d2Abr_dV2
- d2Abr_dV2(d2F_dV2, dF_dV1, dF_dV2, F, V, mu)
d2Abr_dV2()
- Computes 2nd derivatives of|branch flow|^2
w.r.t. V.The derivatives can be take with respect to polar or cartesian coordinates of voltage, depending on the first 3 arguments. Flows could be complex current or complex or real power. Notation below is based on complex power. [H11, H12, H21, H22] = D2ABR_DV2(D2F_DV2, DF_DV1, DF_DV2, F, V, MU) Returns 4 matrices containing the partial derivatives w.r.t. voltage components (angle, magnitude or real, imaginary) of the product of a vector MU with the 1st partial derivatives of the square of the magnitude of branch flows. Takes as inputs a handle to a function that evaluates the 2nd derivatives of the flows (with args V and mu only), sparse first derivative matrices of flow, flow vector, voltage vector V and nl x 1 vector of multipliers MU. Output matrices are sparse. Example: f = branch(:, F_BUS); Cf = sparse(1:nl, f, ones(nl, 1), nl, nb); [Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch); [dSf_dV1, dSf_dV2, dSt_dV1, dSt_dV2, Sf, St] = ... dSbr_dV(branch, Yf, Yt, V); dF_dV1 = dSf_dV1; dF_dV2 = dSf_dV2; F = Sf; d2F_dV2 = @(V, mu)d2Sbr_dV2(Cf, Yf, V, mu, 0); [H11, H12, H21, H22] = ... d2Abr_dV2(d2F_dV2, dF_dV1, dF_dV2, F, V, mu); Here the output matrices correspond to: H11 = d/dV1 (dAF_dV1.' * mu) H12 = d/dV2 (dAF_dV1.' * mu) H21 = d/dV1 (dAF_dV2.' * mu) H22 = d/dV2 (dAF_dV2.' * mu)
See also
dAbr_dV()
,dIbr_dV()
,dSbr_dV()
.For more details on the derivations behind the derivative code used in MATPOWER information, see:
[TN2] R. D. Zimmerman, "AC Power Flows, Generalized OPF Costs and their Derivatives using Complex Matrix Notation", MATPOWER Technical Note 2, February 2010. [Online]. Available: https://matpower.org/docs/TN2-OPF-Derivatives.pdf doi: 10.5281/zenodo.3237866