Preliminaries Overview OPF and Extensions. Convex Optimization. Lecture 8 - Applications in Smart Grids. Instructor: Yuanzhang Xiao

Transcription

1 Convex Optimization Lecture 8 - Applications in Smart Grids Instructor: Yuanzhang Xiao University of Hawaii at Manoa Fall / 32

2 Today s Lecture 1 Generalized Inequalities and Semidefinite Programming 2 Overview of Smart Grids 3 Optimal Power Flow and Extensions 2 / 32

3 Outline 1 Generalized Inequalities and Semidefinite Programming 2 Overview of Smart Grids 3 Optimal Power Flow and Extensions 3 / 32

4 Proper Cones a convex cone K R n is proper if: K is closed K has nonempty interior K is pointed (i.e., contains no line) examples of proper cones: nonnegative orthant K = R n + = {x R n x i 0, i = 1,..., n} positive semidefinite cone K = S n + 4 / 32

5 Generalized Inequalities generalized inequality defined by proper cone K: x K y y x K, x K y y x intk examples of generalized inequalities: component-wise inequality (K = R n +): x R n + y x i y i, i = 1,..., n matrix inequality (K = S n +): X S n + Y Y X positive semidefinite subscripts usually dropped when K = R n + or S n + 5 / 32

6 Some Properties some properties are similar to on R: x K y, u K v x + u K y + v some properties are different: may not have a linear ordering: it is possible that x K y and y K x may not have a minimum element for any subset S: may not exist x such that x y, y S 6 / 32

7 Convexity With Respect To Generalized Inequalities f : R n R m is K-convex if domf is convex and f (θx + (1 θ)y) K θf (x) + (1 θ)f (y) for any x, y domf and θ [0, 1] examples: f : S m S m, f (X ) = X 2 is S m +-convex proof: use the fact that z T X 2 z = Xz 2 2 is convex in X 7 / 32

8 Convex Optimization With Generalized Inequalities convex optimization with generalized inequality constraints: minimize f 0 (x) subject to f i (x) Ki 0, i = 1,..., m Ax = b where f 0 : R n R is convex f i : R n R k i is K i -convex 8 / 32

9 Semidefinite Program (SDP) semidefinite program: minimize where F i, G S k c T x subject to x 1 F 1 + x 2 F x n F n + G 0 Ax = b inequality constraint called linear matrix inequality (LMI) can include multiple LMI constraints x 1 ˆF 1 +x 2 ˆF 2 + +x n ˆF n +Ĝ 0, x 1 F 1 +x 2 F 2 + +x n F n + G 0 is equivalent to x 1 [ ˆF1 0 0 F1 ] +x 2 [ ˆF2 0 0 F2 ] + x n [ ˆFn 0 0 Fn ] [ Ĝ G ] 0 9 / 32

10 Semidefinite Program (SDP) - Standard Form standard form semidefinite program: minimize tr(cx ) subject to tr(a i X ) = b i, i = 1,..., p X 0 where optimization variable X S n C, A 1,..., A p S n tr( ) is the trace of a matrix: tr(x ) n i=1 x ii tr(cx ) = n i=1 n j=1 c ijx ij 10 / 32

11 LP as SDP linear program (LP): minimize subject to c T x Ax b equivalent SDP: minimize c T x subject to diag(ax b) 0 diagonal matrix semidefinite each diagonal element nonnegative 11 / 32

12 SOCP as SDP second-order cone program (SOCP): minimize f T x subject to A i x + b i 2 ci T x + d i, i = 1,..., m equivalent SDP: minimize subject to f [ T ( x ) c T i x + d i I Ai x + b i (A i x + b i ) T c T i x + d i ] 0, i = 1,..., m important fact in SDP: [ A 2 s A T A s 2 si A I A T si ] 0 12 / 32

13 Outline 1 Generalized Inequalities and Semidefinite Programming 2 Overview of Smart Grids 3 Optimal Power Flow and Extensions 13 / 32

14 Traditional Power Systems Smart Grid traditional power systems: Utility Distribution Transmission Utility Distribution smart grid: integration of renewables deregulated electricity market coupling with other infrastructures efficient optimization and computation is key! 14 / 32

15 Transmission Grid Utility Distribution Transmission Utility Distribution transmission grid: high voltage bulk power generators (e.g., coal, hydro-electric generators, wind farms) complicated topology 15 / 32

16 Distribution Grid Utility Distribution Transmission Utility Distribution distribution grid: low voltage small power generators (e.g., residential solar panels) tree topology 16 / 32

