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Performance analysis of perturbation-based methods for real-time optimization.

This paper provides a comprehensive performance analysis approach for Real-Time Optimization (RTO) technologies, which incorporates systematic approaches to estimating bounds on the convergence behaviour and performance effects of on-line experiments used by a given RTO approach. The performance analysis method is illustrated by an investigation of the conventional two-phase approach and representative techniques drawn from the three main classes of perturbation-based RTO methods which attempt to directly compensate for plant/model mismatch through adaptation. The proposed approach is applied to two simulation-based case studies: a heat exchanger system and a continuous bioreactor.

On presente dans cet article une methode complete d'analyse de performance pour les technologies d'optimisation en temps reel (RTO), quicomporte des approches systematiques pour l'estimation des bornes de convergence et les effets de performance sur des experiences en ligne utilisees dans une approche RTO donnee. L'analyse de performance est illustree par une etude de l'approche conventionnelle a deux phases et des techniques representatives issues des trois categories principales de methodes RTO basees sur des perturbations et quitentent de compenser directement l'incompatibilite usine/modele par l'adaptation. La methode proposee est appliquee a deux etudes de cas basees sur des simulations: un systeme d'echangeur de chaleur et un bioreacteur continu.

Keywords: real-time optimization, performance analysis, extended design cost

In the past two decades, increasing economic, quality, safety and environmental pressures have led to a greater need than ever for operating companies to explore possible paths for improving process profitability. The availability of increasingly more powerful computers, improving process modelling techniques, and evolving advanced control strategies have allowed these companies to consider the possibility of optimizing process economics in real-time. Such RTO systems provide the key link between the plant planning/scheduling functions and the advanced control systems layers in any plant automation hierarchy. Thus RTO occupies a crucial position in many Computer Integrated Manufacturing (CIM) strategies and as such has been enjoying considerable industrial interest. A number of successful applications have been published in the literature (e.g. Mudt et al., 1995; Zhang et al., 1995; Besl et al., 1997; Zhang et al., 1997; and Chen, et al., 1998).

A typical structure of model-based RTO system is shown in Figure 1 and comprehensive discussions of RTO technology can be found in Marlin and Hrymak (1997), Perkins (1998), White (1997). The optimization algorithm and model updating scheme are the base on which any RTO system is built. A number of researchers have investigated the design and interaction of these two subsystems, particularly with respect to plant/model mismatch. The Two-Phase approach (Chen and Joseph, 1987) is the most widely used method for model updating and model-based optimization in RTO. In this approach the optimization problem and parameter estimation problem are solved separately. Although this Two-Phase approach attempts to solve the RTO problem by updating the imperfect model, it will not necessarily converge to the correct optimum (Durbeck, 1965; Biegler et al., 1985; Forbes et al., 1994). To address this issue, methods have been proposed to deal explicitly with plant/model mismatch. These methods fall into two distinct classes: (1) those that modify the RTO problem directly; and (2) those that use adaptive control ideas modified to suit RTO applications.

[FIGURE 1 OMITTED]

Roberts (1979) presents an augmentation procedure to compensate for plant/model mismatch that modifies the objective function of the RTO problem with a compensation term. In recent research, this augmentation procedure was extended to develop an iterative technique for solving dynamic optimal control problems (Becerra and Roberts, 1996). Bamberger and Iserman (1978), Garcia and Morari(1981), and McFarlane and Bacon (1989) have extended adaptive control techniques to RTO systems. These approaches use steady-state information extracted from linear or non-linear dynamic models, which have been updated using process measurements, to determine the direction of improving process performance. Golden and Ydstie (1989) extended the work of Bamberger and Iserman (1978) to incorporate first-principles process modelling into the adaptive approach. Although each of these methods use different algorithms for predicting the optimal plant operation, they each use on-line plant experiments (perturbation of the manipulated variables) as a basis for estimating or compensating for the plant/model mismatch. Thus, these approaches are termed perturbation-based methods.

The main contribution of this paper is a critical performance comparison of representative techniques drawn from the main classes of existing perturbation-based RTO methods, based on the Extended Design Cost performance criterion. Furthermore, this work presents systematic methods for developing bounds on the two critical performance characteristics: convergence behaviour and the performance effects of required perturbations.

RTO PERFORMANCE ANALYSIS

In any RTO application, the optimum plant operating conditions may drift as a result of low frequency process changes (e.g. changes in feed quality, ambient conditions, heat transfer coefficients, column tray efficiencies, catalyst activities, and so forth) and the process measurements used by the RTO system are corrupted by high frequency, common cause variation such as sensor noise. The RTO system attempts to track the low frequency changes optimum plant operating conditions and simultaneously reject the high frequency variation to maintain the plant at its most profitable operating point, at any given time.

