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State and Parameter Estimation in Hall-Héroult Cells using Iterated Extended Kalman Filter⁎
Proceedings, 5th IFAC Workshop on Mining, Mineral and Metal Proceedings, Proceedings, 5th 5th IFAC IFAC Workshop Workshop on on Mining, Mining, Mineral Mineral and and Metal Metal Processing Proceedings, 5th IFAC Workshop on Mining, Mineral and Processing Available online at Metal www.sciencedirect.com Processing Shanghai, China, August 23-25, 2018 Proceedings, 5th IFAC Workshop on Mining, Mineral and Metal Processing Shanghai, China, China, August August 23-25, 23-25, 2018 2018 Shanghai, Processing Shanghai, China, August 23-25, 2018 Shanghai, China, August 23-25, 2018
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IFAC PapersOnLine 51-21 (2018) 36–41
State and Parameter Estimation in State and Parameter Estimation in State and Parameter Estimation in State and Parameter Estimation in Hall-H´ e roult Cells using Iterated Extended State and Parameter Estimation in Hall-H´ e roult Cells using Iterated Extended Hall-H´ e roult Cells using Iterated Extended Hall-H´ eroult Cells using Iterated Extended Kalman Filter Hall-H´ eroult Cells using Iterated Extended Kalman Filter Kalman Kalman Filter Filter Kalman Filter Yuchen Yao and Jie Bao
Yuchen Yao and Jie Bao Yuchen Yuchen Yao Yao and and Jie Jie Bao Bao Yuchen Yao and Jie Bao School of Chemical Engineering, University School of Chemical Chemical Engineering, Engineering, University University of of New New South South Wales, Wales, School of New South Wales, Australia (e-mail: [email protected]). School of ofSydney, Chemical Engineering, University of New South Sydney, Australia Australia (e-mail: (e-mail: [email protected]). [email protected]). Wales, Sydney, School ofSydney, Chemical Engineering, University of New South Wales, Australia (e-mail: [email protected]). Sydney, Australia (e-mail: [email protected]). Abstract: The measurement of individual anode currents in the Hall-H´ eeroult process provides Abstract: The measurement of individual anode currents in the Hall-H´ process provides Abstract: The measurement of individual anode currents in the Hall-H´ eeroult roult process provides localized information that can be used to improve cell operation. One promising application is Abstract: The measurement of individual anode currents in the Hall-H´ roult process provides localized information that can be be used used to to improve improve cell cell operation. operation. One One promising promising application application is localized information that can is Abstract: The measurement ofbeindividual anode currents in the Hall-H´ eroult process provides to estimate the local cell conditions, which are impracticable to measure regularly during the localized information that can used to improve cell operation. One promising application is to estimate the local cell conditions, which are impracticable to measure regularly during the to the local cell which are to measure regularly during the localized information thatconditions, can be used to improve cell operation. One promising application is cell operation. presents an approach to estimation localized state variables to estimate estimate theThis localpaper cell conditions, which are impracticable impracticable toof measure regularly during and the cell operation. This paper presents an approach to estimation of localized state variables and cell operation. This paper presents an approach to estimation of localized state variables and to estimate the local cell conditions, which are impracticable to measure regularly during the process variables, including the alumina concentration, anode-cathode distance and bath flow cell operation. This paper presents an approach to estimation of localized state and variables and process variables, including the alumina concentration, anode-cathode distance bath flow process variables, including the alumina concentration, anode-cathode distance and bath cell operation. This paper presents an approach to estimation of localized state variables and rate. This is achieved by using Iterated Extended Kalman Filter (IEKF) with aa discretized process variables, including thethe alumina concentration, anode-cathode distance and bath flow flow rate. This is achieved by using the Iterated Extended Kalman Filter (IEKF) with discretized rate. This is achieved by using the Iterated Extended Kalman Filter (IEKF) with aa discretized process variables, including the alumina concentration, anode-cathode distance and bath flow Hall-H´ e roult cell model. The results show that the proposed IEKF can produce more accurate rate. This is achieved by using the Iterated Extended Kalman Filter (IEKF) with discretized Hall-H´ eeroult cell model. The results show that the proposed IEKF can produce more accurate Hall-H´ cell model. The show the IEKF can produce accurate rate. This isofachieved by usingresults the Iterated Extended Kalman Filter (IEKF) withmore a discretized estimation the state variables and parameters compared to the conventional Extended Kalman Hall-H´ eroult roult cell model. The results show that that the proposed proposed IEKF can produce more accurate estimation of the state variables and parameters compared to the conventional Extended Kalman estimation of the state variables and parameters compared to the conventional Extended Kalman Hall-H´ e roult cell model. The results show that the proposed IEKF can produce more Filter thanks to its robustness to ill-set initiation conditions. estimation of the state variablesto and parameters compared to the conventional Extendedaccurate Kalman Filter thanks to its robustness ill-set initiation conditions. Filter thanks to its robustness to ill-set initiation conditions. estimation of the state variablesto and parameters compared to the conventional Extended Kalman Filter thanks to its robustness ill-set initiation conditions. © 2018,thanks IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Filter to its robustness toindividual ill-set initiation conditions. Keywords: Hall-H´ eeroult process, anode current, state estimation, parameter Keywords: Hall-H´ roult process, individual anode current, state Keywords: Hall-H´ e roult process, individual anode current, state estimation, estimation, parameter parameter estimation, Iterated