Smart Grid. Steven Low. Computing + Math Sciences Electrical Engineering

Transcription

1 Smart Grid Steven Low Computing + Math Sciences Electrical Engineering August 2014

2 Smart grid team (impact of CDS) Caltech! D. Cai, M. Chandy, N. Chen, J. Doyle, K. Dvijotham, M. Farivar, L. Gan, B. Hassibi, J. Ledyard, E. Mallada, M. Nikolai, Q. Peng, T. Teeraratkul, A. Wierman, S. You, C. Zhao Former! S. Bose (Cornell), L. Chen (Colorado), D. Gayme (JHU), J. Lavaei (Columbia), Z. Liu (LBNL/SUNY), L. Li (Harvard), U. Topcu (Upenn)

3 Big picture how should we evolve our energy system (grid)?

4 Watershed moment Power network will undergo similar architectural transformation that phone network went through in the last two decades Tesla: multi-phase AC Deregulation started? 1888 Both started as natural monopolies Both provided a single commodity 1876 Both grew rapidly through two WWs Bell: telephone s s Deregulation started 1969: DARPAnet Convergence to Internet

5 Watershed moment Industries will be destroyed & created AT&T, MCI, McCaw Cellular, Qualcom Google, Facebook, Twitter, Amazon, ebay, Netflix Infrastructure will be reshaped Centralized intelligence, vertically optimized Distributed intelligence, layered architecture What will drive power network transformation?

6 Four drivers Renewables for sustainability Electrification of transportation challenges Advances in power electronics Deployment of sensing, control, comm enablers

7 Area to power the world by solar power: electricity, machines, transportations 2030 usage: 44% greater than 2008 usage solar: 1kW/m 2, 20% efficiency, 2000 hrs/yr

8 DER will reach 30% of Installed US Capacity by 2020 Effectively all incremental growth in capacity will come from customers 30% Backup Generation: 225 GW CHP: 122 GW Demand Response: 90 GW Solar PV: 50 GW Other DG: 25 GW Dist. Storage: 3 GW Potential DER Total: 515 GW Jeff Taft, PNNL, Nov 2013

9 Technical potential of solar power: > 200x world energy demand network of billions of active distributed energy resources (DERs) DER: PV, wind tb, EV, storage, smart bldgs/appls

10 Risk: active DERs introduce rapid random fluctuations in supply, demand, power quality increasing risk of blackouts Opportunity: active DERs enables realtime dynamic network-wide feedback control, improving robustness, security, efficiency Caltech research: distributed control of networked DERs 1. Endpoint based control Self-manage through local sensing, comm, control 2. Local algorithms with global perspective Decompose global objectives into local algorithms 3. CDS tools provide Structure, clarity, systematic algorithm design

11 Caltech research economics and regulations control and optimization voltage regulation freq control demand response (e.g. DC) storage EV charging OPF volt/var DER adoption market power sec min 5 min 60 min day year

12 Caltech research economics and regulations control and optimization freq control voltage regulation demand response (e.g. DC) storage EV charging OPF volt/var DER adoption market power sec min 5 min 60 min day year

13 Key challenges multiple timescales uncertainty large scale nonconvexity

14 Multiple timescale System dynamics and controls at different timescales require different models they interact Sean Meyn, 2010

15 Uncertainty Uncertainty creates difficulty in both control and markets FREQUENCY (Hz) MW Generation Lost SPINNING AND SUPPLEMENTAL RESERVES RESPONSE (10 min) this can be very expensive as uncertainty grows GOVERNOR RESPONSE (1 min) :50 6:00 6:10 6:20 6:30 TIME (pm) Loss of 2 nuclear plants in ERCOT Kirby 2003 [ORNL/TM-2003/19]

16 Uncertainty Imagine when we have 33%+ renewable generation How can load help? ubiquitous continuous fast-acting distributed (10 min) (1 min)

