Flujo de potencia AC/DC unificado (FUBM)

Introducción

El modelo AC/DC unificado (FUBM: Flexible Unified Branch Model) fué desarrollado en [FUBM1] y [FUBM2]

El modelo general considera que todas las ramas de un sistema de potencia se pueden represnetar por el siguiente esquema:

_images/FUBM.png

Sistema de ecuaciones

\[\begin{split}\left[ \begin{matrix} \Delta Va \\ \Delta Vm \\ \Delta \theta_{sh} \\ \Delta m_a \\ \Delta B_{eq} \\ \Delta B_{eq} \\ \Delta m_a \\ \Delta m_a \\ \Delta \theta_{sh} \end{matrix} \right] = \left[ \begin{matrix} J11 & J12 & J13 & J14 & J15 & J16 & J17 & J18 & J19 \\ J21 & J22 & J23 & J24 & J25 & J26 & J27 & J28 & J29 \\ J31 & J32 & J33 & J34 & J35 & J36 & J37 & J38 & J39 \\ J41 & J42 & J43 & J44 & J45 & J46 & J47 & J48 & J49 \\ J51 & J52 & J53 & J54 & J55 & J56 & J57 & J58 & J59 \\ J61 & J62 & J63 & J64 & J65 & J66 & J67 & J68 & J69 \\ J71 & J72 & J73 & J74 & J75 & J76 & J77 & J78 & J79 \\ J81 & J82 & J83 & J84 & J85 & J86 & J87 & J88 & J89 \\ J91 & J92 & J93 & J94 & J95 & J96 & J97 & J98 & J99 \end{matrix} \right]^{-1} \times \left[ \begin{matrix} \Delta P \\ \Delta Q \\ \Delta Pfsh \\ \Delta Qfma \\ \Delta Beqz \\ \Delta Beqv \\ \Delta Vtma \\ \Delta Qtma \\ \Delta Pfdp \end{matrix} \right]\end{split}\]

Si vemos las derivadas que corponden a cada sub-matriz del Jacobiano, observamos que podemos dar un orden contiguo a las variables para que el sistema se computacionalmente mas eficiente al montar la matriz:

