بهینه سازی تکراری از کارائی اقتصادی یک فرایند صنعتی در محدوده صحت مدل گیاهی استاتیک و کاربرد آن به یک خمیر آسیاب

Optimization of the steady state economic efficiency of an industrial process is a specific task because the decision variables of the optimization (setpoints of the control system) affect the process through the control strategy. Thus, the effects of saturation of a control system must be taken into account when the gradient of the objective function is estimated and the necessary optimality conditions are checked. In particular, because the optimality conditions cannot be checked directly in the presence of active constraints on the manipulated variables, approximations of the steady state values of the manipulated variables as functions of the setpoints (static plant model) are needed in order to be able to evaluate the optimality conditions. In this paper an iterative method for optimization of the plant profit rate is proposed avoiding the control saturation and is applied to the Pulp Mill benchmark model optimization. Three different static models describing the steady state values of the manipulated variables are constructed and used in the optimization. The results of the optimization are presented and compared against the straightforward single-step optimization of the plant economic efficiency.

Due to the continuously increasing competition in the pulp and paper industry, there is a need to develop solutions that can increase the economical efficiency of the plants. According to Luyben (1989) it is usually much cheaper, safer and faster to conduct the optimization of plant operations on the basis of a mathematical model rather than experimentally. However, the significant number of studies is still concentrated on the optimization of single unit operations in Pulp Mills: Mcdonough, Uno, Rudie, and Courchene (2008) studied the optimization of the ClO2 requirements for the bleaching process; Tang, Wang, He, and Itoh (2007) optimized the Pulp Washing process; Sarimveis, Angelou, Retsina, Rutherford, and Bafas (2003) optimized the energy management in pulp and paper mills; Smith, Christlmeier, and Van Winkle (1986) studied the possibility of increasing of the recovery boiler throughput; Sidrak (1995) optimized the Kamyr Digester towards significant reduction in the amount of off-specification pulp. Klugman, Karlsson, and Moshfegh (2007) studied the energy consumption and production by the Pulp Mill; Savulescu and Alva-Argaez (2008) minimize the energy consumption through managing the direct heat transfer related to the water streams; Westerlund et al. (1986) solved the equilibrium equations for the white liquor and optimized the lime feed rate; Santos and Dourado (1999) optimized energy consumption and the plant's production; Thibault et al. (2003) concentrated their efforts on the multicriteria optimization of the plant. Nowadays, the trend is to optimize the whole mill with respect to production and quality, minimization of energy, chemical consumption and effluents. In a recent paper Castro and Doyle (2004) have proposed the Pulp Mill benchmark model, having the standard architecture with a Kamyr digester, a bleaching plant and a chemical recovery (see Castro and Doyle (2004) for the details). The benchmark model is well suited for performing of a wide range of the Pulp Mill studies, including the optimization problem of economical efficiency. Recently Mercangöz and Doyle (2008) have performed the optimization of the benchmark model, which deals with the whole Pulp Mill and the optimization criterion is the plant's profit rate including energy costs, cooking and bleaching chemicals costs, final products sales (pulp and steam) and which takes into account delignification and brightness requirements to the final product. However the optimization is performed in a single iteration and the simulation results are relatively far from the prediction of the static model. The attempt to update the model bias and re-optimize the plant (the bias update procedure) is not able to improve the profit rate, even though the updated model promises a significant increase in profits. The inability of the bias update procedure to improve the profit rate may be explained by the fact that the static process model constructed in the area free of control saturation is invalid in the presence of active constraints on the manipulated variables. In addition, convergence of the iterative optimization implemented on the basis of the bias update procedure cannot be ensured and, in practice, convergence may not be reached. An iterative optimization method is proposed in this paper, which is free of the described drawbacks and provides convergence of optimization to the point where the approximated optimality conditions, introduced in the paper, are fulfilled. The article is structured as follows: Section 2 contains a description of the iterative optimization method together with the approximated optimality conditions. Section 3 introduces the Pulp Mill benchmark model and formulation of the profit rate maximization problem. Section 4 contains the results of computations for both one-step and iterative optimizations. Finally, Section 5 contains the discussion and conclusions.

Optimization of the steady state economic efficiency of an industrial process must take into account the fact that the decision variables of the optimization (setpoints of the control system) affect the process through the control strategy. The saturation effects of control system on the steady state of the manipulated variables of the process was briefly studied in the paper, and it was concluded that the finite difference estimation of the objective gradient does not provide correct results in the presence of active manipulated variables constraints. As a result, the necessary optimality conditions cannot usually be checked directly and a static model of the process must therefore be employed to evaluate the approximated optimality conditions introduced in the paper. In fact, the approximated optimality conditions are the most accurate approximation of the original optimality conditions that can feasibly be checked. The gradient-based optimization methods were not used in this study due to problems with the gradient estimation and the impossibility of evaluating the optimality conditions without a static model of the process. An iterative optimization method was proposed that keeps the iterations within the area free of control saturation where the static model of the process is valid. The presented method was able to find a solution satisfying the approximated optimality conditions. The performance of the approximated optimality conditions depends on the accuracy of the static plant model employed by the optimization. Thus three different static plant models were constructed and used to perform both one step and iterative optimization. The multidimensional quadratic model was of satisfactory quality during the optimizations. However, if it is suspected that all the proposed models are not of the proper quality, then the model can be completely re-estimated after a certain number of steps of the iterative method. Since the iterative method keeps all the manipulated variables away from their limits, it is possible to perform some variations in the setpoint without saturation of the control system capacities. The paper presents two examples of degradation of the economic efficiency at the setpoints derived by the single step optimization due to saturation of the control system capacities. The first example is related to limitations on the Lime Kiln fuel flow, while the second example presents the case of saturation of the white liquor flow to the oxygen tower. In both cases, the iterative optimization method was able to avoid saturation, and it also demonstrated the better performance compared to that of the single step optimization method using the same static plant model.