ТЕОРЕТИЧЕСКИЕ ОСНОВЫ ХИМИЧЕСКОЙ ТЕХНОЛОГИИ, 2014, том 48, № 5, с. 486-499
УДК 66.011
NONLINEAR PROGRAMMING STRATEGIES FOR DYNAMIC CHEMICAL
PROCESS OPTIMIZATION © 2014 Lorenz T. Biegler
Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh,
PA 15213, USA lb01@andrew. cmu.edu Received 01.04.2014
Problem formulations and algorithms are considered for optimization problems with differential-algebraic equation (DAE) models. In particular, we provide an overview of direct methods, based on nonlinear programming (NLP), and indirect, or variational, methods. We further classify each method and tailor it to the appropriate applications. For direct methods, we briefly describe current approaches including the sequential approach (or single shooting), multiple shooting method, and the simultaneous collocation (or direct transcription) approach. In parallel to these strategies we discuss NLP algorithms for these methods and discuss optimality conditions and convergence properties. In particular, we present the simultaneous collocation approach, where both the state and control variable profiles are discretized. This approach allows a transparent handling of inequality constraints and unstable systems. Here, large scale nonlinear programming strategies are essential and a novel barrier method, called IPOPT, is described. This NLP algorithm incorporates a number of features for handling large-scale systems and improving global convergence. The overall approach is Newton-based with analytic second derivatives and this leads to fast convergence rates. Moreover, it allows us to consider the extension of these optimization formulations to deal with nonlinear model predictive control and real-time optimization. To illustrate these topics we consider a case study of a low density polyethylene (LDPE) reactor. This large-scale optimization problem allows us to apply off-line parameter estimation and on-line strategies that include state estimation, nonlinear model predictive control and dynamic real-time optimization.
Keywords: nonlinear programming, dynamic chemical process optimization, single shooting, multiple shooting, simultaneous collocation, moving horizon estimation, nonlinear model predictive control, dynamic real-time optimization.
DOI: 10.7868/S0040357114050029
INTRODUCTION For the purpose of this study, we consider the optimization problem stated in the following form:
With growing appreciation of dynamic simulation
in computer aided process engineering, reliable and min W(z(tf))' (1)
efficient optimization tools have also become more zOMO.p
important for these systems. Dynamic optimization dz(t) _ t t , t . _ (2)
studies have been used for a number of offline tasks, in- s . dt fZz't' y()' ( )'p)' Z(0) Z0' ( ) cluding transitions between desired operating conditions, operating profiles for batch process operation,
g(z(t)' y(f)' u(t)' p) = 0' (3)
design and operating studies in response to distur- gf (z(tf)) = 0, (4)
bances and upsets, parameter estimation, model de-
velopment and discrimination for dynamic systems,
UL < u(t) < uv'
and the design of control systems. Online tasks include yL < y(t) < , (5)
the solution of optimization problems for control and < z(j) < ^ identification, particularly in model predictive control
(MPC) and moving horizon estimation (MHE). For The "unknowns" in feis optimization problem are
particularly nonlinear processes, such as polymer pro- the differential state variabtes z(t), a^braw variabtes
cesses, nonlinear models are essential to capture the y(t) control variables u(t), all functions of the scalar
dynamics of the process. As a result, several applica- "time" parameter t e [t0, tf ], as well as time-indepen-
tions of Nonlinear MPC (NMPC) strategies have dent parameters p. As constraints we have the differen-
been reported for these processes. tial and algebraic equations (DAEs) given by (2)—(4)
Fig. 1. Solution strategies for dynamic optimization.
and we assume without loss of generality that the DAE system (2), (3) is index one.