Extended Kalman Filter Keywords: Hall-H´ e roult process, individual anode current, state estimation, parameter estimation, Iterated Extended Kalman Filter estimation, Iterated Extended Kalman Filter Keywords: Hall-H´ e roult process, individual anode current, state estimation, parameter estimation, Iterated Extended Kalman Filter estimation, Iterated Extended Kalman Filter 1. A typical Hall-H´ eeroult cell contains 18 to 40 anodes 1. INTRODUCTION INTRODUCTION A typical Hall-H´ cell contains 18 to 40 anodes 1. A typical Hall-H´ eeroult roult cell contains 18 to 40 anodes connected in parallel, sharing a regulated DC current 1. INTRODUCTION INTRODUCTION A typical Hall-H´ roult cell contains 18 to 40 anodes connected in parallel, sharing a regulated DC current connected in parallel, sharing a regulated DC current 1. INTRODUCTION A typical Hall-H´ e roult cell contains 18 to 40 anodes in the order of several hundred kilo-amps. This connected in parallel, sharing a regulated DC current The Hall-H´ e roult process is the dominant method of proin the the order order of of several several hundred hundred kilo-amps. kilo-amps. This This current The Hall-H´ Hall-H´eeroult roult process process is is the the dominant dominant method method of of propro- in current connected in parallel, sharing a regulated DC current The is normally referred to as the line current, while the in the order of several hundred kilo-amps. This current ducing aluminum and Welch, 1980; Grjotheim The Hall-H´ eroult (Grjotheim process is the dominant method of pro- is normally referred to as the line current, while the ducing aluminum (Grjotheim and Welch, 1980; 1980; Grjotheim is normally referred to as the line current, while the in the order of several hundred kilo-amps. This current ducing aluminum (Grjotheim and Welch, Grjotheim voltage drop across the anode and the cathode is called is normally referred to as the line current, while the The Hall-H´ e roult process is the dominant method of proand Kvande, 1993). It involves the electrolysis of alumina ducing aluminum (Grjotheim and Welch, 1980; Grjotheim voltage drop across the anode and the cathode is called and Kvande, 1993). It involves the electrolysis of alumina voltage drop across the anode and the cathode is called is normally referred to as the line current, while the and Kvande, 1993). It involves the electrolysis of alumina the cell voltage. One of the major limiting factors in voltage drop across the anode and the cathode is called ducing aluminum (Grjotheim and Welch, 1980; Grjotheim (Al O ) dissolved in cryolite (Na AlF ). During the elecand Kvande, 1993).inItcryolite involves the electrolysis of the alumina 3 ) dissolved 3 AlF 6 ). During the cell voltage. One of the major limiting factors in (Al222 O (Na electhe cell voltage. One of the major limiting factors in 3 3 6 voltage drop across the anode and the cathode is called (Al O ) dissolved in cryolite (Na AlF ). During the elecimproving the efficiency of the Hall-H´ e roult process is 3 ) dissolved 3 AlF 6 ).continuously the cell voltage. One of the major limiting factors and2 O Kvande, 1993). It involves the electrolysis of alumina trochemical reaction, carbon anodes are con(Al in cryolite (Na During the elecimproving the the efficiency efficiency of of the the Hall-H´ Hall-H´eeroult roult process process in is 3 3 6 continuously controchemical reaction, carbon anodes are improving is the cell voltage. One of the major limiting factors in trochemical reaction, carbon anodes are continuously conthat usually only the line current and cell voltage are improving the efficiency of the Hall-H´ e roult process is (Al2 O3while ) dissolved in cryolite (Naaluminum During the elecsumed the produced liquid is accumulated 3 AlF 6 ).continuously trochemical reaction, carbon anodes are conthat usually only the line current and cell voltage are sumed while the produced liquid aluminum is accumulated that usually only the line current and cell voltage are improving the efficiency of the Hall-H´ e roult process is sumed while the produced liquid aluminum is accumulated measured continuously online. This means that the cell that usually only the line current and cell voltage are trochemical reaction, carbon anodes are continuously conat of the serving the sumed while the liquid aluminum is accumulated continuously online. This means that the cell at the the bottom bottom of produced the cell, cell, effectively effectively serving as as the cathode. cathode. measured measured continuously online. This means that the cell that usually only the line current and cell voltage are at the bottom of the cell, effectively serving as the cathode. control logics can only be designed based on these two measured continuously online. This means that the cell sumed while the produced liquid aluminum is accumulated The main chemical reaction is shown below: at themain bottom of thereaction cell, effectively serving as the cathode. control logics can only be designed based on these two The chemical is shown below: control logics can be designed based on two continuously online. This cell means thatthese theWith cell The chemical is below: variables that represent conditions. control logics can only only the be averaged designed based on these two at themain bottom of thereaction cell, effectively serving as the cathode. measured The main chemical reaction is→shown shown below: variables that represent the averaged cell conditions. With 2 Al O + 3 C − − 4 Al + 3 CO (1) variables that represent the averaged cell conditions. With 2 O3 + 3 C −−→ 4 Al + 3 CO2 control logics can only be designed based on these two larger cells containing more anodes being built to meet 2 Al (1) variables that represent the averaged cell conditions. With The main chemical is→shown below: 2 3 3reaction 2 2 + 3 C −− + 3 cells containing more anodes being built to meet larger cells containing more anodes being built to meet 2 Al Al22aO Oschematic →4 4 Al Al + Hall-H´ 3 CO CO22 eroult cell.