METHOD OF CONTROLLING THE ALUMINA FEED INTO REDUCTION CELLS FOR PRODUCING ALUMINIUM

Abstract: Method for controlling the alumina feed to reduction cells for the production of aluminium. There is employed an adaptive control with parameter estimation (3) and controller calculation (2) based upon the separation theorem. As a process model there is used a linear model having two inputs and one output. One input (u1) is in the form of alumina feeding minus assumed alumina consumption. Another input (u2) is in the form of movements of the reduction cell anode. The output (y) is in the form of the change in electric resistance across the reduction cell concerned. The model is of the first order in u1 and u2 whereas it is of the order zero in y. An estimated parameter (b1) represents the slope of the curve for resistance as a function of alumina concentration in the electrolytic bath, and the controller (2) controls the addition of alumina to the electrolytic bath in response to the value of b1.

Description: METHOD OF CONTROLLING THE ALUMINA FEED INTO REDUCTION CELLS FOR PRODUCING ALUMINIUM Reduction cells with pre-baked anodes for producing aluminium have become increasingly automated during the last years. The development has passed from cells having side break and feed systems with large and infrequent addition of alumina, to cells having a center break and feed system giving a more frequent addition of alumina and smaller amounts for each addition, and then to cells having point feeders giving fre¬ quent and small additions of alumina. Close control of the alumina concentration in the electrolytic bath is important in order to obtain high current efficiency and good operational regularity of the cell. For obtaining this a good process con¬ trol system is needed. If there is a deficiency of alumina in the bath, an anode effect will occur (the voltage drop increases from appr. 4 V to 20 - 40 V) , and if there is an excess of alumina in the bath, there will be formed sludge in the cell (undissolved alumina and bath deposits at the bottom of the cell) . In order to ob¬ tain a high current efficiency there is employed a bath of low liquidus temperature. Such electrolytic baths having a low liquidus temperature are less capable of dissolving alumina and therefore it is very important to have close control of the feeding of alumina. The process control of reduction cells for producing aluminium is a very difficult task because one has to do with a complicated interaction of many factors. In addition to the alumina concentration and the bath temperature there are, inter alia, important magnitudes such as the anode-cathode distance and the electrical resistance across the cell. Except for the resistance these magnitudes are not easily available for a more or less continuous and sufficiently accurate measure¬ ment during operation, for use in the process control. Thus, there have been proposed various control methods based upon known theories within control engineering. The present invention is directed to all types of alumina reduction cells, including old cells in which the cell resis¬ tance is subject to external disturbances because of magnetic fields or the heat balance. This will be the case in many of the pot lines being today in operation. During many years attempts have been made to feed alu¬ mina into reduction cells for producing aluminium with a certain degree of regularity in order to maintain the alumina content in the bath within certain limits of concentration. In Published British Patent Application 2080830 there is de¬ scribed a method for maintaining the alumina content in the bath within the range 1 - 3,5 %. However, this known method is not so flexible and robust that it may be employed with a good result in plants being subjected to disturbances and noise. Apparently the method is intended for a specific type of cells which inherently have a higher degree of stability than present-day common alumina reduction cells. In this known control method there is added at least one of the following additives to a bath of molten cryolite: 5 to 20 % aluminium fluoride,lithium salts at a concentration equal to or smaller than 1 % in terms of Li, magnesium salts at a concentration equal to or smaller than 2 % in terms of Mg, and alkali metal or alkaline earth metal chloride at a concentration equal to or smaller than 3 % in terms of Cl. The rate of alumina feeding is modulated depending upon the variation in the bath resistance during predetermined time in¬ tervals in alternating cycles of equal duration, for adding alumina at a slower rate and a faster