lecture14

lecture14 - ￿ ￿ ￿ inversion of the bus admittance...

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Unformatted text preview: ￿ ￿ ￿ inversion of the bus admittance matrix is a n3 effort for small and medium size networks, direct building of the matrix is less effort for large size networks, sparse matrix programming with gaussian elimination technique is preferred Direct formation of the matrix −1 Z bus = Ybus Definition Power Systems I ￿ ￿ The Bus Impedance Matrix 1 j 0.4 j0.2 3 j0.8 2 j0.4 j0.4 1 3 4 5 2 selected tree 3 1 0 1 3 extending tree branch 2 1 3 4 0 5 2 2 loop closing co-tree branch bus node = graph vertex line branch = edge Graph theory techniques helps explain the building process Power Systems I ￿ Forming the Bus Impedance Matrix Z m bus Partial Network 0 Reference 1 2 i j Vbus = Z bus I bus Basic construction of the network and the matrix Power Systems I ￿ Forming the Bus Impedance Matrix q 0 Reference Vq = V p + z qp I q Power Systems I Z m bus Partial Network 1 2 p m Adding a Line Z q 0 Reference Vq = 0 + z q 0 I q m bus Partial Network 1 2 p m Power Systems I Ⱥ V1 Ⱥ Ⱥ Z11 ȺV Ⱥ Ⱥ Z Ⱥ 2 Ⱥ Ⱥ 21 Ⱥ ￿ Ⱥ Ⱥ ￿ Ⱥ Ⱥ Ⱥ ȺV p Ⱥ = Ⱥ Z p1 Ⱥ ￿ Ⱥ Ⱥ ￿ Ⱥ Ⱥ Ⱥ ȺVm Ⱥ Ⱥ Z m1 ȺV Ⱥ Ⱥ= Z p1 Ⱥ q Ⱥ Ⱥ Z m2 = Z p2 ￿ Z 21 Z 22 ￿ Z p2 Z1 p Z2 p ￿ Z pp ￿ ￿ ￿ ￿ Z1m Z 2m ￿ Z pm ￿ ￿ ￿ ￿ ￿ Z mp ￿ Z mm ￿ = Z pp ￿ = Z pm ￿ ￿ ￿ ￿ = Z1 p = Z2 p ￿ = Z pp Ⱥ Ⱥ I1 Ⱥ Ⱥ Ⱥ I Ⱥ Ⱥ Ⱥ 2 Ⱥ Ⱥ Ⱥ ￿ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ Ⱥ I p Ⱥ Ⱥ Ⱥ ￿ Ⱥ ￿ Ⱥ Ⱥ Ⱥ = Z mp Ⱥ Ⱥ I m Ⱥ = Z pp + z pq Ⱥ Ⱥ I q Ⱥ Ⱥ Ⱥ Ⱥ Adding a Line to an Existing Line Power Systems I Ⱥ V1 Ⱥ Ⱥ Z11 ȺV Ⱥ Ⱥ Z Ⱥ 2 Ⱥ Ⱥ 21 Ⱥ ￿ Ⱥ Ⱥ ￿ Ⱥ Ⱥ Ⱥ ȺV p Ⱥ = Ⱥ Z p1 Ⱥ ￿ Ⱥ Ⱥ ￿ Ⱥ Ⱥ Ⱥ ȺVm Ⱥ Ⱥ Z m1 ȺV Ⱥ Ⱥ = 0 Ⱥ q Ⱥ Ⱥ ￿ Z1 p ￿ Z1m ￿ Z 2 p ￿ Z 2m ￿￿￿￿ =0 ￿ =0 ￿ =0 Z p 2 ￿ Z pp ￿ Z pm ￿￿￿￿￿ Z m 2 ￿ Z mp ￿ Z mm Z 21 Z 22 ￿ Adding a Line from Reference Ⱥ Ⱥ I1 Ⱥ Ⱥ Ⱥ I Ⱥ Ⱥ Ⱥ 2 Ⱥ Ⱥ Ⱥ ￿ Ⱥ Ⱥ Ⱥ Ⱥ = 0 Ⱥ Ⱥ I p Ⱥ ￿ Ⱥ Ⱥ ￿ Ⱥ Ⱥ Ⱥ Ⱥ = 0 Ⱥ Ⱥ I m Ⱥ = z0 q Ⱥ Ⱥ I q Ⱥ Ⱥ Ⱥ Ⱥ =0 =0 ￿ z pq I l + Vq − V p = 0 Power Systems I z p0 Il − Vp = 0 z pq I l = V p − Vq → 0 Reference Z m bus Partial Network 0 Reference 1 p i m z p0 Il = Vp − 0 → Z m bus Partial Network 1 p q m Closing a Loop T I [ n×1] bus [1×n ] Power Systems I Z ll I bus [1×n ] Ⱥ old ΔZΔZ T Ⱥ = Ⱥ Z bus − Ⱥ I bus Z ll Ⱥ Ⱥ ΔZ T [ n×1] → Il = − I bus [1×n ] Z ll ΔZ[1×n ]ΔZ T [ n×1] + Z ll [1×1] I l old Vbus [1×n ] = Z bus [ n×n ] I bus [1×n ] − 0 = ΔZ Vbus [1×n ] ΔZ[1×n ] Ⱥ Ⱥ I bus [1×n ] Ⱥ Ⱥ Z ll [1×1] Ⱥ Ⱥ I l Ⱥ Ⱥ Ⱥ Ⱥ old = Z bus [ n×n ] I bus [1×n ] + ΔZ[1×n ] I l old ȺVbus [1×n ] Ⱥ Ⱥ Z bus [ n×n ] Ⱥ 0 Ⱥ = ȺΔZ T [ n×1] Ⱥ Ⱥ Ⱥ Ⱥ Eliminating a node from the system Kron Reduction ￿ Z pp Z qp ￿ Z mp ￿ ￿ ￿ ￿ ￿ Power Systems I ￿ ￿ ￿ ￿ ￿ ￿ Z mm Z qm ￿ Z pm ￿ Z1m Z qq −Z pq ￿ Z qm −Z pm Z mq Z qq ￿ Z pq ￿ Z1q Z ll = z pq + Z pp + Z qq − 2 Z pq ￿ Z qp −Z pp Z1 p ￿ Then execute Kron reduction on Zll Ⱥ V1 Ⱥ Ⱥ Z11 Ⱥ ￿ Ⱥ Ⱥ ￿ Ⱥ Ⱥ Ⱥ ȺV p Ⱥ Ⱥ Z p1 Ⱥ Ⱥ Ⱥ ȺVq Ⱥ = Ⱥ Z q1 Ⱥ ￿ Ⱥ Ⱥ ￿ Ⱥ Ⱥ Ⱥ ȺVm Ⱥ Ⱥ Z m1 Ⱥ 0 Ⱥ Ⱥ Z q1−Z p1 Ⱥ Ⱥ Ⱥ Adding a Line between two Lines Z1q −Z1 p Ⱥ Ⱥ I1 Ⱥ Ⱥ Ⱥ Ⱥ ￿ Ⱥ Ⱥ ￿ Ⱥ Z pq −Z pp Ⱥ Ⱥ I p Ⱥ Ⱥ Ⱥ Ⱥ Z qq −Z qp Ⱥ Ⱥ I q Ⱥ Ⱥ Ⱥ ￿ Ⱥ ￿ Ⱥ Ⱥ Ⱥ Z mq −Z mp Ⱥ Ⱥ I m Ⱥ Z ll Ⱥ Ⱥ I l Ⱥ Ⱥ Ⱥ Ⱥ Z1 p ￿ Z pp Z ip ￿ ￿ ￿ Z mp ￿ − Z pp ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ Power Systems I Z1m ￿ Z pm Z im ￿ ￿ ￿ Z mi ￿ Z mm − Z pi ￿ − Z pm Z1i ￿ Z pi Z ii Then execute Kron reduction on Zll Z ll = z p 0 + Z pp Ⱥ V1 Ⱥ Ⱥ Z11 Ⱥ ￿ Ⱥ Ⱥ ￿ Ⱥ Ⱥ Ⱥ ȺV p Ⱥ Ⱥ Z p1 Ⱥ Ⱥ Ⱥ Ⱥ Vi Ⱥ = Ⱥ Z i1 Ⱥ ￿ Ⱥ Ⱥ ￿ Ⱥ Ⱥ Ⱥ ȺVm Ⱥ Ⱥ Z m1 Ⱥ 0 Ⱥ Ⱥ− Z p1 Ⱥ Ⱥ Ⱥ − Z1 p Ⱥ Ⱥ I1 Ⱥ ￿ Ⱥ Ⱥ ￿ Ⱥ Ⱥ Ⱥ Ⱥ −Z pp Ⱥ Ⱥ I p Ⱥ Ⱥ Ⱥ Ⱥ − Z ip Ⱥ Ⱥ I q Ⱥ ￿ Ⱥ Ⱥ ￿ Ⱥ Ⱥ Ⱥ Ⱥ − Z mp Ⱥ Ⱥ I m Ⱥ Z ll Ⱥ Ⱥ I l Ⱥ Ⱥ Ⱥ Ⱥ Adding a Line from a Line to Reference Ⱥ V1 Ⱥ Ⱥ Z11 Ⱥ ￿ Ⱥ Ⱥ ￿ Ⱥ Ⱥ Ⱥ ȺV p Ⱥ Ⱥ Z p1 Ⱥ Ⱥ Ⱥ Ⱥ Vi Ⱥ = Ⱥ Z i1 Ⱥ ￿ Ⱥ Ⱥ ￿ Ⱥ Ⱥ Ⱥ ȺVm Ⱥ Ⱥ Z m1 Ⱥ 0 Ⱥ Ⱥ Z l1 Ⱥ Ⱥ Ⱥ Z new jk =Z ￿ ￿ Z mp ￿ Z lp ￿ ￿ ￿ Z pm ￿ Z im ￿ ￿ Z1m old jk − Z ll Z jl Z lk ￿￿￿ Z mi ￿ Z mm Z li ￿ Z lm Z ii ￿ Z pi ￿ ￿ ￿ Z pp ￿ Z ip Z1i ￿ Z1 p Z1l Ⱥ Ⱥ I1 Ⱥ ￿ Ⱥ Ⱥ ￿ Ⱥ Ⱥ Ⱥ Ⱥ Z pl Ⱥ Ⱥ I p Ⱥ Ⱥ Ⱥ Ⱥ Z il Ⱥ Ⱥ I q Ⱥ ￿ Ⱥ Ⱥ ￿ Ⱥ Ⱥ Ⱥ Ⱥ Z ml Ⱥ Ⱥ I m Ⱥ Z ll Ⱥ Ⱥ I l Ⱥ Ⱥ Ⱥ Ⱥ Kron reduction removes an axis (row & column) from a matrix while retaining the axis’s numerical influence Power Systems I ￿ Kron Reduction ￿ ￿ ￿ ￿ zq0 the new off-diagonal row and column filled with (0) the diagonal element (m+1),(m+1) filled with the element impedance start with existing network matrix [m × m] create a new network matrix [(m+1) × (m+1)] with Rule 1: Addition of a branch to the reference Power Systems I ￿ Z-Bus Building Rules ￿ ￿ ￿ ￿ ￿ the new off-diagonal row and