ITD Network Formulations

There is a diverse number of formulations that can be used to solve the OPFITD, PFITD, and other problem specifications. These can be found in types.jl. A non-exhaustive list of the supported ITD boundary mathematical formulations is presented below.

Sets, Parameters, and (General) Variables

\[\begin{align} % \mbox{sets:} & \nonumber \\ & N \mbox{ - Set of buses}\nonumber \\ & \mathcal{T} \mbox{ - Belongs to transmission network}\nonumber \\ & \mathcal{D} \mbox{ - Belongs to distribution network}\nonumber \\ & \mathcal{B} \mbox{ - Set of boundary links}\nonumber \\ % \mbox{parameters:} & \nonumber \\ & \Re \mbox{ - Real part}\nonumber \\ & \Im \mbox{ - Imaginary part}\nonumber \\ & \Phi = a, b, c \mbox{ - Multi-conductor phases}\nonumber \\ & \chi \rightarrow{\mathcal{T}},{\mathcal{D}} \mbox{ - Belongs to Transmission or Distribution}\nonumber \\ & \beta^{^{\chi}} \mbox{ - Boundary bus}\nonumber \\ % \mbox{variables:} & \nonumber \\ & P_{\beta^{^\mathcal{T}}\beta^{^\mathcal{D}}}^{^\mathcal{T}} \mbox{ - Active power flow from Transmisison boundary bus to Distribution boundary bus}\nonumber \\ & Q_{\beta^{^\mathcal{T}}\beta^{^\mathcal{D}}}^{^\mathcal{T}} \mbox{ - Reactive power flow from Transmisison boundary bus to Distribution boundary bus}\nonumber \\ & P_{\beta^{^\mathcal{D}}\beta^{^\mathcal{T}}}^{{\mathcal{D},\varphi}} \mbox{ - Active power flow from Distribution boundary bus phase $\varphi$ to Transmission boundary bus}\nonumber \\ & Q_{\beta^{^\mathcal{D}}\beta^{^\mathcal{T}}}^{{\mathcal{D},\varphi}} \mbox{ - Reactive power flow from Distribution boundary bus phase $\varphi$ to Transmission boundary bus}\nonumber \\ & V_i^{^\mathcal{T}} \mbox{ - Voltage magnitude at bus $i$}\nonumber \\ & \theta_i^{^\mathcal{T}} \mbox{ - Voltage angle at bus $i$}\nonumber \\ & v_i^{\mathcal{D}, \varphi} \mbox{ - Voltage magnitude at bus $i$ phase $\varphi$}\nonumber \\ & \theta_i^{\mathcal{D}, \varphi} \mbox{ - Voltage angle at bus $i$ phase $\varphi$}\nonumber \\ % \end{align}\]

ACP-ACPU

NLPowerModelITD{ACPPowerModel, ACPUPowerModel}

ACP to ACPU (AC polar to AC polar unbalanced)

Coordinates: Polar
Variables: Power-Voltage
Model(s): NLP-NLP
ITD Boundary Math. Formulation:

\[\begin{align} % \mbox{ITD boundaries: } & \nonumber \\ & \sum_{\varphi \in \Phi} P_{\beta^{^\mathcal{D}}\beta^{^\mathcal{T}}}^{\mathcal{D},\varphi} + P_{\beta^{^\mathcal{T}}\beta^{^\mathcal{D}}}^{^\mathcal{T}} = 0, \ \forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in {\Lambda} \mbox{ - Active power flow at boundary} \\ & \sum_{\varphi \in \Phi} Q_{\beta^{^\mathcal{D}}\beta^{^\mathcal{T}}}^{\mathcal{D},\varphi} + Q_{\beta^{^\mathcal{T}}\beta^{^\mathcal{D}}}^{^\mathcal{T}} = 0, \ \forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in {\Lambda} \mbox{ - Reactive power flow at boundary} \\ & V_{\beta^{^\mathcal{T}}} = v_{\beta^{^\mathcal{D}}}^{^{a}}, \ \forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in {\Lambda} \mbox{ - Voltage mag. equality - phase a} \\ & V_{\beta^{^\mathcal{T}}} = v_{\beta^{^\mathcal{D}}}^{^{b}}, \ \forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in {\Lambda} \mbox{ - Voltage mag. equality - phase b} \\ & V_{\beta^{^\mathcal{T}}} = v_{\beta^{^\mathcal{D}}}^{^{c}}, \ \forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in {\Lambda} \mbox{ - Voltage mag. equality - phase c} \\ & \theta_{\beta^{^\mathcal{T}}} = \theta_{\beta^{^\mathcal{D}}}^{^{a}}, \ \forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in {\Lambda} \mbox{ - Voltage ang. equality - phase a} \\ & \theta_{\beta^{^\mathcal{D}}}^{^{b}} = (\theta_{\beta^{^\mathcal{D}}}^{^{a}} -120^{\circ}), \ \forall \beta^{^\mathcal{D}} \in N^{^\mathcal{B}} \cap N^{^\mathcal{D}} \mbox{ - Voltage ang. equality - phase b} \\ & \theta_{\beta^{^\mathcal{D}}}^{^{c}} = (\theta_{\beta^{^\mathcal{D}}}^{^{a}} +120^{\circ}), \ \forall \beta^{^\mathcal{D}} \in N^{^\mathcal{B}} \cap N^{^\mathcal{D}} \mbox{ - Voltage ang. equality - phase c} \\ % \end{align}\]