rate, respectively, than what is corresponding to the consumption in the cell. This method makes it necessary to operate in the lowest part of the alumina concentration range, because it is required with the method that the bath resistance varies quickly with a change in the alumina concentration. The above known control method takes little account of noise, such as unpredictable variations in the electric vol- tage, and therefore is sensitive to external disturbances. In contrast to the above and other known control methods the present invention is directed to the control of the alumina feed to alumina reduction cells for producing aluminium, by em ¬ ploying adaptive control with parameter estimation and control¬ ler calculation based upon the separation theorem, and in which a linear model is utilized as the process model. The novel and specific features according to the invention are more closely defined in the claims. By means of this invention the above deficiencies and drawbacks of known methods are substantially eliminated. The solution takes into account noise and anode movement and therefore can be used on cells having a somewhat fluctu¬ ating resistance curve. Moreover, good flexibility is ob¬ tained since the method makes it possible to operate in different alumina concentration ranges. For example it may be employed in the range adjacent the minimum of the curve for bath resistance as a function of alumina concentration, being in principle known. The invention is particularly well adapted for use with cells having point feeders and an electrolytic bath having a low liquidus temperature, for the purpose of maintaining the alumina concentration within narrow limits. The invention shall be explained more closely below, referring also to the drawing, in which: Figure 1 shows the curve being in principle known, for elec¬ tric bath resistance in a reduction cell for aluminium production, as a function of the alumina concentration in the bath, and Figure 2 shows the general structure of an adaptive control system for carrying out the method according to the invention. From theory and practice it is known that the electrical resistance across the bath for aluminium electrolysis, the so-called bath resistance, is varying with the content of alumina in the electrolytic bath, broadly as illustrated in Figure 1. The resistance curve has a minimum which may for example be located in the range 3 - 4 % Al-O.,. With decreasing alumina concentrations below this minimum the bath resistance will soon increase very strongly. For a cell during ope¬ ration not only the working point will be displaced back and forth along a portion of the curve illustrated, but also the curve shape itself will actually be displaced both vertically and horizontally in the diagram. Such fluctuation or dis¬ placement may be due to various factors, of which some are in part uncontrollable. An important parameter in this connection is the slope of the resistance curve. Thus, because of the minimum mentioned, this slope will change its sign in a range of the curve which may often be of interest as a working range in the practical operation of reduction cells for aluminium production. Whereas the alumina concentration in an electrolytic bath is not easily available for direct or continuous measurement during operation, the bath resistance on the other hand is well suited for more or less continuous and rather accurate measure¬ ments. Measurement of this bath resistance therefore is incor¬ porated as an important step in the control of reduction cells for producing aluminium. Figure 2 shows the general structure of an adaptive con¬ trol system including the process to be regulated. Block 1 represents the process, i.e. in the present case a reduction cell for producing aluminium, with associated auxiliary equipment, such as feeding devices for alumina, etc. according to design practice being known per se. The controller part is represented by block 2 which may also contain parts and compo¬ nents being known per se, for example for the required control¬ ler calculation. This calculation takes place on the basis of parameter estimation represented by block 3. The func¬ tion thereof is closely related to the principles underlying adaptive control. Thus, it appears from the figure that in addition to a conventional feedback from the process output (Y) to the input of the controller 2 for comparison with a reference value (Y _ f ) , signals are taken from the process output (Y) and the process input (U) , respectively, for use in the parameter estimation in block 3. The estimated parameter or