Learning Notes: Convex Relaxation of Optimal Power Flow
Motivation
- Power flow equations are quadratic and hence OPF can be formulated as a quadratically constrained quadratic program (QCQP). It is generally non-convex and hence NP-hard.
- A popular approximation is a linear program, called DC OPF, obtained through the linearization of the power flow equations. However:
- while DC OPF is useful in a wide variety of applications, it is not applicable in other applications
- a solution of DC OPF may not be feasible (may not satisfy the nonlinear power flow equations). In this case, an operator may tighten some constraints in DC OPF and solve again. This may not only reduce efficiency but also relies on heuristics that are hard to scale to larger systems or faster control in the future.
- when the DC OPF converge, most nonlinear algorithms compute a local optimal usually without assurance on the quality of the solution.
Power Flow Models
Bus Injection Model
\[ {s_j} = \matrix{{\sum\limits_{k:j \sim k} y_{jk}^H\;{V_j}(V_j^H - V_k^H),} & {j \in {N^+}}.} \label{eq-bim-1} \tag{1}\]
\[ {\mathbb{V}(s): = \{V \in {\mathbb{C}^{n+ 1}}\ \vert\ V\;{\rm satisfies}\;\eqref{eq-bim-1}\}.} \]
\[ {V_0}: = 1\angle 0^\circ \]
Branch Flow Model
\[ \begin{align} \sum\limits_{k:j \rightarrow k} {S_{jk}}& = \sum\limits_{i:i \rightarrow j} \left({{S_{ij}} - {z_{ij}}\vert{I_{ij}}{\vert^2}} \right)+ {s_j},\quad j \in {N^+} \\ {I_{jk}} & = {y_{jk}}({V_j} - {V_k}),\quad j \rightarrow k \in \tilde{E} \\ {S_{jk}} & = {V_j} I_{jk}^H,\quad j \rightarrow k \in \tilde{E} \end{align}\label{eq-bfm-2} \tag{2}\]
\[ \tilde{\mathbb{X}}(s): = \{\tilde{x} \in {\mathbb{C}^{2\ m+ n+ 1}}\ \vert\ \tilde{x}\;{\rm satisfies}\;\eqref{eq-bfm-2}\} \]
\[ {V_0}: = 1\angle 0^\circ \]
\(\mathbb{V}(s) \equiv \tilde{\mathbb{X}}(s)\) for any power injections \(s\).
- BIM allows a much cleaner formulation of the SDP relaxation.
- BFM for radial networks has a convenient recursive structure that allows a more efficient computation of power flows and leads to a useful linear approximation of BFM.
Optimal Power Flow
Branch Injection Model
Voltage magnitudes: \[ {\underline v_j} \leq \vert{V_j}{\vert^2} \leq {\overline v_j},\quad j \in {N^+}\label{eq-3} \tag{3}\]
Power injections: \[ {\underline{s}_j} \leq s_j \leq {\overline{s}_j} \label{eq-4} \tag{4}\] \[ {\underline{s}_j} \leq {{\sum\limits_{k:j \sim k} y_{jk}^H\;{V_j}(V_j^H - V_k^H)} \leq {\overline{s}_j} ,\quad {j \in {N^+}}.} \label{eq-5} \tag{5}\]
Branch Flow Model
\(\eqref{eq-bfm-2}, \eqref{eq-3}, \eqref{eq-4}\)
OPF as QCQP
OPF can be written as a standard form QCQP: \[ \begin{align} \min\limits_{{{V}} \in {\mathbb{C}^{n+ 1}}} \ &{V^H}{{CV}}\cr {\rm s.t.}\, \ \quad &{V^H}{\Phi_j}{{V}} \leq {\overline p_j},\quad {V^H}(- {\Phi_j})V \leq - {\underline p_j}& \cr &{V^H}{\Psi_j}V \leq {\overline q_j},\quad {V^H}(- {\Psi_j})V \leq - {\underline q_j} \cr &{V^H}{J_j}V \leq {\overline v_j},\quad {V^H}(- {J_j})V \leq - {\underline v_j} \end{align} \] where:
- \({\Phi_j}: = {1 \over 2}\left({Y_j^H+ {Y_j}} \right)\quad {\rm and}\quad {\Psi_j}: = {1 \over {2{\mathbf{i}}}}\left({Y_j^H - {Y_j}} \right).\)
- \({{{J}}_j}: = {{{e}}_j}e_j^H\) denote the Hermitian matrix with a single 1 in the \((j,j)\) entry and 0 everywhere else.