(1) (1) larger variables that represent the averaged cell conditions. With 3 + 3 C −−plot the growing market demands, it has become more evident Figure 1 depicts of a larger cells containing more anodes being built to meet the growing market demands, it has become more evident Figure 1 depicts a schematic plot of a Hall-H´ e roult cell. 2 Al O + 3 C − − → 4 Al + 3 CO (1) 2a schematic 3 2 eroult cell. the growing market demands, it has become more evident Figure 1 depicts plot of a Hall-H´ larger cells containing more anodes being built to meet that there can be significant spatial variations in the cell the growing market demands, it has become more evident Figure 1 depicts a schematic plot of a Hall-H´eroult cell. that there can be significant spatial variations in the cell that there can be significant spatial variations in the cell the growing market demands, it has become more evident Figure 1 depicts a schematic plot of a Hall-H´eroult cell. (Haupin and Seger, 2001; Keniry et al., 2001; Keniry and that there can be significant spatial variations in the cell (Haupin and Seger, 2001; Keniry et al., 2001; Keniry and (Haupin and Seger, 2001; Keniry et al., 2001; Keniry and that there can be significant spatial variations in the and cell Shaidulin, 2008). Therefore, it has become increasingly (Haupin and Seger, 2001; Keniry et al., 2001; Keniry Shaidulin, 2008). Therefore, it has become increasingly Shaidulin, 2008). Therefore, it has become increasingly (Haupin and Seger, 2001; Keniry et al., 2001; Keniry and important for the localized cell conditions to be properly Shaidulin, 2008). Therefore, it has become increasingly important for the localized cell conditions to be properly important for the localized cell conditions to be Shaidulin, 2008). Therefore, it that, has become dealt with. In order to achieve the measurement of important for the localized cell conditions to increasingly be properly properly dealt with. In order to achieve that, the measurement of dealt with. In order to achieve the measurement of important for the localized cell that, conditions to be(Barnett, properly individual anode currents has been proposed dealt with. In order to achieve that, the measurement of individual anode anode currents currents has has been been proposed proposed (Barnett, (Barnett, individual dealt with. In and order to achieve that, the measurement of 1988; Evans Urata, 2012). The individual anode individual anode currents has been proposed (Barnett, 1988; Evans and Urata, 2012). The individual anode 1988; Evans and Urata, 2012). The individual anode individual anode currents has been proposed (Barnett, current is attractive because it reflects the cell conditions 1988; Evans and Urata, 2012). The individual anode current is attractive because it reflects the cell conditions current is because it cell conditions 1988; Evans and Urata, 2012). Theitthe individual anode in the vicinity of each anode and has found great current is attractive attractive because it reflects reflects the cell conditions in the vicinity of each anode and it has found great in the vicinity of each anode and it has found great current is attractive because it reflects the cell conditions potential in a number of proposed applications (Barber, in the vicinity of eachof anode andapplications it has found great potential in a number proposed (Barber, potential in a number of proposed applications (Barber, in theKeniry vicinity ofShaidulin, eachof anode and it ethas found great 1992; 2008; Rye al., 1998; Dion potential in aand number proposed applications (Barber, 1992; Keniry and Shaidulin, 2008; Rye et al., 1998; Dion 1992; Keniry and Shaidulin, 2008; Rye et al., 1998; Dion potential in aCheung number ofal., proposed applications (Barber, et al., 2015; et 2012, 2013; Yao et al., 2016, 1992; Keniry and Shaidulin, 2008; Rye et al., 1998; Dion et al., 2015; Cheung et al., 2012, 2013; Yao et al., 2016, et al., 2015; Cheung et al., 2012, 2013; Yao et al., 2016, 1992; Keniry and Shaidulin, 2008; Rye et al., 1998; Dion 2017). et al., 2015; Cheung et al., 2012, 2013; Yao et al., 2016, 2017). 2017). et al., 2015; Cheung et al., 2012, 2013; Yao et al., 2016, 2017). The use of state observers in the Hall-H´ eeroult process 2017). The use of state observers in the Hall-H´ process The use of state observers in the Hall-H´ eeroult roult process with the measurement of individual anode currents has The use of state observers in the Hall-H´ roult process with the the measurement measurement of of individual individual anode anode currents currents has with has The use of state observers in itthe Hall-H´ eroult process been proved to be beneficial as provides the estimation with the measurement of individual anode currents has been proved to be beneficial as it provides the estimation been proved to be beneficial as it provides the estimation with the measurement of individual anodethe currents has of localized state variables such as alumina concentration been proved to be beneficial as it provides estimation of localized state variables such as alumina concentration of localized such alumina concentration been provedstate to bevariables beneficial as itas provides the estimation and anode-cathode distance (ACD) (Jakobsen et al., 2001; of localized state variables such as alumina concentration and anode-cathode distance (ACD) (Jakobsen et al., 2001; and anode-cathode distance (ACD) (Jakobsen et al., of localized state variables such as alumina concentration Hestetun and Hovd, 2005; Hestutun and Hovd, Yao Fig. 1. Schematic plot of a Hall-H´ cell. and anode-cathode distance (ACD) (Jakobsen et2006; al., 2001; 2001; Hestetun and Hovd, 2005; Hestutun and Hovd, 2006; Yao Fig. 1. Schematic plot of aa Hall-H´ Hall-H´eeeroult roult cell. Hestetun and Hovd, 2005; Hestutun and Hovd, 2006; Yao Fig. 1. Schematic plot of roult cell. and anode-cathode (ACD) (Jakobsen et2006; al., 2001; et al., 2017). normal operation, the bath flow rates Hestetun andDuring Hovd,distance 2005; Hestutun and Hovd, Yao Fig. 1. Schematic plot of a Hall-H´eroult cell. et al., 2017). During normal operation, the bath flow rates et al., 2017). During normal operation, the bath flow rates Hestetun and Hovd, 2005; Hestutun and Hovd, 2006; Yao Fig. 1. Schematic plot of a Hall-H´ e roult cell. in a Hall-H´ e roult cell often change slowly, thus it is possible et al., 2017). During normal operation, the bath flow rates This work is supported by Australian Research Council Discovery in aa Hall-H´ eeroult cell often change slowly, thus it is possible in Hall-H´ roult cell often change slowly, thus it is possible et al., 2017). During normal operation, the bath flow rates work is supported by Australian Research Council Discovery This to estimate the bath flow rates as process parameters along in a Hall-H´ e roult cell often change slowly, thus it is possible This work is supported by Australian Research Council Discovery to estimate the bath flow rates as process parameters along This work Project DP160101810. is supported by Australian Research Council Discovery to bath as along Project DP160101810. in Hall-H´ethe roult cellflow oftenrates change slowly, parameters thus it is possible to aestimate estimate the bath flow rates as process process parameters along Project DP160101810. This work is supported by Australian Research Council Discovery Project DP160101810. to estimate the bath flow rates as process parameters along