ACR-ACRU

NLPowerModelITD{ACRPowerModel, ACRUPowerModel}

ACR to ACRU (AC rectangular to AC rectangular unbalanced)

Coordinates: Rectangular
Variables: Power-Voltage
Model(s): NLP-NLP
ITD Boundary Math. Formulation:

\[\begin{align} % \mbox{ITD boundaries: } & \nonumber \\ & \sum_{\varphi \in \Phi} P_{\beta^{^\mathcal{D}}\beta^{^\mathcal{T}}}^{\mathcal{D},\varphi} + P_{\beta^{^\mathcal{T}}\beta^{^\mathcal{D}}}^{^\mathcal{T}} = 0, \ \forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in {\Lambda} \mbox{ - Active power flow at boundary} \\ & \sum_{\varphi \in \Phi} Q_{\beta^{^\mathcal{D}}\beta^{^\mathcal{T}}}^{\mathcal{D},\varphi} + Q_{\beta^{^\mathcal{T}}\beta^{^\mathcal{D}}}^{^\mathcal{T}} = 0, \ \forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in {\Lambda} \mbox{ - Reactive power flow at boundary} \\ & \Big({V^\Re_{\beta^{^\mathcal{T}}}}\Big)^2 \!\!\!\! + \! \Big({V^\Im_{\beta^{^\mathcal{T}}}}\Big)^2 \!\!\!\! = \!\Big(v_{\beta^{^\mathcal{D}}}^{^{a,\Re}}\Big)^2 \!\!\!\!+\! \Big(v_{\beta^{^\mathcal{D}}}^{^{a,\Im}}\Big)^2,\forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in \Lambda \mbox{ - Voltage mag. equality - phase a} \\ & \Big({V^\Re_{\beta^{^\mathcal{T}}}}\Big)^2 \!\!\!\!+\! \Big({V^\Im_{\beta^{^\mathcal{T}}}}\Big)^2 \!\!\!\!=\! \Big(v_{\beta^{^\mathcal{D}}}^{^{b,\Re}}\Big)^2 \!\!\!\!+\! \Big(v_{\beta^{^\mathcal{D}}}^{^{b,\Im}}\Big)^2,\forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in \Lambda \mbox{ - Voltage mag. equality - phase b} \\ & \Big({V^\Re_{\beta^{^\mathcal{T}}}}\Big)^2 \!\!\!\!+\! \Big({V^\Im_{\beta^{^\mathcal{T}}}}\Big)^2 \!\!\!\!=\! \Big(v_{\beta^{^\mathcal{D}}}^{^{c,\Re}}\Big)^2 \!\!\!\!+\! \Big(v_{\beta^{^\mathcal{D}}}^{^{c,\Im}}\Big)^2,\forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in \Lambda \mbox{ - Voltage mag. equality - phase c} \\ & \Big({V^\Im_{\beta^{^\mathcal{T}}}}\Big) = \Big(v_{\beta^{^\mathcal{D}}}^{^{a,\Im}}\Big),\ \ \ \forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in \Lambda \mbox{ - Voltage ang. equality - phase a} \\ & \Big(v_{\beta^{^\mathcal{D}}}^{^{b,\Im}}\Big) = tan\Bigg(atan\bigg(\frac{v_{\beta^{^\mathcal{D}}}^{^{a,\Im}}}{v_{\beta^{^\mathcal{D}}}^{^{a,\Re}}} \bigg) -120^{\circ} \Bigg) v_{\beta^{^\mathcal{D}}}^{^{b,\Re}}, \ \forall \beta^{^\mathcal{D}} \in N^{^\mathcal{B}} \cap N^{^\mathcal{D}} \mbox{ - Voltage ang. equality - phase b} \\ & \Big(v_{\beta^{^\mathcal{D}}}^{^{c,\Im}}\Big) = tan\Bigg(atan\bigg(\frac{v_{\beta^{^\mathcal{D}}}^{^{a,\Im}}}{v_{\beta^{^\mathcal{D}}}^{^{a,\Re}}} \bigg) +120^{\circ} \Bigg) v_{\beta^{^\mathcal{D}}}^{^{c,\Re}}, \ \forall \beta^{^\mathcal{D}} \in N^{^\mathcal{B}} \cap N^{^\mathcal{D}} \mbox{ - Voltage ang. equality - phase c} \\ % \end{align}\]