\[\begin{split}\left[ \begin{matrix} \Delta Va \\ \Delta Vm \\ \Delta B_{eq} \\ \Delta m_a \\ \Delta \theta_{sh} \\ \Delta m_a \\ \Delta B_{eq} \\ \Delta m_a \\ \Delta \theta_{sh} \end{matrix} \right] = \left[ \begin{matrix} \Re\left\{\frac{\partial Sbus}{\partial Va}\right\}_{[pvpq,pvpq]} & \Re\left\{\frac{\partial Sbus}{\partial Vm}\right\}_{[pvpq, pq]} & \Re\left\{\frac{\partial Sbus}{\partial Pfsh} \right\}_{[pvpq,:]} & \Re\left\{\frac{\partial Sbus}{\partial Qfma}\right\}_{[pvpq,:]} & \Re\left\{\frac{\partial Sbus}{\partial Beqz}\right\}_{[pvpq,:]} & \Re\left\{\frac{\partial Sbus}{\partial Beqv}\right\}_{[pvpq,:]} & \Re\left\{\frac{\partial Sbus}{\partial Vtma}\right\}_{[pvpq,:]} & \Re\left\{\frac{\partial Sbus}{\partial Qtma}\right\}_{[pvpq,:]} & \Re\left\{\frac{\partial Sbus}{\partial Pfdp}\right\}_{[pvpq,:]} \\ \Im\left\{\frac{\partial Sbus}{\partial Va}\right\}_{[pq, pvpq]} & \Im\left\{\frac{\partial Sbus}{\partial Vm}\right\}_{[pq, pq]} & \Im\left\{\frac{\partial Sbus}{\partial Pfsh}\right\}_{[pq,:]} & \Im\left\{\frac{\partial Sbus}{\partial Qfma}\right\}_{[pq,:]} & \Im\left\{\frac{\partial Sbus}{\partial Beqz}\right\}_{[pq,:]} & \Im\left\{\frac{\partial Sbus}{\partial Beqv}\right\}_{[pq,:]} & \Im\left\{\frac{\partial Sbus}{\partial Vtma}\right\}_{[pq,:]} & \Im\left\{\frac{\partial Sbus}{\partial Qtma}\right\}_{[pq,:]} & \Im\left\{\frac{\partial Sbus}{\partial Pfdp}\right\}_{[pq,:]} \\ \Im\left\{\frac{\partial Sbus}{\partial Va}\right\}_{[iVfBeqbus,pvpq]} & \Im\left\{\frac{\partial Sbus}{\partial Vm}\right\}_{[iVfBeqbus,pq]} & \Im\left\{\frac{\partial Sbus}{\partial Pfsh}\right\}_{[iVfBeqbus,:]} & \Im\left\{\frac{\partial Sbus}{\partial Qfma}\right\}_{[iVfBeqbus,:]} & \Im\left\{\frac{\partial Sbus}{\partial Beqz}\right\}_{[iVfBeqbus,:]} & \Im\left\{\frac{\partial Sbus}{\partial Beqv}\right\}_{[iVfBeqbus,:]} & \Im\left\{\frac{\partial Sbus}{\partial Vtma}\right\}_{[iVfBeqbus,:]} & \Im\left\{\frac{\partial Sbus}{\partial Qtma}\right\}_{[iVfBeqbus,:]} & \Im\left\{\frac{\partial Sbus}{\partial Pfdp}\right\}_{[iVfBeqbus,:]} \\ \Im\left\{\frac{\partial Sbus}{\partial Va}\right\}_{[iVtmabus,pvpq]} & \Im\left\{\frac{\partial Sbus}{\partial Vm}\right\}_{[iVtmabus,pq]} & \Im\left\{\frac{\partial Sbus}{\partial Pfsh}\right\}_{[iVtmabus,:]} & \Im\left\{\frac{\partial Sbus}{\partial Qfma}\right\}_{[iVtmabus,:]} & \Im\left\{\frac{\partial Sbus}{\partial Beqz}\right\}_{[iVtmabus,:]} & \Im\left\{\frac{\partial Sbus}{\partial Beqv}\right\}_{[iVtmabus,:]} & \Im\left\{\frac{\partial Sbus}{\partial Vtma}\right\}_{[iVtmabus,:]} & \Im\left\{\frac{\partial Sbus}{\partial Qtma}\right\}_{[iVtmabus,:]} & \Im\left\{\frac{\partial Sbus}{\partial Pfdp}\right\}_{[iVtmabus,:]} \\ \Re\left\{\frac{\partial Sf}{\partial Va}\right\}_{[iPfsh,pvpq]} & \Re\left\{\frac{\partial Sf}{\partial Vm}\right\}_{[iPfsh,pq]} & \Re\left\{\frac{\partial Sf}{\partial