Project DP160101810. 2405-8963 © 2018 2018, IFAC IFAC (International Federation of Automatic Control) Copyright © 36 Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 36 Peer review© responsibility of International Federation of Automatic Copyright © under 2018 IFAC IFAC 36 Control. Copyright © 2018 IFAC 36 10.1016/j.ifacol.2018.09.389 Copyright © 2018 IFAC 36
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Yuchen Yao et al. / IFAC PapersOnLine 51-21 (2018) 36–41
with the state variables. The dynamics of the Hall-H´eroult cell has very strong output nonlinearity, which should be effectively dealt with by using an Iterated Extended Kalman Filter (IEKF) to achieve better accuracy and robustness to ill-set initial estimation. In this paper, a HallH´eroult cell is discretized into a number of subsystems and an IEKF is developed to estimate the above local state variables and process parameters. This paper is organized as follows. The process model is briefly introduced in the next section, followed by the description of IEKF. The state and parameter estimation results are presented and compared with conventional EKF. A discussion and conclusion is provided at the end of the paper.
37
Fig. 2. Discretization of a Hall-H´eroult cell where accum(Iline ) is the height accumulation rate of molten aluminum at the bottom of the cell, consum(Iline ) is the rate of the anode height reduction through the carbon consumption, D is the ACD and BM is the distance of the busbar movement.
2. PROCESS MODELING
2.3 Cell Voltage
The process model used in this paper is similar to the one described in Yao et al. (2017) and is briefly discussed in this section.
The cell voltage is a nonlinear function of cell conditions, including alumina concentration and ACD, and a number of process parameters that change slowly (Haupin, 1998). The general form of the voltage equation is (6) Vcell = Erev + Eover + Iline Rpath + Vexternal , where Erev and Eover are the reversible and over-potential, Rcell is the ohmic resistance of the cell and Vexternal is the external voltage drop. In this paper, the cell voltage is represented as (7) Vcell (k) = h(Iline (k), D(k), Cd (k), θ), where θ includes cell design and operating parameters.
2.1 Alumina Concentration It is assumed that the dissolution of alumina in the bath are governed by first-order rate equations with two rate constants (Biedler, 2003). The faster dissolution rate constant is 0.099 s−1 , which is equivalent to a residence time of 50 seconds. In order to simplify the model, one minute is chosen as the time step so that the dynamics of the alumina with faster dissolution rate constant can be ignored. The discrete-time equation for alumina dissolution is shown in equation (2). g(k)r Cun (k + 1) = Cun (k) − kdiss Cun (k) + , (2) m where Cun is the concentration of the undissolved alumina, kdiss is the respective dissolution rate constant, g is the amount of alumina fed, r is the weight ratio between the fast and slow dissolving alumina and m is the mass of the bath. It follows that, with alumina consumption governed by Faraday’s Law, the equation for dissolved alumina is g(k)(1 − r) Cd (k + 1) = Cd (k) + kdiss Cun (k) + m F a(Iline ,Al2 O3 (k)) , − m (3) where Cd is the concentration of the dissolved alumina, F a(Iline ,Al2 O3 ) is the alumina consumption rate based on the Faraday’s Law, as shown in equation (4). Iline × MAl2 O3 × η , (4) F a(Iline ,Al2 O3 ) = F ×z where MAl2 O3 is the molar mass of alumina, η is the current efficiency, F is the Faraday constant and z is the number of electrons transferred.
2.4 Cell Discretization Equations (2)-(7) can be slightly modified to describe the subsystems in a discretized Hall-H´eroult cell shown in Figure 2. The cell is discretized into six subsystems according to the location of the feeders. As can be seen, each of the subsystems 2 to 5 contains one feeder and the alumina in subsystems 1 and 6 is supplied by the electrolyte flows from subsystems 2 and 5. It is assumed that the mass transfer is induced by electrolyte flows (bath flows), and the flow rates across any boundary between two subsystems are the same (for example, the bath flow from subsystem 1 to subsystem 2 is the same as the one from subsystem 2 to subsystem 1). Therefore, the process model of, for example subsystem 2, can be written as g2 (k)r (8) Cun,2 (k + 1) = Cun,2 (k) − kdiss Cun,2 (k) + m2 Cd,2 (k + 1) = Cd,2 (k) + kdiss Cun,2 (k) F a(Isum,Al2 O3 ,2 (k)) g2 (k)(1 − r) − + m2 m2 (9) M T1,2 (−Cd,2 (k) + Cd,1 (k)) + m2 M T2,3 (−Cd,2 (k) + Cd,3 (k)) + m2 D2 (k + 1) = D2 (k) − accum(Isum,2 (k)) (10) + consum(Isum,2 (k)) + BM (k) (11) Vcell,2 (k) = h(Isum,2 (k), D2 (k), Cd,2 (k), θ2 ), where M T represents mass transfer rate of bath between two neighboring subsystems and Isum,2 represents the sum of anode currents in subsystem 2, which can be obtained by solving the following system of equations
2.2 Anode-cathode Distance The dynamics of ACD include the accumulation of the liquid aluminum at the bottom of the cell, which gradually reduces the ACD, and the consumption of carbon anodes, which increases the ACD. Thus, the dynamic equation for the ACD is D(k + 1) = D(k) − accum(Iline (k)) (5) + consum(Iline (k)) + BM (k), 37
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Vcell,1 = Vcell,2 Vcell,2 = Vcell,3 ··· Vcell,N −1 = Vcell,N N
∂h |xˆ(k|k)i (19) ∂x ∂h |xˆ(k|k)i (20) M (k)i = ∂v T K(k)i = P (k|k − 1)(H(k))i (H(k)i P (k|k − 1)(H(k)i )T
H(k)i = (12)
+ M (k)i R(k)(M (k)i )T )−1
Isum,j = Iline ,
(21) (22) (23)
j=1
where N is the number of subsystems or anodes.
