Dynamic Network Energy Management via Proximal Message Passing

Transcription

1 Dynamic Network Energy Management via Proximal Message Passing Matt Kraning, Eric Chu, Javad Lavaei, and Stephen Boyd Google, 2/20/2013 1

2 Outline Introduction Model Device examples Algorithm Numerical results Extensions and conclusion Introduction 2

3 Smart grid Á embed intelligence in energy systems to Á do more with less Á reduce CO2 emissions Á handle uncertainties in generation (wind, solar,...) Á exploit new demand response capabilities Á handle shift towards EVs Á extend life of current infrastructure Introduction 3

4 Smart grid Á embed intelligence in energy systems to Á do more with less Á reduce CO2 emissions Á handle uncertainties in generation (wind, solar,...) Á exploit new demand response capabilities Á handle shift towards EVs Á extend life of current infrastructure Á cf. current system Á load is what it is; generation scheduled to match it Á systems built with large margins for max load Introduction 3

5 Smart grid critical technologies: The big picture Á physical layer Á photovoltaics, switches, storage, fuel cells,... Á infrastructure/plumbing Á smart enabled stuff, communication protocols, security,... Á algorithms (our focus) Á real-time decision making Á economics layer Á markets, investment, regulation,... Introduction 4

6 Coordinating devices on the smart grid Á setting: a network of smart devices, that can adjust/change/defer their power consumption/generation Á goal: coordinate device behavior (generation/consumption) over time Introduction 5

7 Coordinating devices on the smart grid Á setting: a network of smart devices, that can adjust/change/defer their power consumption/generation Á goal: coordinate device behavior (generation/consumption) over time Á method: use optimization to coordinate devices Á algorithm: use proximal message passing to solve optimization problem Introduction 5

8 Device coordination via optimization Á devices exchange energy at nodes, in multiple time periods Á generators Á loads (fixed, deferrable, curtailable) Á energy storage systems Á transmission lines Á each device has dynamic constraints, cost function over time Á to coordinate devices, minimize total cost subject to power balance at each node, in each time period Á solving this optimization problem gives Á (optimal) device power schedules Á locational marginal prices at each node in each time period Introduction 6

9 This talk: Proximal message passing algorithm Á decentralized method to solve dynamic energy management problem Á each device schedules its own consumption/generation profile Á devices coordinate via simple message exchanges with neighbors Á can be viewed as sophisticated (location, time varying) price discovery mechanism Á can handle enormous problems Introduction 7

10 Outline Introduction Model Device examples Algorithm Numerical results Extensions and conclusion Model 8

11 Formal network model Á a network consists of Á a set of terminals Ì Á a set of devices Á a set of nets Æ Á and Æ are partitions of Ì, i.e., each terminal is in exactly one device and one net Á can represent network as bipartite graph with Á and Æ the two vertex classes Á Ì as the edges connecting them Model 9

12 Example Á (left) 3 buses, 2 generators, 1 battery, 2 loads, 3 transmission lines Á (middle, right) network model: 11 terminals, 3 nets, 8 devices L 1 G 1 T 1 B G 2 T 2 L 2 T 3 Model 10

13 Terminals Á power flows into or out of terminals on each device (negative power corresponds to power generation) Á each terminal t ¾ Ì has a power schedule p t p t 1µ p t T µµ ¾ R T giving power flow over time periods 1 T Á Ì T set of all terminal power schedules denoted by p ¾ R Model 11

14 Devices Á devices model general power system elements Á generators Á loads (deferrable, curtailable, fixed) Á transmission lines Á energy storage systems Á other energy sinks, sources, and converters Á p d ¾ R d T is the set of d power schedules for terminals in device d Á device objective function f d p d µ R d T R ½ Á ½ used to encode device constraints Á can also have private variables e.g., state of charge for a battery Model 12

15 Nets Á nets are ideal (lossless, uncapacitated) energy exchange points Á p n ¾ R n T is the set of n power schedules for terminals in net n Á semantics of nets: power balance holds at all times t¾n p t µ 0 1 T n ¾ Æ Model 13

