Large problems are not necessarily hard: A case study on distributed NMPC paying off

TL;DR

The paper addresses scaling centralized MPC to large-scale CPS by applying cooperative DMPC with decentralized real-time iterations. It uses a bi-level dSQP framework where an outer convex QP is solved, via ADMM in the inner loop, to coordinate $S$ subsystems with consensus constraints $\sum_i E_i z_i=0$, while leveraging a Gauss-Newton Hessian for efficiency and warm-starting across steps. Through a frequency-control benchmark on power networks, it shows that the required number of optimizer iterations per control step is largely independent of the number of subsystems, with decentralized ADMM and multi-threaded centralized solvers delivering competitive performance; nonlinear cases behave similarly when suboptimal solutions are accepted. The findings indicate that DMPC can scale to large CPS on multi-core CPUs, maintaining real-time feasibility and offering a viable alternative to fully centralized, highly parallelized solvers, especially when per-iteration times can be kept short. The work suggests practical implications for deploying DMPC in large-scale infrastructures and motivates exploring GPU/FPGAs and broader applications to further improve online performance.

Abstract

A key motivation in the development of Distributed Model Predictive Control (DMPC) is to accelerate centralized Model Predictive Control (MPC) for large-scale systems. DMPC has the prospect of scaling well by parallelizing computations among subsystems. However, communication delays may deteriorate the performance of decentralized optimization, if excessively many iterations are required per control step. Moreover, centralized solvers often exhibit faster asymptotic convergence rates and, by parallelizing costly linear algebra operations, they can also benefit from modern multicore computing architectures. On this canvas, we study the computational performance of cooperative DMPC for linear and nonlinear systems. To this end, we apply a tailored decentralized real-time iteration scheme to frequency control for power systems. DMPC scales well for the considered linear and nonlinear benchmarks, as the iteration number does not depend on the number of subsystems. Comparisons with multi-threaded centralized solvers demonstrate competitive performance of the proposed decentralized optimization algorithms.

Figures (7)

• Figure 1: Two 81-bus networks partitioned into nine subsystems of equal size marked in light gray. In Network A, each subsystem contains four generators shown in dark gray and five loads shown in orange. In Network B, three subsystems contain generators and the other subsystems contain loads. Tie lines between coupled buses are shown in black.
• Figure 2: Scalability analysis for convex QPs with $|\mathcal{N}_i| = \{9,16,25\}$ buses per subsystem. The legend on the top left applies to all plots. For each plot, the upper and lower x-axes denote the number of decision variables and the number of subsystems in the network, respectively. Adding more subsystems to the network does not increase the number of necessary iterations, indicating good scalability in decentralized implementations.
• Figure 3: Scalability analysis for non-convex NLPs with varying number of buses per subsystem $|\mathcal{N}_i|$. The legend on the top left applies to all plots. For each plot, the upper and lower x-axes denote the number of decision variables and the number of subsystems in the network, respectively.
• Figure 4: Scalability analysis for dSQP applied to non-convex NLPs with increasing number $| \mathcal{L}_i |$ of loads per subsystem (left) and increasing initial disturbance in the frequency $f_n(0)$ (right). The legend on the left applies to the two plots on the left and the legend on the right applies to the two plots on the right. For each plot, the upper and lower x-axes denote the number of decision variables and the number of subsystems in the network, respectively.
• Figure 5: Closed-loop nonlinear DMPC simulation for the 81-bus Network B with $n_z = 31815$ decision variables. The controller runs dSQP with $k_\mathrm{max} = 1$ and $\l_\mathrm{max} = 10$ iterations per control step. Gray lines show generator trajectories and orange lines show load trajectories.

Theorems & Definitions (3)

• Remark 1: Decentralized implementation
• Remark 2: Closed-loop stability Stomberg2024
• Remark 3: Distributed vs. centralized MPC?