On an Economic Criterion for the Efficiency of Chemical Apparatus Control
Abstract Generated abstract
This paper addresses the formulation of an economic efficiency criterion for controlling individual chemical apparatuses within automated production systems using control computers. It argues that apparatus-level control criteria should be oriented toward the overall profit of production, rather than toward isolated qualitative or quantitative stream indicators, and proposes an objective function based on output stream value, input stream cost, and control-action costs, with stream values dependent on quality. The authors outline a sequential procedure for deriving such criteria, including analysis of the production structure, selection of significant parameters, material-balance analysis, determination of cost functions, mathematical modeling, and verification of the resulting objective function. The method is illustrated for an apparatus separating two acids, leading to a specific optimal-control objective that incorporates product quality, feed cost, and steam consumption.
Full Text
ON AN ECONOMIC CRITERION FOR THE EFFICIENCY OF CONTROL OF A CHEMICAL APPARATUS
R. N. Lyublinskii, E. G. Kleibort, B. Zh. Zhambalnimbuev
(Presented by the scientific-technical seminar of the Department of Automation and Telemechanics)
At the present time, a qualitatively new stage of automation is taking place in the chemical industry. Control is being automated not only at the level of individual apparatuses, but also at the level of technological processes and production facilities. Control systems include control computers (CCs), which possess the ability to process information and solve formalized mathematical problems.
These capabilities of CCs make it possible to approach the control of an apparatus in a new way, namely, to solve formalized problems of optimal control for each apparatus individually. Optimization of the control of apparatuses and the possibility of formalized control of the entire production process as a whole require the formation of criteria for the efficiency of control of each apparatus, oriented toward the general criterion of production operating efficiency.
At present, control is most often carried out according to qualitative or quantitative indicators of individual output streams. These indicators are selected by laboratory modeling or from operating experience with each apparatus, and they are not coordinated with the general criterion of production operating efficiency in the best possible way.
The most widespread criterion of production operating efficiency under ordinary conditions is profit [1]. The present work considers the problem of forming a criterion for the efficiency of control of an individual apparatus, oriented toward the overall profit of production.
The general scheme of the apparatus is presented in Fig. 1.
In this scheme,
\[ \overline{X}=\{x_1,x_2,\ldots,x_k\}\text{ — input streams,} \]
\[ \overline{U}=\{u_1,u_2,\ldots,u_m\}\text{ — control streams,} \]
\[ \overline{Y}=\{y_1,y_2,\ldots,y_n\}\text{ — output streams,} \]
\[ \overline{Z}=\{z_1,z_2,\ldots,z_p\}\text{ — disturbing effects.} \]
Each input, output, and control stream is characterized by a definite set of parameters, which we shall denote by \(x_{ij}\), \(U_{ij}\), \(y_{ij}\), where the indices indicate the \(j\)-th parameter of the \(i\)-th stream.
A special feature of controlling an intermediate apparatus in a chemical-technological scheme is the requirement that the parameters of its out-
the output streams are maximally suitable for further processing, and also so that the losses of raw-material and energy resources in the apparatus are minimal. Orienting the criterion of control efficiency for the apparatus toward production profit requires the formation of a criterion that includes the variable costs of production in the apparatus, as well as the additional costs of ensuring the qualitative and quantitative characteristics of the output products of production when the characteristics of the output products of the apparatus deviate from certain ideal (nominal) ones.
Fig. 1. Diagram of the control object
\[ J=\sum_{i=1}^{n} C_{y_i}(K_{y_i})\cdot Q_{y_i} -\sum_{j=1}^{m} C_{x_j}(K_{x_j})\cdot Q_{x_j} -\sum_{k=1}^{z} S_{u_k}, \tag{1} \]
where \(C_{y_i}(K_{y_i})\) is the cost function for a unit of the output stream as a function of the stream quality \(K_{y_i}\);
\(C_{x_j}(K_{x_j})\) is the cost function for a unit of the input stream as a function of the stream quality \(K_{x_j}\);
\(Q_{y_i}\) is the volume of stream \(y_i\);
\(Q_{x_j}\) is the volume of stream \(x_j\);
\(S_{u_k}\) is the cost of the \(u_k\)-th control action.
Since the quantities \(Q_{y_i}\) and \(K_{y_i}\) are functions of the input and control streams,
\[ Q_{y_i}=f_i(\bar X,\bar U,\bar Z),\quad K_{y_i}=\varphi_i(\bar X,\bar U,\bar Z), \tag{2} \]
the criterion of control efficiency satisfies all the requirements for efficiency criteria given in [1].
