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}\]
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 |