Since the objective of RTO is economic optimization, RTO system performance should be measured in terms of economic benefit. Consider the performance plot for a hypothetical RTO system shown in Figure 2, where it is assumed that there is one degree of freedom for optimization (although the RTO problem may actually contain many variables) and the plant optimum remains constant within the pre-specified performance evaluation period. The top plot in Figure 2 shows the true plant optimum operation given in terms of the value of the independent manipulated variable ([x.sup.*.sub.p]) and the trajectory of RTO predictions ([x.sup.*.sub.m]). Note that initially [x.sup.*.sub.m] approaches the optimal plant operation [x.sup.*.sub.p]; however, in the long-term, the RTO system converges to some neighbourhood of [[xi].sup.*.sub.m]. The operating point [[xi].sup.*.sub.m] is defined as:

[[xi].sup.*.sub.m] [lim.sub.[i[right arrow][infinity]] [[x.sup.sub.m,i]

and denotes the operating policy to which the RTO system converges, on average, where i indicates the RTO interval. Then, after a transient period, the model-based RTO system varies around the operating point [[xi].sup.*.sub.m] due to the propagation of high frequency, common cause variation around the RTO loop.

[FIGURE 2 OMITTED]

The bottom plot in Figure 2 illustrates the performance of the RTO system relative to perfect optimization. The profit derived from perfect optimization is given as P([x.sup.*.sub.p]), and the profit from RTO system is given as P([x.sup.*.sub.m]). Then, the shaded area between the two curves, P([x.sup.*.sub.p]) and P([x.sup.*.sub.m]), represents the performance loss of RTO system for the given period, and is a measure of the RTO system performance. The RTO performance can be considered in terms of behaviour over two different time-scales.

The long-term behaviour considered in Forbes and Marlin (1996) consists of two factors: an offset from the optimal plant operation (i.e., [x.sup.*.sub.p]-E [[x.sup.*.sub.m]]) and the fluctuation of RTO predictions in the neighbourhood of [[xi].sup.*.sub.m] due to common cause variation (e.g. sensor noise) being transmitted around the closed RTO loop. Both factors contribute to RTO performance loss. The initial transient (or convergence) behaviour of an RTO system is also a key contributor to overall system performance. It is particularly important to consider transient behaviour of the RTO system when the process is subject to significant, frequent disturbances that change the optimal operating conditions for the plant.

Thus, the performance of any RTO system can be understood in terms of three factors (Zhang and Forbes, 2000): (1) long-term offset from the optimal plant operation, primarily caused by plant/model mismatch; (2) transmission of measurement noise; and (3) convergence characteristics (transient behaviour) of the RTO system. Each of these factors depends on the process model, the model updating technique and the optimization algorithm. In addition, the transient behaviour of the RTO system can also be influenced by the underlying process control layer, which implements the operating policy determined by the RTO system. For the purposes evaluating RTO techniques the effect of the control system is ignored and the process controllers are assumed to be capable of implementing the RTO moves in a timely fashion.

Extended Design Cost

Extended Design Cost (Zhang and Forbes, 2000) is defined as the total loss of performance relative to perfect optimization during pre-specified performance evaluation period, for a given RTO implementation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where: [x.sup.*.sub.p] are the controller set points at the plant optimum, [x.sup.*.sub.m,i] are the set points predicted by the model-based RTO system at the [i.sup.th] RTO interval, P is the plant profit function, and [k.sub.0], [k.sub.f] indicate the start and end of the RTO performance evaluation period, respectively. Thus, the shaded area in the bottom plot of Figure 2 represents the Extended Design Cost as defined in Equation (2).

Zhang and Forbes (2000) gives a simplified expression for Extended Design Cost and methods for determining the various terms in the expression:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where: kb is the time after which the RTO system can be considered converged to within some neighbourhood of

[[xi].sup.*.sub.m], ie [x.sup.*.sub.m,i] [member of] {x[parallel]x - [[xi].sup.*.sub.m][parallel]< [epsilon]}, I [greater than or equal to] [k.sub.b].

The first term in Equation (3) is the performance loss due to persistent steady-state offset between the average predicted optimum after the RTO system has converged and the true plant optimum. The second term is the performance loss due to variance in the predicted optimum operations and the third term is the performance loss during the initial transient behaviour of the RTO system.