The orientation of the criterion of control efficiency for the apparatus toward production profit is carried out by an appropriate choice of the cost functions \(C_{y_i}(K_{y_i})\) and \(C_{x_j}(K_{x_j})\), which represent the “consumer” cost of a unit of the corresponding stream.
Determining the cost functions constitutes the main difficulty in forming an economic efficiency criterion. Since control of a complex technological scheme requires the formation of efficiency criteria for each apparatus of this scheme, in order to solve the question of decomposing the production efficiency criterion into individual apparatuses (“autonomization” of the apparatus), one should choose a sequential procedure for forming the cost functions.
For this purpose, the following order of formation may be proposed:
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Construct the structural scheme of production and analyze the location of each apparatus within it.
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Study the processes in each apparatus in order to select the most essential parameters characterizing the influence of variation in input streams, control actions, and disturbances on the output parameters of this apparatus.
The selection may be carried out by various methods. The main ones are the following:
a) selection of parameters with the aid of expert assessments;
b) selection of parameters by analyzing the physicochemical processes occurring in the apparatus;
c) selection of parameters according to the sensitivity functions of the output coordinates to deviations of the input parameters;
d) selection of parameters by means of statistical processing of data from passive experiments;
e) selection of parameters by means of carrying out special experiments.
Since the parameters are selected according to the significance of their influence on the output parameters of the apparatus, and the output parameters of each apparatus are input parameters for subsequent ones, it is expedient to begin such selection with the last apparatuses of the technological chain, for which the output parameters are rigidly determined by the technological regulations.
- Compilation of material-balance equations and analysis of losses in each apparatus.
The compilation of material-balance equations and the analysis of losses in the apparatuses are necessary for substantiating the assumptions that may be adopted in determining the efficiency criterion. These assumptions can substantially reduce the dimensionality of the problem of modeling the processes in the apparatus and the dimensionality of the problem of determining the cost functions.
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Selection of the general form of the efficiency criterion.
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Determination of the cost functions for each output stream of the apparatus.
It is also expedient to determine the cost functions beginning with the last apparatus of the scheme. For a complex branched scheme these functions may be nonunique. Each function is determined from a calculation per unit of output stream. Since the cost function is a quantitative characteristic, it can be determined either by processing data on production operation or by setting up special experiments. Determination of the cost function must be preceded by a logical analysis of the possible ways of ensuring the specified quality of the output product of the production process.
- Mathematical modeling of the apparatus.
Mathematical modeling of the apparatus consists in determining equations that relate the parameters of the output streams to the parameters of the input streams and of the control actions. The availability of mathematical models makes it possible to exclude the parameters of the output streams from the expression for the efficiency criterion of the apparatus. In addition, the objective function of the problem of optimal control of the apparatus, in the absence of the indicated model, may fail to satisfy the requirements of controllability, i.e., may not include the control parameters.
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Formation of the specific form of the efficiency criterion.
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Verification of the obtained expression for correspondence to the requirements imposed on the objective function.
The proposed method was used to form an economic criterion for the efficiency of control of an apparatus for separating two acids. Analysis of the material-balance equations showed that losses of acids during separation may be neglected.
The quality of each of the two output streams, corresponding to the outlets of the separated acids, is determined by one indicator—the amount of acid vapor entering the other outlet.
The input stream, representing a mixture of both acids, has one variable parameter—the total flow rate.
The objective function of the optimal separation problem in this case takes the following form:
\[ J=\sum_{i=1}^{2} C_{y_i}(k_{y_i})\cdot Q_{y_i}-C_x(k_x)\cdot Q_x-C_u\cdot U, \tag{3} \]
$C_{y_i}(K_{y_i})$ — cost functions of the output products;
$C_x(K'_x)$ — cost function of the input stream;
$c_u$ — cost of a unit of the control stream;
$Q_{y_i}$ — volume of products at the $i$-th output per unit time;
$Q_x$ — volume of the input stream per unit time;
$\bar U$ — steam consumption per unit time;
$K_{y_i} = \varphi_i(\bar X, \bar U, \bar Z)$ — quality indicators of the output streams.
REFERENCES
- G. G. Li, G. E. Adams, W. M. Gains. Process Control by Means of Computers. Modeling and Optimization. Moscow, “Sovetskoe radio,” 1972.