SDP-OPF
The SDP-OPF model considered in PGLearn is presented below.
Definitions
Let $\im$ be the imaginary unit. Let $\cdot^{*}$ be the complex conjugate operator.
The second order cone of order $n$ is defined as
\[\mathcal{Q}^{n} = \left\{ x \in \mathbb{R}^{n} \ \middle| \ x_{1} \geq \sqrt{x_{2}^{2} + ... + x_{n}^{2}} \right\}.\]
The set of $n \times n$ symmetric matrices is
\[\symmat^{n} = \left\{ X \in \mathbb{R}^{n \times n} \ \middle| \ X = X^{\top} \right\}.\]
The set of $n \times n$ skew-symmetric matrices is
\[\skewmat^{n} = \left\{ X \in \mathbb{R}^{n \times n} \ \middle| \ X = -X^{\top} \right\}.\]
The real positive semidefinite cone of order $n$ is defined as
\[\psdsymmat^{n} = \left\{ X \in \symmat^{n} \ \middle| \ X \succeq 0 \right\}.\]
which is the set of positive semidefinite $n \times n$ symmetric matrices in the real space.
A complex square matrix is Hermitian if it is equal to its conjugate transpose. The set of $n \times n$ Hermitian matrices is
\[\hermat^{n} = \left\{ X \in \mathbb{C}^{n \times n} \ \middle| \ X = X^{H} \right\}.\]
Note that the diagonal entries of a Hermitian matrix are real. The complex positive semidefinite cone of order $n$ is defined as
\[\psdhermat^{n} = \left\{ X \in \hermat^{n} \ \middle| \ X \succeq 0 \right\}.\]
which is the set of positive semidefinite $n \times n$ Hermitian matrices in the complex space.
Mathematical formulation
The SDP-OPF formulation in PGLearn is obtained through the change of variable $\Wsdp = \V \V^{*}$; i.e., $\Wsdp_{ij} = \V_{i} \V_{j}^{*}$. This change allows formulating AC-OPF so that constraints involving $\Wsdp$ are linear in $\Wsdp$. It is known that $\Wsdp = \V \V^{*}$ for some $\V$ if and only if $\Wsdp \in \psdhermat^{N}$ and $rank(\Wsdp) = 1$. By relaxing the rank constraint, a complex semidefinite program (SDP) relaxation of AC-OPF is obtained, where a positive semidefinite constraint requires $\Wsdp \in \psdhermat^{N}$.
The complex SDP can be converted into a real formulation. Define $\Wrsdp \in \symmat^{N}$ and $\Wisdp \in \skewmat^{N}$. Then, it is known that
\[\begin{pmatrix} \Wrsdp & \Wisdp \\ -\Wisdp & \Wrsdp \end{pmatrix} = \begin{pmatrix} \Wrsdp & \Wisdp \\ {\Wisdp}^{\top} & \Wrsdp \end{pmatrix} \in \symmat^{2N}\]
is positive semidefinite if and only if $\Wsdp = \Wrsdp + \im \Wisdp \in \hermat^{N}$ also is positive semidefinite in the complex space. Note that, by definition, $diag(\Wsdp) = diag(\Wrsdp)$.
The resulting SDP-OPF model in the real space is presented below.