x ˆ(k|k)i+1 = x ˆ(k|k − 1) + K(k)i (yk − h(ˆ x(k|k)i ) x(k|k − 1) − x ˆ(k|k)i )), − H(k)i (ˆ until ˆ(k|k)i || ≤ , (24) ||ˆ x(k|k)i+1 − x where is a pre-defined threshold. Therefore, the posterior state estimation can be give as (25) x ˆ(k|k) = x ˆ(k|k)i+1 P (k|k) = (I − K(k)i H(k)i )P (k|k − 1). (26) It is easy to see that the IEKF becomes EKF when i = 0 and the additional iterations provide refining steps to reduce the linearization error.
3. ITERATED EXTENDED KALMAN FILTER The IEKF is a higher order state observer for highly nonlinear system that reduces the linerization error in the EKF (Simon, 2006). Its procedure is outlined in this section. The standard procedure for parameter estimation is used, which treats the bath flow rates M T as state variables that is corrupted by a white noise. M T (k + 1) = M T (k) + ω(k). (13) Equations (7)-(10) can be written in the following form: xj (k + 1) = f (xj (k), uj (k), ωj (k)) (14) Vcell,j (k) = h(xj (k), uj (k), vj (k)).
4. STATE AND PARAMETER ESTIMATION RESULTS 4.1 Simulated operating data
The state vector x ∈ Rn is denoted as [Cun , Cd , D, M T ]T , where n = 4j − 1 and j is the number of subsystems. The vector u = [g, Isum , BM ]T represents the input, and ω and v are the noises in the process model with noise covariance Q and R. The column vector function f (x, u) contains the right-hand side of relevant state equations in equations (7)-(10). Therefore, the problem can be regarded as the estimation of state x with the measurement of u and cell voltage Vcell .
The Hall-H´eroult process is simulated according to equations (7)-(12). In this model, there are 20 anodes in the cell, which is discretized into six subsystems. The feed rates of alumina is taken from the operation log of a real-time application and Figure 3 and 4 show the average anode current, ACD, cell voltage and alumina concentration in subsystem 2 and 6.
For each subsystem j, the priori estimate of state variables, x ˆ(k|k − 1), at the k th step can be obtained from (15)
+ L(k − 1)Q(k − 1)(L(k − 1)) , x ˆ(k|k − 1) = f (ˆ x(k − 1|k − 1), u(k − 1), 0), (16) where F and L,are the partial derivative matrices of state equations with respect to states and noises: ∂f |xˆ(k−1|k−1) F (k − 1) = ∂x (17) ∂f |xˆ(k−1|k−1) ; L(k − 1) = ∂ω
3.7
ACD(cm)
T
3.8
1.045 1.04 1.035 1.03
3.6 3.5
0
200
400
3.4 0
600
Time(min)
Cell voltage(V)
4.05
4
3.95
3.9 0
In the conventional EKF, the output equation h is approximate by the Taylor expansion around the prior estimation x ˆ(k|k − 1). However, the cell voltage equation for the HallH´eroult process is highly nonlinear and a simple Taylor approximation may not provide good accuracy. Therefore, in order to deal with the high nonlinearity in the cell voltage equation, this work proposes the use of the IEKF, in which the linerization error is reduced by re-approximating the output equation around new estimated states, in a iterative manner.
200
400
Time(min)
200
400
600
Time(min)
600
Alumina concentration,Cd (wt%)
T
Current(Amp)
P (k|k − 1) = F (k − 1)P (k − 1|k − 1)(F (k − 1))
10 4
1.05
3.6 3.4 3.2 3 2.8 0
200
400
600
Time(min)
Fig. 3. Simulation results of subsystem 2 The state and parameter estimation is performed based on the simulated data and Figure 5 and 6 show the comparison of the estimated local alumina concentration and ACD with the actual ones. Furthermore, the estimated bath flow rates are shown in Figure 7. It can be seen from Figures 5 to 7 that the proposed IEKF can effectively estimate the state variables and process parameters with fast converging rate and minimum error. In order to illustrate the differences between the IEKF and EKF, the estimation results from the EKF for subsystems 2 and 6
The measurement update of the states are realized by iterations with the following initial conditions: ˆ(k|k − 1) (18) x ˆ(k|k)0 = x For i = 0, 1, 2, · · · , evaluate the following 38
IFAC MMM 2018 Shanghai, China, August 23-25, 2018
10 4
1.025
3.8
1.02 1.015
3.7 3.6
1.01 0
200
400
3.5 0
600
200
Cell voltage(V)
4.05
4
3.95
3.9 0
200
400
600
400
600
Time(min)
3.6 3.4 3.2 3 2.8 2.6 0
200
400
600
160
Actual
3.5
3
300
400
100
200
500
600
120 100
0
100
200
300
400
500
600
4 Estimated Actual
3.5
3
0
100
200
300
400
500
600
400
500
600
Time(min)
ACD(cm)
3.6
3.6 3.5
100
200
300
400
500
3.4 0
600
100
200
Fig. 8. Comparison of estimated and actual state variables of subsystem 2 using EKF
Fig. 5. Comparison of estimated and actual states of subsystem 2
IEKF. To summarize the differences in the performance of the two state estimation methods, Table 1 illustrates different normalized mean-squared errors (NMSE) of the states and parameter estimation, where the differences in initiation condition refer to the NMSE of the initial conditions.