parameters (b) are then delivered to the controller for cal¬ culation and adjustment of the function thereof, including am¬ plification. This takes place according to principles being per se known to experts within the field of control engineering. The functions in block 2 as well as in block 3 may be imple¬ mented by means of digital electronic circuitry and/or com¬ pletely or partly by means of a suitable computer. The tech¬ nology being at disposal at a given time will to a high degree determine which electronic means are resorted to for carrying out the method according to the invention. Thus, as will be seen from Figure 2, the adaptive control algorithm consists of two parts, namely parameter estimation and controller calculation. The separation of these two parts is usually according to the so-called separation theorem, which means that in the controller calculation it is taken as an assumption that the estimated parameters are the real parameters. Therefore, the controller calculation is the same as in conventional control with constant and known parameters, and various controllers may be used depending upon the case concerned. Various techniques may also be employed for parameter estimation. The most popular method is the least squares method. The maximum likelihood method is also frequently used with a somewhat better result, but with a somewhat longer calculating time. The most commonly employed process model for a single input, single output system (SISO) is a linear model of the type: A(Z _1 )y = Z~ d B (Z~ 1 )u + C(Z "1 )v in which ( l 1 \' A(Z _1 ) = 1 + a 1 Z~ 1 + a 2 Z 2 + anZ-n B(Z -1 ) = b 1 Z _1 + b 2 z "2 + + b n z -n CIZ "1 ) = 1 + c^ '1 + c 2 Z -2 + + c n Z Z -1 is a time shift operator, y(k-1) = Z-1 y(k) y is a process output u is a process input v is noise d is dead time n is the order of the model k is a time increment. There is introduced a vector Ψ which contains the earlier outputs, inputs- and noise and a vector Θ which con¬ tains all the parameters. The model may then be written as: y(k) = Ψ (k) Θ(k-1) + v(k) (2) in which ψ T (k) = [-y(k-l) -y(k-n)utk-d-l) ... u(k-d-n)v(k-l) v(k-n)] G T (k-l) = a χ a n , b χ b n , c χ c n ] I . The difference between the real resistance measurement at time k+1 and the one calculated by the model at time k with respect to time k+1, is designated e(k+1), that is e(k+1) = y(k+1) - y est (k+1 |k) = y(k+1) - Ψ (k+1) θ(k) (3) The estimated data vector Ψ is equal to the real data vector Ψ, except that the unknown noise signals v are replaced by the estimate of the noise, being the previous errors e. The vector 0 contains the parameter estimates. The new parameter vector is estimated in a recursive manner by using θ (k+1) = θ(k) + γ(k) e(k+1) in which the gain vector γ is given by γ ι lc >j _ - P (k) — Ψ (k+ 1 ) - (4 ) λ + Ψ (k+ 1 ) P (k) Ψ (k+ 1 ) The matrix P is proportional to the Qøvariance matrix of the parameter estimates and is updated recursively by <\,T . P(k+1) = [I-γ(k) Ψ (k+1)] P(k) (5) λ is a forgetting factor. If λ is equal to 1 , nothing will be forgotten. If λ is smaller than 1, the algorithm forgets the past exponentially. The algorithm may also forget linearly by adding a constant diagonal matrix (R) to the P matrix. The algorithm as described above (4) is valid for esti¬ mation by using the least squares method. Use of the maximum likelihood method requires filtering of the data vector Ψ. According to the invention such filte¬ ring is preferred. A s mentioned previously the control of reduction cells for the production of aluminium is difficult because of the fact that there are not many measurements which can be perfor¬ med in real time (on-line) . Only the resistance or more cor¬ rectly the current and the voltage, are measured on-line. In order to be able to control the aluminaconcentration in the bath it is necessary to have an approximate knowledge thereof at all times. Here it is not sufficient to keep an account of aluminaadded and consumed, since both these are un¬ certain and, besides, the aluminaconcentration may change as a result of the forming or dissolving of sludge and side freeze or additions from the alumina cover of the cell. Thus, the in¬ vention is directed to "measuring" thealumina concentration in¬ directly by estimating the slope (the derivative) . of the re¬ sistancealumina curve in Figure 1. For this purpose there is set up a linear model of the resistance variation as described below, in which the slope is incorporated as a parameter. It is first assumed that the bath resistance (R) is only influenced by the A1 2 0 3 concentration (X) . The bath resistance