Learning Optimal Power Flow with Pointwise Constraints

TL;DR

This work tackles OPF by learning a parameterization that yields $[oldsymbol{s}, oldsymbol{v}]$ from demand $oldsymbol{r}$ while enforcing constraints pointwise for every realization. The authors formulate a dual-domain training approach using an augmented Lagrangian and propose several multiplier schemes (pointwise, shared, and hybrid) to balance feasibility with computational efficiency. Empirical results across five power systems show that pointwise dual training substantially reduces constraint violations, especially in larger networks and corner cases, albeit with occasional trade-offs in optimality and increased memory for storing per-realization multipliers. The approach leverages Graph Attention Networks to capture network structure and provides a principled alternative to average-constraint or supervised methods for fast, feasible OPF predictions in large-scale grids.

Abstract

Training learning parameterizations to solve optimal power flow (OPF) with pointwise constraints is proposed. In this novel training approach, a learning parameterization is substituted directly into an OPF problem with constraints required to hold over all problem instances. This is different from existing supervised learning methods in which constraints are required to hold across the average of problem instances. Training with pointwise constraints is undertaken in the dual domain with the use of augmented Lagrangian and dual gradient ascent algorithm. Numerical experiments demonstrate that training with pointwise constraints produces solutions with smaller constraint violations. Experiments further demonstrated that pointwise constraints are most effective at reducing constraint violations in corner cases - defined as those realizations in which constraints are most difficult to satisfy. Gains are most pronounced in power systems with large numbers of buses.

Figures (13)

• Figure 1: Circuit Diagram of a $\pi$-Section Branch Model with Transformer. A branch $(i,j)$ from bus $i$ to bus $j$ is shown. Variables $v_i \in \mathbb{C}$ and $v_j \in \mathbb{C}$ denote voltages at buses $i$ and $j$, whereas variables $f_{ij}\in \mathbb{C}$ and $f_{ji}\in \mathbb{C}$ represent power flowing from bus $i$ to bus $j$ and to bus $i$ from bus $j$, respectively. The branch is characterized by a transformer ratio $t_{nm} \in \mathbb{C}$, line admittance $y_{ij} \in\mathbb{C}$, and shunt admittances $y^{\text{C}}_{ij} \in \mathbb{C}$ and $y^{\text{C}}_{ji} \in \mathbb{C}$.
• Figure 2: Optimal Power Flow. We seek optimal power generation $\mathbf{s}^{\dagger}(\mathbf{r})$ and voltage $\mathbf{v}^\dagger(\mathbf{r})$ values that minimize the cost $C(\mathbf{s})$ [cf. \ref{['eqn_cost']}] to meet demand $\mathbf{r}$ while satisfying electrical circuit equations [cf. \ref{['eqn_branch_from_elementwise']}, \ref{['eqn_branch_to_elementwise']} and \ref{['eqn_power_balance']}] as well as physical limits on voltages and power flows [cf. \ref{['eqn_generator_limit']}, \ref{['eqn_voltage_limit']}, \ref{['eqn_power_limit']} and \ref{['eqn_voltage_angle_difference']}].
• Figure 3: Electrical Grid Variables. Power demands $r_i$ at each bus are the input variables of the optimal power flow (OPF) problem of Figure \ref{['fig_OPF']} while generated powers $s_i$ and voltages $v_{i}$ are output variables. As stated in Figure \ref{['fig_OPF']}, the flow variables $f_{ij}$ and $f_{ji}$ at each branch are substituted for their explicit expressions in \ref{['eqn_branch_from_elementwise']} and \ref{['eqn_branch_to_elementwise']} to solve OPF.
• Figure 4: Electrical Grid Parameters. An electrical grid $\mathbb{W}$ is defined by the paramters in this table associated with a list of $N$ buses $i$ and $M$ directed branches $(i,j)$. These parameters include electrical properties of buses and branches -- $y_{i}^{\text{S}}$, $y_{ij}$, $y^{\text{C}}_{ij}$, $y^{\text{C}}_{ji}$ and $t_{ij}$ -- as well as operating range bounds -- $s^{\text{G}}_{i, \min}$, $s^{\text{G}}_{i,\max}$, $v_{i,\min}$, $v_{i, \max}$, $f_{ij, \max}$, $f_{ji, \max}$, $\theta_{ij,\min}$ and $\theta_{ij, \max}$-- and parameters of the cost function -- $c_{0i}$, $c_{1i}$ and $c_{2i}$.
• Figure 5: Dual Gradient Ascent Iteration. The primal step (P) determines parameters $\mathbf{A}_{t+1} = \mathbf{A}(\bm\lambda_{t}, \bm\mu_{t})$ that minimize the Lagrangian $L(\mathbf{A}, \bm\lambda_{t}, \bm\mu_{t})$ [cf. \ref{['eqn_lagrangian_minimizers']}]. The dual step (D) updates dual variables by following constraint slacks [cf. \ref{['eqn_supergradient_ascent']}]. When we exit dual gradient ascent after $T$ iterations we adopt $\mathbf{A}_{T}$ as a proxy for the optimal argument $\mathbf{A}^\dagger$ [cf. \ref{['eqn_opf_learning_pointwise']}].

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3