diff --git a/CHANGELOG.md b/CHANGELOG.md index 8cca336..2ba658e 100755 --- a/CHANGELOG.md +++ b/CHANGELOG.md @@ -5,6 +5,7 @@ - Bumped PMITD compatibility of `PMD` to the latest versions (i.e., v0.14.7). - Added journal citing information to `README.md`. - Added journal citing information to `index` in docs. +- Added ITD boundary network mathematical formulations to `docs`. ## v0.7.6 diff --git a/docs/make.jl b/docs/make.jl index 14210d0..254cdab 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -24,6 +24,7 @@ makedocs( "Manual" => [ "Getting Started" => "manual/quickguide.md", "File Formats" => "manual/fileformat.md", + "Formulations" => "manual/formulations.md", ], "Tutorials" => [ "Beginners Guide" => "tutorials/Beginners Guide.md", diff --git a/docs/src/manual/formulations.md b/docs/src/manual/formulations.md new file mode 100644 index 0000000..7ead2c6 --- /dev/null +++ b/docs/src/manual/formulations.md @@ -0,0 +1,265 @@ +# 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 + +```math +\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}`](@ref NLPowerModelITD) + +ACP to ACPU (AC polar to AC polar unbalanced) + +- **Coordinates**: Polar +- **Variables**: Power-Voltage +- **Model(s)**: NLP-NLP +- **ITD Boundary Math. Formulation**: + +```math +\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}`](@ref NLPowerModelITD) + +ACR to ACRU (AC rectangular to AC rectangular unbalanced) + +- **Coordinates**: Rectangular +- **Variables**: Power-Voltage +- **Model(s)**: NLP-NLP +- **ITD Boundary Math. Formulation**: + +```math +\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}`](@ref IVRPowerModelITD) + +IVR to IVRU (AC-IV rectangular to AC-IV rectangular unbalanced) + +- **Coordinates**: Rectangular +- **Variables**: Current-Voltage +- **Model(s)**: NLP-NLP +- **ITD Boundary Math. Formulation**: + +```math +\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}`](@ref LPowerModelITD) + +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**: + +```math +\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} \\ +% +\end{align} +``` + +## ACR-FBSUBF + +[`NLBFPowerModelITD{ACRPowerModel, FBSUBFPowerModel}`](@ref NLBFPowerModelITD) + +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**: + +```math +\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}`](@ref NLFOTPowerModelITD) + +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**: + +```math +\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}`](@ref NLFOTPowerModelITD) + +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**: + +```math +\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} +``` + +## BFA-LinDist3Flow + +[`BFPowerModelITD{BFAPowerModel, LinDist3FlowPowerModel}`](@ref BFPowerModelITD) + +BFA to LinDist3Flow (Branch flow approximation to LinDist3Flow approximation) + +- **Coordinates**: W-space +- **Variables**: Power-Voltage (W) +- **Model(s)**: Apprx.-Apprx. +- **ITD Boundary Math. Formulation**: + +```math +\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({w_{\beta^{^\mathcal{T}}}} \Big) = \Big(w^{a}_{\beta^{^\mathcal{D}}}\Big), \ \forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in \Lambda \mbox{ - W equality - phase a} \\ +& \Big({w_{\beta^{^\mathcal{T}}}} \Big) = \Big(w^{b}_{\beta^{^\mathcal{D}}}\Big), \ \forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in \Lambda \mbox{ - W equality - phase b} \\ +& \Big({w_{\beta^{^\mathcal{T}}}} \Big) = \Big(w^{c}_{\beta^{^\mathcal{D}}}\Big), \ \forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in \Lambda \mbox{ - W equality - phase c} \\ +% +\end{align} +``` + +## SOCBF-LinDist3Flow + +[`BFPowerModelITD{SOCBFPowerModel, LinDist3FlowPowerModel}`](@ref BFPowerModelITD) + +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**: + +```math +\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({w_{\beta^{^\mathcal{T}}}} \Big) = \Big(w^{a}_{\beta^{^\mathcal{D}}}\Big), \ \forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in \Lambda \mbox{ - W equality - phase a} \\ +& \Big({w_{\beta^{^\mathcal{T}}}} \Big) = \Big(w^{b}_{\beta^{^\mathcal{D}}}\Big), \ \forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in \Lambda \mbox{ - W equality - phase b} \\ +& \Big({w_{\beta^{^\mathcal{T}}}} \Big) = \Big(w^{c}_{\beta^{^\mathcal{D}}}\Big), \ \forall (\beta^{^\mathcal{T}},\beta^{^\mathcal{D}}) \in \Lambda \mbox{ - W equality - phase c} \\ +% +\end{align} +``` diff --git a/docs/src/reference/data_models.md b/docs/src/reference/data_models.md index 7e12146..bf2c0b3 100644 --- a/docs/src/reference/data_models.md +++ b/docs/src/reference/data_models.md @@ -27,4 +27,6 @@ assign_boundary_buses! resolve_units! replicate sol_data_model! +calc_transmission_branch_flow_ac! +transform_pmitd_solution_to_eng! ```