Tradeoffs Between Ef/iciency and Risk in the Future Grid. Munther A. Dahleh

Transcription

1 Tradeoffs Between Ef/iciency and Risk in the Future Grid Munther A. Dahleh 1

2 Collaborators on Power Grid Risk QinqQing Huang: MIT Mardavij Roozbehani: MT Control Architecture Donatello Materassi Saverio Bolognani Mardavij Roozbehani: MT 2

3 Power Grid: Ef/iciency- Reliability- CP Security Efficiency: Environmental, Economic Reliability: Power outages cost US economy $80B -150B annually(1 % of GDP) CP Security: Problem of the future Outage>100MW Outages affecmng 50K or more users 3

4 Generators Utilities Consumers Transmission Distribution Real Time Demand Response The Smart Grid ISO Real Time Demand Pricing and Response Market Real Time Pricing 4

5 Ef/iciency and Reliability Efficiency and reliability at the component Level Efficiency and Reliability at the system level InformaMon and Decision Architecture Modeling and simulamon CommunicaMon and Control Architectures Market Architecture IncenMves for InnovaMon RegulaMon 5

6 Ef/icient Components Energy usage divided among sectors ResidenMal ~ 37% Commercial ~34% Industrial ~26% TransportaMon ~ (currently) small Efficient power electronic Components More effecmve lighmng (e.g., LED lighmng), air condimoning (e.g., high- efficiency inverter- driven AC), appliances (reduced standby power, higher efficiency), etc. Improved electronic controls (FACTS), sensing and communicamon (PMU), and management offer substanmal demand- side control 6

7 Ef/iciency and Reliability at System Level Robust communicamon and control architecture Caveats: Feedback (e.g., Real- Mme demand response) CoordinaMon vs Strategic (e.g., demand shib) Network effect 7

8 Price Volatility Locational Marginal Prices at PJM ISO 07-April :40 EST 8

9 Price Volatility Locational Marginal Prices at PJM ISO 07-April :45 EST 9

10 Price Volatility Locational Marginal Prices at PJM ISO 07-April :50 EST 10

11 Price Volatility Locational Marginal Prices at PJM ISO 07-April :55 EST 11

12 Price Volatility One Hour Later... 12

13 Price Volatility Locational Marginal Prices at PJM ISO 07-April :45 EST 13

14 Price Volatility Locational Marginal Prices at PJM ISO 07-April :50 EST 14

15 Price Volatility Locational Marginal Prices at PJM ISO 07-April :55 EST 15

16 Price Volatility: It does get even more dramatic... There are many examples like this. 16

17 Dynamic Retail Pricing of Electricity 1. Time of use Pricing 2. Critical Peak Pricing 3. Real Time Pricing Borenstein et al, : We conclude by advocating much wider use of dynamic retail pricing, under which prices faced by end-use customers can be adjusted frequently and on short notice to reflect changes in wholesale prices. Such price-responsive demand holds the key to mitigating price volatility in wholesale electricity spot markets. W. Hogan, : any consumer who is paying the RTP for energy is charged the full LMP for its consumption and avoids paying the full LMP when reducing consumption. Expanding the use of dynamic pricing, particularly real-time pricing, to provide smarter prices for the smart grid would be a related priority... 17

18 Real- Time Demand Response Closing the Loop wind ISO Dispatch Supply Schedule Locational marginal price Demand prediction d t Consumer Model 18

19 Stability vs. Volatility price price Deferred consumption Synchronized consumption time time 19

20 RTP Increases Volatility 20

21 Network Effect Three Bus Network Upperbound Simulation Heterogeneity amplifies the "network effects Incremental L2 Gain / Volatility "More interconnection" can lead to more volatility L12 Capacity of line 1-2 The elasticity of consumers and producers is such that the interactions at each node are very stable. Due to heterogeneity, interactions between a consumer and a producer at a different bus may be unstable or less stable. Increasing the transmission capacity between 1 and 2 amounts to interconnecting dynamical systems. 21

22 Value of Anarchy Price of Anarchy: Loss in efficiency due to strategic interacmons in contrast to a coordinamon Simple model: one agent with shibable demand and another with instantaneous demand Contrast opmmal efficient solumon to a Stackelberg game of strategic behavior A new tradeoff: CooperaMon can increase endogenous risk 22