may then be differenciated as follows: dR dR dx dt 3χ dt (6 ) dR is the slope of the curve in Figure 1 and dx dx is the change in Al 2 0 3 concentration which is due to con¬ sump If is designated u, equation R(k) - R(k-1) = Tb.,u (k-1) ( 7 ) in which T '= the sampling time. If the change in resistance which is due to anode move¬ ment is taken into account, equation (7) may be expressed as : R ( k) - R ( k-1) = b lUl (k-1) + b 2 u 2 (k-1) (8 ) in which u. . = T u and b 2 is change of resistance per mm anode movement. If R ( k ) - R ( k-1) = y(k), and a first order noise contri ¬ bution is taken into account, equation (8) may be expressed as: y ( k ) = lUl ( k-1) + b 2 u 2 (k-1) + v(k) + c, v(k-1 ) ( 9 ) in which v ( k ) is an independent white noise sequence. When comparing this model and the general model (1) it is seen that ( 9 ) is a two-input, one-output model of the first order in u.,,u 2 and v and of zero order in y. It follows from this that the algorithm being developed for linear models can be employed without change with a data vector and a parameter vector. ψ (k) = [ 1 (k-1) u 2 (k-1) v(k-1)] or " Ψ- (k) = [u. j k-U u 2 (k-1) e(k-1)]T x and Θ (k) = [b.. b 2 c,]^ The data vector is filtered according to a particular embodiment: φ (k) = [u 1 ' (k-1) u 2 '(k-1) e'(k-1)] in which u'^k-1) = Ul (k-1) - c. (k) 1 ' (k-2) u' 2 (k-1) = u 2 (k-1) - c. (k) u 2 ' (k-2) e'(k-1) = e(k-1) - c, (k) e' (k-2) The model estimates the resistance measurement at time (k) for the time (k+1) y k+1) = Ψ (k+1) θ (k) (10) J est The estimation error is calculated by e(k+1) = y(k+1) - Y est (Jc+1) ( 11 ) in which y(k+1) is the value being actually measured at time (k+1). The parameter vector is estimated recursively by using 0 (k+1) = 0 (k) + γ(k) e(k+1)/r (12) in which the gain vector is given by γ (k) = -- P(k] Ψ(k+1) 1 + Ψ (k+1) P(k) Ψ (k+1) in which P is proportional to the covariance matrix of the parameter estimates and is updated recursively by P(k+1) = [I - γ(k) Ψ (k+1)]P (k) (13) in which I is the identity matrix. In the above particular embodiment with filtering one puts (k+1) = φ(k+1) . The magnitude T has been introduced in order to reduce the gain at large, separate excursions of the estimation error resulting from sudden changes in the resistance which are not caused by changes in the alumina concentration or anode regula¬ tion. r = max 1, le (k+1) I e filt (k + lϊ in which e filt (k+ ) = se fiit ( * + t d" s) l e(k) I s and t are suitably chosen constants. The input signal u, (k) must be calculated from the sum of the Al 2 0 3 additions in the time interval from (k-1) to k and the estimated Al_0 3 consumption during the same interval, i.e. u_. (k) = (rate of addition - rate of consumption) T/bath volume. When thealumina concentration in the bath varies, the esti¬ mator will adjust the value of the slope b, correspondingly. In order to obtain reliable estimates it is necessary that the process is excited to a sufficient degree. According to the invention this is attended to by having the controller vary the additions of oxide so that b, will oscillate about a working point at the curve of Fig. 1 being favourable to the process. The parameter b ? depends upon the resistivity in the bath and therefore will vary with bath composition and bath tempe ¬ rature. In order to be sure that the estimator is "awake" a con ¬ stant diagonal matrix R is added to the covariance matrix P in equation (5) . This is not absolutely necessary, but constitutes a preferred feature according to the invention. The advantage of doing this instead of using a simple forgetting factor λ is that a different forgetting may be associated with the various parameters. If the feeding involves large changes in the Al 2 0 3 concentration, there will be ex ¬ pected a significant change of b... The element in R which corresponds to b.. must be relatively large if the estimator shall be able to follow these variations. The drawback of applying a quick forgetting is that the estimator will be sen ¬ sitive to noise and gives a poorer filtering of the data. As mentioned the regulator provides for keeping the alumina concentration within a favourable range. In practice this is implemented thereby that the controller selects between a small number of feed frequencies based upon the estimate of para¬ meter b χ . In addition thereto the controller provides for the keeping of the bath resistance close to a reference value. Also this function is associated with the value of b, in order to avoid frequent regulations as a result of the deliberate fluctuation of the resistance with the alumina concentration.

