# (560a) Graph Representation and Decomposition of Diffusion-Convection-Reaction Processes for Distributed Control

#### AIChE Annual Meeting

#### 2018

#### 2018 AIChE Annual Meeting

#### Computing and Systems Technology Division

#### Dynamics, Reduction, and Control of Distributed Parameter Systems

#### Wednesday, October 31, 2018 - 3:30pm to 3:48pm

However, most DPS's, in reality, are diffusion-convection-reaction systems, whose dynamics are captured by parabolic PDEs (PPDEs). In this work, we propose an equation graph representation of the variables of PPDE systems. The manipulated inputs in a PPDE system are of three types - velocity manipulation, distributed manipulation, and boundary manipulation. The outputs of the PPDE system are of three types - boundary outputs (e.g., stream composition or temperature at the exit of a column), distributed outputs (e.g., concentration or temperature profiles in the column), and spatially varying time-dependent outputs (e.g., hot spot temperature in a column). For the equation graph representation, we incorporate the boundary manipulations in the PPDEs using a Dirac delta function, and define modified output variables as functions of the weighted averages over the spatial domain of the state variables, where the weight is a piece-wise smooth function of the spatial coordinate that attains values close to 1 where the original outputs are located (e.g., at the boundary, at a finite number of locations along the spatial coordinate, or at the hot-spot location), and a small non-zero value at the location of the boundary manipulations (if any). The spatial averaging converts the distributed output variables into vectors with elements that are functions of time alone. The weight function ensures that the values of the modified outputs are close to those of the original outputs while collocating the manipulated variables with the modified outputs. The collocation ensures that the strength of the interaction between a manipulated variable and a modified output can be captured in the form of an explicit dependence of the time-derivatives of the modified output on the manipulated variable. In the equation graph, we represent each input variable, distributed state variable, and the modified output variable with a node. An edge connects an input or a state node to another state node if the former variable appears explicitly in the PPDE corresponding to the latter variable's dynamics. An edge connects a state node to an output node if they are algebraically related. We show that in this graph, the length of an input-output path, i.e., the smallest number of edges joining the two variables, is directly related to the strength of the structural interaction between the variables as captured by the lowest order time-derivative of the output that explicitly depends on the input variable. This is analogous to the equation graphs of LPS's [5] and hyperbolic PDE systems. Thus, this representation allows us to capture on a common equation graph all the variables and interconnections of a generic chemical network to determine the optimal network decomposition for distributed control.

As a case study, we consider a typical reaction-separation plant which comprises four tubular reactors with interstage cooling using three heat exchangers, and two flash columns, with material recycle and heat integration. A first-order, reversible, exothermic (in the forward direction) reaction occurs in the tubular reactors, which have significant convection and diffusion phenomena, and thus are described by quasi-linear PPDEs. The heat exchangers and flash columns are modeled as LPS's. This model is used for implementing model predictive control (MPC). The network decomposition algorithm identifies three sub-networks (communities) in the plant's optimal decomposition, while the edges across the three communities on the equation graph represent the communication among the local controllers of each sub-network. We demonstrate the optimality of the decomposition by comparing the control performance of distributed MPC implemented on the optimal decomposition to the MPC based on other architectures such as the sub-optimal decompositions determined from the algorithm and the conventional centralized and decentralized architectures.

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