23 Setup t Inflexible load Flexible load t+1 t+2 t+3 23

24 Model System state: Aggregate unshiftable loads Consumer arrival with shiftable load Load shifting decision: Only 1 decision maker at : the new arrival with shiftable load Split load into two periods based on

25 Problem Formulation Deadline constraints on demands: " Endogenous prices couple individual decisions: " Non-cooperative decision making: " Minimize individual cost Cooperative decision making:" Minimize aggregate cost 25

26 Solution: Strategic Symmetric Markov Perfect equilibrium in dynamic stochastic game t t+1 t+2 Overlapping type 2 consumers Flavor of Stackelberg competition

27 Solution: Strategic Symmetric Markov Perfect equilibrium in dynamic stochastic game Equilibrium strategy

28 Solution: Cooperative Bellman equation for infinite horizon average cost MDP Optimal stationary policy

29 Welfare impacts Under linear stationary policy _ Efficiency Tail probability Risk Variance Strategic Cooperative

30 Price of Anarchy: what about risk? Aggregate demand sample path! spikes small time scale large time scale Cooperative Non-cooperative 30

31 Example I: L = 2 Aggregate demand stationary distribution! Low variance spikes Y axis Linear scale Y axis Log scale Cooperative Non-cooperative

32 Tradeoff Introduce congestion pricing to adjust individual payoff (internalize the externality)" Parameterize by " Efficiency" (- variance)" " Robustness" (1- tail probability)" α Non-cooperative" é Externality ê cooperative 32

33 Mitigating Consumption Peaks: Control Architecture Agents behaving in a coopera've or synchronized manner may lead to spikes in demand Tasks are deferrable with hard deadlines MiMgaMng the frequency of consumpmon peaks can be achieved by imposing addimonal constraints on aggregate consumpmons Both hard and sob bounds How to smll keep the benefits (efficiency) of a price system? 33

34 Pure Price Mechanisms: Challenges Two identical tasks with demand equal to 2 and deadline equal to 4 Assume a global capacity constrain of 1 per time unit If a low price materializes at k=0,1, both tasks would consume If a high price materializes at k=0,1, both tasks would defer Maximum aggregate rate U=1 Infeasible requests, Contention to be resolved Why not forget, temporarily, about prices and look for feasible solutions only? Maximum aggregate rate U=1 Infeasible requests, Missed Deadline 34

35 Modeling as a Scheduling Problem Each consumer has tasks defined by a deadline and a demand Each task has a maximum rate at which it can be served In aggregate the tasks have a maximum rate U at which they can be served At each time, which tasks can be served guaranteeing feasibility? Maximum aggregate rate U=3 35

36 Global characterization for feasibility A feasible set of tasks Deadlines and demands and rates of service At each Mme, the set of tasks that can be served keeping the problem feasible is described by a POset defined on the slacks This is a universal characteriza'on of all the possible architectures that provide a solumon of the congesmon problem. 36

37 Feasibility Characterization A Feasible set of tasks: B (i) Variables; Deadline, Demand, Rate of Service: { } {T (i),d (i),r (i) } { Define Slacks: {S (i) = T (i) D(i) R (i) } Feasibility at each instant is described as a POset on the Set of slacks 37

38 Description of the feasible actions The set of tasks that can be served at time without compromising the feasibility of the problem belong to a POset with a unique maximal element Assume 1. n tasks with idenmcal maximum service rate 1 and integer demand 2. integer aggregate capacity U 3. each task is either served at max rate or not served { Demands served are denoted by u 2 {0, 1} n 2 { } Ordered vector of slacks for such task is denoted by 2 S(u) 2 N U S(u) apple Feasible actions: (Maximal Element in POset) 38

39 Feasible actions As an example, consider =(0,2,3) 1. if we serve three tasks with slacks (0,2,2) the scheduling remains feasible, since (0,2,2) (0,2,3) 2. if we serve three tasks with slacks (1,2,3) the scheduling loses its feasibility, since (1,2,3) (0,2,3) The vector of slacks may also contain terms equal to By this notamon we mean that less than U tasks are served As an example, consider =(2,3, ) 1. if we serve two tasks with slacks (2,3) the scheduling remains feasible, since (0,2, ) (2,3, ) 2. if we serve three tasks with slacks (3,3,3) the scheduling loses its feasibility, since (3,3,3) (2,3, ) 39