17 Uncertainty Real-time price can be more than 100x the average price! Illinois, July 1998 California, July Purchase Price $/MWh Previous week Spinning reserve prices PX prices $/MWh Mon Tues Weds Thurs Fri Weds Thurs Fri Sat Sun Mon Tues Weds Ontario, November 2005 APX Power NL, April 23, 2007 Forecast Prices Forecast Demand Demand in MW Last Updated 11:00 AM Predispatch Dispatch Hourly Ontario Energy Price $/MWh Last Updated 11:00 AM Predispatch Dispatch Time Tues Weds Thurs Volume (MWh) Current Week Volume Current Week Price Prev. Week Volume Prev. Week Price Tues Wed Thus Fri Sat Sun Mon Price (Euro/MWh) Figure: Real-world price dynamics Sean Meyn, 2010

18 Large scale Example: Southern California Edison! 4-5 million customers SCE Rossi feeder circuit! #houses: 1,407; #commercial/industrial: 131! #transformers: 422! #lines: 2,064 (multiphase, inc. transfomers)! peak load: 3 6 MW! #optimization variables: 50,000 SCE has 4,500 feeders! ~100M variables United States! 131M customers, 300K miles of transmission & distr lines, 3,100 utilities much more DERs in the future

19 Caltech research economics and regulations control and optimization freq control voltage regulation demand response (e.g. DC) storage EV charging OPF volt/var DER adoption market power sec min 5 min 60 min day year

20 Optimal power flow (OPF) OPF is solved routinely to determine! How much power to generate where! Parameter setting, e.g. taps, VARs! Market operation & pricing Non-convex and hard to solve! Huge literature since 1962! Common practice: DC power flow (LP)

21 OPF: bus injection model min tr CVV * subject to s j tr ( Y j VV * ) s j v j V j 2 v j nonconvex (QCQP) due to Kirchhoff s laws cannot be designed away should exploit hidden convexity structure not just for speed and scale

22 Feasible sets min tr CVV * subject to s j tr ( Y j VV * ) s j v j V j 2 v j Equivalent problem: quadratic in V linear in W!! min tr CW subject to s j tr ( Y j W ) s j v i W ii vi W 0, rank W =1 convex in W except this constraint

23 But SDP is not scalable enough

24 Feasible set dec # vars Consider full matrix W partial matrix partial matrix W c(g) W G defined on a chordal ext of G defined on G C1: C2: C3: W 0, rank W = 1 W c(g) 0, rank W c(g) = 1 W G ( j,k) 0, rank W G ( j,k)=1, ( j,k) 2 E, Â ( j,k)2c \W jk = 0 mod 2p

25 Feasible set dec # vars Consider full matrix W partial matrix partial matrix W c(g) W G defined on a chordal ext of G defined on G C1: C2: C3: W 0, rank W = 1 W c(g) 0, rank W c(g) = 1 W G ( j,k) 0, rank W G ( j,k)=1, ( j,k) 2 E, Â ( j,k)2c \W jk = 0 mod 2p

26 Feasible set dec # vars Consider full matrix W partial matrix partial matrix W c(g) W G defined on a chordal ext of G defined on G C1: C2: C3: W 0, rank W = 1 W c(g) 0, rank W c(g) = 1 W G ( j,k) 0, rank W G ( j,k)=1, ( j,k) 2 E, Â ( j,k)2c \W W Gjk = 0 mod 2p 2x2 rank-1 [ ] jk cycle condition

27 Feasible set Theorem C1 = C2 = C3 C1: C2: C3: W 0, rank W = 1 W c(g) 0, rank W c(g) = 1 W G ( j,k) 0, rank W G ( j,k)=1, ( j,k) 2 E, Â ( j,k)2c \W W Gjk = 0 mod 2p [ ] jk cycle condition

28 Feasible set Theorem C1 = C2 = C3 Moreover, given W G that satisfies C3, there is a unique completion W that satisfies C1 C1: C2: C3: W 0, rank W = 1 W c(g) 0, rank W c(g) = 1 W G ( j,k) 0, rank W G ( j,k)=1, ( j,k) 2 E, Â ( j,k)2c \W W Gjk = 0 mod 2p [ ] jk cycle condition

29 W G := Implication: feasible sets! W jj,w jk : ( j, k) in G " # satisfy linear constraints $ % & idea: W G = ( VV * only on G)! W ( j, k) 0 rank-1, " # cycle cond on W jk $ % & W c(g) :=! W jj,w jk : ( j, k) in c(g) $ " % # satisfy linear constraints & W c(g) 0 rank-1 idea: W c(g) = VV * on c(g) ( ) { } matrix completion [Grone et al 1984] W:= { W: satisfies linear constraints } W 0 rank-1 idea: W = VV * { }

30 Feasible sets V W W c(g) W G W + + W c(g) W G + Theorem! Radial G :! Mesh G : V W + + W c(g) V W + + W c(g) + W G + W G

31 But even SOCP is not scalable enough

32 Examples: radial unbalanced network BIM-SDP BFM-SDP time ratio time ratio 13-bus 1.7s 5.7e s 8.2e bus 3.1s 6.6e bus 4.6s 1.0e s 3.8e bus 9.3s 9.5e-8 6.8s 6.1e bus 320s 4.9e-8 Table 1: Simulation results using convex programming solver sedumi. network BIM-SDP BFM-SDP time ratio time ratio 13-bus 2.6s 1.1e-8 2.6s 4.6e-8 34-bus 4.6s 1.2e-8 37-bus 4.6s 1.9e-8 5.5s 4.5e bus 8.4s 1.1e-8 8.1s 4.0e bus 398s 5.0e-11 Table 2: Simulation results using convex programming solver sdpt3.

33 OPF: branch flow model min f x ( ) over x := (S, I,V, s) s. t. s j s j s j v j v j v j branch'flow' model' ( 2 S ij z ij I ) ij S jk i j V j = V i z ij I ij j k = s j S ij = V i I ij * numerically more stable better linear approximation for tree networks

34 SOCP in branch flow model V W W c(g) W G X W + + W c(g) W G + X + Theorem W G X and W G + X +

35 Examples: radial unbalanced network BIM-SDP BFM-SDP time ratio time ratio 13-bus 1.7s 5.7e s 8.2e bus 3.1s 6.6e bus 4.6s 1.0e s 3.8e bus 9.3s 9.5e-8 6.8s 6.1e bus 320s 4.9e-8 Table 1: Simulation results using convex programming solver sedumi. network BIM-SDP BFM-SDP time ratio time ratio 13-bus 2.6s 1.1e-8 2.6s 4.6e-8 34-bus 4.6s 1.2e-8 37-bus 4.6s 1.9e-8 5.5s 4.5e bus 8.4s 1.1e-8 8.1s 4.0e bus 398s 5.0e-11 Table 2: Simulation results using convex programming solver sdpt3. Branch flow model is much more numerically stable, but more variables!

36 Examples: radial balanced solution time (sec) 180 CVX IPM radial network balanced, BFM # buses

37 OPF2socp' OPF2ch' OPF2sdp' OPF2socp' W G * * W c(g) W * x * 2x2'rank21' Y,'mesh' Y' radial' rank21' Y' Y' radial' equality' Y,'mesh' cycle' condi6on' Y' Recover'V * Y' cycle' condi6on' OPF'solu6on'

38 OPF2socp' OPF2ch' OPF2sdp' OPF2socp' W G * * W c(g) W * x * 2x2'rank21' Y,'mesh' Y' radial' rank21' Y' Y' radial' equality' Y,'mesh' cycle' condi6on' Y' Recover'V * Y' cycle' condi6on' OPF'solu6on'

39 OPF2socp' OPF2ch' OPF2sdp' OPF2socp' W G * * W c(g) W * x * 2x2'rank21' Y,'mesh' Y' radial' rank21' Y' Y' radial' equality' Y,'mesh' cycle' condi6on' Y' Recover'V * Y' cycle' condi6on' OPF'solu6on'

