As ThreePhasePowerModels implements a variety of power network optimization problems, the implementation is the best reference for precise mathematical formulations. This section provides a complex number based mathematical specification for a prototypical unbalanced AC Optimal Power Flow problem, to provide an overview of the typical mathematical models in ThreePhasePowerModels.
ThreePhasePowerModels implements a generalized version of the AC Optimal Power Flow problem, from Matpower but extended to take into account phase unbalance [1]. These generalizations make it possible for ThreePhasePowerModels to more accurately capture real-world distribution network datasets. The core generalizations are,
- Support for multiple load and shunt components on each bus
- Line charging (shunt) that supports a conductance and asymmetrical values
In the mathematical description below,
- Bold typeface indicates a vector ($\in \mathbb{C}^c$) or matrix ($\in \mathbb{C}^{c\times c}$)
- Operator $diag$ takes the diagonal (vector) from a square matrix
- The set of complex numbers is $\mathbb{C}$ and real numbers is $\mathbb{R}$
- Superscript $H$ indicates complex conjugate transpose (Hermitian adjoint)
- Note that complex power is defined as $\mathbf{S}_{ij} = \mathbf{V}_{i} \mathbf{I}_{ij}^H$ and is therefore a complex matrix of dimension $c \times c$
- The line $\mathbf{Y}^c_{ij}, \mathbf{Y}^c_{ji}$ and bus $\mathbf{Y}^s_{k}$ shunt matrices do not need to be diagonal
The definitions of the sets involved remain unchanged w.r.t. the balanced OPF problem definition, except for the addition of the conductor set:
\[\begin{align}
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\mbox{sets:} & \nonumber \\
& N \mbox{ - buses}\nonumber \\
& R \mbox{ - references buses}\nonumber \\
& E, E^R \mbox{ - branches, forward and reverse orientation} \nonumber \\
& G, G_i \mbox{ - generators and generators at bus $i$} \nonumber \\
& L, L_i \mbox{ - loads and loads at bus $i$} \nonumber \\
& S, S_i \mbox{ - shunts and shunts at bus $i$} \nonumber \\
& C \mbox{ - conductors} \nonumber \\
%
\end{align}\]
where the set of conductors $C$ typically equals $\{ a,b,c\}$.
\[\begin{align}
\mbox{data:} & \nonumber \\
& S^{gl}_{k,c}, S^{gu}_{k,c} \in \mathbb{C} \;\; \forall k \in G, \forall c \in C \nonumber; \mathbf{S}^{gl}_{k}:= [S^{gl}_{k,c}]_{c \in C}, \mathbf{S}^{gu}_{k} := [S^{gu}_{k,c}]_{c \in C} \\
& c_{2k}, c_{1k}, c_{0k} \in \mathbb{R} \;\; \forall k \in G \nonumber \\
& v^l_{i,c}, v^u_{i,c} \in \mathbb{R} \;\; \forall i \in N, \forall c \in C \nonumber; \mathbf{v}^l_{i} := [v^l_{i,c}]_{c \in C}, \mathbf{v}^u_{i} := [v^u_{i,c}]_{c \in C} \\
& S^d_{k,c}\in \mathbb{C} \;\; \forall k \in L, \forall c \in C \nonumber; \mathbf{S}^d_{k} := [S^d_{k,c}]_{c \in C} \\
& \mathbf{Y}^s_{k}\in \mathbb{C}^{c\times c} \;\; \forall k \in S \nonumber \\
& \mathbf{Y}_{ij}, \mathbf{Y}^c_{ij}, \mathbf{Y}^c_{ji}\in \mathbb{C}^{c\times c} \;\; \forall (i,j) \in E \nonumber \\
& {s^u}_{ij,c}, \theta^{\Delta l}_{ij,c}, \theta^{\Delta u}_{ij,c} \in \mathbb{R}\;\; \forall (i,j) \in E, \forall c \in C \nonumber, {\mathbf{s}^u}_{ij} := [{s^u}_{ij,c}]_{c \in C} \\
& V^{\text{ref}}_{i,c} \in \mathbb{C} \;\; \forall r \in R; \mathbf{V}^{\text{ref}}_{i} = [V^{\text{ref}}_{i,c}]_{c \in C} \\
%
\end{align}\]
where the notation $\mathbf{v}^l_{i} := [v^l_{i,c}]_{c \in C}$ reflects that the vector $\mathbf{v}^l_{i}$ is constructed by putting the individual phase values $v^l_{i,c}$ in a vector (in order $a,b,c$).