IVR-IVRU

IVRPowerModelITD{IVRPowerModel, IVRUPowerModel}

IVR to IVRU (AC-IV rectangular to AC-IV rectangular unbalanced)

Coordinates: Rectangular
Variables: Current-Voltage
Model(s): NLP-NLP
ITD Boundary Math. Formulation:

\[\begin{align} % \mbox{ITD boundaries: } & \nonumber \\ & {V^\Re_{\beta^{^\mathcal{T}}}} \Re\Big(I_{\beta^{^\mathcal{T}}\beta^{^\mathcal{D}}}^{^\mathcal{T}}\Big) + {V^\Im_{\beta^{^\mathcal{T}}}} \Im\Big(I_{\beta^{^\mathcal{T}}\beta^{^\mathcal{D}}}^{^\mathcal{T}}\Big) = \!\!-\!\!\Bigg[\sum_{\varphi \in \Phi} \Bigg( \Big(v_{\beta^{^\mathcal{D}}}^{^{\varphi,\Re}}\Big) \Re\Big(I_{\beta^{^\mathcal{D}}\beta^{^\mathcal{T}}}^{\mathcal{D},\varphi}\Big) \!\!+\!\! \Big(v_{\beta^{^\mathcal{D}}}^{^{\varphi,\Im}}\Big) \Im\Big(I_{\beta^{^\mathcal{D}}\beta^{^\mathcal{T}}}^{\mathcal{D},\varphi}\Big) \Bigg) \Bigg], \ \forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in \Lambda \mbox{ - Active power flow at boundary} \\ & {V^\Im_{\beta^{^\mathcal{T}}}} \Re\Big(I_{\beta^{^\mathcal{T}}\beta^{^\mathcal{D}}}^{^\mathcal{T}}\Big) - {V^\Re_{\beta^{^\mathcal{T}}}} \Im\Big(I_{\beta^{^\mathcal{T}}\beta^{^\mathcal{D}}}^{^\mathcal{T}}\Big) = \!\!-\!\! \Bigg[\sum_{\varphi \in \Phi} \Bigg( \Big(v_{\beta^{^\mathcal{D}}}^{^{\varphi,\Im}}\Big) \Re\Big(I_{\beta^{^\mathcal{D}}\beta^{^\mathcal{T}}}^{\mathcal{D},\varphi}\Big) \!\!-\!\! \Big(v_{\beta^{^\mathcal{D}}}^{^{\varphi,\Re}}\Big) \Im\Big(I_{\beta^{^\mathcal{D}}\beta^{^\mathcal{T}}}^{\mathcal{D},\varphi}\Big) \Bigg) \Bigg], \ \forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in \Lambda \mbox{ - Reactive power flow at boundary} \\ & \Big({V^\Re_{\beta^{^\mathcal{T}}}}\Big)^2 \!\!\!\! + \! \Big({V^\Im_{\beta^{^\mathcal{T}}}}\Big)^2 \!\!\!\! = \!\Big(v_{\beta^{^\mathcal{D}}}^{^{a,\Re}}\Big)^2 \!