The finite element orthogonal collocation method is widely used in the discretization of differential algebraic equations (DAEs), while the discrete strategy significantly affects the accuracy and efficiency of the results. In this work, a finite element meshing method with error estimation on noncollocation point is proposed and several cases were studied. Firstly, the simultaneous strategy based on the finite element is used to transform the differential and algebraic optimization problems (DAOPs) into large scale nonlinear programming problems. Then, the state variables of the reaction process are obtained by simulating with fixed control variables. The noncollocation points are introduced to compute the error estimates of the state variables at noncollocation points. Finally, in order to improve the computational accuracy with less finite element, moving finite element strategy was used for dynamically adjusting the length of finite element appropriately to satisfy the set margin of error. The proposed strategy is applied to two classical control problems and a large scale reverse osmosis seawater desalination process. Computing result shows that the proposed strategy can effectively reduce the computing effort with satisfied accuracy for dynamic optimization problems.
The direct transcription method is an important method to solve the problem of optimal control. By discretization of differential and algebraic optimization problems (DAOPs), the state variables and control variables are completely discretized. The discrete method uses the finite element orthogonal collocation, and generally the number of finite elements is empirically selected and the length of each finite element is equally divided. This results in low discretization accuracy for state and control variables, and to guarantee the satisfactory accuracy for some problems, the calculation time is too long to accept. Moving finite element strategy is a good idea for the solution. With the need of discrete differential algebraic equations, moving finite element is becoming the popular and practical technique for chemical process.
At present, we are concerned with calculation accuracy not only of the material, but also of the time in the process parameters for the chemical process attention. Modeling methods based on first principle and datadriven are used for practical control. With the development of solving technology, we can better understand the changes of state variables in the whole process of chemical reaction.
Betts and Kolmanovsky proposed a refinement procedure for nonlinear programming for discrete processes and estimating the discretization error for state variables [
The above work is quite helpful to quick and stable solution of the dynamic optimization problem, but most of them put emphasis on method of choice and do not propose a specific operational process. As we know, the results of the optimization problem are often dependent on the specific algorithm.
This paper is a mesh refinement strategy based on the variable finite element mesh method [
According to the whole chemical reaction process, we want to know the changes in the material. Discrete solution of the model is the key part for computing process. Here, we introduce the orthogonal polynomials based on Lagrange’s use of the Radau collocation points on the finite element [
On the selection of collocation points on finite elements, if the mathematical expression is configured according to the Gauss point, the algebraic precision of the numerical integration is the highest.
In order to improve the accuracy of the solution, Vasantharajan and Biegler [
Generally, in the solution of problem (
From (
As the error estimation gotten with (
Equally spaced method for finite element mesh.
On the contrary, this approach will take more time and get larger
Divide the time domain
When each finite element error
Once the error of each finite element
However, the finite element mesh gotten above has not satisfied accuracy yet. So we first fix the mesh
Through this way, seasonable initial values for optimization are provided. What is more, it also offers some freedom to search the breakpoint of control profiles.
From now on, we can summarize the approach described with program flow chart in Figure
Finite element mesh refinement algorithm chart.
Through the above grid division and GAMS platform simulation and optimization, the optimized result is obtained finally. However, the optimality of the above result should be verified. According to the optimal control theory, the value of the Hamiltonian function comprising the objective function and the constraints should be kept as constant along the time axis. According to the optimal control theory, the discrete Hamiltonian function is denoted as
When the solution is optimal, the last two terms of the Hamiltonian are zero. According to the optimal control theory, when a system is optimal, the Hamiltonian must be a constant:
Therefore, if the finite element and the configuration point are not constant, the result is not optimal, and the meshing needs to readd the finite element. Here we use method of Tanartkit and Biegler [
When
This numerical experiment is based on Intel (R) Core (TM) i32350M 2.3 GHz processor with 4 GB RAM; the nonlinear programming solver IPOPT [
To illustrate the above strategies, we consider two classical process dynamic optimization problems and a large scale reverse osmosis seawater (SWRO) desalination process. At the same time, we use the error analysis based on the equipartition finite element method for comparison. For the following example, we use the 3order Radau collocation points; each finite element has a noncollocation point. Three of these collocation points are
This example is a chemical process, considering a nonisothermal reactor with first order
The problem has been solved in many pieces of literature. In [
Error of differential profiles at finite elements with equally spaced method.
As shown in Figure
Error of differential profiles at finite elements with new method.
In Figure
Size of finite element in simulation and optimization.
Figures
Optimal control for the batch reaction problem.
State profiles for the batch reaction problem.
Figure
Profile of Hamiltonian function.
From Table
Batch reaction numerical results.
Strategy 

Var  Eq  Iter  Accuracy  CPU(s) 