Calculation of Extended Design Cost in Equation (3) directly depends on the evaluation times, [k.sub.0], [k.sub.b] and [k.sub.f]. The evaluation period for Extended Design Cost calculations should be chosen to reflect the frequency of major plant disturbances, which affect the optimal plant operations. If the major plant disturbances occur on a similar time-scale to that of the transient behaviour of an RTO system, then a short evaluation period that reflects the disturbance frequency should be chosen. In this case, the transient behaviour of the RTO system is expected to dominate the Extended Design Cost. On the contrary, when process disturbances occur infrequently, the long-term RTO system behaviour will dominate Extended Design Cost and a longer evaluation period should be considered for RTO system performance evaluation. Thus, any RTO system design is a trade-off between the cost associated with persistent steady-state offset, predicted optimum variance and transient behaviour.

PERTURBATION-BASED METHODS FOR RTO SYSTEM

The particular techniques chosen in this study were selected as representatives of the main classes of perturbation-based RTO methods and the intent here is to investigate the relative merits of these classes, not to rate or comment on the individual methods. Furthermore, this work does not purport to give an exhaustive comparison of all available RTO techniques, but concentrates on the main classes of RTO methods.

Perturbation-based methods for RTO system studied in this work include: (1) the Integrated System Optimization and Parameter Estimation (ISOPE) approach (Roberts, 1979); (2) a Linear Adaptive On-line Optimization (LAOO) approach (McFarlane and Bacon, 1989); and (3) a Quadratic Adaptive Online Optimization (QAOO) approach (Golden and Ydstie, 1989). A conventional RTO method, the Two-Phase approach (Chen and Joseph, 1987), which is widely used in current commercial RTO systems, is considered as a benchmark in this work.

In the Two-Phase approach, the model is updated and is then used in a model-based optimization step. Typically the process model is updated by estimating some unknown model parameters using the available process measurements. There are a variety of possible formulations for the parameter estimation problem in any RTO system design. The Least-Squares Regression formulation (Box, 1970) adopted for this paper is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where: [PHI] is an objective function of the parameter estimation problem, [epsilon] is a residual error vector, f is a set of process model equations (equality constraints), [??], [??] are process measurements of both independent and dependent variables, and alpha], [beta]are the fixed and adjustable model parameters, respectively. Then in the second phase, the updated process model is used to determine the optimum operating conditions. The optimization phase of the RTO cycle can be stated as follows:

max P(x,u)

subject to

f(x,u,[alpha],[beta]) = 0

g(x,u,[alpha],[beta]) [less than or equal to] 0 (5)

where: P is the profit function, and g is a set of operating constraints (inequality constraints).

In the Two-Phase approach, the optimization problem and parameter estimation problem are solved separately. As first discussed by Durbeck (1965) and more formally by Biegler et al. (1985), the Two-Phase approach will not necessarily converge to the correct optimum conditions unless the derivatives of the real process outputs with respect to the manipulated variables are matched exactly with the corresponding derivatives in the model. Forbes et al. (1994) give a necessary condition for zero-offset from the true plant optimum for a model-based RTO system, which can be used to determine whether a given process model is adequate for a specific RTO application.

ISOPE

A modified Two-Phase approach, termed Integrated System Optimization and Parameter Estimation (ISOPE), was developed by Roberts (1979) and has seen a number of extensions by Roberts and co-workers. The core idea in this method is to augment the objective function within the model-based optimization problem to include a first-order correction term, which is an approximation of the plant/model mismatch as estimated by the model updater. Then, the ISOPE approach is representative of the class of RTO approaches that attempt to compensate for the effect of plant/model mismatch on gradient information.

In the ISOPE approach the Karush-Kuhn-Tucker optimality conditions are used to form an augmentation term, which is introduced into the objective function of Equation (5) to compensate for plant/model mismatch:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

subject to

f(x,u,[alpha],[beta]) = 0

g(x,u,[alpha],[beta]) [less than or equal to] 0 (6)

The parameter estimation problem defined in Equation (4) remains unchanged in ISOPE approach. The augmentation term [lambda] in Equation (6) is given in Roberts (1979) to be:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where: [[nabla].sub.x][??] are the real plant derivatives with respect to x, and [[nabla].sub.[beta]] P, [[nabla].sub.x]u and [nabla].sub.[beta]] u are sensitivity matrices determined using the process models.

The RTO cycle for the ISOPE approach is as in the conventional Two-Phase approach, but has two additional steps that are completed prior to the optimization phase: (1) perturbation step--perturb each independent manipulated variable individually around the current operating point to get derivative matrix [[nabla].sub.x][??], (note that a new steady-state must be attained for each perturbation in order to evaluate the plant derivatives); and (2) calculate the augmentation term [lambda] using Equation (7).

