The PowerModelsDistribution Mathematical Model

As PowerModelsDistribution 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 PowerModelsDistribution.

Unbalanced AC Optimal Power Flow

PowerModelsDistribution implements a generalized version of the AC Optimal Power Flow problem, taking into account phase unbalance. ^[1] These generalizations make it possible for PowerModelsDistribution 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

Sets

The definitions of the sets involved remain unchanged w.r.t. the balanced OPF problem definition, except for the addition of the conductor set:

\[\begin{align} % \mbox{sets:} & \nonumber \\ & N \mbox{ - buses}\nonumber \\ & R \mbox{ - references buses}\nonumber \\ & E, E^R \mbox{ - branches, forward and reverse orientation} \nonumber \\ & G, G_i \mbox{ - generators and generators at bus $i$} \nonumber \\ & L, L_i \mbox{ - loads and loads at bus $i$} \nonumber \\ & S, S_i \mbox{ - shunts and shunts at bus $i$} \nonumber \\ & C \mbox{ - conductors} \nonumber \\ % \end{align}\]

where the set of conductors $C$ typically equals $\{ a,b,c\}$.

Data

\[\begin{align} \mbox{data:} & \nonumber \\ & S^{gl}_{k,c}, S^{gu}_{k,c} \in \mathbb{C} \;\; \forall k \in G, \forall c \in C \nonumber; \mathbf{S}^{gl}_{k}:= [S^{gl}_{k,c}]_{c \in C}, \mathbf{S}^{gu}_{k} := [S^{gu}_{k,c}]_{c \in C} \\ & c_{2k}, c_{1k}, c_{0k} \in \mathbb{R} \;\; \forall k \in G \nonumber \\ & v^l_{i,c}, v^u_{i,c} \in \mathbb{R} \;\; \forall i \in N, \forall c \in C \nonumber; \mathbf{v}^l_{i} := [v^l_{i,c}]_{c \in C}, \mathbf{v}^u_{i} := [v^u_{i,c}]_{c \in C} \\ & S^d_{k,c}\in \mathbb{C} \;\; \forall k \in L, \forall c \in C \nonumber; \mathbf{S}^d_{k} := [S^d_{k,c}]_{c \in C} \\ & \mathbf{Y}^s_{k}\in \mathbb{C}^{c\times c} \;\; \forall k \in S \nonumber \\ & \mathbf{Y}_{ij}, \mathbf{Y}^c_{ij}, \mathbf{Y}^c_{ji}\in \mathbb{C}^{c\times c} \;\; \forall (i,j) \in E \nonumber \\ & {s^u}_{ij,c}, \theta^{\Delta l}_{ij,c}, \theta^{\Delta u}_{ij,c} \in \mathbb{R}\;\; \forall (i,j) \in E, \forall c \in C \nonumber, {\mathbf{s}^u}_{ij} := [{s^u}_{ij,c}]_{c \in C} \\ & V^{\text{ref}}_{i,c} \in \mathbb{C} \;\; \forall r \in R; \mathbf{V}^{\text{ref}}_{i} = [V^{\text{ref}}_{i,c}]_{c \in C} \\ % \end{align}\]

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.

Variables for a Bus Injection Model

\[\begin{align} & S^g_{k,c} \in \mathbb{C} \;\; \forall k\in G, \forall c \in C \nonumber; \mathbf{S}^g_{k} := [S^g_{k,c}]_{c \in C} \\ & V_{i,c} \in \mathbb{C} \;\; \forall i\in N, \forall c \in C \nonumber; \mathbf{V}_{i} := [V_{i,c}]_{c \in C} \\ & \mathbf{S}_{ij} \in \mathbb{C}^{c\times c} \;\; \forall (i,j) \in E \cup E^R \\ % \end{align}\]

