Constrained real-time optimization of a grinding circuit using steady-state linear programming supervisory control

doi:10.1016/S0032-5910(02)00028-1

Powder Technology

Volume 124, Issue 3, 29 April 2002, Pages 254-263

https://doi.org/10.1016/S0032-5910(02)00028-1 Get rights and content

Abstract

This paper presents an application of real-time optimization (RTO) to a simulated ore grinding plant. The control and optimization methods are based on a dynamic linear model of the process. A linear programming (LP) method is used on-line to find the optimum controller set-point as a function of the process-operating constraints. The optimizer selects set-point values that maximize circuit throughput subject to constraints on circulating load, pump box level and hydrocyclone overflow and underflow densities. At the regulatory control level, performances of unconstrained and constrained multivariable predictive controllers are compared and discussed.

Introduction

Ore comminution processes are huge consumers of energy, which determine the performance of the subsequent ore beneficiation processes. Their optimization is then an important issue in the mineral processing industry [21], [28]. As the concept of optimization is rather vague, optimization studies require a clear definition of the goals to be aimed at and of the level of optimization that is to be applied. Let us first define the four different optimization levels that may be considered for a grinding circuit.

At a first level, optimization can be performed at the design stage of the process for the selection of the flow-sheet configuration and the equipment size [19]. The objective is to achieve the metallurgical specifications while minimizing the investments and operating costs, and maximizing the reliability, flexibility, and expandability of the system. A second optimization level is the optimal tuning of the operating conditions, such as the selection of the mill RPM, the grinding media load, the classifier settings (number of hydrocyclones, vortex finder and apex diameters…). The objective is to generate a particle size distribution, a slurry density, and a production capacity, which are optimal for the economic performance of the subsequent separation process, usually assessed by the net smelter return (NSR). At a third level, optimization is performed on the operating set-points: feed-rate, slurry densities, particle size distribution of the product, circulating load. Most of the time the objective is to maximize the throughput while keeping the product fineness at a target value. Finally, the lower level in the hierarchy of process optimization is optimal control [9], [23], [31], [32]. The objective of optimal control is to keep the process at its set-point, while minimizing a performance criterion containing the squared deviations to the set-points and the squared amplitudes of the control actions.

The present study considers mainly the third optimization level, i.e. the selection of the optimal set-points of the controller. However, to illustrate the method on a simulated grinding circuit, the fourth level, optimal control, will be also simulated, although this is not the primary objective of the study. Fig. 1 shows the organization of the third level optimizer and the fourth level optimal controller. There are various approaches to real-time optimization (RTO) at this third level. The qualities of the process model used as well as the evaluation of the economic benefits associated to RTO are important issues of these approaches [8], [16], [17], [29]. The RTO scheme proposed in this study of grinding circuit optimization is based on a linear programming (LP) technique. The method has already been successfully applied to various processes [20], [22], [30], [35], [36]. The steady-state part of the grinding process is linearized around nominal operating conditions, thus allowing to apply an LP method to maximize a linear efficiency criterion subject to equality and inequality constraints.

The set-points selected by the LP–RTO are then distributed to the control loops at the lower level. The latter may use conventional controllers or model-based controllers. In the case of LP optimization, the process is to be operated on constraints. The controllers have to manage these constraints, which should also be enforced during transient regimes. An optimal method is proposed in this study to solve this problem. It is based on a constrained predictive control algorithm.

The paper is organized into three parts. In the first one, a closed-loop grinding circuit is described. In the second part, an LP–RTO method is presented in a sufficiently general form to allow its application to any other process. The paper goes through the model expression, the constant factor updating, the formulation of the objective function and the constraints, the calculation of the set-points, a discussion on the feasibility and, finally, the presentation of the optimal constrained controller for the lower level of the hierarchical optimization strategy. In the third part, the application of the above steps to the closed-loop grinding circuit is detailed. Simulated results are then discussed, showing the efficiency of this hierarchical method based on an LP steady-state optimal supervision of the set-points of an optimal multivariable controller.

Section snippets

Process description

A typical rod-mill ball-mill grinding circuit is depicted in Fig. 2. The ore is fed directly to the rod-mill, which discharges into a pump box. The pump feeds a group of hydrocyclone classifiers. The hydrocyclone overflow is the finished product directed to the flotation stage. The hydrocyclone underflow product is recycled to the pump box after being reground in a ball mill. In some plants, a ball mill, a semi-autogenous mill or any other adequate comminution device can replace the rod mill.

Linear programming optimizer for constrained processes

A real process is necessarily restricted by constraints. Depending upon the structure of the optimization criterion and the location of the constraints, the optimum can be located on constraints or can be unconstrained. An optimization criterion linear with respect to the manipulated variables leads necessarily to an optimum on constraints. Finding an optimum subject to constraints is a task very different from finding an unconstrained optimum. When the optimal process operation is located on

Optimal control

The primary objective of the control strategy is to bring the process to its optimal condition $u$ * and $y$ *. The usual control objective of a predictive controller is to minimize squared deviations to set-point while minimizing the quadratic sum of the variations of the actions. This calculation is made on the predicted trajectory of the process model over the prediction horizon N: $min u (k),Λ u (k+N−1) I(k)= ∑ i=1 N (u (k+i−1)− u (k+i−2))^{T} R (u (k+i−1)− u (k+i−2))+(y (k+i)− y *(k))^{T} Q (y (k+i)− y *(k))$ where the diagonal

Application of the LP–RTO method to a simulated grinding process

For the purpose of demonstrating the LP–RTO method, the process of Fig. 2 is simulated using the software DYNAFRAG [10], [13]. The dynamic simulator is based on a kinetic model of the breakage process involving a fragment distribution function and a rate function [1]. The transport and mixing properties in the mill are represented by a series of interactive perfectly mixed tanks. The hydrocyclone model is a semi-empirical model describing the classifier efficiency as a function of particle size

Conclusion

Real-time optimization is used to find the operating conditions that give the best performances achievable as disturbances occur in a process. Various optimization structures exist in order to achieve this goal. LP optimization is a simple method that can be used when the optimum is located on process constraints and when the objective function, as well as the process model, is linear or linearized. For small non-linearities, bias update is a simple method that gives a continuous correction to

Acknowledgements

This study is part of a generic project on the application of automatic control to the mineral processing and extractive metallurgy industries. It is supported by the Natural Resources Departments of the Quebec and Canada Governments as well as by a consortium of 14 Canadian mining companies.

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View full text

Constrained real-time optimization of a grinding circuit using steady-state linear programming supervisory control

Abstract

Introduction

Section snippets

Process description

Linear programming optimizer for constrained processes

Optimal control

Application of the LP–RTO method to a simulated grinding process

Conclusion

Acknowledgements

Int. J. Miner. Process.

Miner. Eng.

Comput. Chem. Eng.

Comput. Chem. Eng.

Int. J. Miner. Process.

Comput. Chem. Eng.

Chem. Eng. Sci.

Chem. Eng. Sci.

Miner. Eng.

Int. J. Miner. Process.

Process Engineering of Size Reduction: Ball Milling

The prediction of metallurgical performances as a function of fineness of grind

Tuning flotation circuit operation as a function of metal prices

Practical approach to digital control of a grinding circuit at Brenda mines

SME/AIME Trans.

Linear Programing

Methods and Applications of Linear Programming

Optimized multivariable control of an industrial run-of-mine milling circuit

J. S. Afr. Inst. Min. Metall.

On-line optimization of complex process units

Chem. Eng. Prog.