, t > T 1, i K 1 K l i (x)a i (x, t)dx p t i=1 17 / 21

18 Adequacy Aggregate demand for t slots from tasks 1,, K 1 is d 1 t = 1 K 1 0 1{h i (x) t}l i (x)dx kw for t slots i=1 and demand for t slots from tasks K 1 + 1,, K is d 2 t = 1 0 K i=k {h i (x) t}l i (x)dx kw for t slots Theorem p is adequate for demands d 1, d 2 with deadlines T 1, T iff T 1 i=t 1 d 1 i + i=t 1 +t 2 d 2 i T 1 i=t 1 p i + i=t 1 +t 2 p i, 0 < t 1 T 1, 0 < t 2 T T 1 Extension for many deadlines T 1 < T 2 < < T. 18 / 21

19 Market implementation The DDD services market consists of Prices: π 1 h, π2 h per kw of service of duration h and deadline T 1, T Consumers: x maximizes net benefit by purchasing (l (x), h (x)) arg max l,h πh 2 h=1 k=k 1 +1 U(x, l, h) K T 1 K 1 πh 1 h=1 k=1 1{h k = h}l k Supplier: uses supply (p 1,..., p T ) to produce services L = {(L h, h), h = 1,, T } to maximize revenue L arg max π h L h L s.t. i=t h=i h L h p i, t = 1, T i=t 1{h k = h}l k 19 / 21

20 Equilibrium Theorem There exist {πh 1, π2 h ; l (x), h (x), L } such that (l (x), h (x)) maximizes net benefits Supplier maximizes revenues Market clears Properties of equilibrium πh 1 π2 h : shorter deadline service is more expensive. π i (h 1 + h 2 ) π i (h 1 ) + π i (h 2 ): can convert (h 1 + h 2 )-service into h 1 -service and h 2 -service. Generalization Extensions for DD-services with different start times and different deadlines, i.e. need for h slots during [S i,, T i ], i = 1,, K. 20 / 21

21 Concluding Remarks New types of energy services can be used to tailor demand to variable renewable supply. Services that incorporate demand flexibility can provide the right incentives for consumers. Duration and demand-differentiated services may offer a useful model. Future work: uncertainty in supply; price discovery. Thank you 21 / 21