Linear Adaptive On-Line Optimization Approach

One empirical strategy for determining the real plant input-output sensitivities [[nabla].sub.x][??] during transitions between RTO cycles, proposed by McFarlane and Bacon (1989), is based on on-line identification of a linear ARX model (Soderstrom and Stoica, 1989) and does not require any prior knowledge of process (i.e., first-principles models are unnecessary). In this approach, Equation (5) is modified to an unconstrained steady-state optimization problem by using a desirability function (McFarlane and Bacon, 1989). Then, a discrete linear dynamic model relating [P.sup.*](x,u) to the manipulated variables x may be given in ARX form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The steady-state gradient of [P.sup.*] with respect to the manipulated variables x can be obtained by applying Final Value Theorem to Equation (8). Thus, a new estimate of the optimal plant operations is determined in terms of the steepest descent direction as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where [mu] is a tuning parameter used to limit the step size taken at any RTO interval. In this LAOO approach, a pseudo-random binary sequence (PRBS) signal is superimposed on each of the inputs to identify the ARX approximation of the process dynamics.

The RTO cycle for LAOO approach is: (1) perturbation step--add independent PRBS signals to each of manipulated variables and gather dynamic process data to identify Equation (8) during RTO cycle; (2) calculate the steady-state gradient [[nabla].sub.x][??] [P.sup.*]; and (3) obtain the new optimal operating policy using Equation (9).

Quadratic Adaptive On-Line Optimization Approach

The two classes of RTO methods, which were discussed in the previous two sub-sections, focused on the accuracy of first-order derivative information. If accurate second-order or curvature information is available and is used within the RTO system, then the system should be expected to show higher performance. The adaptive optimization framework used in the LAOO approach can be extended to encompass both first- and second-order information, which is termed the Quadratic Adaptive On-line Optimization (QAOO) approach in this paper.

The work of Golden and Ydstie (1989), although limited to the single-input-and-single-output case, uses a first-principles process model:

u = [??] (x,[x.sub.c], [alpha], [beta]) (10)

where [??] is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

and [x.sub.c] is the current process operating point, [??]([x.sub.c]) is the process measurement, [alpha] is the set of fixed model parameters and [beta] is the set of adjustable model parameters. The adjustable model parameters, [beta], are chosen to ensure that the process model in Equation (10) has the same local geometric characteristics (both gradient and curvature) at the current operating point as the real plant (i.e., [[nabla].sup.2.sub.x][??] = [[nabla].sup.2.sub.x][??] and [[nabla].sub.x][??], where [??] denotes the real plant). Then [[beta].sub.1] and [[beta].sub.2] can be calculated as:

[[beta].sub.1] = [[nabla].sup.2.sub.x][??] - [[nabla].sup.2.sub.x]h (12)

[[beta].sub/.2] = [[nabla].sup.2.sub.x][??] [[nabla].sub.x]h - [[beta].sub.1]x (13)

Since the real plant derivatives [[nabla].sup.j.sub.x][??] (j = 1,2), are not known

exactly, they are determined by applying Final Value Theorem to the following non-linear Hammerstein model:

A([q.sup.-1)u(k) = B([q.sup.-1])x(k - 1) + C([q.sup.-1)[x.sup.2] (k - 1) + d + v(k) (14)

where the parameter A, B and C are identified on-line using a persistent excitation signal added to the manipulated variable. The excitation signal used for this approach is a Pseudo-Random-Ternary-Sequence (PRTS). As a result, the QAOO approach can be expressed as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Subject to

U = [??](x, [x.sub.c],[alpha],[beta]) (15)

and RTO cycle for the QAOO approach is: (1) perturbation step--add independent PRTS signals to each of the manipulated variables and gather dynamic process data to identify Hammerstein model; (2) calculate the derivative and curvature of the real plant; (3) calculate the model parameters [[beta].sub.1] and [[beta].sub.2] in Equation (10) and construct the modified process model [??]; and (4) solve the optimization Equation (15) to obtain the new optimal operating policy.

Although the QAOO approach can be extended to multiple-input-and multiple-output systems, a key difficulty is that as the dimensionality of the RTO problem increases, the computational load for estimating the second-order or curvature term increases approximately as the square of the number of variables.

COMPARISON OF RTO METHODS

In this section, the various RTO approaches are compared in terms of: (1) process modelling and model updating; (2) plant/model mismatch compensation; (3) required plant perturbations; (4) expected convergence rates; and (5) approach in dealing with inequality constraints.

Process Modelling and Model Updating

Model structure is a key issue in the design of any RTO system, which is very crucial to a successful implementation (Forbes et al., 1994). Since no process model can possibly be a perfect representation of the plant, the first problem in any RTO system design is usually selection of an appropriate model structure. A comparison of the candidates approaches are presented in Table 1.

Plant/Model Mismatch Compensation

In examining each RTO approach, with respect to plant/model mismatch compensation, it is necessary to consider: (1) the model updating approach used; (2) how the method characterizes the plant/model mismatch; and (3) how the mismatch is compensated in terms of the reduced properties of the model-based optimization problem.

