Branch Flow Model. Computing + Math Sciences Electrical Engineering

Branch Flow Model Masoud Farivar Steven Low Computing + Math Sciences Electrical Engineering Arpil 014

TPS paper Farivar and Low Branch flow model: relaxations and convexification IEEE Trans. Power Systems, 8(3), Aug. 013 This talk will only motivate why branch flow model

Outline High-level summary Branch flow model (BFM) Advantages BFM for radial networks Equivalence Recursive structure Linearization and bounds Application: OPF and SOCP relaxation

Branch flow model s j i -1 j k z ij = y ij graph model G: directed

Branch flow model V i -V j = z ij I ij for all i j Ohm s law S ij = V i I ij * for all i j power definition å ( S ij - z ij I ) ij + s j = å S jk for all j i j j k power balance sending end pwr loss sending end pwr S ij : branch power I ij : branch current s j V j : voltage

Bus injection model I = YV s j = V j I j * for all j Ohm s law power balance admittance matrix: Y ij := ìå y ik if i = j ï k~i ï í-y ij if i ~ j ï ï0 else î I j : nodal current V j : voltage s j

Recap Bus injection model Branch flow model s j = å H y jk V j V j -V k k:k~ j ( ) H V i -V j = z ij I ij S ij = V i I ij * ( ) å S jk = å S ij - z ij I ij j k i j + s j solution set (V, s) Î C (n+1) V (S, I,V, s) Î C (m+n+1) X

Advantages of BFM It models directly branch power and current flows Easier to use for certain applications e.g. line limits, cascading failures, network of FACTS Much more numerically stable for large-scale computation

Advantages of BFM Recursive structure for radial networks [BaranWu1989] Simplifies power flow computation Forward-backward sweep is very fast and numerically stable Linearized model for radial networks Much more useful than DC approx. for distribution systems Provide simple bounds on branch powers and voltages

Comparison of linearized models Linear DistFlow Includes reactive power and voltage magnitudes useful for volt/var control and optimization Explicit expression in terms of injections Provides simple bounds to nonlinear BFM vars Applicable only for radial networks DC power flow Ignores reactive power and fixes voltage magnitudes Unclear relation with nonlinear BIM vars

Outline High-level summary Branch flow model (BFM) Advantages of BFM BFM for radial networks Equivalence Recursive structure Linearization and bounds Application: OPF & SOCP relaxation

Relaxed BFM Branch flow model Relaxed model ( ) å S jk = å S ij - z ij I ij j k i j + s j V i -V j = z ij I ij V i I ij * = S ij (S, I,V, s) Î C (m+n+1) X ( ) å P jk = å P ij - r ij I ij j k i j ( ) å Q jk = å Q ij - x ij I ij j k i j + p j + q j

Relaxed BFM Branch flow model Relaxed model ij := I ij v i := V i ( ) å S jk = å S ij - z ij I ij j k i j + s j å S jk = å S ij - z ij ij j k i j ( ) + s j V i -V j = z ij I ij V i I ij * = S ij v i - v j = Re( z * ij S ) ij - z ij v i ij = S ij ij (S, I,V, s) Î C (m+n+1) (S,, v, s) Î R 3(m+n+1) X these solution sets are generally not equivalent X nc

Branch flow model ANS. ON CONTROL OF NETWORK SYSTEMS, 014, TO APPEAR Branch flow model C (m+n+1) Relaxed model R 3(m+n+1) h equivalent model of BFM X h -1 X X X nc nc X + The relaxed model is in general different from BFM but they are equivalent for radial networks +! Feasible sets X of OPF (9) in BFM, its equivalent set

BFM for radial networks power flow solutions: x := S,, v, s ( ) satisfy å S jk = S ij - z ij ij + s j j k v i - v j = Re( z * ij S ) ij - z ij ij ijv i = S ij Advantages Recursive structure Linearized model & bounds ij := I ij v i := V i DiskFlow equations Baran and Wu 1989 for radial networks

BFM for radial networks Recursive structure allows very fast & stable computation initialization Baran and Wu 1989 for radial networks

BFM for radial networks Accurate & versatile linearized model å S jk = S ij - z ij ij + s j j k v i - v j = Re( z * ij S ) ij - z ij ij ijv i = S ij Linearization: ignores line loss reasonable if line loss is much smaller than branch power ij := I ij v i := V i Linear DiskFlow Baran and Wu 1989 for radial networks

BFM for radial networks Accurate & versatile linearized model P ij lin = - å kît j p k, Q lin ij = - å kît j q k v lin i - v lin j = ( r ij P lin lin ij + x ij Q ) ij S ij lin s j å kît j - s k

