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A novel two-layer predictive control scheme for a continuous biodiesel transesterification reactor is presented. Based on a validated mechanistic model, the least squares (LS) algorithm is used to identify the finite step response (FSR) process model adapted in the controller. The two-layer predictive control method achieves the steady-state optimal setpoints and resolves the multivariable dynamic control problems synchronously. Simulation results show that the two-layer predictive control strategy leads to a significant improvement of control performance in terms of the optimal set-points tracking and disturbances rejection, as compared to conventional PID controller within a multiloop framework.

With the depletion of fossil fuels and global environmental degradation, the development of alternative fuels from renewable resources has received considerable attention. Biodiesel has become the foremost alternative fuel to those refined from petroleum products. It can be produced from renewable sources, such as vegetable and animal oils, as well as from wastes, such as used cooking oil. Transesterification is the primary method of converting these oils to biodiesel [

Biodiesel production by transesterification.

A modern transesterification plant is continuous instead of batch. A continuous biodiesel production leads to better heat economization, better product purity from phase separation by removing only the portion of the layer furthest from the interface, better recovery of excess methanol in order to save on methanol cost and regulatory issues, minimal operator interference in adjusting plant parameters, and lower capital costs per unit of biodiesel produced. The same technology can also be applied to other biofuels production [

Biodiesel transesterification reactor is the most crucial operation unit to be controlled because any drift in standard operating condition may lead to significant changes in process variable and production quality specification [

Recently, a number of reports have appeared on the controller design and dynamic optimization in continuous and batch biodiesel reactors. Mjalli et al. developed a rigorous mechanistic model of a continuous biodiesel reactor and proposed a multimodel adaptive control strategy which realized the set-point tracking and disturbance rejection [

This work considers the advanced control strategy of biodiesel continuous transesterification reactor. Model predictive control (MPC) is one of the most popular advanced control strategies. It is a class of model-based control algorithm, which has become a complex standard process industry solving complicated constrained multivariable control problems, and widely used in the chemical and petrochemical processes [

In recent years, there has been an integrated steady-state optimization of the two-layer predictive control strategy in MPC industry technology [

Although two-layer predictive control strategy has been widely used in many applications of chemical reactors, hardly any work was done on the biodiesel transesterification reactor. In this paper, a two-layer predictive control strategy is designed, tested, and simulated on a continuous biodiesel transesterification reactor. The scheme can amplify the advantages of both technologies in terms of process stability, and optimal and improved performances. Section

The modeling of transesterification reactors starts with understanding the complex reaction kinetic mechanism. The stoichiometry of vegetable oil methanolysis reaction requires three mol of methanol (A) and one mol of triglyceride (TG) to give three mol of fatty acid methyl ester (E) and one mol of glycerol (G) [

The methanolysis, in turn, consists of three consecutive reversible reactions, where a mole of fatty acid methyl ester is released in each step, and monoglycerides (MG) and diglycerides (DG) are intermediate products. The stepwise reactions are

The stepwise reactions can be termed as pseudo-homogeneous catalyzed reactions, following second-order kinetics. The second-order kinetic model can be explained through the following set of differential equations [

The previously selected kinetic model can be formulated in terms of a general reaction equation

The catalyst concentration remained constant because the sidereactions that consume the catalyst were supposed to be negligible. Therefore, each effective rate constant includes the catalyst concentration (

The temperature influence on the reaction rate was studied from the Arrhenius equation (

In order to realize the optimization and control of continuous biodiesel production process, the model used in the paper on the basis of the second-order kinetic model jointing the material and energy balance equations as well as the dynamic equation of the coolant temperature. The material balance for each component is expressed as follows [

The reactor energy balance is expressed as

The coolant fluid energy balance is expressed as

The function equation of heat transfer coefficient is approximately expressed as

In modern process industries, the MPC controller is part of a multilevel hierarchy of optimization and control functions. Typically it is three-layer structure; that is, an RTO block is at the top layer, a MPC block is at the middle, and a PID block is at the bottom [

Reference [

Framework of two-layer predictive control of industrial processes.

Mathematical description of the two-layer predictive control include establishing steady-state mathematical model, steady-state target calculation, and a dynamic controller design [

Assume an MIMO plant with

Under the control increment

abbreviated as

The system can be written at the steady-state time

To meet the requirements of steady-state target calculation, model (

Steady-state target calculation is to maximize economic benefits for the purpose of self-optimization under MPC existing configuration mode according to the process conditions. According to the production process characteristics and objectives, the basic problem of steady-state target calculation is the optimization process, which controlled input as cost variables, controlled output as steady-state variables. A common description of the objective function is as follows [

Since

Given the steady-state constraints of input and output variables, global-optimization problem of steady-state target calculation can be described as the following linear program (

The global-optimization problem of steady-state target calculation can be described as the following quadratic program (

Mathematically, optimization feasibility is the existence problem of the optimal solution. Feasibility of steady-state target calculation means that optimal steady state of input-output should meet their operating constraints; if feasible solution does not exist, the optimization calculation has no solution. The solving process is as follows: first, judge the existence of space domain formed by the constraints and if there is in it for optimization, if does not exist, then through the soft constraints adjustment to obtain the feasible space domain, and then to solve.

Soft constraints adjustment is an effective way to solve infeasible optimization [

Engineering standards of the priority strategy of soft constraints adjustment are the following: give priority to meet the highly important operating constraints, and allow less important operating constraints to be violated appropriately under the premise of satisfying the engineering constraints.