4 Estimated Actual
3.5
3
100
300
Time(min)
Time(min)
Alumina concentration, Cd(wt%)
600
3.7
3.5
200
300
400
500
600
4.2 Real operating data
600
In order to show that the proposed IEKF can be applied to real-time applications, Figure 10 shows the estimated local alumina concentration, ACD and bath flow rate for one of the subsystems from the data collected from an operating cell. The figure shows that the estimation is converging and the estimated states and parameter are in the reasonable operating range.
Time(min) 3.9 3.8
ACD(cm)
500
3.8
3.7
0
400
140
Time(min)
3.4 0
300
Time(min) Bath flow rate between subsystem 5 and 6
160
3.8
ACD(cm)
0
Fig. 7. Comparison of estimated and actual bath flow rates
Estimated
200
Estimated Actual
180
Alumina concentration, Cd (wt%)
Alumina concentration, Cd (wt%)
180
Time(min)
4
100
200
Time(min)
Fig. 4. Simulation results of subsystem 6
0
39
Bath flow rate between subsystem 1 and 2
220
Time(min) Alumina concentration, Cd (wt%)
Time(min)
Bath flow rate(kg/min)
1.005
Bath flow rate(kg/min)
3.9
ACD(cm)
Current(Amp)
1.03
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3.7 3.6 3.5 0
100
200
300
400
500
Time(min)
Table 1. Comparison of NMSE of state and parameter estimation at different initiation conditions
Fig. 6. Comparison of estimated and actual states of subsystem 6
Difference in initiation 0.2173 0.1228 0.0576 0.0106
are shown in Figures 8 and 9. It should be noted that the compared methods have exactly the same initiation conditions and tuning parameters Q and R, while the estimation results are significantly worse than that of the 39
EKF 2.1613 1.0929 0.8316 0.1948
IEKF 0.0316 0.0256 0.0110 0.0021
Alumina concentration, Cd (wt%)
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dissolved alumina, which contributes to a significant factor in determining the efficiency of the process.
4 Estimated Actual
3.5
The further work of this paper will include the verification of the proposed IEKF by analysing bath samples taking from the operating cell. In addition, the correlation between the anode currents and bath flow rates should be explored so that the state and parameter estimation can provide reasonable results even when the bath flow rates vary rapidly.
3
0
100
200
300
400
500
600
Time(min) 3.9
ACD(cm)
3.8 3.7
REFERENCES
3.6
Barber, G. (1992). The Impact of Anode-related Process Dynamics of Cell Behaviour During Aluminium Electrolysis. Ph.D. thesis, Department of Chemical and Materials Engineering, School of Engineering, The University of Auckland, New Zealand. Barnett, W. (1988). Measuring current distribution in an aluminium reduction cell, U.S. Patent: 4786379. Biedler, P. (2003). Modeling of an Aluminium Reudction Cell for the Development of a State Estimator. Ph.D. thesis, Department of Mechanical and Aerospace Engineering, West Virginia University. Cheung, C., Menictas, C., Bao, J., Skyllas-Kazacos, M., and Welch, B.J. (2013). Characterization of individual anode current signals in aluminum reduction cells. Industrial and Engineering Chemistry Research, 52(28), 9632 – 9644. Cheung, C., Menictas, C., Bao, J., Skyllas-Kazacos, M., and Welch, B. (2012). Spatial temperature profiles in an aluminium reduction cell under different anode current distributions. AIChE Journal, 59(5), 1544–1556. Dion, L., Lagace, C., Evans, J.W., Victor, R., and Kiss, L.I. (2015). On-line monitoring of individual anode currents to understand and improve the process control at Alouette. In Proceedings of TMS Light Metals, Orlando, FL., volume 2015, 723 – 728. Evans, J. and Urata, N. (2012). Wireless and noncontacting measurement of individual anode currents in Hall-H´eroult pots; experience and benefits. In Proceedings of TMS Light Metals, Orlando, FL., 939 – 942. Grjotheim, K. and Kvande, H. (eds.) (1993). Introduction to Aluminium Electrolysis: Understanding the HallH´eroult Process. D¨ usseldorf: Aluminium-Verlag. Grjotheim, K. and Welch, B.J. (1980). Aluminium Smelter Technology: a Pure and Applied Approach. D¨ usseldorf : Aluminium-Verlag. Haupin, W. and Seger, E. (2001). Aiming for zero anode effects. In Proceedings of TMS Light Metals, New Orleans, LA., 329 – 336. Haupin, W. (1998). Interpreting the components of cell voltage. In Proceedings of TMS Light Metals, San Antonio, TX., 531 – 537. Hestetun, K. and Hovd, M. (2005). Detecting abnormal feed rate in aluminium electrolysis using extended Kalman filter. In Proceedings of the 16th IFAC World Congress, Prague, Czech republic, volume 16, 85 – 90. Hestutun, K. and Hovd, M. (2006). Detection of abnormal alumina feed rate in aluminium electrolysis cells using state and parameter estimation. In 16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering, Garmisch-Partenkirchen, Germany.