17 Example Topology of Transmission Grids transmission grid (IEEE 118-bus system): complicated with many cycles 17 / 32

18 Example Topology of Distribution Grids distribution grid: tree / radial networks 18 / 32

19 Outline 1 Generalized Inequalities and Semidefinite Programming 2 Overview of Smart Grids 3 Optimal Power Flow and Extensions 19 / 32

20 Graph Model of The Grid a power system is usually modeled by a (undirected) graph (N, E) N : set of nodes representing generator and/or load E: set of edges representing transmission lines key elements: (complex) power injection at node j: s j C admittance of line (i, j) E: y ij C usually y ij = y ji Kirchhoff s law: s j = k:(j,k) E ( yjk H V j Vj H Vk H ) 20 / 32

21 Optimal Power Flow optimal power flow (vanilla version): minimize C(V ) subject to v j V j 2 v j, j N s j ( ) yjk H V j Vj H Vk H s j, j N k:(j,k) E where V C n : the vector of voltages at each bus C(V ): cost of generation, loss of power in transmission, etc. voltage magnitude constraints: stability of transmission lines power injection constraints: physics of generators usually solved by system operators: extremely important, solved every 5-15 minutes nonconvex 21 / 32

22 QCQP Formulation cost is usually quadratic: C(V ) = V H CV admittance matrix Y C n n : k:(i,k) E y ik if i = j Y ij = y ij if i j and (i, j) E 0 otherwise Ohm s law: (I is the current injections to each bus) I = YV 22 / 32

23 QCQP Formulation power injections: ( ) ) s j = V j Ij H = ej H V (I H e j ) (( = tr (ej H VV H Y H e j = tr Y H e j ej H ( ) = tr V H Y j V = V H Y j V ) VV H) tr (Y j VV H) similarly, V j V H j = V H J j V, where J j = e j e H j optimal power flow as nonconvex QCQP: minimize subject to V H CV v j V H J j V v j, j N s j V H Y j V s j, j N 23 / 32

24 Equivalent Formulation observe: ( V H MV = tr MVV H) = tr (MW ) where W VV H C n n equivalent problem: minimize tr (CW ) subject to v j tr (J j W ) v j, j N s j tr (Y j W ) s j, j N W 0, rank(w ) = 1 only nonconvexity comes from rank(w ) = 1 24 / 32

25 Semidefinite Programming Relaxation SDP relaxation by discarding rank constraint: minimize tr (CW ) subject to v j tr (J j W ) v j, j N s j tr (Y j W ) s j, j N W 0 convex, can be efficiently solved solution to SDP relaxation: W sdp ; solution to OPF: V if W sdp is rank-1, then W sdp = V (V ) H when is relaxation exact? how tight is the relaxation? 25 / 32

26 Exactness and Tightness of Relaxation relaxation is exact if the network is tree rank of solution W sdp treewidth of the network link exactness / tightness to the network topology 26 / 32

27 Tree Decomposition of Graph tree decomposition of a graph: a tree with nodes X 1,..., X m, where X i is subset of N 1 union of all sets X i is N 2 if X i and X j both contain k N, all nodes in the path between X i and X j contain k 3 if (k, l) N, there is a set X i that contain k and l example: 27 / 32

28 Treewidth of Graph tree decomposition is not unique a trivial tree decomposition for any graph: tree with 1 node N width of a tree decomposition: size of largest set X i minus one max X i 1 treewidth of a graph: minimum width among all tree decompositions 28 / 32

29 Examples of Treewidth treewidth of a tree is 1: original graph 1,3 1,2 1,4 3,5 4,6 tree decomposition each subset X i is an edge (i.e., two nodes) in the original tree 29 / 32

30 Examples of Treewidth a fully-connected graph: original graph 1,2 2,3 3,1 invalid tree decomposition 1,2,3 tree decomposition for a fully-connected graph: unique tree decomposition: the trivial one treewidth: N 1 30 / 32

31 Power System State Estimation power of electricity flow on each transmission line l: V H H l V = tr (H l W ) according to physics measurements at a few selected lines: z l = tr (H l W ) + noise power system state estimation: find V that minimizes the estimation error minimize [z l tr (H l W )] 2 l L subject to W 0, rank(w ) = 1 31 / 32

32 Power System State Estimation SDP relaxation: minimize subject to l L x l [ x l z l tr (H l W ) z l tr (H l W ) 1 W 0 ] 0 32 / 32