Demand Response for Flat Nonlinear MIMO Processes using Dynamic Ramping Constraints

Abstract

Volatile electricity prices make demand response (DR) attractive for processes that can modulate their production rate. However, if nonlinear dynamic processes must be scheduled simultaneously with their local multi-energy system, the resulting scheduling optimization problems often cannot be solved in real time. For single-input single-output processes, the problem can be simplified without sacrificing feasibility by dynamic ramping constraints that define a derivative of the production rate as the ramping degree of freedom. In this work, we extend dynamic ramping constraints to flat multi-input multi-output processes by a coordinate transformation that gives the true nonlinear ramping limits. Approximating these ramping limits by piecewise affine functions gives a mixed-integer linear formulation that guarantees feasible operation. As a case study, dynamic ramping constraints are derived for a heated reactor-separator process that is subsequently scheduled simultaneously with its multi-energy system. The dynamic ramping formulation bridges the gap between rigorous process models and simplified process representations for real-time scheduling.

Figures (11)

• Figure 1: A MIMO process that can vary its production rate $\rho$ and thus its energy demand while additional control inputs $\mathbf{u}$ are available to control further process outputs variables $\mathbf{y}$.
• Figure 2: Visualization of flatness-based coordinate transformation showing original nonlinear MIMO process model as introduced in assumption 1 (left), transformations $\boldsymbol{\phi}, \boldsymbol{\psi}_1, \boldsymbol{\psi}_2$ as introduced in assumption 3 (gray arrows), and linear MIMO process model in transformed coordinate space as introduced in assumption 3 (right).
• Figure 3: Overview of steps to derive dynamic ramping constraints performed in the respective sections. For the nonlinear MIMO model and the linear MIMO model, all variable symbols are omitted for clarity as they are identical to Figure 2. The variables of the ramping SISO model are the production rate $\rho$, its derivatives $\dot{\rho},...,\rho^{(\delta-1)}$, and the ramping degree of freedom $\nu$.
• Figure 4: Constraints for ramping degree of freedom $\nu$ as function of production rate $\rho$ for an illustrative case with first-order ramping constraints and two limiting inputs $u_1$, $u_2$. For first-order ramping constraints, the ramping state vector $\boldsymbol{\varphi}$ is of dimension one and equal to the production rate $\rho$. Consequently, the limits on the ramping degree of freedom $\nu$ only depend on $\rho$. The true nonlinear limits caused by the minimum and maximum values of the two inputs $u_1$, $u_2$ are compared to static limits (dotted), linear limits (dashed) and piecewise-affine (PWA) limits (dashed-dotted). Moreover, a linear scale-bridging model (SBM) is visualized (compare to discussion in Section 3.3)
• Figure 5: Case study of reactor-separator process with recycle: States $\mathbf{x}$ are the concentrations of component $A$ and $B$, $c_{A}$, and $c_{B}$, respectively, and the temperature $T$ in the reactor (1) and in the flash (2). Manipulated control inputs $\mathbf{u}$ are the bottom stream $F_B$, the purge stream $F_P$, the heat input to the reactor $Q_1$, and the heat input to the flash $Q_2$. The scheduling degree of freedom is the production rate $\rho$. All other material flow rates are given as functions of $F_B$, $F_p$, and $\rho$, e.g., the reactant stream is equal to the purge $F_p$ plus the production rate $\rho$ as no accumulation of material occurs.