16 Average net power imbalance Á for terminal t corresponding to net n, we define p t 1 n t ¼ ¾n i.e., p t averages terminal power profiles over its net Á pd p t t ¾ d Á net power balance can be written as p 0 p t ¼ Model 14

17 Dynamic optimal power flow problem dynamic optimal power flow problem (D-OPF): minimize f pµ subject to p 0 È fd pd µ d¾ Á Ì T variables are terminal power schedules p ¾ R Á net power balance equality constraints are linear Á other constraints, objective terms contained in device objectives Á optimal dual variables give (scaled) locational marginal prices (LMP), which are time-varying Á when all device objective functions are convex, D-OPF can be solved globally and effeciently (in principle) Model 15

18 Outline Introduction Model Device examples Algorithm Numerical results Extensions and conclusion Device examples 16

19 Generator Á single terminal device with power schedule p gen È T Á cost function gen pgen µµ 1 Á min/max power constraints: P min p gen P max Á ramp-rate constraints: R min p gen 1µ p gen µµ R max Á can include other costs and constraints, e.g., Á turning on and off Á power change costs Device examples 17

20 Transmission line Á two terminal device with power schedules p 1 and p 2 Á zero cost function Á capacity constraint: p 1 p 2 C max Á line loss constraint: p 1 p 2 p 1 p 2µ Á p1 p2µ 0 is loss function ( 0 0µ 0, typically convex) Device examples 18

21 Energy storage system Á single terminal device with power schedule p ess Á zero cost function Á charging/discharging rate limits D max p ess C max Á local storage state variables q µ q init t 1 p ess tµ 1 T Á capacity limits 0 q µ Q max, 1 T Á more sophisticated models can include storage cycling penalty, state-dependent charging and discharging rate limits, efficiencies Device examples 19

22 Loads Á single terminal device with power schedule p load Á fixed (non-smart) load: p load l, l ¾ R T is given load profile Á deferrable load: total energy consumption E in the time interval A D : D A p load µ E 0 p load L max Á curtailable load: pay penalty for failing to meet load profile l: «T 1 l µ p load µµ Device examples 20

23 Electric vehicle charging Á single terminal device with power schedule p ev Á desired minimum state of charge profile q des ¾ R T Á can only be charged in time interval A D Á charging constraints 0 p ev C max Á charge level given by q µ q init Á shortfall cost function «D A ¼ A p ev ¼ µ q des µ q µµ Á can add terminal constraint, q Dµ Q cap Device examples 21

24 Outline Introduction Model Device examples Algorithm Numerical results Extensions and conclusion Algorithm 22

25 Proximal message passing algorithm repeat until convergence: 1. Proximal power schedule update. p k 1 d argmin p d in parallel, for each device f d p d µ 2 pd pd k p d k ud k µ ( 0; RHS is proximal operator of f d at p k d p k d u k d ) 2. Scaled price update. in parallel, for each net un k 1 un k p n k Algorithm 23

26 Devices compute new tentative power profiles Algorithm 24

27 Devices send tentative power profiles to neighboring nets Algorithm 25

28 Nets compute power imbalance; update prices Algorithm 26

29 Nets send updated prices, power imbalance to neighboring devices Algorithm 27

30 Proximal message passing algorithm Á each device only has knowledge of its own objective function Á for each device class, need to implement prox operator Á all message passing is local, between devices and adjacent nets Á no global coordination other than iteration synchronization Algorithm 28

31 Convergence if device objectives are closed convex proper and D-OPF has solution Á residual convergence: p k 0 Á objective convergence: È d¾ fd pk d µ f Á dual variable convergence: u k y (power balance achieved) (operation is optimal) (optimal prices found) Algorithm 29

32 Outline Introduction Model Device examples Algorithm Numerical results Extensions and conclusion Numerical results 30

33 Numerical examples Á 140 examples: 20 each of 7 different sizes Á Æ ranges from 100 to Á ranges from 200 to Á Ì ranges from 300 to Á T 96 (24 hour period, 15-minute intervals) Á number of variables in D-OPF ranges from 30k to 30M Á network topology (transmission line connections) chosen as random geometric graph, plus some long lines Numerical results 31