For the purposes of this study, the mismatch will be viewed in the reduced space (Gill et al., 1981) of the model-based optimization problem. Consider an RTO system that includes both optimization problem and parameter estimation problem. The system may converge to the true plant optimum, if the reduced gradient of the real plant and the model-based optimization problems, denoted as ([[nabla].sub.r]Z) and ([[nabla].sub.r]P), match exactly at the operating point. Then, the plant/model mismatch can be quantified as:

[delta] = [[nabla].sub.r]Z - [[nabla].sub.r]P (16)

where [delta] represents additive mismatch. The diverse means to deal with this [delta] lead to the different performance of RTO methods in eliminating the steady-state offset and a comparison of the compensation strategies for each of the candidate methods is presented in Table 2.

It can be conclude that: (1) the Two-Phase approach assumes that the plant/model mismatch exists in a form that does not affect the reduced gradient of the optimization problem, as this method has no means to compensate for such a mismatch; (2) the LAOO approach of McFarlane and Bacon (1989) attempts to directly determine the reduced gradient of the plant profit surface through plant experiments; and (3) the ISOPE and QAOO approaches assume specific and different structural forms for the plant/model mismatch in the reduced space and estimate the mismatch using plant experiments.

Required Plant Perturbations

In order to obtain the geometric characteristics of the real plant profit surface, two different procedures are adopted for plant experimentation. The ISOPE approach uses steady-state plant experiments. In this approach, the process is perturbed at each new steady-state and the process must reach steady-state again before any RTO calculation can proceed. Each independent manipulated variable must be perturbed in turn and steady-state waited for after each perturbation. This decreases the realized convergence rate and results in a significant performance loss. The adaptive RTO methods (e.g. LAOO and QAOO) use dynamic plant experiments. These methods ensure identifiability using a persistent input excitation signal (e.g. pseudo-random binary and ternary sequences), which are designed to ensure that high-quality static information can be gathered. Furthermore, these two classes of plant perturbations, as well as the corresponding parameter estimation procedures, have the significantly different effects on the variance of RTO predictions. Table 3 summarizes the variance effects of the required plant perturbations for each approach.

Then, it is difficult to draw meaningful general conclusions as to which perturbation-based RTO method has the smaller covariance of the estimated parameters. Thus, any comparisons must be carried out for a specific example and the conclusions drawn there from will be limited to the example only.

Convergence Characteristics

The convergence or transient behaviour of an RTO system is a key contributor to the Extended Design Cost associated with a particular design. As previously discussed, such transient behaviour is particularly important when significant process disturbances occur frequently. One definition for convergence rate of an optimization algorithm is (Fletcher, 1987):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

Where 0 < [gamma] < 1, with [p.sup.th] order convergence. Thus, for linear convergence p = 1, quadratic convergence requires p = 2, and so on. Further, for super-linear convergence p = 1 and the local convergence rate constant [gamma] tends to zero as k [right arrow] [infinity].

For a given local convergence rate (?), assuming [parallel][x.sup.*.sub.m,o] - [[xi].sup.*.sub.m][parallel] < 1, the RTO interval after which the system can be considered to have converged ([k.sub.b]) can be estimated as follows:

Pr[x.sup.*.sub.m,i] - [[xi].sup.*.sub.m][parallel] [less than or equal to] kappa] [x.sup.*.sub.m,o] - [[xi].sup.*.sub.m][parallel] [greater than or equal to] [theta], [for all]I > [k.sub.b] (18)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

where 0 < [kappa] < 1 is a convergence level and [theta[ is the probability level at which the system will be considered to have converged around the point [[xi].sup.*.sub.m]. Inequality (18) states that the RTO system will be considered to have "converged" after [k.sub.b] RTO intervals, if the distance to the optimum [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has been reduced to some fraction (k) of the starting distance [parallel][x.sup.*.sub.m,o] - [[xi].sup.*.sub.m][parallel]. The fraction (k) would normally be chosen with respect to the expected variation in the predicted optimal operation ([x.sup.*.sub.m]). Then, by repeated application of the definition of convergence rate to Inequality (18),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

In the case of linear convergence (p = 1), the resulting upper bound for [k.sub.b] is:

[k.sup.(1).sub.b] [less than or equal to] [log.sub.[gamma]] [kappa] (21)

Otherwise, if p > 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

where [parallel][x.sup.*.sub.m,o] - [[xi].sup.*.sub.m][parallel] > 0. Since for any p > 1 and Natural number [k.sub.b] [less than or equal to] 1, Bernoulli's Inequality is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

