Iterative ant-colony algorithm and its application to dynamic optimization of chemical process

Abstract

For solving dynamic optimization problems of chemical process with numerical methods, a novel algorithm named iterative ant-colony algorithm (IACA), the main idea of which was to iteratively execute ant-colony algorithm and gradually approximate the optimal control profile, was developed in this paper. The first step of IACA was to discretize time interval and control region to make the continuous dynamic optimization problem be a discrete problem. Ant-colony algorithm was then used to seek the best control profile of the discrete dynamic system. At last, the iteration based on region reduction strategy was employed to get more accurate results and enhance robustness of this algorithm. Iterative ant-colony algorithm is easy to implement. The results of the case studies demonstrated the feasibility and robustness of this novel method. IACA approach can be regarded as a reliable and useful optimization tool when gradient is not available.

Introduction

Chemical process is often described by a group of very complex nonlinear differential equations. Dynamic optimization, which is often employed for the chemical industry productions and managements, is to make a performance index optimal by controlling operation variables (Emil & Jens, 2001; Eva et al., 2001; Kvamsdal et al., 1999). A typical dynamic optimization problem of continuous process is described as follows (Roubos et al., 1997).

$$\min J(u)=\Phi (x(t_{\text{f}}),t_{\text{f}})+{\int }_0^{t_{\text{f}}}\Psi (x(t),u(t),t)\text{d}t$$

$$\text{s}.\text{t}.\frac{\text{d}x}{\text{d}t}=f(x(t),u(t),t)$$

$$x(0)=x_0;u_{\min }\le u(t)\le u_{\max }$$

where x and u are, respectively, called state variables and control variables.

Several methods to solve the dynamic optimization problem have been reported in the literatures. In gradient algorithms based on the Hamiltonian function (Roubos et al., 1997), firstly the time interval is divided into a number of stages and then for each time stage the local gradient of the objective function with respect to changes in the values of the control variables is calculated. Subsequently, the local sensitivities are used to adjust the control trajectories in order to improve the objective function. Gradient algorithms proved to be reliable (Roubos, van Straten, & van Boxtel, 1999), but the gradient computing is not an easy job and moreover, the gradient is not available at all times. More importantly, using these algorithms one can only find a local optimum in many cases.

Dynamic programming (DP) was developed based on Bellman's principle of optimality (Chen, Sun, & Chang, 2002) and has been proved to be a feasible method when the gradient based on the Hamiltonian function is difficult to compute. Both time interval and control variables are discretized to a predefined number of values. Then, a systematic backward search method in cooperation with system simulation models is used to find the optimal path through the defined grid. To have a reasonable result, DP needs a large number of grid values for the state variables and the control variables. Therefore, numerous integrations are needed at each time stage. Obviously, with an increase on dimensions of a concerned problem, the well-known curse of dimensionality will become unavoidable.

To avoid this difficulty, iterative dynamic programming (IDP), a modified DP, was proposed. Overcoming the problem of curse of dimensionality through the usage of coarse grid points and region-reduction strategy, IDP not only promotes the efficiency of computation but also increases numerical accuracy. It should be noted that the basic principle of IDP is still to optimize one time stage in turn instead of optimizing all stages simultaneously (Luus, 1993; Dadebo & Mcauley, 1995; Rusnák et al., 2001). This principle is just the same as that of DP and the approach also needs discretization of states variables just as DP does. When there are more state variables, it will still be troublesome.

Genetic algorithm (GA) has become very popular for nonlinear dynamic optimization in recent years (Pham, 1998, Roubos et al., 1999). Roubos et al. used a real-coded chromosome to represent a feasible control profile. With cross-over and mutation operators, control profiles at all time stages are optimized simultaneously. This is its advantage over IDP. Continuous ant-colony algorithm (CACA), the integration of a derivation of ACA and GA, also has this advantage (Rajesh, Gupta, & Kusumakar, 2001). Nevertheless searching optimum in continuous regions using either GA or CACA is troublesome.

Comparison of stochastic algorithm (GA or CACA) and IDP for dynamic optimization will give some suggestions. The advantage of IDP lies in discretization of control region and iteration process that make a complex continuous problem be tandem and simple discrete problem. Anyway searching among finite candidates is easier and simpler than in a continuous region. The merit of stochastic algorithm is that control profile at all time stages is optimized simultaneously, and also it is easier to compute states variables and performance index. That is to say, stochastic algorithm does not need discretization of states variables. So the integration of iteration and stochastic algorithm is a good choice.

In this paper, a novel algorithm named iterative ant-colony algorithm (IACA) is developed, the main idea of which is to iteratively execute ant-colony algorithm and gradually approximate the optimal control profile. IACA is more concise than IDP because it does not need discretization of states variables and control profiles at all time stages can be optimized simultaneously. It is easy to implement and is more efficient than GA and CACA because of searching optimum among finite candidates. IACA has demonstrated its feasibility and robustness with the successful applications to various case studies.