\[\begin{align} \min \quad & \label{eq:SDPOPF:objective} \sum_{i \in \NODES} \sum_{j \in \GENERATORS_{i}} c_j \PG_j + c_0 \\ \textrm{s.t.} \quad & \label{eq:SDPOPF:kcl_p} \sum_{j \in \GENERATORS_{i}} \PG_{j} - \sum_{e \in \EDGES^{+}_{i}} \PF_{e} - \sum_{e \in \EDGES^{-}_{i}} \PT_{e} - \GS_i \Wrsdp_{ii} = \sum_{j\in\LOADS_i} \PD_j & \forall i & \in \NODES % && [\lambda^{p}] \\ & \label{eq:SDPOPF:kcl_q} \sum_{g \in \mathcal{G}_{i}} \QG_{g} - \sum_{e \in \EDGES^{+}_{i}} \QF_{e} - \sum_{e \in \EDGES^{-}_{i}} \QT_{e} + \BS_{i} \Wrsdp_{ii} = \sum_{j\in\LOADS_i} \QD_j & \forall i & \in \NODES % && [\lambda^{q}] \\ % Ohm's law & \label{eq:SDPOPF:ohm_pf} \gff_{e} \Wrsdp_{ii} + \gft_{e} \Wrsdp_{e} + \bft_{e} \Wisdp_{e} - \PF_{e} = 0 & \forall e = (i,j) & \in \EDGES % && [\lambdaPf] \\ & \label{eq:SDPOPF:ohm_qf} -\bff_{e} \Wrsdp_{ii} - \bft_{e} \Wrsdp_{e} + \gft_{e} \Wisdp_{e} - \QF_{e} =0 & \forall e = (i, j) & \in \EDGES % && [\lambdaQf] \\ & \label{eq:SDPOPF:ohm_pt} \gtt_{e} \Wrsdp_{jj} + \gtf_{e} \Wrsdp_{e} - \btf_{e} \Wisdp_{e} - \PT_{e} = 0 & \forall e = (i, j) & \in \EDGES % && [\lambdaPt] \\ & \label{eq:SDPOPF:ohm_qt} -\btt_{e} \Wrsdp_{jj} - \btf_{e} \Wrsdp_{e} - \gtf_{e} \Wisdp_{e} - \QT_{e} =0 & \forall e = (i, j) & \in \EDGES % && [\lambdaQt] \\ % Thermal limits & \label{eq:SDPOPF:sm_f} (\bar{s}_{e}, \PF_{e}, \QF_{e}) \in \mathcal{Q}^{3} & \forall e & \in \EDGES % && [\nuThermalfr] \\ & \label{eq:SDPOPF:sm_t} (\bar{s}_{e}, \PT_{e}, \QT_{e}) \in \mathcal{Q}^{3} & \forall e & \in \EDGES % && [\nuThermalto] \\ % Voltage angle deviation & \label{eq:SDPOPF:va_diff} \tan(\dvamin_{e}) \Wrsdp_{e} \leq \Wisdp_{e} \leq \tan(\dvamax_{e}) \Wrsdp_{e} & \forall e & \in \EDGES % && [\muAngleDiff] \\ % Variable bounds & \label{eq:SDPOPF:pg_bounds} \pgmin_{i} \leq \PG_{i} \leq \pgmax_{i} & \forall i & \in \GENERATORS % && [\muPg] \\ & \label{eq:SDPOPF:qg_bounds} \qgmin_{i} \leq \QG_{i} \leq \qgmax_{i} & \forall i & \in \GENERATORS % && [\muQg] \\ & \label{eq:SDPOPF:wm_bounds} \vmmin_{i}^{2} \leq \Wrsdp_{ii} \leq \vmmax_{i}^{2} & \forall i & \in \NODES % && [\muWm] \\ & \label{eq:SDPOPF:pf_bounds} {-}\overline{S}_{e} \leq \PF_{e} \leq \overline{S}_{e} & \forall e & \in \EDGES % && [\mu^{pf}] \\ & \label{eq:SDPOPF:qf_bounds} {-}\overline{S}_{e} \leq \QF_{e} \leq \overline{S}_{e} & \forall e & \in \EDGES % && [\mu^{qf}] \\ & \label{eq:SDPOPF:pt_bounds} {-}\overline{S}_{e} \leq \PT_{e} \leq \overline{S}_{e} & \forall e & \in \EDGES % % && [\mu^{pt}] \\ & \label{eq:SDPOPF:qt_bounds} {-}\overline{S}_{e} \leq \QT_{e} \leq \overline{S}_{e} & \forall e & \in \EDGES % && [\mu^{qt}] \\ % PSD constraint & \label{eq:SDPOPF:psd} \begin{pmatrix} \Wrsdp & \Wisdp \\ -\Wisdp & \Wrsdp \end{pmatrix} \in \psdsymmat^{2N} & % && [\Ssdp] \end{align}\]
If there are multiple branches $e_{1}, e_{2}, ...$ from bus $i$ to $j$, then $\Wrsdp_{e_{1}}, \Wrsdp_{e_{2}},...$ will all refer to the same $\Wrsdp_{ij}$ entry in the matrix. Similarly for $\Wisdp$.