Inequality (23) can be re-written as,

[log.sub.[gamma]] [kappa] [greater than or equal to] [k.sub.b] (1 + (p - 1)[log.sub.[gamma]] [parallel][x.sup.*.sub.m,o] - [[xi].sup.*.sub.m][parallel] (25)

and an upper bound of [k.sub.b] for the [p.sup.th] order convergence is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

or,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](27)

Note that the ratio given in Inequality (27) is defined on the closed interval [0, 1] as p [less than or equal to] 1 and [log.sub.[gamma]] [parallel][x.sup.*.sub.m,o] - [[xi].sup.*.sub.m][parallel]. Generally, this ratio becomes smaller for a larger p and depends on the starting point of RTO system for p > 1.

Table 4 presents expressions for the upper bounds on convergence period for each perturbation approach. In both the ISOPE and LAOO approaches, the reduced gradient is matched. Thus, these two approaches are analogous to steepest descent methods and are expected to converge linearly. Furthermore, the ISOPE approach may deviate from linear convergence due to the time required to perform steady-state plant perturbation experiments (i.e., at least dim (x) steady-state perturbation experiments are required to identify the process derivatives within one RTO interval, where x is the vector of decision variables). The QAOO approach uses second-order (curvature) information and as such can be considered analogous to the Quasi-Newton methods for optimization. Thus, the QAOO approach should be expected to show super-linear convergence, with an upper bound of quadratic convergence.

Based on this analysis, the QAOO approach should have the smallest transient behaviour contribution to Extended Design Cost, at the cost of a larger number of model update calculations. It should be noted that this analysis assumes an imperfect model. Should the process model be perfect, each of the approaches will yield the same result.

Inequality Constraints

Another important difference between the various RTO approaches is the manner in which they deal with inequality constraints (i.e., bounds on the process variables and other operating constraints). In the Two-Phase and Roberts' ISOPE approaches the inequality constraints form a natural part of the model-based optimization problem. Thus no special procedures are required to incorporate such constraints. A key issue for these methods is that, should sufficient plant/model mismatch exist, a feasible operating policy calculated based on the process model may not be feasible with respect to actual plant operation. The LAOO approach of McFarlane and Bacon (1989) uses an approach analogous to the Barrier Function methods used in many primal Interior Point algorithms (Wright, 1997) to deal implicitly with inequality constraints. A problem with this approach to inequality constrained optimization problems is that the active constraints are only approached as the number of iterations becomes large. Thus in an RTO application, the McFarlane and Bacon (1989) approach may ensure feasible operation at the cost of being a distance from the optimal operation. In Golden and Ydstie (1989) the problem of inequality constraints is not addressed. These could be incorporated via either Penalty or Barrier Function methods (Gill et al., 1981).

CASE STUDIES

In these case studies, the perturbation-based RTO methods are applied to simulated process systems involving a heat exchanger system and a continuous bioreactor. The Extended Design Cost discussed in the RTO Performance Analysis section is used to evaluate the performance for each approach against a known optimal operating policy.

Example 1: Heat Exchanger System

Forbes (1994) presented a simple heat exchanger for illustrating the necessary conditions of zero-offset for RTO system, which is shown in Figure 3. It is assumed that all process variables can be measured and these measurements are noise free. The latter assumption is made so that a clear comparison of convergence behaviour of the various approaches is possible. The optimization objective of this process is maximization of the outlet temperature ([T.sub.out]) and the fraction (r) of constant feed flow [F.sub.in] that passes into [F.sub.1,in] is the only manipulated variable available for optimization. The flow fraction, r = 0.2904, is the optimal operation. The heat transfer coefficients are not known exactly and as a result, they are considered as adjustable parameters ([beta]) for the purposes of RTO (i.e., [[beta].sub.i]= [U.sub.i]). Note that plant/model mismatch is introduced into this case study as the flow dependence of the heat transfer coefficients is ignored.

[FIGURE 3 OMITTED]

Figure 4 shows the RTO trajectory of the manipulated variable in the heat exchanger network with no process noise. It is clear that Two-Phase approach converges to another fixed point r = 0.3029 and results in a persistent steady-state offset from the true plant optimum due to plant/model mismatch. The perturbation-based RTO methods successfully converge to the true plant optimum; however, their initial transient behaviour are significantly different due to differences in dealing with plant/model mismatch.

[FIGURE 4 OMITTED]

Figure 4 confirms what is expected from the theoretical analysis presented in the last section. The QAOO approach was the most rapid and accurate in finding the optimal fl ow split. ISOPE proved faster than the LAOO approach, due to the formers use of a more structurally accurate model; however, the ISOPE approach exhibits longer term oscillations, which are associated with the plant perturbation scheme used by the method. An analysis the transient data shows that the QAOO approach converges super-linearly; whereas both LAOO and ISOPE converge linearly.