\!\!\!+\! \Big(v_{\beta^{^\mathcal{D}}}^{^{a,\Im}}\Big)^2,\forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in \Lambda \mbox{ - Voltage mag. equality - phase a} \\ & \Big({V^\Re_{\beta^{^\mathcal{T}}}}\Big)^2 \!\!\!\!+\! \Big({V^\Im_{\beta^{^\mathcal{T}}}}\Big)^2 \!\!\!\!=\! \Big(v_{\beta^{^\mathcal{D}}}^{^{b,\Re}}\Big)^2 \!\!\!\!+\! \Big(v_{\beta^{^\mathcal{D}}}^{^{b,\Im}}\Big)^2,\forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in \Lambda \mbox{ - Voltage mag. equality - phase b} \\ & \Big({V^\Re_{\beta^{^\mathcal{T}}}}\Big)^2 \!\!\!\!+\! \Big({V^\Im_{\beta^{^\mathcal{T}}}}\Big)^2 \!\!\!\!=\! \Big(v_{\beta^{^\mathcal{D}}}^{^{c,\Re}}\Big)^2 \!\!\!\!+\! \Big(v_{\beta^{^\mathcal{D}}}^{^{c,\Im}}\Big)^2,\forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in \Lambda \mbox{ - Voltage mag. equality - phase c} \\ & \Big({V^\Im_{\beta^{^\mathcal{T}}}}\Big) = \Big(v_{\beta^{^\mathcal{D}}}^{^{a,\Im}}\Big),\ \ \ \forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in \Lambda \mbox{ - Voltage ang. equality - phase a} \\ & \Big(v_{\beta^{^\mathcal{D}}}^{^{b,\Im}}\Big) = tan\Bigg(atan\bigg(\frac{v_{\beta^{^\mathcal{D}}}^{^{a,\Im}}}{v_{\beta^{^\mathcal{D}}}^{^{a,\Re}}} \bigg) -120^{\circ} \Bigg) v_{\beta^{^\mathcal{D}}}^{^{b,\Re}}, \ \forall \beta^{^\mathcal{D}} \in N^{^\mathcal{B}} \cap N^{^\mathcal{D}} \mbox{ - Voltage ang. equality - phase b} \\ & \Big(v_{\beta^{^\mathcal{D}}}^{^{c,\Im}}\Big) = tan\Bigg(atan\bigg(\frac{v_{\beta^{^\mathcal{D}}}^{^{a,\Im}}}{v_{\beta^{^\mathcal{D}}}^{^{a,\Re}}} \bigg) +120^{\circ} \Bigg) v_{\beta^{^\mathcal{D}}}^{^{c,\Re}}, \ \forall \beta^{^\mathcal{D}} \in N^{^\mathcal{B}} \cap N^{^\mathcal{D}} \mbox{ - Voltage ang. equality - phase c} \\ % \end{align}\]