As shown in Fig. 1, a number of approaches can be taken to solve (1)—(5). Currently, DAE optimization problems are solved using a variational approach or by various strategies that apply nonlinear programming (NLP) solvers to the DAE model. Until the 1970s, these problems were solved using an indirect or varia-tional approach, based on the first order necessary conditions for optimality obtained from Pontryagin's Maximum Principle [1, 2]. For problems without inequality constraints, these conditions can be written as a set of DAEs. Obtaining a solution to these equations requires careful attention to the boundary conditions. Often the state variables have specified initial conditions and the adjoint variables have final conditions; the resulting two-point boundary value problem (TPBVP) can be addressed with different approaches, including single shooting, invariant embedding, multiple shooting or some discretization method such as collocation on finite elements or finite differences. A review of these approaches can be found in [3]. On the other hand, if the problem requires the handling of active inequality constraints, finding the correct switching structure as well as suitable initial guesses for state and adjoint variables is often very difficult. Early approaches to deal with these problems can be found in [2].
Methods that apply NLP solvers can be separated into two groups, sequential and the simultaneous strategies. In the sequential methods, also known as control vector parameterization, only the control variables are discretized. In this formulation the control variables are represented as piecewise polynomials [4—6] and optimization is performed with respect to the polyno-
mial coefficients. Given initial conditions and a set of control parameters, the DAE model is embedded within an inner loop controlled by an NLP solver; parameters representing the control variables are updated by the NLP solver itself. Gradients of the objective function with respect to the control coefficients and parameters are calculated either from direct sensitivity equations of the DAE system or by integration of the adjoint equations; several efficient codes have been developed for both sensitivity methods.
Sequential strategies are relatively easy to construct and to apply as they incorporate the components of reliable DAE and NLP solvers. On the other hand, repeated numerical integration of the DAE model is required, which may become time consuming for large scale problems. Moreover, it is well known that sequential approaches have properties of single shooting methods and cannot handle open loop instability [7, 8]. Finally, path constraints can be handled only approximately, within the limits of the control parameterization. More information on these approaches can be found in [3].
Multiple shooting is a simultaneous approach that inherits many of the advantages of sequential approaches. Here the time domain is partitioned into smaller time elements and the DAE models are integrated separately in each element [9—11]. Control variables are parametrized as in the sequential approach and gradient information is obtained for both the control variables as well as the initial conditions of the states variables in each element. Finally, equality constraints are added to the NLP to link the elements and ensure that the states are continuous across each element. As with the sequential approach, inequality
constraints for states and controls can be imposed directly at the grid points.
In the simultaneous approach, also known as direct transcription, we discretize both the state and control profiles in time using collocation of finite elements. This approach corresponds to a particular implicit Runge—Kutta method with high order accuracy and excellent stability properties. Also known as fully implicit Gauss forms, these methods are usually too expensive (and rarely applied) as initial value solvers. However, for boundary value problems and optimal control problems, which require implicit solutions anyway, this discretization often requires far fewer time steps to obtain accurate solutions. On the other hand, the simultaneous approach does lead to large-scale NLP problems that require efficient optimization strategies [12—14]. As a result, these methods directly couple the solution of the DAE system with the optimization problem; the DAE system is solved only once, at the optimal point, and therefore can avoid intermediate solutions that may not exist or may require excessive computational effort.
In the next section we formulate the simultaneous approach and summarize its main advantages and characteristics. Section 3 then reviews NLP strategies that solve the resulting problem. Section 4 provides a parameter estimation study of a low density polymerization reactor to demonstrate the performance of our dynamic optimization approach. Section 5 then extends this approach to deal with on-line optimization for state estimation and model predictive control and explores the advanced step concept for this purpose. Section 6 provides a case study with the LDPE reactor that demonstrates the performance of this approach. Finally, conclusions and directions for future work are given in Section 7.
FORMULATION AND CHARACTERISTICS OF THE SIMULTANEOUS APPROACH
The DAE optimization problem can be converted into an NLP by approximating state and control profiles by a family of polynomials on finite elements (t0 < t1 < ... < tN = 0). These polynomials can be represented as power series, sums of orthogonal polynomials or in Lagrange form. Here, we use the following monomial basis representation for the differential profiles, which is popular for Runge—Kutta discretizations:
z(t) = zt-i + h X Ц
q=i
t - tt-i 1 dz
ht J dt t,q
(6)
at the collocation point q, and Q is a polynomial of order K, satisfying
Qq(0) = 0 for q = 1,...,K, QiPr)
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