BFM for radial networks Accurate & versatile linearized model P ij lin = - å kît j p k, Q lin ij = - å kît j q k v lin i - v lin j = ( r ij P lin lin ij + x ij Q ) ij linear functions of injections s

Relaxed BFM Branch flow model Relaxed model ( ) å + s å j S jk = S ij - z ij ij å S jk = S ij - z ij I ij j k i j j k å i j ( ) + s j V i -V j = z ij I ij V i I ij * = S ij v i - v j = Re( z * ij S ) ij - z ij v i ij = S ij ij S ij ³ S ij lin v i v i lin

Outline High-level summary Branch flow model (BFM) Advantages of BFM BFM for radial networks Equivalence Recursive structure Linearization and bounds Application: OPF & SOCP relaxation

OPF & relaxation: examples With PS

OPF: branch flow model min å r ij I ij + i~ j over (S, I,V, s g, s c ) å ai V i + c i i iîg å p i g real power loss CVR (conservation voltage reduction) generation cost

OPF: branch flow model min f x ( ) over x := (S, I,V, s g, s c ) s. t. s i g s i g s i g s i s i c s i v i v i v i

OPF: branch flow model min f x ( ) over x := (S, I,V, s g, s c ) s. t. s i g s i g s i g s i c s i c s i c v i v i v i

OPF: branch flow model min branch flow model f x ( ) over x := (S, I,V, s g, s c ) s. t. s i g s i g s i g å( S ij - z ij I ) ij - S jk i j V j =V i - z ij I ij s i c s i c s i c å j k = s j c - s j g S ij = V i I ij * v i v i v i generation, volt/var control Branch flow model is more convenient for applications

OPF: branch flow model min branch flow model f x ( ) over x := (S, I,V, s g, s c ) s. t. s i g s i g s i g å( S ij - z ij I ) ij - S jk i j V j =V i - z ij I ij s i c s i c s i c å j k = s j c - s j g S ij = V i I ij * Challenge: nonconvexity! v i v i v i demand response

Branch flow model Branch flow model Relaxed model ( ) å S jk = å S ij - z ij I ij j k i j + s j V i -V j = z ij I ij V i I ij * = S ij (S, I,V, s) Î C (m+n+1) X ( ) å P jk = å P ij - r ij I ij j k i j ( ) å Q jk = å Q ij - x ij I ij j k i j + p j + q j

Branch flow model Branch flow model Relaxed model ij := I ij v i := V i ( ) å S jk = å S ij - z ij I ij j k i j + s j å S jk = å S ij - z ij ij j k i j ( ) + s j V i -V j = z ij I ij V i I ij * = S ij v i - v j = Re( z * ij S ) ij - z ij v i ij = S ij ij (S, I,V, s) Î C (m+n+1) (S,, v, s) Î R 3(m+n+1) X these solution sets are generally not equivalent X nc

Branch flow model power flow solutions: x := S,, v, s ( ) satisfy å S jk = S ij - z ij ij + s j j k v i - v j = Re( z * ij S ) ij - z ij ijv i = S ij Advantages Recursive structure (radial networks) Variables represent physical quantities ij ij := I ij v i := V i Baran and Wu 1989 for radial networks

Branch flow model ANS. ON CONTROL OF NETWORK SYSTEMS, 014, TO APPEAR Branch flow model C (m+n+1) Relaxed model R 3(m+n+1) h restrict to get an equivalent set X h -1 X X nc X + relax to get a second-order cone Feasible is sets nonconvex X of OPF and effective (9) in superset BFM, its of equivalent X set X nc +

Branch flow model power flow solutions: x := S,, v, s ( ) satisfy å S jk = å S ij - z ij ij k:j k i:i j ( ) v i - v j = Re( z * ij S ) ij - z ij + s j ij linear ijv i = S ij nonconvexity ij := I ij v i := V i Baran and Wu 1989 for radial networks

Branch flow model power flow solutions: x := S,, v, s ( ) satisfy å S jk = å S ij - z ij ij k:j k i:i j ( ) v i - v j = Re( z * ij S ) ij - z ij + s j ij linear ijv i ³ S ij second-order cone ij := I ij v i := V i Baran and Wu 1989 for radial networks

Cycle condition A relaxed solution condition if x satisfies the cycle $q s.t. Bq = b(x) mod p incidence matrix; depends on topology x := (S,, v, s) b jk (x) := Ð( v j - z H jk S ) jk