Alternatively, the series impedance of a line can be written in impedance form:
\[\mathbf{Z}_{ij} \in \mathbb{C}^{c\times c} \;\; \forall (i,j) \in E \nonumber, \mathbf{Y}_{ij} = ( \mathbf{Z}_{ij})^{-1}\]
where superscript $-1$ indicates the matrix inverse. Note that $\mathbf{Y}_{ij}$ or $\mathbf{Z}_{ij}$ may not be invertible, e.g. in case of single-phase branches in a three-phase grid. In this case the pseudo-inverse can be used.
\[\begin{align}
& S^g_{k,c} \in \mathbb{C} \;\; \forall k\in G, \forall c \in C \nonumber; \mathbf{S}^g_{k} := [S^g_{k,c}]_{c \in C} \\
& V_{i,c} \in \mathbb{C} \;\; \forall i\in N, \forall c \in C \nonumber; \mathbf{V}_{i} := [V_{i,c}]_{c \in C} \\
& \mathbf{S}_{ij} \in \mathbb{C}^{c\times c} \;\; \forall (i,j) \in E \cup E^R \\
%
\end{align}\]
A complete mathematical model is as follows,
\[
\begin{align}
\mbox{minimize: } & \sum_{k \in G} c_{2k} \left( \sum_{c \in C} \Re(S^g_{k,c}) \right)^2 + c_{1k} \sum_{c \in C} \Re(S^g_{k,c}) + c_{0k} \\
%
\mbox{subject to: } & \nonumber \\
& \mathbf{V}_{i} = \mathbf{V}^{\text{ref}}_{i} \;\; \forall r \in R \\
& S^{gl}_{k,c} \leq S^g_{k,c} \leq S^{gu}_{k,c} \;\; \forall k \in G, \forall c \in C \\
& v^l_{i,c} \leq |V_{i,c}| \leq v^u_{i,c} \;\; \forall i \in N, \forall c \in C \\
& \sum_{\substack{k \in G_i}} \mathbf{S}^g_k - \sum_{\substack{k \in L_i}} \mathbf{S}^d_k - \sum_{\substack{k \in S_i}} \mathbf{V}_i \mathbf{V}^H_i (\mathbf{Y}^s_k)^H = \sum_{\substack{(i,j)\in E_i \cup E_i^R}} diag(\mathbf{S}_{ij}) \;\; \forall i\in N \\
& \mathbf{S}_{ij} = {\mathbf{V}_i \mathbf{V}_i^H} \left( \mathbf{Y}_{ij} + \mathbf{Y}^c_{ij}\right)^H - {\mathbf{V}_i \mathbf{V}^H_j} \mathbf{Y}^H_{ij} \;\; \forall (i,j)\in E \\
& \mathbf{S}_{ji} = \mathbf{V}_j \mathbf{V}_j^H \left( \mathbf{Y}_{ij} + \mathbf{Y}^c_{ji} \right)^H - {\mathbf{V}^H_i \mathbf{V}_j} \mathbf{Y}^H_{ij} \;\; \forall (i,j)\in E \\
& |diag(\mathbf{S}_{ij})| \leq \mathbf{s}^u_{ij} \;\; \forall (i,j) \in E \cup E^R \\
& \theta^{\Delta l}_{ij,c} \leq \angle (V_{i,c} V^*_{j,c}) \leq \theta^{\Delta u}_{ij,c} \;\; \forall (i,j) \in E, \forall c \in C
%
\end{align}\]
\[\begin{align}