40 Feasible actions What are the tasks that can be served at k=0 keeping the following problem feasible? Maximum aggregate rate U=3 40

41 POset of the Feasible actions The maximal element can be computed and is =(0,2,4) We can visualize all the actions in a Hasse diagram 41

42 Computation of the Maximal Element The POset of feasible acmons is described by its unique maximal element If there were mulmple maximal elements they would have to be computed by exhausmon Instead, the property of uniqueness guarantees that the POset can be efficiently computed 42

43 Computation of the Maximal Element We begin from set of all possible acmons 43

44 Computation of the Maximal Element From the least slack element we move to the minimal element that has a larger minimal first component 44

45 Computation of the Maximal Element If such an element is not an admissible slack vector for the scheduling, we start considering the second component 45

46 Computation of the Maximal Element From the least slack element we move to the minimal element that has a larger minimal second component 46

47 Computation of the Maximal Element If this slack vector is admissible for the scheduling problem, it becomes our new esmmate for the maximal element 47

48 Computation of the Maximal Element We try to increase again the second component 48

49 Computation of the Maximal Element If we reach a non- admissible slack vector we start considering the third component 49

50 Computation of the Maximal Element We check if a slack vector with larger third component is admissible 50

51 Computation of the Maximal Element If such a slack vector is admissible for the scheduling problem it becomes the new esmmate for the maximal element 51

52 Computation of the Maximal Element The procedure is repeated unml no more updates are possible 52

53 Constrained Auction Mechanism The result suggests the implementamon of a constrained aucmon system 1. Each consumer bids for a slot of energy 2. Slots are assigned to the highest combinamon of bidders within the Poset, possibly with a reserve price Assume =(0,2,5, ) with a reserve price of 5 Bids associated with 8 tasks along with their slacks 53

54 Constrained Auction Mechanism The result suggests the implementamon of a constrained aucmon system 1. Each consumer bids for a slot of energy 2. Slots are assigned to the highest combinamon of bidders within the Poset, possibly with a reserve price Assume =(0,2,5, ) with a reserve price of 5 Tasks with Slack less or equal to 0 (namely the 1st element of ) Winner is in red. 54

55 Constrained Auction Mechanism The result suggests the implementamon of a constrained aucmon system 1. Each consumer bids for a slot of energy 2. Slots are assigned to the highest combinamon of bidders within the Poset, possibly with a reserve price Assume =(0,2,5, ) with a reserve price of 5 Tasks with Slack less or equal to 2 (namely the 2nd element of ) Winner is in red. Previous winners in blue 55

56 Constrained Auction Mechanism The result suggests the implementamon of a constrained aucmon system 1. Each consumer bids for a slot of energy 2. Slots are assigned to the highest combinamon of bidders within the Poset, possibly with a reserve price Assume =(0,2,5, ) with a reserve price of 5 Tasks with Slack less or equal to 5, (namely the third element of ) Winner is in red. Previous winners in blue 56

57 Constrained Auction Mechanism The result suggests the implementamon of a constrained aucmon system 1. Each consumer bids for a slot of energy 2. Slots are assigned to the highest combinamon of bidders within the Poset, possibly with a reserve price Assume =(0,2,5, ) with a reserve price of 5 Tasks with Slack less or equal to, Fourth element of No winner because of the reserve price. Previous winners in blue 57

58 Advantages of the Characterization Universality: a policy samsfies the congesmon constraints along with the task deadlines iff at each Mme it picks an acmon from the computed POset. Computability: the presence of a unique maximal element facilitates the computamon of the POset Scalability: a hierarchy of congesmon constrained can be handled in a distributed manner Design: the result suggests the implementamon of a constrained aucmon system, but other mechanisms can be checked, as well 58

59 Conclusions Research in OpMmality vs Risk Next step from robust control theory Two characterismcs Many decision makers Network effects Value of Anarchy Classical tradeoff between opmmality and robustness CharacterizaMon of decision architecture 59

60 Thank You 60