Pfsh}\right\}_{[iPfsh,:]} & \Re\left\{\frac{\partial Sf}{\partial Qfma}\right\}_{[iPfsh,:]} & \Re\left\{\frac{\partial Sf}{\partial Beqz}\right\}_{[iPfsh,:]} & \Re\left\{\frac{\partial Sf}{\partial Beqv}\right\}_{[iPfsh,:]} & \Re\left\{\frac{\partial Sf}{\partial Vtma}\right\}_{[iPfsh,:]} & \Re\left\{\frac{\partial Sf}{\partial Qtma}\right\}_{[iPfsh,:]} & \Re\left\{\frac{\partial Sf}{\partial Pfdp}\right\}_{[iPfsh,:]} \\ \Im\left\{\frac{\partial Sf}{\partial Va}\right\}_{[iQfma,pvpq]} & \Im\left\{\frac{\partial Sf}{\partial Vm}\right\}_{[iQfma,pq]} & \Im\left\{\frac{\partial Sf}{\partial Pfsh}\right\}_{[iQfma,:]} & \Im\left\{\frac{\partial Sf}{\partial Qfma}\right\}_{[iQfma,:]} & \Im\left\{\frac{\partial Sf}{\partial Beqz}\right\}_{[iQfma,:]} & \Im\left\{\frac{\partial Sf}{\partial Beqv}\right\}_{[iQfma,:]} & \Im\left\{\frac{\partial Sf}{\partial Vtma}\right\}_{[iQfma,:]} & \Im\left\{\frac{\partial Sf}{\partial Qtma}\right\}_{[iQfma,:]} & \Im\left\{\frac{\partial Sf}{\partial Pfdp}\right\}_{[iQfma,:]} \\ \Im\left\{\frac{\partial Sf}{\partial Va}\right\}_{[iBeqz,pvpq]} & \Im\left\{\frac{\partial Sf}{\partial Vm}\right\}_{[iBeqz,pq]} & \Im\left\{\frac{\partial Sf}{\partial Pfsh}\right\}_{[iBeqz,:]} & \Im\left\{\frac{\partial Sf}{\partial Qfma}\right\}_{[iBeqz,:]} & \Im\left\{\frac{\partial Sf}{\partial Beqz}\right\}_{[iBeqz,:]} & \Im\left\{\frac{\partial Sf}{\partial Beqv}\right\}_{[iBeqz,:]} & \Im\left\{\frac{\partial Sf}{\partial Vtma}\right\}_{[iBeqz,:]} & \Im\left\{\frac{\partial Sf}{\partial Qtma}\right\}_{[iBeqz,:]} & \Im\left\{\frac{\partial Sf}{\partial Pfdp}\right\}_{[iBeqz,:]} \\ \Im\left\{\frac{\partial St}{\partial Va}\right\}_{[iQtma,pvpq]} & \Im\left\{\frac{\partial St}{\partial Vm}\right\}_{[iQtma,pq]} & \Im\left\{\frac{\partial St}{\partial Pfsh}\right\}_{[iQtma,:]} & \Im\left\{\frac{\partial St}{\partial Qfma}\right\}_{[iQtma,:]} & \Im\left\{\frac{\partial St}{\partial Beqz}\right\}_{[iQtma,:]} & \Im\left\{\frac{\partial St}{\partial Beqv}\right\}_{[iQtma,:]} & \Im\left\{\frac{\partial St}{\partial Vtma}\right\}_{[iQtma,:]} & \Im\left\{\frac{\partial St}{\partial Qtma}\right\}_{[iQtma,:]} & \Im\left\{\frac{\partial St}{\partial Pfdp}\right\}_{[iQtma,:]} \\ \frac{\partial Pfdp}{\partial Va}_{[iPfdp, pvpq]} & \frac{\partial Pfdp}{\partial Vm}_{[iPfdp,pq]} & \frac{\partial Pfdp}{\partial Pfsh}_{[iPfdp,:]} & \frac{\partial Pfdp}{\partial Qfma}_{[iPfdp,:]} & \frac{\partial Pfdp}{\partial Beqz}_{[iPfdp,:]} & \frac{\partial Pfdp}{\partial Beqv}_{[iPfdp,:]} & \frac{\partial Pfdp}{\partial Vtma}_{[iPfdp,:]} & \frac{\partial Pfdp}{\partial Qtma}_{[iPfdp,:]} & \frac{\partial Pfdp}{\partial Pfdp}_{[iPfdp,:]} \end{matrix} \right]^{-1} \times \left[ \begin{matrix} \Delta P \\ \Delta Q \\ \Delta Pfsh \\ \Delta Qfma \\ \Delta Beqz \\ \Delta Beqv \\ \Delta Vtma \\ \Delta Qtma \\ \Delta Pfdp \end{matrix} \right]\end{split}\]

