Bethany L. Nicholson

Parallel cyclic reduction decomposition for dynamic optimization problems

Abstract Direct transcription of dynamic optimization problems, with differential-algebraic equations discretized and written as algebraic constraints, can create very large nonlinear optimization problems. When this discretized optimization problem is solved with an NLP solver, such as IPOPT, the dominant computational cost often lies in solving the linear system that generates Newton steps for the KKT system. Computational cost and memory constraints for this linear system solution raise many challenges as the system size increases. On the other hand, the linear KKT system for our dynamic optimization problem is sparse and structured, and can be permuted to form a block tridiagonal matrix. This study explores a parallel decomposition strategy for block tridiagonal systems that is based on cyclic reduction (CR) factorization of the KKT matrix. The classical CR method has good observed performance, but its numerical stability properties need further study for our KKT system. Finally, we discuss modifications to the CR decomposition that improve performance, and we apply the approach to four industrially relevant case studies. On the largest problem, a parallel speedup of a factor of four is observed when using eight processors.

Uncovering New Opportunities from Frequency Regulation Markets with Dynamic Optimization and Pyomo.DAE

Abstract Real-time energy pricing has caused a paradigm shift for process operations with flexibility becoming a critical driver of economics. As such, incorporating real-time pricing into planning and scheduling optimization formulations has received much attention over the past two decades (Zhang and Grossman, 2016). These formulations, however, focus on 1-hour or longer time discretizations and neglect process dynamics. Recent analysis of historical price data from the California electricity market (CAISO) reveals that a majority of economic opportunities come from fast market layers, i.e., real-time energy market and ancillary services (Dowling et al., 2017). We present a dynamic optimization framework to quantify the revenue opportunities of chemical manufacturing systems providing frequency regulation (FR). Recent analysis of first order systems finds that slow process dynamics naturally dampen high frequency harmonics in FR signals (Dowling and Zavala, 2017). As a consequence, traditional chemical processes with long time constants may be able to provide fast flexibility without disrupting product quality, performance of downstream unit operations, etc. This study quantifies the ability of a distillation system to provide sufficient dynamic flexibility to adjust energy demands every 4 seconds in response to market signals. Using a detailed differential algebraic equation (DAE) model (Hahn and Edgar, 2002) and historic data from the Texas electricity market (ECROT), we estimate revenue opportunities for different column designs. We implement our model using the algebraic modeling language Pyomo (Hart et al., 2011) and its dynamic optimization extension Pyomo.DAE (Nicholson et al., 2017). These software packages enable rapid development of complex optimization models using high-level modelling constructs and provide flexible tools for initializing and discretizing DAE models.

A General Framework for Sensitivity-Based Optimal Control and State Estimation

Abstract New modelling and optimization platforms have enabled the creation of frameworks for solution strategies that are based on solving sequences of dynamic optimization problems. This study demonstrates the application of the Python-based Pyomo platform as a basis for formulating and solving Nonlinear Model Predictive Control (NMPC) and Moving Horizon Estimation (MHE) problems, which enables fast on-line computations through large-scale nonlinear optimization and Nonlinear Programming (NLP) sensitivity. We describe these underlying approaches and sensitivity computations, and showcase the implementation of the framework with large DAE case studies including tray-by-tray distillation models and Bubbling Fluidized Bed Reactors (BFB).

Benchmarking ADMM in nonconvex NLPs

Abstract We study connections between the alternating direction method of multipliers (ADMM), the classical method of multipliers (MM), and progressive hedging (PH). The connections are used to derive benchmark metrics and strategies to monitor and accelerate convergence and to help explain why ADMM and PH are capable of solving complex nonconvex NLPs. Specifically, we observe that ADMM is an inexact version of MM and approaches its performance when multiple coordination steps are performed. In addition, we use the observation that PH is a specialization of ADMM and borrow Lyapunov function and primal-dual feasibility metrics used in ADMM to explain why PH is capable of solving nonconvex NLPs. This analysis also highlights that specialized PH schemes can be derived to tackle a wider range of stochastic programs and even other problem classes. Our exposition is tutorial in nature and seeks to to motivate algorithmic improvements and new decomposition strategies

Mathematical Modeling and Optimization

This chapter provides a primer on optimization and mathematical modeling. It does not provide a complete description of these topics. Instead, this chapter provides enough background information to support reading the rest of the book. For more discussion of optimization modeling techniques see, for example, Williams [86]. Implementations of simple examples of models are shown to provide the reader with a quick start to using Pyomo.

Mathematical Programs with Equilibrium Constraints

This chapter documents how to formulate mathematical programs with equilibrium constraints (MPECs), which naturally arise in a wide range of engineering and economic systems. We describe Pyomo components for complementarity conditions, and transformation capabilities that automate the reformulation of MPEC models, and meta-solvers that leverage these transformations to support global and local optimization of MPEC models.

Differential Algebraic Equations

This chapter documents how to formulate and solve optimization problems with differential and algebraic equations (DAEs). The pyomo.dae package allows users to easily incorporate detailed dynamic models within an optimization framework and is flexible enough to represent a wide variety of differential equations. We also demonstrate several automated solution techniques included in pyomo.dae that apply a simultaneous discretization approach to solve dynamic optimization problems.

Structured Modeling with Blocks

This chapter documents how to express hierarchically-structured models using Pyomo’s Block component. Many models contain significant hierarchical structure; that is, they are composed of repeated groups of conceptually related modeling components. Pyomo allows the modeler to define fundamental building blocks, and then construct the overall problem by connecting these building blocks together in an object-oriented manner. In this chapter, we describe the fundamental Block component along with common examples of its use, including repeated components and managing model scope.

Generalized Disjunctive Programming

This chapter documents how to express and solve Generalized Disjunctive Programs (GDPs). GDP models provide a structured approach for describing logical relationships in optimization models.We show how Pyomo blocks provide a natural base for representing disjuncts and forming disjunctions, and we how to solve GDP models through the use of automated problem transformations.

Pyomo Models and Components: An Introduction

This chapter describes the core classes that are used to define optimization models in Pyomo. Most of the discussion focuses on modeling components that are used to declare parts of a model. We include a discussion of the options that can be used when declaring the components and information about key component attributes and methods.