NFA-NFAU

LPowerModelITD{NFAPowerModel, NFAUPowerModel}

NFA to NFAU (Linear Network flow approximation to Linear Network flow approximation unbalanced)

Coordinates: N/A
Variables: N/A
Model(s): Apprx.-Apprx.
ITD Boundary Math. Formulation:

ACR-FBSUBF

NLBFPowerModelITD{ACRPowerModel, FBSUBFPowerModel}

ACR to FBSUBF (AC rectangular to forward-backward sweep unbalanced branch flow approximation)

Coordinates: Rectangular
Variables: Power-Voltage
Model(s): NLP-Apprx.
ITD Boundary Math. Formulation:

\[\begin{align} % \mbox{ITD boundaries: } & \nonumber \\ & \sum_{\varphi \in \Phi} P_{\beta^{^\mathcal{D}}\beta^{^\mathcal{T}}}^{\mathcal{D},\varphi} + P_{\beta^{^\mathcal{T}}\beta^{^\mathcal{D}}}^{^\mathcal{T}} = 0, \ \forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in {\Lambda} \mbox{ - Active power flow at boundary} \\ & \sum_{\varphi \in \Phi} Q_{\beta^{^\mathcal{D}}\beta^{^\mathcal{T}}}^{\mathcal{D},\varphi} + Q_{\beta^{^\mathcal{T}}\beta^{^\mathcal{D}}}^{^\mathcal{T}} = 0, \ \forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in {\Lambda} \mbox{ - Reactive power flow at boundary} \\ & \Big({V^\Re_{\beta^{^\mathcal{T}}}}\Big)^2 \!\!\!\! + \! \Big({V^\Im_{\beta^{^\mathcal{T}}}}\Big)^2 \!\!\!\! = \!\Big(v_{\beta^{^\mathcal{D}}}^{^{a,\Re}}\Big)^2 \!\!\!\!+\! \Big(v_{\beta^{^\mathcal{D}}}^{^{a,\Im}}\Big)^2,\forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in \Lambda \mbox{ - Voltage mag. equality - phase a} \\ & \Big({V^\Re_{\beta^{^\mathcal{T}}}}\Big)^2 \!\!\!\!+\! \Big({V^\Im_{\beta^{^\mathcal{T}}}}\Big)^2 \!\!\!\!=\! \Big(v_{\beta^{^\mathcal{D}}}^{^{b,\Re}}\Big)^2 \!\!\!\!+\! \Big(v_{\beta^{^\mathcal{D}}}^{^{b,\Im}}\Big)^2,\forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in \Lambda \mbox{ - Voltage mag. equality - phase b} \\ & \Big({V^\Re_{\beta^{^\mathcal{T}}}}\Big)^2 \!\!\!\!+\! \Big({V^\Im_{\beta^{^\mathcal{T}}}}\Big)^2 \!\!\!\!=\! \Big(v_{\beta^{^\mathcal{D}}}^{^{c,\Re}}\Big)^2 \!\!\!\!+\! \Big(v_{\beta^{^\mathcal{D}}}^{^{c,\Im}}\Big)^2,\forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in \Lambda \mbox{ - Voltage mag. equality - phase c} \\ & \Big({V^\Im_{\beta^{^\mathcal{T}}}}\Big) = \Big(v_{\beta^{^\mathcal{D}}}^{^{a,\Im}}\Big),\ \ \ \forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in \Lambda \mbox{ - Voltage ang. equality - phase a} \\ & \Big(v_{\beta^{^\mathcal{D}}}^{^{b,\Im}}\Big) = tan\Bigg(atan\bigg(\frac{v_{\beta^{^\mathcal{D}}}^{^{a,\Im}}}{v_{\beta^{^\mathcal{D}}}^{^{a,\Re}}} \bigg) -120^{\circ} \Bigg) v_{\beta^{^\mathcal{D}}}^{^{b,\Re}}, \ \forall \beta^{^\mathcal{D}} \in N^{^\mathcal{B}} \cap N^{^\mathcal{D}} \mbox{ - Voltage ang. equality - phase b} \\ & \Big(v_{\beta^{^\mathcal{D}}}^{^{c,\Im}}\Big) = tan\Bigg(atan\bigg(\frac{v_{\beta^{^\mathcal{D}}}^{^{a,\Im}}}{v_{\beta^{^\mathcal{D}}}^{^{a,\Re}}} \bigg) +120^{\circ} \Bigg) v_{\beta^{^\mathcal{D}}}^{^{c,\Re}}, \ \forall \beta^{^\mathcal{D}} \in N^{^\mathcal{B}} \cap N^{^\mathcal{D}} \mbox{ - Voltage ang. equality - phase c} \\ % \end{align}\]

ACR-FOTRU

NLFOTPowerModelITD{ACRPowerModel, FOTRUPowerModel}

ACR to FOTRU (AC rectangular to first-order Taylor rectangular unbalanced approximation)

Coordinates: Rectangular
Variables: Power-Voltage
Model(s): NLP-Apprx.
ITD Boundary Math. Formulation:

