NEO-Grid: A Neural Approximation Framework for Optimization and Control in Distribution Grids

Technical Summary: NEO-Grid

Problem Statement
The proliferation of distributed energy resources (DERs), such as solar photovoltaics and battery storage, is transforming distribution grids, introducing significant variability and heterogeneity that challenge voltage stability. Effective management requires accurate modeling of the nonlinear relationship between power injections and voltage magnitudes (the power flow or PF model). Traditional approaches rely on linearized approximations (e.g., LinDistFlow), which often fail to capture critical nonlinearities under heavy loading or in highly resistive feeders. Furthermore, while Volt-Var Optimization (VVO) seeks centralized solutions and Volt-Var Control (VVC) seeks local autonomous policies, both are hindered by the computational complexity of solving nonlinear PF equations and the recursive, closed-loop nature of VVC dynamics. Existing methods for VVC, such as unrolled recurrent neural networks (RNNs), suffer from memory burdens and gradient vanishing issues.

Methodology
The paper proposes NEO-Grid, a unified learning-based framework that replaces traditional linear approximations with neural network surrogates for both optimization and control.

1. Neural Power Flow Surrogate (NEO-Grid Model):

   • The authors construct a neural network (specifically a single-hidden-layer network with ReLU activations) to approximate the nonlinear AC power flow mapping $v = f(p, q)$.
   • This surrogate is trained on data generated by accurate AC-PF simulations under diverse operating conditions, including reactive power compensation scenarios.
   • Because ReLU networks produce piecewise-linear (PWL) functions, the surrogate can be embedded into Mixed-Integer Linear (MIL) or Mixed-Integer Quadratic (MIQP) formulations, making the nonlinear physics tractable for standard solvers.

2. VVO via Mixed-Integer Programming:

   • For Volt-Var Optimization, the trained neural surrogate replaces the nonlinear PF constraints in the optimization problem.
   • The objective is to minimize voltage deviation from the nominal 1.0 p.u. profile. The PWL nature of the network allows the problem to be solved as a Mixed-Integer Quadratic Program (MIQP) or relaxed to a Mixed-Integer Linear Program (MILP).

3. VVC via Deep Equilibrium Models (DEQ):

   • For Volt-Var Control, the authors address the challenge of designing local control rules $q_i = g(v_i)$ that operate in a closed loop.
   • Instead of unrolling the system dynamics over time (as in RNNs), NEO-Grid models the system as a Deep Equilibrium Model (DEQ).
   • The control rule is parameterized as a structured ReLU-based function (mimicking the IEEE 1547 standard's piecewise-linear droop curves).
   • The system state is defined as a fixed point $v^* = F_\phi(v^*)$, where $F_\phi$ composes the neural PF surrogate and the VVC rule.
   • Training utilizes implicit differentiation and root-finding algorithms (e.g., Anderson acceleration) to compute gradients with respect to the control parameters $\phi$ directly at the equilibrium point. This eliminates the need to store long trajectories, significantly reducing memory usage and computational cost.

Key Contributions
The paper outlines three primary contributions:

1. Unified PF Design: The construction of a single neural power flow surrogate that serves as a tractable "neuro-simulator" for both centralized optimization (VVO) and decentralized control (VVC).
2. Generalizability: The framework is designed as a "universal" neuro-simulator capable of supporting diverse tasks, including data fusion, predictive analysis, and digital-twin applications within distribution systems.
3. Equilibrium-Based VVC Learning: The integration of PWL control rule representations with the NEO-Grid surrogate, optimized via DEQ-based implicit differentiation. This approach allows for the efficient optimization of closed-loop control performance without the computational overhead of unrolled recurrent architectures.

Experimental Results
The framework was evaluated on the IEEE 33-bus radial distribution system.

• Power Flow Approximation: The NEO-Grid neural surrogate achieved a Mean Squared Error (MSE) of $1.81 \times 10^{-6}$ on the test set. This was an order of magnitude better than linear regression ($3.16 \times 10^{-5}$) and over 100 times more accurate than the LinDistFlow model ($1.95 \times 10^{-4}$).
• VVO Performance: In voltage regulation tasks, the NEO-Grid-based optimization achieved the lowest average voltage deviation (0.94%) and the fewest large violations (>3% deviation) compared to linear regression and LinDistFlow baselines.
• VVC Performance: The DEQ-trained NEO-Grid controller achieved an average voltage deviation of 3.63%, outperforming linear and heuristic baselines. Crucially, it eliminated all voltage entries with deviations exceeding 7%, whereas other optimization-based methods still exhibited significant large deviations.
• Comparison: While VVO (centralized) naturally achieved better regulation than VVC (decentralized) due to greater flexibility, the NEO-Grid VVC approach demonstrated superior robustness and consistency compared to existing linear/RNN-based methods.

Significance and Claims
The authors claim that NEO-Grid establishes a scalable, accurate, and interpretable solution for learning-based voltage regulation. By bridging the gap between the accuracy of nonlinear physics and the tractability required for optimization and control, the framework offers a practical alternative to linear approximations. The use of DEQs specifically addresses the computational bottlenecks of closed-loop control design, enabling efficient training on single-layer policies without the memory burden of trajectory unrolling. The paper positions NEO-Grid not just as a control algorithm, but as a foundational "neuro-simulator" that can be extended to broader digital-twin use cases, sensitivity analysis, and "what-if" studies in distribution grids.