(162f) Approximation of Closed-Loop Dynamics for Dynamic Real-Time Optimization Calculations

Competitive global market conditions and demand-driven production are characteristics of the present day environment that industries face.  In order to operate in a cost-optimal fashion, industrial plant economic optimization has conventionally been addressed via a multilevel process automation architecture with a time-scale separation.  At the upper level, a real-time optimization (RTO) system performs economic optimization in response to demand changes or slow-varying disturbances affecting profit (Marlin and Hrymak, 1997; Darby et al., 2011).  RTO interacts with the lower level process automation activities in a cascade fashion by providing economically optimal set-points for the underlying regulatory control system, which is typically model predictive control (Qin and Badgwell, 2003; Darby and Nikolaou, 2012).

 The conventional RTO strategy is designed based on a steady-state model, which suffers from a limited execution frequency because economic optimization can only be executed once the process is deemed to have reached steady state.  Such an approach generally does not account for the economic performance of the plant during its transition from one steady-state to another, and thus is suboptimal for processes that exhibit slow dynamics and those that experience frequent transitions.  Recent advances have transformed the steady-state RTO to dynamic RTO (DRTO) based on a dynamic prediction model, thus substantially increasing the frequency at which economic optimization can be performed.  Proposed DRTO strategies perform economic optimization in an open-loop fashion without taking into account the presence of the plant control system (Tosukhowong et al., 2004, Wurth et al., 2011), denoted here as an open-loop DRTO approach.  In this approach, the set-points prescribed to the underlying control system are constructed from the optimal DRTO open-loop trajectories under an expectation that the closed-loop response dynamics at the plant level will follow the trajectories obtained at the DRTO level.  An alternative to the multilevel configuration is a single-level, economic model predictive control (EMPC) approach that optimizes the plant economics at the controller sampling frequency. Such a strategy aims to address the issues of model inconsistencies and conflicting objectives between the traditional RTO system and the MPC control layer. In the single-level approach, the objective function could be based purely on economics (Amrit et al., 2013), or a hybrid between cost and control performance (Ellis and Christofides, 2014).

In previous work (Jamaludin and Swartz, 2014), we proposed a closed-loop DRTO strategy that fits well within the two-level industrial process automation architecture.  The original closed-loop DRTO problem structure is in the form of a multilevel programming formulation, as it contains a sequence of embedded MPC controller optimization subproblems to be solved in order to generate the closed-loop response dynamics at the DRTO level.  The closed-loop DRTO formulation computes the controller set-point trajectories that determine the best economics of the predicted closed-loop response over a finite optimization horizon.  The multilevel problem is subsequently reformulated as a single-level mathematical program with complementarity constraints (MPCC) by transforming the MPC subproblems to constraint sets using their first-order, Karush-Kuhn-Tucker (KKT) optimality conditions.  For a constrained MPC problem formulated as a convex quadratic programming (QP) problem as considered in this study, such a transformation is valid as the KKT conditions are necessary and sufficient for optimality.  Details of this construct can be found in Baker and Swartz (2008).  Also in our work, the complementarity constraints arising from the KKT conditions of the MPC subproblems are handled using an exact penalty formulation (Ralph and Wright, 2004) by imposing them as an additional weighted penalty term in the DRTO economic objective.  With an appropriate choice of a penalty parameter, the complementarity penalty term will be negligible at the optimum, hence recovering the solution of the original MPCC problem. We have shown that the closed-loop DRTO approach outperforms the open-loop DRTO approach in terms of economics and control performance. Despite its advantages, the multilevel programming formulation significantly increases the size and solution time of the DRTO problem.  Simulation studies also show that significant computational effort is spent on handling the complementarity variables. 

The aim of this work is to compare different techniques for approximating the MPC closed-loop dynamics embedded at the DRTO level to allow the economic optimization problem to remain manageable and computationally tractable.  We have identified and tested possible approximation techniques, which include: (i) a bilevel programming formulation, (ii) a hybrid closed/open-loop formulation, and (iii) an unconstrained MPC approximation with input clipping.  In the bilevel formulation, only a single MPC optimization subproblem is needed where the prediction and control horizons are extended to match the DRTO optimization horizon, which is significantly longer in order to generate economically viable operating policies.  Input trajectories from the MPC subproblem are assumed to be fully implemented over the entire horizon (Jamaludin and Swartz, 2015).  The hybrid closed/open-loop formulation involves the application of a closed-loop prediction applied over a limited DRTO optimization horizon, after which an open-loop prediction is used.  This means that input trajectories needed to generate the closed-loop response dynamics are obtained from the solution of a reduced number of MPC subproblems, whereas the open-loop response dynamics are generated by a full set of input trajectories from another MPC subproblem.  On the other hand, closed-loop approximation via input clipping involves use of an unconstrained MPC algorithm where an input saturation mechanism is applied over the DRTO optimization horizon using complementarity constraints.  The proposed approximation techniques result in a significantly reduced problem size and faster solution time in comparison to the original multilevel DRTO formulation.  We found that the number of complementarity variables for the hybrid closed/open-loop formulation moderately increases when the MPC control horizon becomes longer, while other approximation techniques are insensitive to the length of control horizon.  The relative performance of these methods is demonstrated via case studies. Excellent approximation of closed-loop prediction is obtained from all methods, with the economic performance largely retained. 

Future work includes application of the closed-loop DRTO approach to more complex systems, and as a higher-level, centralized supervisory controller in a distributed MPC environment where coordination of multiple local MPC controllers can be addressed while making economically optimal decisions.

References

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