If branch $e \in \EDGES$ is connected from bus $i$ to $j$, then $\Wisdp_{e}$ refers to $\Wisdp_{ij}$. If instead branch $e$ is connected from bus $j$ to $i$, then $\Wisdp_{e}$ refers to $\Wisdp_{ji}$, which by definition is equal to $-\Wisdp_{ij}$.
More details of the model can be found in Qiu, G., Tanneau, M., & Van Hentenryck, P. (2025). Dual Conic Proxy for Semidefinite Relaxation of AC Optimal Power Flow. arXiv preprint arXiv:2502.06978.
Variables
- $\PG \in \mathbb{R}^{G}$: active power dispatch
- $\QG \in \mathbb{R}^{G}$: reactive power dispatch
- $\Wrsdp \in \symmat^{N}$: real part of voltage product
- $\Wisdp \in \skewmat^{N}$: imaginary part of voltage product
- $\PF \in \mathbb{R}^{E}$: active power flow "from"
- $\QF \in \mathbb{R}^{E}$: reactive power flow "from"
- $\PT \in \mathbb{R}^{E}$: active power flow "to"
- $\QT \in \mathbb{R}^{E}$: reactive power flow "to"
Objective
The objective function $\eqref{eq:SDPOPF:objective}$ minimizes the cost of active power generation.
PGLearn currently supports only linear cost functions. Support for quadratic functions is planned for a later stage; please open an issue if you would like to request this feature.
Constraints
- $\eqref{eq:SDPOPF:kcl_p}-\eqref{eq:SDPOPF:kcl_q}$: Kirchhoff's current law for active and reactive power
- $\eqref{eq:SDPOPF:ohm_pf}-\eqref{eq:SDPOPF:ohm_qt}$: Ohm's law for active/reactive power flows in from and to directions
- $\eqref{eq:SDPOPF:sm_f}-\eqref{eq:SDPOPF:sm_t}$: thermal limits
- $\eqref{eq:SDPOPF:va_diff}$: voltage angle deviation constraints
- $\eqref{eq:SDPOPF:pg_bounds}-\eqref{eq:SDPOPF:qg_bounds}$: active/reactive generation limits
- $\eqref{eq:SDPOPF:wm_bounds}$: bounds on squared voltage magnitude
- $\eqref{eq:SDPOPF:pf_bounds}-\eqref{eq:SDPOPF:qt_bounds}$: power flow bounds, derived from thermal limits
- $\eqref{eq:SDPOPF:psd}$: positive semidefinite constraint
Although power flow variable bounds$\eqref{eq:SDPOPF:pf_bounds}-\eqref{eq:SDPOPF:qt_bounds}$ are redundant with thermal limits $\eqref{eq:SDPOPF:sm_f}-\eqref{eq:SDPOPF:sm_t}$, their inclusion improves the performance of interior-point solvers like Ipopt.
Data format
Primal solution
| Variable | Data | Size | Description |
|---|---|---|---|
| $diag(\Wrsdp)$ | w | $N$ | Squared nodal voltage magnitude, the diagonal entries of $\Wrsdp$ |
| $\PG$ | pg | $G$ | Active power generation |
| $\QG$ | pg | $G$ | Reactive power generation |
| $\Wrsdp_{\EDGES}$ | wr | $E$ | Voltage product variable (real part) corresponding to branches |
| $\Wisdp_{\EDGES}$ | wi | $E$ | Voltage product variable (imaginary part) corresponding to branches |
| $\PF$ | pf | $E$ | Active power flow (from) |
| $\PT$ | pt | $E$ | Active power flow (to) |
| $\QF$ | qf | $E$ | Reactive power flow (from) |
| $\QT$ | qt | $E$ | Reactive power flow (to) |
Since $\Wrsdp$ and $\Wisdp$ can be dense, only the off-diagonal entries that correspond to branches are extracted to save space. Other entries only appear in constraints the positive semidefinite constraint $\eqref{eq:SDPOPF:psd}$ and not other constraints. Therefore, values of these entries can be obtained by solving a positive semidefinite matrix completion problem to recover a full feasible matrix.
