Identification of realistic distillation column using hybrid particle swarm optimization and NARX based artificial neural network

Abstract

Nonlinear identification of a distillation column is a challenging problem in the process industry. The performance of the controller of nonlinear and dynamic columns can be viewed or analyzed using this type of identification. In this work, a novel method is proposed for the identification of a distillation column using hybrid PSO (particle swarm optimization) and ANN (artificial neural network). Since the real distillation column is dynamic in nature, this hybrid system is used as a nonlinear function in NARX (nonlinear autoregressive with exogenous input) structure. This hybrid NARX model is called PSO-NARX-ANN. In PSO-NARX-ANN, NARX-ANN is trained by using the PSO algorithm. The PSO training process has the advantage of training neural network without getting trapped at local optimal points. Reflux rate and reboiler temperature were used as variable inputs while the top and the bottom compositions (mole fractions) were used as variable outputs. The column was realistically simulated in HYSYS process simulation software and data was generated. To ensure robustness and accuracy, 750 of the 1000 samples of data collected from HYSYS were used for training, and the remaining 250 samples of data were used for validation of the proposed model. The performance of proposed model compared with (back propagation) BP-ANN, NARX-BP-ANN, and PSO-ANN. The results showed that PSO-NARX-ANN outperformed all others.

Appendices

Appendix

A Dynamic mathematical model

Distillation column trays are numbered from top to bottom. In the following equation, $L_{R}$, D, V, L, M, B, $x_{i}$, and $y_{i}$ are used for reflux flow, distillate flow, vapor flow, liquid flow, hold up, the liquid mole fraction of the component, and vapor mole fraction of component respectively. Mass balance and component mass balance at different stages are given below.

At the condenser,

$$\begin{aligned} V_{2} = & {} L_{R}+D,\end{aligned}$$

(17)

$$\begin{aligned} \frac{dx_{1}}{dt}= & {} \frac{V_{2}}{M_{C}}(y_{2}-x_{1}). \end{aligned}$$

(18)

At the all stages except condenser, feed stage and reboiler is given by

$$\begin{aligned} \frac{dM_{i}x_{i}}{dt}=L_{i-1}x_{i-1}+V_{i+1}y_{i+1}-L_{i}x_{i}-V_{i}y_{i}, \end{aligned}$$

(19)

$$\begin{aligned} \frac{dx_{i}}{dt}=\frac{L_{i-1}x_{i-1}+V_{i+1}y_{i+1}-L_{i}x_{i}-V_{i}y_{i}}{M_{T}}. \end{aligned}$$

(20)

At the feed stage,

$$\begin{aligned} \begin{aligned} \frac{dM_{nf}x_{nf}}{dt} = &\,\, {} L_{nf-1}x_{nf-1}+V_{nf+1}y_{nf+1} \\ &-L_{nf}x_{nf}-V_{nf}y_{nf}+Fz_{f}, \end{aligned} \end{aligned}$$

(21)

$$\begin{aligned} \begin{aligned} \frac{dx_{nf}}{dt}= &{} \frac{L_{nf-1}x_{nf-1} + V_{nf+1}y_{nf+1}}{M_{T}} \\ &\,\,\frac{-L_{nf}x_{nf} -V_{nf}y_{nf}+Fz_{f}}{M_{T}}. \end{aligned} \end{aligned}$$

(22)

At the reboiler,

$$\begin{aligned} L_{n-1}=B+V_{1}, \end{aligned}$$

(23)

$$\begin{aligned} \frac{dx_{n}}{dt}=\frac{L{n-1}x_{n-1}-Bx_{n}-V_{1}y_{n}}{M_{R}}. \end{aligned}$$

(24)

Vapor and liquid flow in all stages except feed stage is represented by

$$\begin{aligned} L_{i}=L_{i+1}, \end{aligned}$$

(25)

$$\begin{aligned} V_{i}=V_{i-1}. \end{aligned}$$

(26)

Vapor and liquid flow in the feed stage is given by

$$\begin{aligned} V_{nf}=V{nf}+F(1-q) \quad for \quad i=nf, \end{aligned}$$

(27)

$$\begin{aligned} L_{nf}=L{nf}+F(q) \quad for \quad i=nf, \end{aligned}$$

(28)

where q is given by equation

$$\begin{aligned} q=\frac{L_{nf}-L{nf-1}}{F}. \end{aligned}$$

(29)

Vapour liquid equilibrium relationship of component is given by Eq. 30

$$\begin{aligned} y_{i}=\frac{\alpha x_{i}}{1+(\alpha -1)x_{i}}. \end{aligned}$$

(30)