A Multi-Disciplinary Optimization Approach for Process and Energy Systems

Abstract

A hierarchical multi-disciplinary optimization (MDO) approach for Process and Energy Systems (PES) is outlined and its efficiency is discussed. The mathematical models of PES consist of systems of equations (differential, PDE, and algebraic) that represent the relevant physics. The models could be lumped or distributed, dynamic or steady state. The equations typically link together state variables with parameters such as performance parameters (e.g., controls), geometry parameters (sizes), and operating conditions e.g., loads or environmental parameters. The PES has to be optimized for performance and cost/weight/volume at different operating conditions (points), by varying the performance and geometry parameters. In general, the full system may be too large to be optimized by an All-at-Once approach, mainly due to simulation difficulties, e.g., the system simulation often fails if initial values and parameter ranges are not set properly. In contrast, the methodology proposed here is very robust, even for large scale systems. The multi-disciplinary optimization approach presented in this paper combines features from major MDO techniques in use. It has four sequential stages: optimization of individual subsystems; coupling of optimizations of subsystems; system level optimization; and multi-point optimization. In the first stage, individual subsystems are analyzed, the essential optimization parameters found and the feasible regions delimited. These are necessary for the second stage, when subsystem optimizations are coupled sequentially and subsystems are optimized for performance and geometry parameters. Central optimization difficulties have to be treated here, for example, those that arise from the subsystem interactions via inputs/outputs or from the competition between the objectives of the different subsystems, e.g., the optimal solution of one subsystem may lead to infeasible solutions for other subsystem. When the second stage optimization stabilizes in a feasible region, it is used in the third stage, a hierarchical optimization, when the geometry is optimized at system level while the performance is optimized at subsystem level. The system level optimization starts with a global optimization, as for example one performed by a global search or genetic type algorithm. For each fixed set of geometry parameters, the subsystems are optimized locally for performance parameters. The best system level solutions of the global optimization are used for a system level local optimization both for geometry and performance parameters. Finally, in the fourth stage, a multi-point optimization for various operating conditions of the full system is performed using the solutions obtained from the third stage optimization as starting points. The four stage approach will produce useful (optimal) results at all stages. Additionally, it provides significant inputs at each stage for effective execution of the next (higher level) stage, and it ensures multi-level convergence of the full problem, which is one of the major shortcomings of many conventional hierarchical approaches. The four stage MDO approach was implemented for the global optimization for weight/volume/performance of a relatively large industrial processing system incorporating more than 10000 variables coupled in partial differential, algebraic equations (PDAE). Successful optimization runs illustrated the capability of the approach to overcome major computational difficulties that lead to failures of many other approaches. The difficulties that had to be solved include: relatively long solver time, solver difficulties as initialization and convergence, vastness of the optimization domain, nonlinearity, feasibility at component and system level, large percent of solver crashes due to un-physical parameter combinations, and major MDO specific difficulties related to subsystem coupling and decoupling.