Vector de error

\[ \begin{align}\begin{aligned}\Delta P = \Re \left\{\Delta S_{[pvpq]}\right\}\\\Delta Q = \Im \left\{\Delta S_{[pq]}\right\}\\\Delta Pfsh = \Re \left\{Sf_{[iPfsh]}\right\} - Pfset_{[iPfsh]}\\\Delta Qfma = \Im \left\{Sf_{[iQfma]}\right\} - Qfset_{[iQfma]}\\\Delta Beqz = \Im \left\{Sf_{[iBeqz]}\right\} - 0\\\Delta Beqv = \Im \left\{\Delta S_{[VfBeqbus]}\right\}\\\Delta Vtma = \Im \left\{\Delta S_{[Vtmabus]}\right\}\\\Delta Qtma = \Im \left\{St_{[iQtma]}\right\} - Qtset_{[iQtma]}\\\Delta Pfdp = -\Re \left\{Sf_{[iPfdp]}\right\} + Pfset_{[iPfdp]} + Kdp_{[iPfdp]} \cdot ( Vm_{[busF_{[iPfdp]}]} - Vmfset_{[iPfdp]} )\end{aligned}\end{align} \]

Dónde:

\[\Delta S = V \cdot (Ybus \times V)^{*} - Sbus(Vm)\]

Jacobiano

\[ \begin{align}\begin{aligned}j11 = \Re\left\{\frac{\partial Sbus}{\partial Va}\right\} [pvpq,pvpq]\\j12 = \Re\left\{\frac{\partial Sbus}{\partial Vm}\right\}[pvpq, pq]\\j13 = \Re\left\{\frac{\partial Sbus}{\partial Pfsh} \right\}[pvpq,:]\\j14 = \Re\left\{\frac{\partial Sbus}{\partial Qfma}\right\}[pvpq,:]\\j15 = \Re\left\{\frac{\partial Sbus}{\partial Beqz}\right\}[pvpq,:]\\j16 = \Re\left\{\frac{\partial Sbus}{\partial Beqv}\right\}[pvpq,:]\\j17 = \Re\left\{\frac{\partial Sbus}{\partial Vtma}\right\}[pvpq,:]\\j18 = \Re\left\{\frac{\partial Sbus}{\partial Qtma}\right\}[pvpq,:]\\j19 = \Re\left\{\frac{\partial Sbus}{\partial Pfdp}\right\}_{[pvpq,:]}\end{aligned}\end{align} \]
\[ \begin{align}\begin{aligned}j21 = \Im\left\{\frac{\partial Sbus}{\partial Va}\right\}[pq, pvpq]]\\j22 = \Im\left\{\frac{\partial Sbus}{\partial Vm}\right\}[pq, pq]\\j23 = \Im\left\{\frac{\partial Sbus}{\partial Pfsh}\right\}[pq,:]\\j24 = \Im\left\{\frac{\partial Sbus}{\partial Qfma}\right\}[pq,:]\\j25 = \Im\left\{\frac{\partial Sbus}{\partial Beqz}\right\}[pq,:]\\j26 = \Im\left\{\frac{\partial Sbus}{\partial Beqv}\right\}[pq,:]\\j27 = \Im\left\{\frac{\partial Sbus}{\partial Vtma}\right\}[pq,:]\\j28 = \Im\left\{\frac{\partial Sbus}{\partial Qtma}\right\}[pq,:]\\j29 = \Im\left\{\frac{\partial Sbus}{\partial Pfdp}\right\}[pq,:]\end{aligned}\end{align} \]

