Permutation-equivariant deep equilibrium networks for real-time single-bus DER dispatch with O(1) parameter scaling

Abstract Real-time dispatch of heterogeneous distributed energy resources (DERs)—electric vehicles, battery storage, and flexible loads—under a single-bus real-time price requires solving a high-dimensional equilibrium every dispatch interval. We cast the problem as a competitive equilibrium with device-specific dynamics and constraints, and learn the equilibrium policy together with the clearing price using a permutation-equivariant Transformer Deep Equilibrium Network (DER-DEQN) whose parameter count is O (1) in the number of devices N , in contrast to MLP-DEQN baselines that scale O ( N ). We redesign the positional encoding to inject device-type, temporal-window, and feeder priors as the symmetry structure, and train with an unsupervised N -component equilibrium loss (an intertemporal marginal-value Euler analogue, three families of KKT complementarities, and the market-clearing residual that pins the real-time price)—requiring no labeled data. To avoid conflating an architectural guarantee with an equilibrium-quality guarantee—which probe different computational limits—we evaluate the two separately and at the scale appropriate to each. Experiments on two NVIDIA RTX 3090 GPUs establish two distinct claims at their relevant scales. The architecture-level properties hold to $$N{=}5000$$ : O (1) parameter scaling (a flat $$1.19\\times 10^{5}$$ parameters versus $$8.6\\times 10^{7}$$ for an MLP), machine-precision permutation equivariance ( $$\\sim \\!2\\times 10^{-6}$$ ), and a real-time inference latency of 8.8 ms at $$N{=}5000$$ . The equilibrium-quality results—stable label-free training in which the market-clearing and KKT residuals converge to $$\\sim \\!10^{-4}$$ and a competitive welfare comparison to the centralized convex optimum—are validated at $$N{=}50$$ , the largest N at which the unsupervised residual loss is observed to converge to small residuals under our training budget; we report an honest limitation that the intertemporal marginal-value residual settles to a nonzero floor, so exact welfare parity with centralized solvers remains open.