AC Optimal Power Flow Problem Overview

• AC Optimal Power Flow (AC-OPF) is a nonlinear, NP-hard optimization framework that minimizes generation costs while satisfying full AC transmission constraints.
• Modern AC-OPF formulations incorporate robust and chance-constrained strategies to handle uncertainties from renewable resources and load variations.
• Advances such as convex relaxations, homotopy methods, and learning-based surrogates have significantly improved solution efficiency and reliability.

The AC Optimal Power Flow (AC-OPF) problem is the canonical nonlinear optimization framework at the heart of secure and economic operation of power grids. It seeks to determine power generation schedules, bus voltages, and power flows that minimize operational costs subject to the full nonlinear AC transmission network constraints, device operating limits, and, increasingly, operational reliability under uncertainty from renewable resources and loads. AC-OPF is fundamentally nonconvex and NP-hard, resulting in a spectrum of deterministic and robust solution methodologies, convex relaxations, algorithmic refinements, and learning-based surrogates found in contemporary literature.

1. Rigorous Formulation of the AC-OPF and Its Characteristic Properties

The general AC-OPF problem for a power grid (with buses $\mathcal{N}$, generators $\mathcal{G}$, branches/lines $\mathcal{L}$) is stated as a nonlinear program. Key components include:

• Variables: Real and reactive generator powers ($P^G_k, Q^G_k$), bus voltages (magnitudes $|V_k|$ and angles $\theta_k$), branch flows.
• Objective: Minimize the total generation cost,

$\min \sum_{k\in\mathcal{G}} f_k(P^G_k)$

typically quadratic or piecewise-linear.

• Power-Flow Constraints:
  • Real and reactive nodal power balances (for each bus $k$), incorporating full nonlinear power flow relations,

  $$\begin{align*} P^G_k - P^L_k &= \sum_{l\in N_k} \text{Re}\{ V_k (V_k - V_l)^* y_{kl}^* \} \ Q^G_k - Q^L_k &= \sum_{l\in N_k} \text{Im}\{ V_k (V_k - V_l)^* y_{kl}^* \} \end{align*}$$

• Device and Network Operating Limits:

  • Generator: $\underline P_{k} \le P^G_{k} \le \overline P_{k}$, $\underline Q_{k}\le Q^G_{k} \le \overline Q_{k}$
  • Voltage magnitudes: $\underline V_{k} \le |V_{k}| \le \overline V_{k}$
  • Branch (thermal/voltage drop) limits: e.g., $|V_l(V_l - V_m)^* y_{lm}^*| \le S_{lm}^{\max}$ or $|V_l - V_m| \le \Delta V_{lm}^{\max}$

The problem is nonconvex (due to trigonometric and bilinear voltage terms), NP-hard in general, and frequently admits multiple local optima or infeasible regions (Pandey et al., 2020, Repiso et al., 17 Oct 2025).

2. Robust AC-OPF and Uncertainty Modeling

Modern operational contexts require risk-averse solutions that account for uncertain renewable injections and load forecasts. Approaches include:

• Chance-Constrained AC-OPF: Control variables are optimized to minimize cost while bounding the probability of operational limit violations to a small $\epsilon$,

$$\min_{u} f(u) \quad \text{s.t. } \Pr_{\delta\sim\mu}\left\{\exists x : g(u, x, \delta) = 0, \; h(u, x, \delta) \le 0\right\} \le \epsilon$$

where $u$ collects nominal generator setpoints, $g$ encodes power-balance, $h$ encodes inequalities, and $\delta$ captures uncertainty in renewable and load injections, with a known (potentially non-Gaussian) law $\mu$ (Chamanbaz et al., 2017).

• Robust AC-OPF: Ensures feasibility for all uncertainty realizations in a prescribed set (e.g., confidence ellipsoid). Solution approaches include robust convex restrictions (inner approximations), guaranteeing that for each $w$ in the uncertainty set, there exists a feasible power flow and all operational limits are satisfied (Lee et al., 2020). Affine recourse policies and SDP inner approximations provide tractable direct mechanisms for policy selection in two-stage robust AC-OPF (Louca et al., 2017).

3. Advances in Solution Algorithms

3.1 Classical and Robust Nonlinear Programming

• Interior-Point Methods (IPM): The de facto choice for deterministic AC-OPF, employing Newton-type iterates on KKT conditions, with logarithmic barrier terms for inequalities. Modern solvers (using polar or rectangular coordinates) feature convergence on practical networks up to 100+ buses, particularly when initialized with data-driven warm starts (Repiso et al., 17 Oct 2025).
• Homotopy/Continuation: The Incremental Model Building (IMB) homotopy approach transitions from a trivial, relaxed grid to the true AC-OPF by stepping a scalar parameter $v$ from $1 \rightarrow 0$, with all physics interpolated. At each $v$, a well-conditioned primal-dual step is solved via interior point, with network slack variables ensuring feasibility throughout (Pandey et al., 2020). This yields superior convergence robustness on large or congested networks.