Claims: C l a i m s : 1. Method of controlling the alumina feed to reduction cells for producing aluminum, characterized in that the non-linear function relation between resistance and alumina concentration in the electrolytic bath (Fig. 1) is utilized for obtaining a knowledge of the alumina concentration at all times, that the derivative of this function is estimated, and that the alumina supply in order to secure reliable estimates, is controlled so that the value of this derivative oscillates about a freely selected working point corresponding to an alumina concentration being favourable to the process, and wherein there is applied for this estimation a linear model of the relationship between resistance change, alumina concentration change and a change in the anode position of the form: y(k) = b 1 u 1 (k-l) + b 2 u 2 (k-1) + v(k) + c 1 v(k-l) in which y(k) is an output in the form of electrical resistance change across the cell in the time increment k-1 to k, u, is an input variable 1 in the form of the difference between alumina added and assumed alumina consumption in the time increment concerned ( in terms of concentration change) , u~ is an input variable 2 in the form of move¬ ment of the anode position in the time incre¬ ment concerned, v is a white noise sequence which drives the noise model: v(k) + c^fk-l) , k is the time increment no. and in which parameter b, represents said derivative of the curve for resistance as a function of alumina concentration in the electrolytic bath, in which parameter b 2 gives resistance change per length unit movement of the anode and in which parameter c^ is incorporated in a first order model of the noise which influences the resistance, that parametersb, , b ? and c, are estimated recursively by relating predicted resistance change for the instant k+1 to the actually occurring resistance change, and the deviation is utilized for updating the parameter estimates as follows: e(k+l) = y(k+l) - ψ (k+1) &(k) and θ ( k+l) = θ( ) + γ(k) e(k+l) / r in which Ψ(k+1) is the data u 2 (k) e(k)J , 9(k) is the parameter vector j. b, b- c, J and T designates transposed, in which the gain vector γ is given by in which P is proportional to the covariance matrix of the parameter estimate and is updated recursively by P(k+1) = Cl - γ(k) Ψ (k+1) JP(k), in which I is the identity matrix and in which r, which reduces the gain at large estimation errors caused by sudden excursions of the resistance, is given by r - - « E - . ' -f • il'*t; J in which e f i lt (k+1) = s e fϋ (k) + ~ (1 " s) I e(k) with s and t as suitably chosen constants. 2. Method according to claim 1, characterized in that in order to avoid the effect of random disturbances, the data vector is filtered as follows: Φ (k) = [u l ' (k-1) u 2 ' (k-1) e' (k-1)] in which u. j Mk-D = u^k-1) - c 1 (k) u 1 • (k-2) u 2 ' (k-1) = u 2 (k-1) - c 1 (k) u 2 ' (k-2) e' (k-1) = e(k-1) - c, (k) e' (k-2) and putting Ψ(k) = φ(k) . 3. Method according to claim 1 or 2, characterized in that the algorithm is adapted to forget linearly by adding a constant diagonal matrix R to the P matrix: P(k+1) = P(k+1) + R.

Inventors: AALBU JOHANNES, AALBU MARIT STANGELAND, BORG PETER, MOEN TOM, AALBU, JOHANNES, AALBU, MARIT, STANGELAND, BORG, PETER, MOEN, TOM
Applicants: ARDAL OG SUNNDAL VERK, AORDAL OG SUNNDAL VERK A.S.
Application year: 1986
Main classification: G05B13
IPC codes: A22C21, C22B21, C25C3, C25C7, G05B13, G05B17