34 Example network with Æ 100 (30k variables) Numerical results 32

35 Devices Á to each net, we attach a randomly chosen single terminal device Á generator Á battery Á fixed load Á deferrable load Á curtailable load Á device parameters chosen so that problem is feasible but challenging Numerical results 33

36 Prox functions Á prox functions are easy to implement when f d is separable in time Á fixed load Á curtailable load Á transmission line (prox evaluation times measured in ns) Á for others, use CVXGEN to generate custom C code to solve QPs Á generator Á battery Á deferrable load (prox evaluation times measured in s) Numerical results 34

37 Serial multithreaded implementation Á examples run on 32-core 2 2Ghz Xeon with 64 (hyper)threads Á each prox function assigned to one of 64 threads using OpenMP Á maximum time for prox function evaluation in each iteration is 1 ms, so we can estimate fully decentralized run time Numerical results 35

38 Convergence for Æ 3000 (1M variables) f k f f p k 2 Ô Ì T iter k iter k Numerical results 36

39 Solve time scaling (serial) 1000 time (seconds) Æ Á serial multi-threaded implementation on 32-core machine with 64 independent threads Á fit exponent is Á with fully decentralized computation, sub second solve time for any size network Numerical results 37

40 Outline Introduction Model Device examples Algorithm Numerical results Extensions and conclusion Extensions and conclusion 38

41 Handling uncertainty via receding horizon control Á in every time period Á each device forecasts its own future costs/constraints over some horizon Á devices coordinate (optimize) using forecasts to obtain consumption/generation plan Á devices execute first period consumption/generation in plan Extensions and conclusion 39

42 Handling uncertainty via receding horizon control Á in every time period Á each device forecasts its own future costs/constraints over some horizon Á devices coordinate (optimize) using forecasts to obtain consumption/generation plan Á devices execute first period consumption/generation in plan Á reacts to changes in constraint/objective forecasts Á same method used in chemical process control, supply chain optimization,... Á forecasts do not need to be accurate Extensions and conclusion 39

43 Handling AC power flow Á assume voltage magnitudes are fixed Á introduce voltage phase angle profile t for each terminal Á add phase angle consistency constraint for each net n t 1 t n : t1 t2 t n Á local device objectives include phase angle constraints Á proximal message passing readily extended to include phase angles Extensions and conclusion 40

44 Handling non-convexities Á with non-convex device objectives, D-OPF is (nominally) hard Á one approach: form convex relaxation of D-OPF (RD-OPF) minimize subject to f env pµ p 0 Èd¾ f env d p d µ where f env d is convex envelope of f d Á RD-OPF is convex optimization problem Á readily solved Á gives lower bound on D-OPF optimal value Á provides good starting point for local optimization Á in some cases, relaxation is tight Extensions and conclusion 41

45 gen pgenµ Relaxed generator Á left: (nonconvex) generator with power range, option to turn off Á right: its relaxation p gen gen pgenµ env p gen Extensions and conclusion 42

46 Relaxed transmission line p 2 p 2 p 1 p 1 black: lossless, capacitated line; gray: AC power loss Extensions and conclusion 43

47 Summary and vision Á we ve developed a completely decentralized method for optimal power exchange/consumption/generation on a smart grid Á decentralized computation allows for sub second solve times independent of network size Extensions and conclusion 44

48 Summary and vision Á we ve developed a completely decentralized method for optimal power exchange/consumption/generation on a smart grid Á decentralized computation allows for sub second solve times independent of network size Á when combined with receding horizon control, can be used for real-time network operation Extensions and conclusion 44

49 Summary and vision Á we ve developed a completely decentralized method for optimal power exchange/consumption/generation on a smart grid Á decentralized computation allows for sub second solve times independent of network size Á when combined with receding horizon control, can be used for real-time network operation Á we envision a plug-and-play system that is robust, self-healing (internet of power) Extensions and conclusion 44