3.2 Convex Relaxations and Approximations

• Semidefinite Relaxation (SDR), Chordal SDP, and Second-Order Cone Relaxation (SOCR): The AC-OPF quadratic constraints are lifted to matrix variables ($V=vv^H$), and either the rank-constraint is dropped (SDR), replaced by clique-wise PSDs (chordal SDP), or further weakened to $2\times2$ cones per branch (SOCR) (Bingane et al., 2019, Kocuk et al., 2017). The "tight-and-cheap conic relaxation" (TCR) strengthens SOCR with $3\times3$ branch-wise PSD blocks and RLT cuts, achieving nearly SDR-quality bounds at SOCP computational cost in large networks (Bingane et al., 2019). Tight LP relaxations via polyhedral approximations of SOCs and trigonometric envelopes preserve bounds on voltage and reactive variables, facilitating the use of fast LP/MILP solvers (Mhanna et al., 2016).
• Strengthened QC Relaxations: Convex envelopes of trilinear terms ($v_\ell v_m \cos\theta_{\ell m}$ and $v_\ell v_m \sin\theta_{\ell m}$) are built using vertex enumeration, curvature-aware tangent planes, and bus-specific phase rotations to sharpen the relaxation. The Linear Rotated QC (LRQC) approach improves optimality gaps by $1.31$pp on average, compared to classical QC (Narimani et al., 2023).

3.3 Mixed-Integer and Combinatorial AC-OPF

Radial distribution systems with discrete demand constraints (switchable inelastic loads) render the problem strongly NP-hard. SOCP relaxations on branch-flow models admit approximation schemes:

• Polynomial-Time Approximation Schemes (PTAS) exploit enumeration-plus-rounding on a small subset of discrete users, with subsequent SOCP and small LP rounding steps, achieving $(1+\epsilon)$-approximations under mild technical conditions (Khonji et al., 2017).

4. Data-Driven and Learning-Based AC-OPF Methods

Recent interest has surged in machine learning surrogates for AC-OPF, to enable ultra-fast or online grid management.

4.1 Physics-Informed and Interpretable Learning

• KT-AC (KNN+Taylor-AC): Combines K-nearest-neighbor regression on load profile, lifted to a linearized Taylor expansion of the full AC physics, solved as a single LP. KT-AC matches or outperforms neural networks (NN) in terms of dispatch cost, infeasibility rate ($\Delta C <0.02\%$ and $VIOL(q)<0.25\%$ on 14-bus IEEE) but retains full interpretability and runs in order-of-magnitude faster time (Pineda et al., 30 Jul 2024).
• Residual Correction Models: Use DC-OPF as a baseline, learning only the necessary nonlinear corrections via topology-aware Graph Neural Networks (GNNs) and physics-informed losses. Such approaches reduce mean-squared error and feasibility errors significantly, and maintain accuracy under unseen N-1 contingencies, scaling to $2000$-bus systems (Za'ter et al., 17 Oct 2025).

4.2 Deep Learning (Supervised and Unsupervised)

• End-to-End Unsupervised Learning: Output partial variables via DNN, recovering all others through fast (decoupled) AC power flow, then train using an augmented Lagrangian which penalizes constraint violations but does not require solver-based labels. Accelerates computation by two orders of magnitude, with $>99\%$ feasibility and $<0.2\%$ optimality loss (Chen et al., 2022, Pan et al., 2020).
• Holomorphic Embedding Methods: Learning policies are trained end-to-end by differentiating through a holomorphic embedding (HELM) solver, circumventing issues with spurious power-flow solutions and enabling gradient-based learning of entire OPF policies, including unit-commitment binaries. Such solvers show $12\times$ speedups and $40\%$ robustness improvement when compared to interior-point solvers on 200-bus test cases (Lange et al., 2020).

5. Large-Scale, Distributed, and Security-Constrained AC-OPF

• Distributed Optimization (ADMM and Component-wise Decomposition): Grid decomposition enables massively parallel solution of AC-OPF by dividing the network into component-wise subproblems. ADMM variants, with GPU acceleration, allow tracking AC-OPF states over grids up to $70,000$ buses with sub-second update times per period and real-time contingency re-dispatch (Kim et al., 2021).
• Security-Constrained Optimization and N-1 Feasibility: Security-constrained AC-OPF (SC-ACOPF) must secure feasible post-event operation for a planning set of generator, line, or transformer outages. Recent advances include smoothing disjunctive recourse constraints, parallel two-level ADMM, and contingency screening heuristics, enabling solution of industry-scale problems ($30,000$ buses, $22,000$ contingencies) within operational time windows (Gholami et al., 2022).

6. Theoretical Guarantees, Practical Performance, and Comparative Evaluation

• Convex Relaxation Exactness: For lossless, weakly-cyclic (at-most-one-cycle per line, all cycles size 3, $Q_{k,\min}=-\infty$), the convexified (relaxed) SDP robust AC-OPF is exact---the solution is always rank-one for all uncertainty realizations (Chamanbaz et al., 2017).
• Sample Complexity for Probabilistic Approaches: The scenario-with-certificates method ensures, for $$N=\frac{e}{\epsilon(e-1)}(\ln(1/\beta)+n_\theta - 1)$$, the probability of operational-constraint violation is no more than $\epsilon$ with $1-\beta$ confidence, $n_\theta$ the number of design variables (Chamanbaz et al., 2017).
• Empirical Assessment: On test systems (e.g., IEEE 39-bus, 14/30/57/118-bus, PGLib up to 6515 buses, synthetic >10k), robust/scenario-based, convexified, and learning-based surrogates all routinely achieve $<1\%$ optimality gaps, $>99\%$ empirical feasibility, and order-of-magnitude acceleration over classical NLP solvers. For learning-boosted and hybrid methods with a physics-informed structure, even when training data is limited, data efficiency and empirical reliability surpass black-box NNs (Pineda et al., 30 Jul 2024, Za'ter et al., 17 Oct 2025).
• Practical Recommendations: Classical IPM solvers remain the routine workhorse up to several hundreds of buses (sub-second solves), especially with data-boosted warm starts. For guaranteed global optimality, spatial branching with strong relaxations is indispensable but computationally demanding ($10$