Branch flow model relaxed solution: x := S,, v, s ( ) X := ì í î x :satisfies linear constraints ìx :satisfies linear X + := í î constraints ü ý þ Ç ì v = S ï üï jk j í ý îï cycle cond on xþï ü ý þ Ç jk v j ³ S { } Theorem X º X Í X + second-order cone (convex)

Feasible sets X of OPF (9) in BFM, its equivalent set + Branch flow model ANS. ON CONTROL OF NETWORK SYSTEMS, 014, TO APPEAR Branch flow model C (m+n+1) Relaxed model R 3(m+n+1) h X h -1 X X nc X +

Branch flow model power flow solutions: x := S,, v, s ( ) X := ì í î x :satisfies linear constraints ìx :satisfies linear X + := í î constraints ü ý þ Ç ì v = S ï üï jk j í ý îï cycle cond on xþï ü ý þ Ç jk v j ³ S { } OPF: min xîx f x ( ) SOCP: min xîx + f x ( )

OPF-socp OPF-ch OPF-sdp OPF-socp W G * * W c(g) W * x * x rank-1 Y, mesh Y radial rank-1 Y Y radial equality Y, mesh cycle condition Y Recover V * Y cycle condition OPF solution

Exact relaxation Definition A relaxation is exact if an optimal solution of the original OPF can be recovered from every optimal solution of the relaxation

OPF-socp OPF-ch OPF-sdp OPF-socp W G * * W c(g) W * x * x rank-1 Y, mesh Y radial rank-1 Y Y radial equality Y, mesh Definition Every optimal matrix or partial matrix is (x) rank-1 cycle condition Y Recover V * Definition Every optimal relaxed solution attains equality Y cycle condition OPF solution

1. QCQP over tree ( ) QCQP C,C k min over x * Cx x Î C n s.t. x * C k x b k k Î K graph of QCQP G( C,C ) k has edge (i, j) Û C ij ¹ 0 or [ C ] k ij ¹ 0 for some k QCQP over tree G( C,C ) k is a tree

1. Linear separability Im ( ) QCQP C,C k min over x * Cx x Î C n s.t. x * C k x b k k Î K Re Key condition ( ) lie on half-plane through 0 i ~ j : C ij,[ C ] k ij, "k Theorem SOCP relaxation is exact for QCQP over tree

1. Linear separability no lower bounds removes these C k [ ] ij

1. Linear separability sufficient cond remove these Ck [ ] ij

Outline Radial networks 3 sufficient conditions Mesh networks with phase shifters

e n r. l h l - ). ). o across a line, and has no impedance nor limits on the shifted Phase angles. shifter Specifically, consider an idealized phase shifter parametrized by φ ij across line (i,j ), as shown in Figure 4. As before, let V i denote the sending-end i! ij k z ij ideal phase shifter Fig. 4: Model of a phase shifter in line (i,j ). voltage. Define I ij to be the sending-end current leaving node i towards node j. Let k be the point between j

s ij ij k A. and Review: I Phase k be model the voltage without at phase k and shifters current from k to j respectively. Then shifter the effect of the idealized phase shifter For ease of reference, we reproduce the branch flow is summarized by the following modeling assumption: model BFM of without [] here: phase V k = V i e iφ shifters: ij and I k = I ij e iφ ij I ij = y ij (V i V j ) (1) The power transferred from nodes i to j is still (defined to be) S ij S= V ij := i I ij V i I ij which, as expected, is equal to() the power s V k I k j = X from S jk nodes X ks ij to j zsince ij I ij the + phase yj V j shifter (3) is assumed to k:j be! k lossless. i :i! japplying Ohm s law across z ij, we define the branch flow model with phase shifters as Recall the set X(s) of branch flow solutions given s the following BFM with phase set of shifters: equations: defined in []: I ij = y ij V i V j e iφ ij (9) X(s) := { x := (S, S ij = V i I ij I, V, s 0 ) x solves (1) (3) given s} (10) s j = X S jk X (4) S ij z ij I ij + yj V j (11) and the set X of all branch flow solutions: k:j! k i :i! j there (, φ for with W mini the 1) resu (S, ` solu netw bran requ

Convexification of mesh networks OPF OPF-ar OPF-ps min x min x min x,f f h(x) ( ) s.t. x Î X f h(x) ( ) s.t. x Î Y f h(x) ( ) s.t. x Î X optimize over phase shifters as well Theorem X = Y Need phase shifters only outside spanning tree X X X Y

Convexification of mesh networks OPF-ps min x,f f h(x) ( ) s.t. x Î X optimize over phase shifters as well Optimization of f Min # phase shifters (#lines - #buses + 1) f Min : NP hard (good heuristics) Given existing network of PS, min # or angles of additional PS X X

Examples With PS