& S^g_{k,c} \in \mathbb{C}\;\; \forall k\in G, \forall c \in C \nonumber; \mathbf{S}^g_{k} := [S^g_{k,c}]_{c \in C} \\
& V_{i,c} \in \mathbb{C} \;\; \forall i\in N, \forall c \in C \nonumber; \mathbf{V}_{i} := [V_{i,c}]_{c \in C} \\
& I^{s}_{ij,c} \in \mathbb{C}\;\; \forall e \in E, \forall c \in C \nonumber; \mathbf{I}^{s}_{ij} := [{I}^{s}_{ij,c}]_{c \in C} \\
& \mathbf{S}_{ij} \in \mathbb{C}^{c\times c} \;\; \forall (i,j) \in E \cup E^R \\
%
\end{align}\]
A complete mathematical model is as follows,
\[\begin{align}
\mbox{minimize: } & \sum_{k \in G} c_{2k} \left( \sum_{c \in C} \Re(S^g_{k,c}) \right)^2 + c_{1k} \sum_{c \in C} \Re(S^g_{k,c}) + c_{0k} \\
%
\mbox{subject to: } & \nonumber \\
& \mathbf{V}_{i} = \mathbf{V}^{\text{ref}}_{i} \;\; \forall r \in R \\
& S^{gl}_{k,c} \leq S^g_{k,c} \leq S^{gu}_{k,c} \;\; \forall k \in G, \forall c \in C \\
& v^l_{i,c} \leq |V_{i,c}| \leq v^u_{i,c} \;\; \forall i \in N, \forall c \in C \\
& \sum_{\substack{k \in G_i}} \mathbf{S}^g_k - \sum_{\substack{k \in L_i}} \mathbf{S}^d_k - \sum_{\substack{k \in S_i}} \mathbf{V}_i \mathbf{V}^H_i (\mathbf{Y}^s_k)^H = \sum_{\substack{(i,j)\in E_i \cup E_i^R}} diag(\mathbf{S}_{ij}) \;\; \forall i\in N \\
& \mathbf{S}_{ij} + \mathbf{S}_{ji} = \mathbf{V}_i \mathbf{V}_i^H (\mathbf{Y}^c_{ij})^H + \mathbf{Z}_{ij} \mathbf{I}^{s}_{ij}(\mathbf{I}^{s}_{ij})^H + \mathbf{V}_j \mathbf{V}_j^H (\mathbf{Y}^c_{ji})^H \;\; \forall (i,j)\in E \\
& \mathbf{S}^{s}_{ij} = \mathbf{S}_{ij} + \mathbf{V}_i \mathbf{V}_i^H (\mathbf{Y}^c_{ij})^H \;\; \forall (i,j) \in E \cup E^R \\
& \mathbf{S}^{s}_{ij} = \mathbf{V}_i (\mathbf{I}^{s}_{ij})^H \;\; \forall (i,j) \in E \cup E^R\\
& \mathbf{V}_i = \mathbf{V}_j - \mathbf{Z}_{ij} \mathbf{I}^{s}_{ij} \forall (i,j)\in E \\
& |diag(\mathbf{S}_{ij})| \leq \mathbf{s}^u_{ij} \;\; \forall (i,j) \in E \cup E^R \\
& \theta^{\Delta l}_{ij,c} \leq \angle (V_{i,c} V^*_{j,c}) \leq \theta^{\Delta u}_{ij,c} \;\; \forall (i,j) \in E, \forall c \in C
%
\end{align}\]
[1] Gan, L., & Low, S. H. (2014). Convex relaxations and linear approximation for optimal power flow in multiphase radial networks. In PSSC (pp. 1–9). Wroclaw, Poland. https://doi.org/10.1109/PSCC.2014.7038399