If there are multiple parallel branches $e_{1}, e_{2}, ...$ from bus $i$ to $j$, the same entries $\Wrsdp_{ij}$ and $\Wisdp_{ij}$ are extracted for each of the branches.
Dual solution
| Constraint | Data | Size |
|---|---|---|
| $\eqref{eq:SDPOPF:kcl_p}$ | kcl_p | $N$ |
| $\eqref{eq:SDPOPF:kcl_q}$ | kcl_q | $N$ |
| $\eqref{eq:SDPOPF:ohm_pf}$ | ohm_pf | $E$ |
| $\eqref{eq:SDPOPF:ohm_qf}$ | ohm_qf | $E$ |
| $\eqref{eq:SDPOPF:ohm_pt}$ | ohm_pt | $E$ |
| $\eqref{eq:SDPOPF:ohm_qt}$ | ohm_qt | $E$ |
| $\eqref{eq:SDPOPF:sm_f}$ | sm_fr | $E \times 3$ |
| $\eqref{eq:SDPOPF:sm_t}$ | sm_to | $E \times 3$ |
| $\eqref{eq:SDPOPF:va_diff}$ | va_diff | $E$ |
| $\eqref{eq:SDPOPF:pg_bounds}$ | pg | $G$ |
| $\eqref{eq:SDPOPF:qg_bounds}$ | qg | $G$ |
| $\eqref{eq:SDPOPF:wm_bounds}$ | w | $N$ |
| $\eqref{eq:SDPOPF:pf_bounds}$ | pf | $E$ |
| $\eqref{eq:SDPOPF:qf_bounds}$ | qf | $E$ |
| $\eqref{eq:SDPOPF:pt_bounds}$ | pt | $E$ |
| $\eqref{eq:SDPOPF:qt_bounds}$ | qt | $E$ |
| $\eqref{eq:SDPOPF:psd}$ | s, sr, si | $N$, $E$, $E$ |
The dual variable corresponding to the positive semidefinite constraint $\eqref{eq:SDPOPF:psd}$ of the real SDP-OPF is $\Ssdp$ which is constrained to be in $\psdsymmat^{2N}$.
By deriving the dual of the complex SDP formulation, it can be shown that any feasible $\Ssdp$ in the real formulation has the same 4-block structure as the primal matrix, and each block has a sparsity structure corresponding to $\EDGES$. Without considering the complex formulation, the dual problem of the real formulation also implies a transformation to obtain such a feasible sparse 4-block matrix from any feasible $\Ssdp$ easily. Therefore, without loss of generality,
\[\Ssdp = \begin{pmatrix} \Srsdp & \Sisdp \\ -\Sisdp & \Srsdp \end{pmatrix}\]
where the off-diagonal entries of $\Srsdp \in \symmat^{N}$ and $\Sisdp \in \skewmat^{N}$ that don't correspond to any branch are 0.
Currently, there is no necessity to compute the aforementioned transformation, since interior point solvers and splitting cone solvers are observed to produce $\Ssdp$ with the sparse 4-block structure when solving SDP-OPF, even though sparsity is not explicitly exploited.
s, sr and si are the diagonal entries of $\Srsdp$, the off-diagonal entries of $\Srsdp$ that correspond to branches, and the off-diagonal entries of $\Sisdp$ that correspond to branches, respectively. Unlike in the primal case, these values are sufficient to directly reconstruct the original $\Ssdp$.