Considering the following constraints (

The algorithm steps of feasibility judgment and soft constraint adjustment based on the priority strategy are as follows.

Initialization: according to the characteristics of the output variables and process conditions, set the upper and lower output constraints priority ranks, the same priority rank setting adjustments according to actual situation constraint weights.

According to the priority ranks, judge the feasibility and adjust the soft constraints in accordance with the ranks from large to small. Under a larger priority rank if cannot find a feasible solution, the constraints of the rank will be relaxed to hard constraints, and then consider less priority rank constraints, until we find a feasible solution.

Then the steady-state target calculation entered the stage of economy optimization or target tracking.

For Step

Solving (

Go to the procedure of judging rank

For (

In engineering applications, dynamic matrix control (DMC) algorithm is one of the most widely used MPC algorithms based on the step response model of the plant. This paper adopts DMC and steady-state target calculation integration strategy.

The difference is that the general DMC algorithms have no requirements on the steady-state position of the control input, and they only require the controlled output as close as possible to arrive at its set point. However, the integration strategy DMC requires both input and output variables to approach their steady-state targets (

Based on system process step response model, at the current time

In the receding horizon optimization process, control increment can be obtained in every execution cycle by minimizing the following performance index:

Subject to the model

Subject to bound constraints

Through the necessary conditions of extreme value

The instant increment can be calculated as follows:

The difference between the process sample values by the present moment

In the biodiesel reactor control, multiloops are necessary to stabilize the plant. One loop is needed to maintain the set point of specifying the product purity, and another loop is needed to ensure an optimal yield of biodiesel and to minimize the generation of unwanted by-products even in the presence of disturbances.

To achieve these goals, the control loop configurations analysis is meaningful. Based on the analysis of Mjalli et al. [

Consequently, the two-layer predictive controller is designed to handle a 2 × 2 system of inputs and outputs. The controlled output variables include biodiesel concentration (

The design of the control loop based on the two-layer predictive control strategy for the biodiesel reactor is shown in Figure

Two-layer predictive framework of biodiesel process.

For the two-layer predictive control scheme to be successful, process modeling plays a key role in capturing the varying dynamics of the system. Section

Firstly, generalized binary noise (GBN) signal is selected as the excitation signal. GBN signals switch between

Next, least squares (

Consider experimental tests of collecting input sequence

Consider matching between data and models; the introduction of residuals for each output can be independently expressed as follows:

Matrix form is written as

Minimize the squared residuals

Obtain the optimal estimate

For the model predictive controller design, the

Coefficients matrix of

Finally, (

In the work, GBN as the excitation signal was added to the model input to produce output data. The parameters of GBN signal applied to the first input are

Under the action of two inputs, reactant flow rate

Biodiesel concentration prediction result and relative error under reactor flow rate

Reactor temperature prediction result and relative error under reactor flow rate

Biodiesel concentration prediction result and relative error under reactor flow rate

Reactor temperature prediction result and relative error under reactor flow rate

Figures

Step response curve of biodiesel concentration and reactor temperature, respectively, under

Step response curve of biodiesel concentration and reactor temperature, respectively, under

To validate the effectiveness and immunity in two-layer predictive control, the models obtained in Section

The reaction rate constants come from [

The economic optimization method described in (^{3}/s, the input ^{3}/s, and the output ^{3} and 3.196 kmol/m^{3}, the output

The parameters of the dynamic control layer adopted the unconstrained DMC algorithm: the modeling time domain

Conventional PID controller has also been designed in this simulation for comparison of performance to two-layer predictive controller. The parameters of PID controller for

Biodiesel concentration and controller moves of two-layer predictive controller and PID controller.

Reactor temperature and controller moves of two-layer predictive controller and PID controller.

As Figures

The optimized values as the setpoints were send to the lower layer DMC. In the beginning, the closed loop response of the two-layer predictive controller was a little sluggish in bringing the biodiesel concentration back the optimum steady-state values, this is because that the algorithm enter the constraint adjustment stage based on the priority strategy which adjusting the upper limit and lower limit to be handled. About At the time

Considering the actual application, the control input is also an important indicator of good or bad controller. From Figures

To challenge the stability of two-layer predictive controller, some disturbances were exerted alone and at the same time. The chosen disturbance variables include coolant input temperature (

Biodiesel concentration and controller moves of four individual disturbance variables effects.

Reactor temperature and controller moves of four individual disturbance variables effects.

Figures ^{3}. For the reactor temperature loop, the feed temperature

Biodiesel transesterification reactor control has become very important in recent years due to the difficulty in controlling the complex and highly nonlinear dynamic behavior. In this paper, a novel two-layer predictive control scheme for a continuous biodiesel transesterification reactor has been proposed. The SSO layer achieved optimal output setpoints according to the local economic optimization goal of the actual production process, and the MPC layer realized the dynamic tracking control. The main aim was to optimize and control the biodiesel concentration and reactor temperature in order to obtain the product of the highest quality at the lower cost. With steady-state optimum target calculation and DMC algorithm implement, the performance of the two-layer predictive controller was superior to that of a conventional PID controller. The two-layer predictive control is not only stable but also tracks set points more efficiently with minimal overshoots and shorter settling times. Moreover, it exhibits good disturbance rejection characteristics.

This work is supported by the National Natural Science Foundation of China (61034008) and the Science Research Foundation of Liaoning Provincial Department of Education (L2012145).