column filled with a copy of row p and column p the diagonal element (m+1),(m+1) filled with the element impedance zpq plus the diagonal impedance Zpp connecting to existing bus p start with existing network matrix [m × m] create a new network matrix [(m+1) × (m+1)] with Rule 2: Addition of a branch to an existing bus Power Systems I ￿ Z-Bus Building Rules ￿ ￿ ￿ ￿ the new off-diagonal row and column filled with a copy of row q minus row p and column q minus column p the diagonal element (m+1),(m+1) filled with zpq + Zpp + Zqq - 2 Zpq perform Kron reduction on the m+1 row and column ￿ ￿ connecting to existing buses p and q start with existing network matrix [m × m] create a new network matrix [(m+1) × (m+1)] with Rule 3: Addition of a linking branch Power Systems I ￿ Z-Bus Building Rules ￿ ￿ ￿ ￿ the new off-diagonal row and column filled with a copy of the negative of row p and the negative of column p the diagonal element (m+1),(m+1) filled with zp0 + Zpp perform Kron reduction on the m+1 row and column ￿ ￿ connecting to existing bus p and reference start with existing network matrix [m × m] create a new network matrix [(m+1) × (m+1)] with Rule 4: Addition of a linking branch Power Systems I ￿ Z-Bus Building Rules Network j0.8 j0.4 j0.4 2 1 3 4 2 3 Graph 1 0 Line adding order: 1-0, 2-0, 1-3, 1-2, then 2-3 3 Power Systems I 1 j 0.4 j0.2 Example 5 2 [ j 0.2] 0 j 0. 4 0 0 Ⱥ j 0.4Ⱥ Ⱥ Power Systems I Ⱥ j 0.2 2. Ⱥ Ⱥ 0 Ⱥ j 0.2 3. Ⱥ 0 Ⱥ Ⱥ j 0.2 Ⱥ 0. 1. Example j 0.2Ⱥ 0 Ⱥ Ⱥ j 0.6Ⱥ Ⱥ Ⱥ j 0.171 Ⱥ j 0.057 Ⱥ Ⱥ j 0.171 Ⱥ 0.285 j 0.057 j 0.057 j 0.2 j 0.171Ⱥ j 0.057 Ⱥ Ⱥ j 0.571Ⱥ Ⱥ j 0.2 Ⱥ − j 0.4Ⱥ 0 Ⱥ j 0.6 j 0.2 Ⱥ Ⱥ j 0.2 j1.4 Ⱥ ( j 0.2)( j 0.2) = j 0.17 Z11 = j 0.2 − j1.4 0 Ⱥ j 0.2 Ⱥ 0 j 0.4 Ⱥ 4. Ⱥ j 0.2 0 Ⱥ Ⱥ j 0.2 − j 0.4 Power Systems I Ⱥ j 0.171 Ⱥ j 0.057 5. Ⱥ Ⱥ j 0.171 Ⱥ Ⱥ j 0.114 Ⱥ j 0.16 Ⱥ j 0.08 Ⱥ Ⱥ j 0.12 Ⱥ Example j 0.171 j 0.114 Ⱥ j 0.285 j 0.057 − j 0.228Ⱥ Ⱥ j 0.057 j 0.571 j 0.514 Ⱥ Ⱥ j1.14 Ⱥ − j 0.228 j 0.514 j 0.08 j 0.12Ⱥ j 0.24 j 0.16Ⱥ Ⱥ j 0.16 j 0.34Ⱥ Ⱥ j 0.057 ...
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This note was uploaded on 02/01/2012 for the course EEL 4213 taught by Professor Thomasbaldwin during the Fall '11 term at FSU.

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