Only Pf control elements iPfsh:

\[ \begin{align}\begin{aligned}j31 = \Re\left\{\frac{\partial Sf}{\partial Va}\right\}[iPfsh,pvpq]\\j32 = \Re\left\{\frac{\partial Sf}{\partial Vm}\right\}[iPfsh,pq]\\j33 = \Re\left\{\frac{\partial Sf}{\partial Pfsh}\right\}[iPfsh,:]\\j34 = \Re\left\{\frac{\partial Sf}{\partial Qfma}\right\}[iPfsh,:]\\j35 = \Re\left\{\frac{\partial Sf}{\partial Beqz}\right\}[iPfsh,:]\\j36 = \Re\left\{\frac{\partial Sf}{\partial Beqv}\right\}[iPfsh,:]\\j37 = \Re\left\{\frac{\partial Sf}{\partial Vtma}\right\}[iPfsh,:]\\j38 = \Re\left\{\frac{\partial Sf}{\partial Qtma}\right\}[iPfsh,:]\\j39 = \Re\left\{\frac{\partial Sf}{\partial Pfdp}\right\}[iPfsh,:]\end{aligned}\end{align} \]

Only Qf control elements iQfma:

\[ \begin{align}\begin{aligned}j41 = \Im\left\{\frac{\partial Sf}{\partial Va}\right\}[iQfma,pvpq]\\j42 = \Im\left\{\frac{\partial Sf}{\partial Vm}\right\}[iQfma,pq]\\j43 = \Im\left\{\frac{\partial Sf}{\partial Pfsh}\right\}[iQfma,:]\\j44 = \Im\left\{\frac{\partial Sf}{\partial Qfma}\right\}[iQfma,:]\\j45 = \Im\left\{\frac{\partial Sf}{\partial Beqz}\right\}[iQfma,:]\\j46 = \Im\left\{\frac{\partial Sf}{\partial Beqv}\right\}[iQfma,:]\\j47 = \Im\left\{\frac{\partial Sf}{\partial Vtma}\right\}[iQfma,:]\\j48 = \Im\left\{\frac{\partial Sf}{\partial Qtma}\right\}[iQfma,:]\\j49 = \Im\left\{\frac{\partial Sf}{\partial Pfdp}\right\}[iQfma,:]\end{aligned}\end{align} \]

Only Qf control elements iQfbeq:

\[ \begin{align}\begin{aligned}j51 = \Im\left\{\frac{\partial Sf}{\partial Va}\right\}[iBeqz,pvpq]\\j52 = \Im\left\{\frac{\partial Sf}{\partial Vm}\right\}[iBeqz,pq]\\j53 = \Im\left\{\frac{\partial Sf}{\partial Pfsh}\right\}[iBeqz,:]\\j54 = \Im\left\{\frac{\partial Sf}{\partial Qfma}\right\}[iBeqz,:]\\j55 = \Im\left\{\frac{\partial Sf}{\partial Beqz}\right\}[iBeqz,:]\\j56 = \Im\left\{\frac{\partial Sf}{\partial Beqv}\right\}[iBeqz,:]\\j57 = \Im\left\{\frac{\partial Sf}{\partial Vtma}\right\}[iBeqz,:]\\j58 = \Im\left\{\frac{\partial Sf}{\partial Qtma}\right\}[iBeqz,:]\\j59 = \Im\left\{\frac{\partial Sf}{\partial Pfdp}\right\}[iBeqz,:]\end{aligned}\end{align} \]

Only Vf control elements iVfbeq:

\[ \begin{align}\begin{aligned}j61 = \Im\left\{\frac{\partial Sbus}{\partial Va}\right\}[VfBeqbus,pvpq]\\j62 = \Im\left\{\frac{\partial Sbus}{\partial Vm}\right\}[VfBeqbus,pq]\\j63 = \Im\left\{\frac{\partial Sbus}{\partial Pfsh}\right\}[VfBeqbus,:]\\j64 = \Im\left\{\frac{\partial Sbus}{\partial Qfma}\right\}[VfBeqbus,:]\\j65 = \Im\left\{\frac{\partial Sbus}{\partial Beqz}\right\}[VfBeqbus,:]\\j66 = \Im\left\{\frac{\partial Sbus}{\partial Beqv}\right\}[VfBeqbus,:]\\j67 = \Im\left\{\frac{\partial Sbus}{\partial Vtma}\right\}[VfBeqbus,:]\\j68 = \Im\left\{\frac{\partial Sbus}{\partial Qtma}\right\}[VfBeqbus,:]\\j69 = \Im\left\{\frac{\partial Sbus}{\partial Pfdp}\right\}[VfBeqbus,:]\end{aligned}\end{align} \]

