Multistage Economic MPC for Systems with a Cyclic Steady State: A Gas Network Case Study

TL;DR

The paper addresses robust operation of gas networks under cyclical demand by developing a multistage economic NMPC (E-NMPC) framework that directly minimizes an economic objective while enforcing a cyclic steady state (CSS) via endpoint constraints. Stability is guaranteed through a Lyapunov-descent constraint, extended to the multistage setting by enforcing the descent of the expected Lyapunov function $\mathbb{E}_{\mathbb{C}}(V)$ with a slack to maintain feasibility, yielding Input-to-State practical Stability (ISpS). The approach is tested on two natural gas network case studies (a small test network and GasLib-40), showing that the multistage E-NMPC can handle uncertain demands and prevent constraint violations, at the expense of increased computation and modest energy penalties relative to nominal E-NMPC. The results demonstrate improved constraint satisfaction and robustness to uncertainty, with the framework achieving CSS convergence properties in the presence of disturbances and model misspecifications, and providing a practical pathway to robust economic operation of gas networks. The work highlights the trade-off between robustness and computation, and identifies directions for scalability, such as scenario selection and parallelization to reduce solve times.

Abstract

Multistage model predictive control (MPC) provides a robust control strategy for dynamic systems with uncertainties and a setpoint tracking objective. Moreover, extending MPC to minimize an economic cost instead of tracking a pre-calculated optimal setpoint improves controller performance. In this paper, we develop a formulation for multistage economic MPC which directly minimizes an economic objective function. The multistage economic MPC framework is extended for systems with a cyclic steady state (CSS) and stability is guaranteed by employing a Lyapunov-based stability constraint. The multistage economic MPC framework is validated on two natural gas network case studies to minimize the net energy consumption during gas transmission. In both instances, the multistage economic MPC effectively manages uncertain demands by preventing constraint violations and guides the network to its optimal cyclic operating conditions. The Lyapunov function remains bounded in both instances, validating the robust stability of the controller.

Figures (18)

• Figure 1: Fully branched multistage NMPC controller with states $z$, controls $v$ and uncertain parameter $w$. The red dotted lines indicate non-anticipativity constraints on the control variables
• Figure 2: Multistage NMPC controller with a robust horizon of length 1. The red dotted lines indicate non-anticipativity constraints on the control variables within the robust horizon
• Figure 3: Test network schematic
• Figure 4: Gas flow at sink nodes and the corresponding optimal compressor pressure ratios when there is no uncertain parameter in the test network and the system is controlled using a nominal E-NMPC
• Figure 5: Pressures at sink nodes in the test network, the dashed black line indicates the lower bound

Theorems & Definitions (8)

• Definition 1: $\mathcal{K}$ function
• Definition 2: $\mathcal{KL}$ function
• Definition 3: $\mathcal{K}_\infty$ function
• Definition 4: Robust positive invariant (RPI)
• Definition 5: ISpS Lyapunov function
• Definition 6: ISpS stability
• Lemma 1
• proof