Figure 5 is a plot of Extended Design Cost versus RTO interval for the four RTO methods considered in this paper. Since the measurements are noise free in this case study, Extended Design Cost consists solely of costs associated with the initial transient and steady-state behaviour of the closed-loop RTO system. Figure 5 clearly illustrates the performance advantages of the QAOO approach in this case study. There was a significant difference in the Extended Design Cost for the LAOO and ISOPE approaches, yet little difference between theses two methods in terms of their manipulated variable trajectories (cf. Figures 4 and 5). This is due to the significant economic performance loss incurred by the ISOPE approach during the first steady-state plant experiment and the subsequent need for ISOPE to execute steady-state plant experiments.

[FIGURE 5 OMITTED]

As discussed in the RTO Performance Analysis section, RTO performance evaluation must be considered over some pre-specified performance evaluation period that reflects the expected frequency of changes to the optimum plant operations. Figure 5 shows that for this case study, when the major unmeasured plant disturbances occur more frequently than every two or three RTO intervals, then there is little performance advantage in using any of the perturbation-based RTO approaches and the Two-Phase approach may give the best combination of performance and ease of implementation. As the frequency of major plant disturbances increases to eight intervals, the QAOO approach yields a distinctively superior performance in comparison to all the other methods and the Two-Phase approach yields equivalent or better performance than the remaining two perturbation approaches. Beyond a disturbance frequency of approximately eight RTO intervals, the LAOO approach yields better performance than the conventional Two-Phase approach. The ISOPE approach only outperforms the conventional Two-Phase approach in this case study, when very long-term steady-state behaviour is the chief consideration.

Example 2: Continuous Bioreactor

The process considered in this example is a continuous bioreactor presented by Golden and Ydstie (1989). The optimization objective for this bioreactor is maximization of biomass production and the steady-state optimization problem is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

subject to

X -f(D) = 0

where: the dilution rate D is the process input, the exit biomass concentration X is the process output, and f (D) is the approximate steady-state model used for RTO:

f(D) = Y'(S' - [K'.sub.s] D/[[mu]'.sub.max] - D) (29)

The parameters in Equation (29) are given as: [[mu]'.sub.max] = 0.42[h.sup.-1], [K'.sub.s] = 0.19g/L, S' = 5.0g/L and Y' = 0.4. In the simulation studies, Y is considered to be an adjustable parameter in both the Two-Phase and IOSPE approaches.

Measurements of the dilution rate and exit biomass concentration are considered to be contaminated with zero mean, normally distributed noise. The standard deviation is assumed to be 5% of nominal measurement value. The performance evaluation period is assumed to be fifteen RTO intervals for this case study, based on the expected frequency of significant process disturbances. The simulation results are summarized in Table 5 and Figure 6 gives the trajectory of the dilution rate for each RTO approach.

[FIGURE 6 OMITTED]

Table 5 shows that the conventional Two-Phase approach exhibits offset and this constitutes the largest contribution to its Extended Design Cost. All of the perturbation-based methods eventually converged to the optimal plant operations. The price of eliminating offset (or compensating for plant/model mismatch) is an inflation of the variance cost ([C.sub.V]) caused by the required on-line experiments. The ISOPE approach inflates the variance cost significantly more than either the LAOO or QAOO approaches. This is due to the advantages of identifying steady-state behaviour via dynamic model identification procedures as in the LAOO and QAOO approaches, rather than direct steady-state identification as in the ISOPE method.

Figure 6 illustrates the susceptibility of the various methods to the effects of measurement noise. At approximately 3500 min from the simulation start both the ISOPE and QAOO approach diverge significantly from the optimal process operations, and the convergence rate of the LAOO approach is significantly slowed. In each of these cases, process derivatives are being estimated from noisy plant measurements and at this point in the simulation, the measurement noise is sufficiently large to corrupt the comparatively smaller gradient information. Of course, larger signal to noise ratio could be used for the on-line perturbations to improve identifiability at the expense of further inflating the variance cost. Finally, each of the perturbation-based methods must continually run their on-line experiments after the optimal operation has been reached to ensure that any changes in optimal operation can be efficiently tracked. Again, this inflates the long-term variance cost of the RTO design.