Mathematical Formulation of a Bus Injection Model

A complete mathematical model is as follows,

\[ \begin{align} \mbox{minimize: } & \sum_{k \in G} c_{2k} \left( \sum_{c \in C} \Re(S^g_{k,c}) \right)^2 + c_{1k} \sum_{c \in C} \Re(S^g_{k,c}) + c_{0k} \\ % \mbox{subject to: } & \nonumber \\ & \mathbf{V}_{i} = \mathbf{V}^{\text{ref}}_{i} \;\; \forall r \in R \\ & S^{gl}_{k,c} \leq S^g_{k,c} \leq S^{gu}_{k,c} \;\; \forall k \in G, \forall c \in C \\ & v^l_{i,c} \leq |V_{i,c}| \leq v^u_{i,c} \;\; \forall i \in N, \forall c \in C \\ & \sum_{\substack{k \in G_i}} \mathbf{S}^g_k - \sum_{\substack{k \in L_i}} \mathbf{S}^d_k - \sum_{\substack{k \in S_i}} \mathbf{V}_i \mathbf{V}^H_i (\mathbf{Y}^s_k)^H = \sum_{\substack{(i,j)\in E_i \cup E_i^R}} diag(\mathbf{S}_{ij}) \;\; \forall i\in N \\ & \mathbf{S}_{ij} = {\mathbf{V}_i \mathbf{V}_i^H} \left( \mathbf{Y}_{ij} + \mathbf{Y}^c_{ij}\right)^H - {\mathbf{V}_i \mathbf{V}^H_j} \mathbf{Y}^H_{ij} \;\; \forall (i,j)\in E \\ & \mathbf{S}_{ji} = \mathbf{V}_j \mathbf{V}_j^H \left( \mathbf{Y}_{ij} + \mathbf{Y}^c_{ji} \right)^H - {\mathbf{V}^H_i \mathbf{V}_j} \mathbf{Y}^H_{ij} \;\; \forall (i,j)\in E \\ & |diag(\mathbf{S}_{ij})| \leq \mathbf{s}^u_{ij} \;\; \forall (i,j) \in E \cup E^R \\ & \theta^{\Delta l}_{ij,c} \leq \angle (V_{i,c} V^*_{j,c}) \leq \theta^{\Delta u}_{ij,c} \;\; \forall (i,j) \in E, \forall c \in C % \end{align}\]

Variables for a Branch Flow Model

\[\begin{align} & S^g_{k,c} \in \mathbb{C}\;\; \forall k\in G, \forall c \in C \nonumber; \mathbf{S}^g_{k} := [S^g_{k,c}]_{c \in C} \\ & V_{i,c} \in \mathbb{C} \;\; \forall i\in N, \forall c \in C \nonumber; \mathbf{V}_{i} := [V_{i,c}]_{c \in C} \\ & I^{s}_{ij,c} \in \mathbb{C}\;\; \forall e \in E, \forall c \in C \nonumber; \mathbf{I}^{s}_{ij} := [{I}^{s}_{ij,c}]_{c \in C} \\ & \mathbf{S}_{ij} \in \mathbb{C}^{c\times c} \;\; \forall (i,j) \in E \cup E^R \\ % \end{align}\]

Mathematical Formulation of a Branch Flow Model

A complete mathematical model is as follows,

\[\begin{align} \mbox{minimize: } & \sum_{k \in G} c_{2k} \left( \sum_{c \in C} \Re(S^g_{k,c}) \right)^2 + c_{1k} \sum_{c \in C} \Re(S^g_{k,c}) + c_{0k} \\ % \mbox{subject to: } & \nonumber \\ & \mathbf{V}_{i} = \mathbf{V}^{\text{ref}}_{i} \;\; \forall r \in R \\ & S^{gl}_{k,c} \leq S^g_{k,c} \leq S^{gu}_{k,c} \;\; \forall k \in G, \forall c \in C \\ & v^l_{i,c} \leq |V_{i,c}| \leq v^u_{i,c} \;\; \forall i \in N, \forall c \in C \\ & \sum_{\substack{k \in G_i}} \mathbf{S}^g_k - \sum_{\substack{k \in L_i}} \mathbf{S}^d_k - \sum_{\substack{k \in S_i}} \mathbf{V}_i \mathbf{V}^H_i (\mathbf{Y}^s_k)^H = \sum_{\substack{(i,j)\in E_i \cup E_i^R}} diag(\mathbf{S}_{ij}) \;\; \forall i\in N \\ & \mathbf{S}_{ij} + \mathbf{S}_{ji} = \mathbf{V}_i \mathbf{V}_i^H (\mathbf{Y}^c_{ij})^H + \mathbf{Z}_{ij} \mathbf{I}^{s}_{ij}(\mathbf{I}^{s}_{ij})^H + \mathbf{V}_j \mathbf{V}_j^H (\mathbf{Y}^c_{ji})^H \;\; \forall (i,j)\in E \\ & \mathbf{S}^{s}_{ij} = \mathbf{S}_{ij} + \mathbf{V}_i \mathbf{V}_i^H (\mathbf{Y}^c_{ij})^H \;\; \forall (i,j) \in E \cup E^R \\ & \mathbf{S}^{s}_{ij} = \mathbf{V}_i (\mathbf{I}^{s}_{ij})^H \;\; \forall (i,j) \in E \cup E^R\\ & \mathbf{V}_i = \mathbf{V}_j - \mathbf{Z}_{ij} \mathbf{I}^{s}_{ij} \forall (i,j)\in E \\ & |diag(\mathbf{S}_{ij})| \leq \mathbf{s}^u_{ij} \;\; \forall (i,j) \in E \cup E^R \\ & \theta^{\Delta l}_{ij,c} \leq \angle (V_{i,c} V^*_{j,c}) \leq \theta^{\Delta u}_{ij,c} \;\; \forall (i,j) \in E, \forall c \in C % \end{align}\]