3.5 3.4 0
100
200
300
400
500
600
Time(min)
Alumina concentration, Cd (wt%)
Fig. 9. Comparison of estimated and actual state variables of subsystem 6 using EKF 3.5 3 2.5 2
0
50
100
150
200
250
300
50
100
150
200
250
300
50
100
150
200
250
300
ACD(cm)
4.2 4.1 4 3.9
Bath flow rate(kg/min)
3.8 0
400 300 200 100 0
0
Time(min)
Fig. 10. Estimation of local states and parameters based on real operation data 5. DISCUSSIONS AND CONCLUSIONS This paper proposes the IEKF as the state observer for the estimation of local alumina concentration, ACD and bath flow rate. It has been shown in the comparisons between the estimated variables and the simulated actual ones that the IEKF is effective in estimating the localized process variables and process parameters with small estimation error and fast converging rate. Comparing the IEKF with EKF, it is evident that IEKF is superior to the EKF in that it provides robustness to ill-set initiation conditions. In an extreme case, the MSE of the state and parameter estimation can be 10 times smaller than that of the EKF. This is because the iteration steps in the IEKF can greatly reduce the linearization error. The robustness to different initiation conditions is rather important as the initiation conditions can be crucial in state estimation and it can be difficult to set as the states are normally not measurable. This work is different from Yao et al. (2017) in that the bath flow rate is directly estimated as a process parameter, under the premise that it does not change significantly during the process. The flow rate is a crucial factor in the Hall-H´eroult cells as it is related to the uniformity of 40
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Jakobsen, S., Hestetun, K., Hovd, M., and Solberg, I. (2001). Estimating alumina concentration distribution in aluminium electrolysis cells. In Automation in Mining, Mineral, and Metal Processing: a proceedings volume from the 10th IFAC symposium, Tokyo, Japan. Keniry, J. and Shaidulin, E. (2008). Anode signal analysis: the next generation in reduction cell control. In Proceedings of TMS Light Metals, Warrendale, PA., 287 – 92. Keniry, J., Barber, G.C., Taylor, M.P., and Welch, B.J. (2001). Digital processing of anode current signals: An opportunity for improved cell diagnosis and control. In Proceedings of TMS Light Metals, New Orleans, LA., 1225 – 1232. Rye, K., K¨ onigsson, M., and Solberg, I. (1998). Current redistribution among individual anode carbons in a Hall-H´eroult prebake cell at low alumina concentrations. In Proceedings of TMS Light Metals, San Antonio,TX., 241 – 246. Simon, D. (2006). Optimal state estimation. John Wiley & Sons. Yao, Y., Cheung, C., Bao, J., Skyllas-Kazacos, M., Welch, B., and Akhmetov, S. (2016). Detection of local cell condition based on individual anode current measurements. In Proceedings of TMS Light Metals, Nashville,TN. Yao, Y., Cheung, C.Y., Bao, J., Skyllas-Kazacos, M., Welch, B.J., and Akhmetov, S. (2017). Estimation of spatial alumina concentration in an aluminum reduction cell using a multilevel state observer. AIChE Journal, 63(7), 2806–2818.
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State and Parameter Estimation in State and Parameter Estimation in State and Parameter Estimation in State and Parameter Estimation in Hall-H´ e roult Cells using Iterated Extended State and Parameter Estimation in Hall-H´ e roult Cells using Iterated Extended Hall-H´ e roult Cells using Iterated Extended Hall-H´ eroult Cells using Iterated Extended Kalman Filter Hall-H´ eroult Cells using Iterated Extended Kalman Filter Kalman Kalman Filter Filter Kalman Filter Yuchen Yao and Jie Bao
Yuchen Yao and Jie Bao Yuchen Yuchen Yao Yao and and Jie Jie Bao Bao Yuchen Yao and Jie Bao School of Chemical Engineering, University School of Chemical Chemical Engineering, Engineering, University University of of New New South South Wales, Wales, School of New South Wales, Australia (e-mail: [email protected]). School of ofSydney, Chemical Engineering, University of New South Sydney, Australia Australia (e-mail: (e-mail: [email protected]). [email protected]). Wales, Sydney, School ofSydney, Chemical Engineering, University of New South Wales, Australia (e-mail: [email protected]). Sydney, Australia (e-mail: [email protected]). Abstract: The measurement of individual anode currents in the Hall-H´ eeroult process provides Abstract: The measurement of individual anode currents in the Hall-H´ process provides Abstract: The measurement of individual anode currents in the Hall-H´ eeroult roult process provides localized information that can be used to improve cell operation. One promising application is Abstract: The measurement of individual anode currents in the Hall-H´ roult process provides localized information that can be be used used to to improve improve cell cell operation. operation. One One promising promising application application is localized information that can is Abstract: The measurement ofbeindividual anode currents in the Hall-H´ eroult process provides to estimate the local cell conditions, which are impracticable to measure regularly during the localized information that can used to improve cell operation. One promising application is to estimate the local cell conditions, which are impracticable to measure regularly during the to the local cell which are to measure regularly during the localized information thatconditions, can be used to improve cell operation. One promising application is cell operation. presents an approach to estimation localized state variables to estimate estimate theThis localpaper cell conditions, which are impracticable impracticable toof measure regularly during and the cell operation. This paper presents an approach to estimation of localized state variables and cell operation. This paper presents an approach to estimation of localized state variables and to estimate the local cell conditions, which are impracticable to measure regularly during the