ESM  14  294  252  11 

9.841 
MFE  7  217  190  39 

5.330 
Here we consider another optimal control problem which can be described as
In this example, the number of finite elements is often estimated by trial and error. Here, we use 104 equally spaced finite elements to make the two differential equations meet the error requirements, as shown in Figure
Error of differential profiles at finite elements with equally spaced method.
As can be seen from Figure
Error of differential profiles at finite elements with new method.
Figure
Size of finite element in simulation and optimization.
Figures
Optimal control for Rayleigh problem.
State profiles for Rayleigh problem.
The Hamiltonian function image of Figure
Profile of Hamiltonian function.
In Table
Rayleigh reaction numerical results.
Strategy 

Var  Eq  Iter  Accuracy  CPU(s) 

ESM  104  36  31  8 

67.277 
MFE  8  248  217  71 

5.352 
The feasibility of the proposed method is illustrated by two classical chemical reaction control problems. The following is an operational optimization study of a large scale reverse osmosis seawater (SWRO) desalination process. The model equation can be expressed as
Here
The optimization proposition also needs to satisfy the RO process model, the reservoir model, the operation cost model, and the inequality constraint equation. The system cost model is shown as follows:
Here,
From the optimization program mainly to the day, there are peaks and valleys of electricity prices, resulting in different periods of water production costs. The system can be done through the cistern reservoir volume, regulating the water system and the user’s demand for water to reduce the overall cost.
The reverse osmosis process model mainly includes three differential variables: membrane concentration, flow rate, and pressure. In this paper, the number of finite elements is divided into discrete errors based on the steadystate simulation process, so as to reduce the computational complexity of the optimization model.
Figure
Error of differential profiles at finite elements with equally spaced method. The black dotted line represents the maximum error of each finite element at the noncollocation point.
Error of differential profiles at finite elements with new method. The black dotted line represents the maximum error of each finite element at the noncollocation point.
Figure
Size of finite element in simulation and optimization.
Figures
Membrane channel within one hour of the velocity changes.
Membrane channel within one hour of the concentration changes.
Membrane channel within one hour of the pressure changes.
Table
SWRO numerical results.
Strategy 

Var  Eq  Iter  Accuracy  CPU(s) 

ESM  22  44044  43996  42 

163.775 
MFE  14  23836  23778  33 

154.924 
It can be seen that the number of finite elements required for the model discretization is reduced to 14 when using the moving finite element method. Due to the finite element method, the finite element length can be adjusted to a small extent, which greatly increases the accuracy of the solution.
In this paper, we proposed a meshpartitioning strategy based on the direct transcription method to solve the optimal control problem. This method discretizes the differential algebraic equation (DAE) using the Radau collocation point based on the variable finite element and finally transforms into a nonlinear programming problem. Here, the initialization variable finite element uses a valid termination criterion. At the same time, the computation time can be saved under the condition of satisfying certain precision.
Cases study of two classical control problems and a large scale reverse osmosis process optimization problem was carried with our proposed method and conventional method. The computing results show that the proposed method can effectively reduce the number of finite element and thus reduce the computing efforts with permitted discrete accuracy.
Terminal time (s)
The scalar objective function
Differential state variables
Algebraic state variables
Constraint equation for the state and control variables
Control variable
Time independent optimization variable
The length of element
Number of collocation points
First derivative in element
Constant depending on collocation points
Differential state equation at
Default set point error estimation limit
The setting error
Number of finite elements
Number of variables
Number of equations
The number of iterations calculated
Equally spaced method
Moving finite elements
Membrane water permeability (m·s^{−1}·Pa^{−1})
Intrinsic membrane water permeability (m·s^{−1}·Pa^{−1})
Membrane TDS permeability (m/s)
Intrinsic membrane TDS permeability (m/s)
Constant parameter
Constant parameter
Constant parameter
Feed (operational) temperature (K)
Feed pressure (bar)
Pressure drop along RO spiral wound module (bar)
Solvent flux (kg/m^{2}·s)
Pressure loss of osmosis pressure (bar)
Solute flux (kg/m^{2}·s)
Salt concentration of membrane surface (kg/m^{3})
Permeate concentration of RO unit (kg/m^{3})
Bulk concentration along feed channel (kg/m^{3})
Mass transfer coefficient (m/s)
Reynolds number (dimensionless)
Schmidt number (dimensionless)
Hydraulic diameter of the feed spacer channel (m)
Dynamic viscosity (m^{2}/s)
Kinematic viscosity
Density of permeate water (kg/m^{3})
Friction factor
Empirical parameters
Axial velocity in feed channel (m/s)
Height of the feed spacer channel (m)
Operational cost of entire process
Energy cost of RO process
Energy cost for intake and pretreatment
Cost for chemicals
Other costs
Specific energy consumption (kw·h/m^{3})
Permeate flow rate (m^{3}/h)
Electricity price (CNY/kw·h)
Output pressure of the intake pump (bar)
Load coefficient
Mechanical efficiency of intake pump.
The authors declare no conflicts of interest.
This work is supported by National Natural Science Foundation (NNSF) of China (no. 61374142), the Natural Science Foundation of Zhejiang (LY16F030006), the Public Projects of Zhejiang Province, China (no. 2017C31065), Research and Innovation Fund of Hangzhou Dianzi University (CXJJ2016043), and Hangzhou Dianzi University Graduate Core Curriculum Construction Project (GK168800299024012).