\[\begin{align} % \mbox{ITD boundaries: } & \nonumber \\ & \sum_{\varphi \in \Phi} P_{\beta^{^\mathcal{D}}\beta^{^\mathcal{T}}}^{\mathcal{D},\varphi} + P_{\beta^{^\mathcal{T}}\beta^{^\mathcal{D}}}^{^\mathcal{T}} = 0, \ \forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in {\Lambda} \mbox{ - Active power flow at boundary} \\ & \sum_{\varphi \in \Phi} Q_{\beta^{^\mathcal{D}}\beta^{^\mathcal{T}}}^{\mathcal{D},\varphi} + Q_{\beta^{^\mathcal{T}}\beta^{^\mathcal{D}}}^{^\mathcal{T}} = 0, \ \forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in {\Lambda} \mbox{ - Reactive power flow at boundary} \\ & \Big({V^\Re_{\beta^{^\mathcal{T}}}}\Big)^2 \!\!\!\! + \! \Big({V^\Im_{\beta^{^\mathcal{T}}}}\Big)^2 \!\!\!\! = \!\Big(v_{\beta^{^\mathcal{D}}}^{^{a,\Re}}\Big)^2 \!\!\!\!+\! \Big(v_{\beta^{^\mathcal{D}}}^{^{a,\Im}}\Big)^2,\forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in \Lambda \mbox{ - Voltage mag. equality - phase a} \\ & \Big({V^\Re_{\beta^{^\mathcal{T}}}}\Big)^2 \!\!\!\!+\! \Big({V^\Im_{\beta^{^\mathcal{T}}}}\Big)^2 \!\!\!\!=\! \Big(v_{\beta^{^\mathcal{D}}}^{^{b,\Re}}\Big)^2 \!\!\!\!+\! \Big(v_{\beta^{^\mathcal{D}}}^{^{b,\Im}}\Big)^2,\forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in \Lambda \mbox{ - Voltage mag. equality - phase b} \\ & \Big({V^\Re_{\beta^{^\mathcal{T}}}}\Big)^2 \!\!\!\!+\! \Big({V^\Im_{\beta^{^\mathcal{T}}}}\Big)^2 \!\!\!\!=\! \Big(v_{\beta^{^\mathcal{D}}}^{^{c,\Re}}\Big)^2 \!\!\!\!+\! \Big(v_{\beta^{^\mathcal{D}}}^{^{c,\Im}}\Big)^2,\forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in \Lambda \mbox{ - Voltage mag. equality - phase c} \\ & \Big({V^\Im_{\beta^{^\mathcal{T}}}}\Big) = \Big(v_{\beta^{^\mathcal{D}}}^{^{a,\Im}}\Big),\ \ \ \forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in \Lambda \mbox{ - Voltage ang. equality - phase a} \\ & \Big(v_{\beta^{^\mathcal{D}}}^{^{b,\Im}}\Big) = tan\Bigg(atan\bigg(\frac{v_{\beta^{^\mathcal{D}}}^{^{a,\Im}}}{v_{\beta^{^\mathcal{D}}}^{^{a,\Re}}} \bigg) -120^{\circ} \Bigg) v_{\beta^{^\mathcal{D}}}^{^{b,\Re}}, \ \forall \beta^{^\mathcal{D}} \in N^{^\mathcal{B}} \cap N^{^\mathcal{D}} \mbox{ - Voltage ang. equality - phase b} \\ & \Big(v_{\beta^{^\mathcal{D}}}^{^{c,\Im}}\Big) = tan\Bigg(atan\bigg(\frac{v_{\beta^{^\mathcal{D}}}^{^{a,\Im}}}{v_{\beta^{^\mathcal{D}}}^{^{a,\Re}}} \bigg) +120^{\circ} \Bigg) v_{\beta^{^\mathcal{D}}}^{^{c,\Re}}, \ \forall \beta^{^\mathcal{D}} \in N^{^\mathcal{B}} \cap N^{^\mathcal{D}} \mbox{ - Voltage ang. equality - phase c} \\ % \end{align}\]

ACP-FOTPU

NLFOTPowerModelITD{ACPPowerModel, FOTPUPowerModel}

ACP to FOTPU (AC rectangular to first-order Taylor polar unbalanced approximation)

Coordinates: Polar
Variables: Power-Voltage
Model(s): NLP-Apprx.
ITD Boundary Math. Formulation:

BFA-LinDist3Flow

BFPowerModelITD{BFAPowerModel, LinDist3FlowPowerModel}

BFA to LinDist3Flow (Branch flow approximation to LinDist3Flow approximation)

Coordinates: W-space
Variables: Power-Voltage (W)
Model(s): Apprx.-Apprx.
ITD Boundary Math. Formulation:

SOCBF-LinDist3Flow

BFPowerModelITD{SOCBFPowerModel, LinDist3FlowPowerModel}

SOCBF to LinDist3Flow (Second-order cone branch flow relaxation to LinDist3Flow approximation)

Coordinates: W-space
Variables: Power-Voltage (W)
Model(s): Relax.-Apprx.
ITD Boundary Math. Formulation: