Optimal Power Flow. S. Bose, M. Chandy, M. Farivar, D. Gayme S. Low. C. Clarke. Southern California Edison. Caltech. March 2012

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1 Optimal Power Flow over Radial Networks S. Bose, M. Chandy, M. Farivar, D. Gayme S. Low Caltech C. Clarke Southern California Edison March 2012

2 Outline Motivation Semidefinite relaxation Bus injection model Conic relaxation Branch flow model

3 Challenge: uncertainty mgt High Levels of Wind and Solar PV Will Present an Operating Challenge! Source: Rosa Yang, EPRI

4 Optimal power flow (OPF) OPF is solved routinely to determine How much power to generate where Market operation & pricing Parameter setting, e.g. taps, VARs Non-convex and hard to solve Huge literature t since 1962 In practice, operators often solve linearized model and verify using AC power flow model Core of many problems OPF, LMP, Volt/VAR, DR, EV, planning

5 Worldwide energy demand: 16 TW Wind power over land (exc. Antartica) TW electricity demand: 2.2 TW wind capacity (2009): 159 GW grid-tied PV capacity (2009): 21 GW Solar power over land 340 TW Source: Renewable Energy Global Status Report, 2010 Source: M. Jacobson, 2011

6 Implications Current control paradigm works well today Low uncertainty, few active assets to control Centralized, open-loop, human-in-loop, worst-case preventive Schedule supplies to match loads Future needs Fast computation to cope with rapid, random, large fluctuations in supply, demand, voltage, freq Simple algorithms to scale to large networks of active DER Real-time data for adaptive control

7 Implications Must close the loop Real-time feedback control, risk-limiting Driven by uncertainty of renewables Must be scalable Distributed & decentralized optimization Orders of magnitude more endpoints that can generate, compute, communicate, actuate Control and optimization framework Theoretical foundation for a holistic framework that integrates engineering + economics Systematic algorithm design, understandable global behavior Clarify ideas, explore structures, suggest direction

Application: Volt/VAR control Motivation

8 Application: Volt/VAR control Motivation Static capacitor control cannot cope with rapid random fluctuations of PVs on distr circuits Inverter control Much faster & more frequent IEEE 1547 d t ti i IEEE 1547 does not optimize VAR currently (unity PF)

9 Load and Solar Variation c p i p g i Empirical distribution of (load, solar) for Calabash

10 Improved reliability g c p i, p i for which problem is feasible Implication: reduced likelihood of violating voltage limits or VAR flow constraints

11 Energy savings

12 Summary More reliable operation Energy savings

13 Key message Radial networks computationally simple Exploit tree graph & convex relaxation Real-time scalable control promising

14 Outline Motivation Semidefinite relaxation Bus injection model Conic relaxation Branch flow model

15 Optimal power flow (OPF) Problem formulation Carpentier 1962 Computational techniques: Dommel & Tinney 1968 Surveys: Huneault et al 1991, Momoh et al 2001, Pandya et al 2008 SDP formulation (bus injection model): Bai et al 2008, 2009, Lavaei et al 2010 Bose et al 2011, Sojoudi et al 2011, Zhang et al 2011 Lesieutre et al 2011 Branch flow model Baran & Wu 1989 Chiang & Baran 1990 Farivar et al Baran & Wu 1989, Chiang & Baran 1990, Farivar et al 2011

16 Models g s j c s j i j k branch S j S flow S ij k S jk bus injection S jk i z j ij r ij, x ij k V i I ij V j I j k I jk

17 Models: Kirchoff s law S i * S ij V i I i j linear relation: I YV V i V j V k S ij V i I ij * V * V * 2 VV * S ij V i V j i * z V i iv j * * ij z ij z ij

18 Bus injection model Nodes i and j are linked with an admittance Kirchhoff's Law:

19 Classical OPF min f g k P k over kg subject to P k g P g k,q g k, V k Q k g V k P k g Q k g g Pk Q k g V k V k KVL/KCL power balance Generation cost Generation power constraints Voltage magnitude constraints nonconvexity

20 Classical OPF In terms of V: kg P k tr k VV * min tr M k VV Q * k tr k VV over V * k := Y k * Y k 2 Y k k : = 2i * Y k s.t. P g d * P g d k P k tr k VV Pk P k g d Q tr VV * g d Q Qk k Q k k Q k Q k V k 2 tr J k VV * 2 V k Key observation [Bai et al 2008]: Key observation [Bai et al 2008]: OPF = rank constrained SDP

21 Classical OPF min kg tr M k W over W positive semidefinite matrix s.t. P k tr k W Pk Q k tr k W Q k V k 2 W 0, 2 tr J k W V k rank W 1 convex relaxation: SDP

22 Semi-definite relaxation Non-convex QCQP Rank-constrained SDP Relax the rank constraint and solve the SDP Does the optimal solution satisfy the rank-constraint? yes no We are done! Solution may not be meaningful

23 SDP relaxation of OPF min kg tr M k W over W positive semidefinite matrix s.t. P k tr k W Pk k, k Q k tr k W Q k k, k V k 2 W 0 2 tr J k W V k k, k Lagrange multipliers A M k, k, k J kg k := M k k k k k k J k

24 Sufficient condition Theorem opt If A has rank n-1 then W opt has rank 1, SDP relaxation is exact Duality gap is zero A globally optimal V opt can be recovered IEEE test systems (essentially) satisfy the condition! J. Lavaei and S. H. Low: Zero duality gap in optimal power flow problem. Allerton 2010, TPS 2011

25 OPF over radial networks Suppose tree (radial) network no lower bounds on power injections Theorem opt A always has rank n-1 W opt always has rank 1 (exact relaxation) OPF always has zero duality gap V opt Globally optimal solvable efficiently S. Bose, D. Gayme, S. H. Low and M. Chandy, OPF over tree networks. Allerton 2011

26 OPF over radial networks Suppose tree (radial) network no lower bounds on power injections Theorem opt A always has rank n-1 W opt always has rank 1 (exact relaxation) OPF always has zero duality gap V opt Globally optimal solvable efficiently Also: B. Zhang and D. Tse, Allerton 2011 S. Sojoudi and J. Lavaei, submitted 2011

27 QCQP over tree QCQP C,C k min x * Cx over x C n s.t. x * C k x b k k K graph of QCQP GC,C k C ij 0 or QCQP over tree GC,C k hasedge (i, j) C k is a tree ij 0 for some k

28 QCQP over tree QCQP C,C k min x * Cx over x C n s.t. x * C k x b k k K Semidefinite relaxation min tr CW over W 0 s. t. tr C k W b k k K

29 QCQP over tree QCQP C,C k min x * Cx over x C n s.t. x * C k x b k k K Key assumption (i, j) GC,C k : 0 int conv hull C ij, C k ij, k Theorem Semidefinite relaxation is exact for QCQP over tree S. Bose, D. Gayme, S. H. Low and M. Chandy, submitted March 2012

30 OPF over radial networks Theorem opt A always has rank n-1 W opt always has rank 1 (exact relaxation) OPF always has zero duality gap V opt no lower bounds removes these C k Globally optimal solvable efficiently ij

31 OPF over radial networks Theorem opt A always has rank n-1 W opt always has rank 1 (exact relaxation) OPF always has zero duality gap V opt bounds on constraints remove these C k Globally optimal solvable efficiently ij

32 Outline Motivation Semidefinite relaxation Bus injection model Conic relaxation Branch flow model

33 Model g s j c s j i j k power S k injection S ij S jk i z j ij r ij, x ij k V i I ij V j radial (tree) network

34 Model g s j c s j i j k power S k injection S ij S jk Kirchoff s Law: S ij S jk z ij I ij 2 sj c s j g k:j~k line loss load gen

35 Model g s j c s j i j k power S k injection S ij S jk Kirchoff s Law: S ij S jk z ij I ij 2 sj c s j g k:j~k Ohm s Law: V S * j V i z ij I ij ij V i I ij

36 OPF l 2 ij : I ij min over r ij l ij i v i i~ j i (P,Q, v,q g,l) s. real t. power P ij loss P jk k:j~k CVR (conservation voltage reduction) r ij l ij p j c p j g x l c g v 2 Q ij Q jk ij ij q j q i : V i j k:j~k r 2 2 ij x ij v i v j 2 r ij P ij x ij Q ij 2 2 l ij P ij Q ij v i q i q j g q i l l ij

37 OPF using branch flow model min r l ij ij i v i i~ j i over (S, I,V, s g, s c ) s. t. s i g s i g s i g v i v i v i s i s i c i i i Kirchoff s Law: S ij S jk z ij I ij 2 sj c s j g k:j~k Ohm s Law: V S * j V i z ij I ij ij V i I ij

38 Solution strategy OPF branch flow model nonconvex Angle relaxation eliminate phases nonconvex exact for tree always exact SOCP relaxation convex program

39 1. Angle relaxation Angles of I i, V i eliminated! Points relaxed to circles P ij Q ij k:j~k P jk 2 r ij I ij 2 pj c p j g 2 Q jk x ij I ij k:j~k q j c q j g demands 2 V i V j 2 rij r ij P ij x ij Q ij V i 2 I 2 ij P 2 2 ij QQ ij 2 V i 2 2 r ij x ij 2 I ij Baran and Wu 1989 for radial networks

40 1. Angle relaxation Angles of I i, V i eliminated! Points relaxed to circles P ij Q ij V i 2 2 P jk r ij I ij c g pj p j generation k:j~k 2 Q jk x ij I ij q c g j q VAR control j k:j~k 2 2 r ij x ij 2 V i V j 2 rij r ij P ij x ij Q ij I 2 ij P 2 2 ij QQ ij 2 V i 2 I ij g g g c s i s i s i s i s i v i v i v i

41 1. Angle relaxation l 2 ij : I ij min r ij l ij i~ j i v i i over (S,l, v, s g, s c ) s. t. P ij P jk k:j~k r ij l ij p j c p j g c g v 2 Q i : V i ij Q jk x ij l ij q j q j Linear objective Linear constraints Quadratic equality k:j~k r 2 2 ij x ij v i v j 2 r ij P ij x ij Q ij 2 2 l ij P ij Q ij v i 2 2 l, s i s g i s i v i v i v i, s i s i c l ij

42 2. SOCP min r ij l ij i v i relaxation i~ j i Quadratic inequality over (S,l,, v,, s g, s c ) s. t. P ij P jk Q ij Q ij k:j~k r ij l ij p j c p j g Q jk ij ij q j q j k:j~k x ij l ij q j c q j g r 2 2 ij x ij l v i v j 2 r ij P ij x ij Q ij 2 2 ij l, s s g s ij P ij Q ij v i i i i v i v i v i, s i s i c l ij

43 OPF over radial networks Theorem Both relaxation steps are exact SOCP relaxation is (convex and) exact Phase angles can be uniquely determined Original OPF problem has zero duality gap M. Farivar, C. Clarke, S. H. Low and M. Chandy, Inverter VAR control for distribution systems with renewables. SmartGridComm 2011

44 What about mesh networks?? M. Farivar and S. H. Low, submitted March 2012

45 Solution strategy OPF branch flow model nonconvex Angle relaxation eliminate phases nonconvex SOCP relaxation convex program exact for tree always exact??

46 OPF using branch flow model min r l ij ij i v i i~ j i over (S, I,V, s g, s c ) s. t. s i g s i g s i g v i v i v i s i s i c i i i Kirchoff s Law: S ij S jk z ij I ij 2 sj c s j g k:j~k Ohm s Law: V S * j V i z ij I ij ij V i I ij

47 Convexification of mesh networks OPF min f ĥ(x) s.t. x X, s S x,s OPF-ar min f ĥ(x) h(x) x,s s.t. x Y, s S OPF-ps min x,s, f ĥ(x) s.t. x X, s S Theorem X Y Need phase shifters only outside spanning tree X X X Y

48 Convexification of mesh networks OPF min f ĥ(x) s.t. x X x,s OPF-ar min f ĥ(x) h(x) x,s s.t. x Y OPF-ps min x,s, f ĥ(x) s.t. x X Theorem X Y Need phase shifters only outside spanning tree X X X Y

49 Key message Radial networks computationally simple Exploit tree graph & convex relaxation Real-time scalable control promising Mesh networks can be convexified Design for simplicity Need few phase shifters (sparse topology)

Optimal Demand Response

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