SUMMARY AND CONCLUSIONS

The objectives of this work was to develop insight into the performance characteristics of a broad range of RTO approaches and to compare the performance of three classes of perturbation-based methods for RTO to the widely used Two-Phase approach in terms of: (1) the process modelling techniques employed; (2) the adjustable model parameters that were estimated on-line and the approaches to estimate them; (3) the plant/model mismatch compensation techniques employed; (4) the required plant perturbations; (5) the expected convergence rates; and (6) the methods used to deal with inequality constraints. Approaches were developed for estimating bounds on the performance degradation due to the on-line perturbations required by each class of RTO and the expected convergence rates for the class. Two different case studies were used to illustrate and compare the performance of the RTO approaches. From this work, it is possible to draw some useful guidelines. The principal conclusion is that the perturbation methods are most applicable to RTO problems where there are few degrees of freedom (independent manipulated variables) for optimization and process measurement noise is not a significant factor. The calculations required for on-line identification may become prohibitive for large plants with many manipulated variables. Furthermore, since on-line experiments degrade the performance of any closed-loop RTO system, a balance must be struck between the accuracy of the identified parameters and the economic cost associated with these plant experiments.

A wide range of opportunities for further research exist in the area of perturbation-based RTO methods. These include: (1) modification of existing methods through the incorporation of inequality constraints into the QAOO approach or integrating the dynamic identification methods of McFarlane and Bacon (1989) into Roberts' ISOPE approach; and (2) tailoring statistical Design of Experiments methods and adaptive control approaches to RTO.

ACKNOWLEDGMENTS

The authors are grateful for the financial support of Imperial Oil Ltd., Sunoco Inc. and the Natural Sciences and Engineering Research Council of Canada in the completion of this research.

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Manuscript received December 8, 2004; revised manuscript received December 16, 2005; accepted for publication January 23, 2006.

Yale Zhang (1) and J. Fraser Forbes (2 *)

(1.) Process Automation, Dofasco Inc., Hamilton, ON, Canada L8N 3J5

(2.) Department of Chemical and Materials Engineering, University of Alberta, Edmonton, AB, Canada T6G 2G6

* Author to whom correspondence may be addressed.

E-mail address: fraser.forbes@ualberta.ca
Table 1. Model/updating comparison

 Two-Phase ISOPE

Model type Mechanistic Mechanistic
Model fidelity Developer specified Developer specified
Adjustable Plant parameters Plant parameters
 parameters

 LAOO QAOO

Model type Linear ARX Mechanistic + Hammerstein
Model fidelity Local approximation Developer specified
Adjustable Estimated coeffcients Estimated coeffcients
 parameters

Table 2. Plant/model mismatch compensation comparison

 Two-Phase ISOPE

Model updating Yes Yes
Mismatch ([delta]) N/A [delta] = [lambda]
[[nabla].sub.r]F No Yes
 Compensation
[[nabla].sup.2.sub No No
 .r]F Compensation

 LAOO QAOO

Model updating N/A No
Mismatch ([delta]) [delta] = 0 [delta] = ([[beta].sub.1]x +
 [[beta].sub.2]) [[nabla].sub.u]P
[[nabla].sub.r]F Yes Yes
 Compensation
[[nabla].sup.2.sub No Yes
 .r]F Compensation

Table 3. Comparison of required perturbations

 Two-Phase ISOPE

Perturbation None Steady-state
type steps
Variance N/A [MATHEMATICAL EXPRESSION
inflation NOT REPRODUCIBLE IN
 ASCII]

 LAOO QAOO

Perturbation PRBS PRTS
type
Variance [MATHEMATICAL EXPRESSION [MATHEMATICAL EXPRESSION
inflation NOT REPRODUCIBLE IN NOT REPRODUCIBLE IN
 ASCII] ASCII]

Table 4. Comparison of convergence periods

 Convergence bound

ISOPE [k.sub.b,ISOPE] [less than or equal to](1 + dim(x))[log.sub
 .[gamma]][kappa]
LAOO [k.sub.b,LAOO] [less than or equal to] [log.sub.[gamma]]
 [kappa]
QAOO [k.sub.b,QAOO] / [k.sub.b,LAOO] [less than or equal to] 1 /
 [(1 + (p - 1) [log.sub.[gamma]] [parallel][x.sup.*.sub.m,o] -
 [[xi].sup.*.sub.m][parallel]).sup.2] 1 < p < 2

Table 5. Extended design cost for the continuous bioreactor case study

 [[xi].sup.
Approach *.sub.m] [C.sub.B] [C.sub.T]

Two-Phase 0.3349 0.6801 0.4782
ISOPE 0.3027 0.0016 1.3889
LAOO 0.3014 0.0057 2.2598
QAOO 0.3038 1.1546 x [10.sup.-4] 0.8318

Approach [C.sub.V] [C.sub.E]

Two-Phase 2.1403 x [10.sup.-4] 1.1585
ISOPE 1.6536 3.0441
LAOO 0.2029 2.4684
QAOO 0.1429 0.9745
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