40 When will SOCP be exact?

41 Exactness For tree networks, SOCP always exact practically For general networks, often exact empirically but no theory Bus injection model! Jabr 2006, Bai et al 2008, Lavaei & Low 2012! Bose et al 2011, Zhang & Tse 2011, Sojoudi & Lavaei 2012, Bose et al 2012,! Lesieutre et al 2011, Branch flow model! Baran & Wu 1989, Chiang & Baran 1990, Taylor 2011, Farivar et al SGC2011,! Farivar et al TPS2013, Gan et al TAC2014, Bose et al TAC2014

42 Caltech research economics and regulations control and optimization freq control voltage regulation demand response (e.g. DC) storage EV charging OPF volt/var DER adoption market power sec min 5 min 60 min day year

43 Frequency control Frequency control is traditionally done on generation side secondary freq control primary freq control economic dispatch unit commitment sec min 5 min 60 min day year dynamic model e.g. swing eqtn power flow model e.g. DC/AC power flow

44 Can household Grid Friendly appliances follow its own PV production? 60,000 AC avg demand ~ 140 MW wind var: +- 40MW temp var: 0.15 degc Dynamically adjust thermostat setpoint Fig. 7. Load control example for balancing variability from intermittent renewable generators, where the end-use functionvin this case, thermostat setpointvis used as the input signal. Callaway, Hiskens (2011) Callaway (2009)

45 Network model Generator bus (may contain load): ω i = 1 $ d i + D i ω i P m i + P ij P ji M & i % i j j i ' ) ( Load bus (no generator): i j 0 = d i + D i ω i P i m + P ij P ji Real branch power flow: j i P ij = b ij ( ω i ω ) j i j swing dynamics

46 Frequency control ω i = 1 $ d i + D i ω i P m i + P ij P ji M & i % i j j i i j j i 0 = d i + D i ω i P i m + P ij P ji ' ) ( P ij = b ij ( ω i ω ) j i j Suppose the system is in steady state and suddenly ω i = 0 Pij = 0

47 Frequency control Given: disturbance in gens/loads Current: adapt remaining generators P i m! to re-balance power! restore nominal freq and inter-area flows (zero ACE) Our goal: adapt controllable loads! same as above d i! while minimizing disutility of load control

48 Frequency control ω i = 1 $ d i + D i ω i P m i + P ij P ji M & i % i j j i i j j i 0 = d i + D i ω i P i m + P ij P ji ' ) ( P ij = b ij ( ω i ω ) j i j proposed approach current approach

49 Load-side controller design ω i = 1 $ d i + D i ω i P m i + P ij P ji M & i % i j j i i j j i 0 = d i + D i ω i P i m + P ij P ji ' ) ( P ij = b ij ( ω i ω ) j i j How to design feedback control law d i = F i ( ω(t), P(t) )

50 Load-side controller design ω i = 1 $ d i + D i ω i P m i + P ij P ji M & i % i j j i i j j i 0 = d i + D i ω i P i m + P ij P ji ' ) ( P ij = b ij ( ω i ω ) j i j Control goals! Rebalance power! Resynchronize/stabilize frequency! Restore nominal frequency! Restore scheduled inter-area flows

51 Load-side controller design ω i = 1 $ d i + D i ω i P m i + P ij P ji M & i % i j j i i j j i 0 = d i + D i ω i P i m + P ij P ji ' ) ( P ij = b ij ( ω i ω ) j i j Design approach: forward engineering! formalize control goals as OLC! derive local control as distributed solution

52 Optimal load control (OLC) min d, ˆd,P,R i! # " c i ( d ) i + 1 2D i e E s. t. d i + ˆd i = P i m C ie 2 ˆdi d i = P m i L i v j Ĉv = ˆP j $ & % P e demand = supply per bus restore nominal frequency restore inter-area flows

53 Summary Primary frequency control! Completely decentralized load control works! network dynamics + active load control = primal-dual algorithm for OLC! Feedback system is GAS Secondary frequency control! Each load maintains internal dynamic vars and communicates with neighbors! same as above d i = F i ( ω i (t)) Load-side frequency control works!

54 Simulations load-side participation improves transient as well as steady state Hz ERCOT threshold for freq control

55 Simulations AGC only OLC only AGC+OLC 0.1 Power (pu) load-side participation improves transient as well as steady state Time (s)