Only Vt control elements iVtma:

\[ \begin{align}\begin{aligned}j71 = \Im\left\{\frac{\partial Sbus}{\partial Va}\right\}[Vtmabus,pvpq]\\j72 = \Im\left\{\frac{\partial Sbus}{\partial Vm}\right\}[Vtmabus,pq]\\j73 = \Im\left\{\frac{\partial Sbus}{\partial Pfsh}\right\}[Vtmabus,:]\\j74 = \Im\left\{\frac{\partial Sbus}{\partial Qfma}\right\}[Vtmabus,:]\\j75 = \Im\left\{\frac{\partial Sbus}{\partial Beqz}\right\}[Vtmabus,:]\\j76 = \Im\left\{\frac{\partial Sbus}{\partial Beqv}\right\}[Vtmabus,:]\\j77 = \Im\left\{\frac{\partial Sbus}{\partial Vtma}\right\}[Vtmabus,:]\\j78 = \Im\left\{\frac{\partial Sbus}{\partial Qtma}\right\}[Vtmabus,:]\\j79 = \Im\left\{\frac{\partial Sbus}{\partial Pfdp}\right\}[Vtmabus,:]\end{aligned}\end{align} \]

Only Qt control elements iQtma:

\[ \begin{align}\begin{aligned}j81 = \Im\left\{\frac{\partial St}{\partial Va}\right\}[iQtma,pvpq]\\j82 = \Im\left\{\frac{\partial St}{\partial Vm}\right\}[iQtma,pq]\\j83 = \Im\left\{\frac{\partial St}{\partial Pfsh}\right\}[iQtma,:]\\j84 = \Im\left\{\frac{\partial St}{\partial Qfma}\right\}[iQtma,:]\\j85 = \Im\left\{\frac{\partial St}{\partial Beqz}\right\}[iQtma,:]\\j86 = \Im\left\{\frac{\partial St}{\partial Beqv}\right\}[iQtma,:]\\j87 = \Im\left\{\frac{\partial St}{\partial Vtma}\right\}[iQtma,:]\\j88 = \Im\left\{\frac{\partial St}{\partial Qtma}\right\}[iQtma,:]\\j89 = \Im\left\{\frac{\partial St}{\partial Pfdp}\right\}[iQtma,:]\end{aligned}\end{align} \]

Only Droop control elements iPfdp:

\[ \begin{align}\begin{aligned}j91 = \frac{\partial Pfdp}{\partial Va}[iPfdp, pvpq]\\j92 = \frac{\partial Pfdp}{\partial Vm}[iPfdp,pq]\\j93 = \frac{\partial Pfdp}{\partial Pfsh}[iPfdp,:]\\j94 = \frac{\partial Pfdp}{\partial Qfma}[iPfdp,:]\\j95 = \frac{\partial Pfdp}{\partial Beqz}[iPfdp,:]\\j96 = \frac{\partial Pfdp}{\partial Beqv}[iPfdp,:]\\j97 = \frac{\partial Pfdp}{\partial Vtma}[iPfdp,:]\\j98 = \frac{\partial Pfdp}{\partial Qtma}[iPfdp,:]\\j99 = \frac{\partial Pfdp}{\partial Pfdp}[iPfdp,:]\end{aligned}\end{align} \]

Derivadas

Derivadas necesarias:

\[\frac{\partial Y}{\partial Pfsh} = ...\]

Derivadas de potencias nodales:

\[ \begin{align}\begin{aligned}\frac{\partial Sbus}{\partial Va} = j[V] \times (Y \times [V])^* + [V] \times (Y \times j[V])^*\\\frac{\partial Sbus}{\partial Vm} = [E] \times (Y \times [V])^* + [V] \times (Y \times [E])^*\\\frac{\partial Sbus}{\partial Pfsh} = [V] \times \left( \frac{\partial Y}{\partial Pfsh} \times [V]\right)^*\\\frac{\partial Sbus}{\partial Qfma} = ...\\\frac{\partial Sbus}{\partial Beqz} = ...\\\frac{\partial Sbus}{\partial Beqv} = ...\\\frac{\partial Sbus}{\partial Vtma} = ...\\\frac{\partial Sbus}{\partial Qtma} = ...\\\frac{\partial Sbus}{\partial Pfdp} = ...\end{aligned}\end{align} \]

Derivadas de potencias de rama desde el lado «from»:

\[ \begin{align}\begin{aligned}\frac{\partial Sf}{\partial Va} = ...\\\frac{\partial Sf}{\partial Vm}\ = ...\\\frac{\partial Sf}{\partial Pfsh} = ...\\\frac{\partial Sf}{\partial Qfma} = ...\\\frac{\partial Sf}{\partial Beqz} = ...\\\frac{\partial Sf}{\partial Beqv} = ...\\\frac{\partial Sf}{\partial Vtma} = ...\\\frac{\partial Sf}{\partial Qtma} = ...\\\frac{\partial Sf}{\partial Pfdp} = ...\end{aligned}\end{align} \]

Derivadas de potencias de rama desde el lado «to»:

\[ \begin{align}\begin{aligned}\frac{\partial St}{\partial Va} = ...\\\frac{\partial St}{\partial Vm} = ...\\\frac{\partial St}{\partial Pfsh} = ...\\\frac{\partial St}{\partial Qfma} = ...\\\frac{\partial St}{\partial Beqz} = ...\\\frac{\partial St}{\partial Beqv} = ...\\\frac{\partial St}{\partial Vtma} = ...\\\frac{\partial St}{\partial Qtma} = ...\\\frac{\partial St}{\partial Pfdp} = ...\end{aligned}\end{align} \]

Derivadas de la potencia «droop»:

\[ \begin{align}\begin{aligned}\frac{\partial Pfdp}{\partial Va} = -\Re\left\{\frac{\partial Sf}{\partial Va}\right\}\\\frac{\partial Pfdp}{\partial Vm} = -\Re\left\{\frac{\partial Sf}{\partial Vm}\right\} + diag(Kdp) \times Cf\\\frac{\partial Pfdp}{\partial Pfsh} = -\Re\left\{\frac{\partial Sf}{\partial Pfsh}\right\}\\\frac{\partial Pfdp}{\partial Qfma} = -\Re\left\{\frac{\partial Sf}{\partial Qtma}\right\}\\\frac{\partial Pfdp}{\partial Beqz} = -\Re\left\{\frac{\partial Sf}{\partial Beqz}\right\}\\\frac{\partial Pfdp}{\partial Beqv} = -\Re\left\{\frac{\partial Sf}{\partial Beqv}\right\}\\\frac{\partial Pfdp}{\partial Vtma} = -\Re\left\{\frac{\partial Sf}{\partial Vtma}\right\}\\\frac{\partial Pfdp}{\partial Qtma} = -\Re\left\{\frac{\partial Sf}{\partial Qtma}\right\}\\\frac{\partial Pfdp}{\partial Pfdp} = -\Re\left\{\frac{\partial Sf}{\partial Pfdp}\right\}\end{aligned}\end{align} \]
[FUBM1]Flexible General Branch Model Unified Power Flow Algorithm for future flexible AC/DC Networks, Abraham Álavarez Bustos and Behzah Kazemtabrizi, IEEE, 2018
[FUBM2]Universal branch model for the solution of optimal power flows in hybrid AC/DC grids, Abraham Álavarez Bustos, Behzah Kazemtabrizi, Mahmoud Shahbazi and Enrique Acha-Daza, International Journal of Electrical and power and Energy Systems, 2021