process variables, including the alumina concentration, anode-cathode distance and bath flow cell operation. This paper presents an approach to estimation of localized state and variables and process variables, including the alumina concentration, anode-cathode distance bath flow process variables, including the alumina concentration, anode-cathode distance and bath cell operation. This paper presents an approach to estimation of localized state variables and rate. This is achieved by using Iterated Extended Kalman Filter (IEKF) with aa discretized process variables, including thethe alumina concentration, anode-cathode distance and bath flow flow rate. This is achieved by using the Iterated Extended Kalman Filter (IEKF) with discretized rate. This is achieved by using the Iterated Extended Kalman Filter (IEKF) with aa discretized process variables, including the alumina concentration, anode-cathode distance and bath flow Hall-H´ e roult cell model. The results show that the proposed IEKF can produce more accurate rate. This is achieved by using the Iterated Extended Kalman Filter (IEKF) with discretized Hall-H´ eeroult cell model. The results show that the proposed IEKF can produce more accurate Hall-H´ cell model. The show the IEKF can produce accurate rate. This isofachieved by usingresults the Iterated Extended Kalman Filter (IEKF) withmore a discretized estimation the state variables and parameters compared to the conventional Extended Kalman Hall-H´ eroult roult cell model. The results show that that the proposed proposed IEKF can produce more accurate estimation of the state variables and parameters compared to the conventional Extended Kalman estimation of the state variables and parameters compared to the conventional Extended Kalman Hall-H´ e roult cell model. The results show that the proposed IEKF can produce more Filter thanks to its robustness to ill-set initiation conditions. estimation of the state variablesto and parameters compared to the conventional Extendedaccurate Kalman Filter thanks to its robustness ill-set initiation conditions. Filter thanks to its robustness to ill-set initiation conditions. estimation of the state variablesto and parameters compared to the conventional Extended Kalman Filter thanks to its robustness ill-set initiation conditions. © 2018,thanks IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Filter to its robustness toindividual ill-set initiation conditions. Keywords: Hall-H´ eeroult process, anode current, state estimation, parameter Keywords: Hall-H´ roult process, individual anode current, state Keywords: Hall-H´ e roult process, individual anode current, state estimation, estimation, parameter parameter estimation, Iterated Extended Kalman Filter Keywords: Hall-H´ e roult process, individual anode current, state estimation, parameter estimation, Iterated Extended Kalman Filter estimation, Iterated Extended Kalman Filter Keywords: Hall-H´ e roult process, individual anode current, state estimation, parameter estimation, Iterated Extended Kalman Filter estimation, Iterated Extended Kalman Filter 1. A typical Hall-H´ eeroult cell contains 18 to 40 anodes 1. INTRODUCTION INTRODUCTION A typical Hall-H´ cell contains 18 to 40 anodes 1. A typical Hall-H´ eeroult roult cell contains 18 to 40 anodes connected in parallel, sharing a regulated DC current 1. INTRODUCTION INTRODUCTION A typical Hall-H´ roult cell contains 18 to 40 anodes connected in parallel, sharing a regulated DC current connected in parallel, sharing a regulated DC current 1. INTRODUCTION A typical Hall-H´ e roult cell contains 18 to 40 anodes in the order of several hundred kilo-amps. This connected in parallel, sharing a regulated DC current The Hall-H´ e roult process is the dominant method of proin the the order order of of several several hundred hundred kilo-amps. kilo-amps. This This current The Hall-H´ Hall-H´eeroult roult process process is is the the dominant dominant method method of of propro- in current connected in parallel, sharing a regulated DC current The is normally referred to as the line current, while the in the order of several hundred kilo-amps. This current ducing aluminum and Welch, 1980; Grjotheim The Hall-H´ eroult (Grjotheim process is the dominant method of pro- is normally referred to as the line current, while the ducing aluminum (Grjotheim and Welch, 1980; 1980; Grjotheim is normally referred to as the line current, while the in the order of several hundred kilo-amps. This current ducing aluminum (Grjotheim and Welch, Grjotheim voltage drop across the anode and the cathode is called is normally referred to as the line current, while the The Hall-H´ e roult process is the dominant method of proand Kvande, 1993). It involves the electrolysis of alumina ducing aluminum (Grjotheim and Welch, 1980; Grjotheim voltage drop across the anode and the cathode is called and Kvande, 1993). It involves the electrolysis of alumina voltage drop across the anode and the cathode is called is normally referred to as the line current, while the and Kvande, 1993). It involves the electrolysis of alumina the cell voltage. One of the major limiting factors in voltage drop across the anode and the cathode is called ducing aluminum (Grjotheim and Welch, 1980; Grjotheim (Al O ) dissolved in cryolite (Na AlF ). During the elecand Kvande, 1993).inItcryolite involves the electrolysis of the alumina 3 ) dissolved 3 AlF 6 ). During the cell voltage. One of the major limiting factors in (Al222 O (Na electhe cell voltage. One of the major limiting factors in 3 3 6 voltage drop across the anode and the cathode is called (Al O ) dissolved in cryolite (Na AlF ). During the elecimproving the efficiency of the Hall-H´ e roult process is 3 ) dissolved 3 AlF 6 ).continuously the cell voltage. One of the major limiting factors and2 O Kvande, 1993). It involves the electrolysis of alumina trochemical reaction, carbon anodes are con(Al in cryolite (Na During the elecimproving the the efficiency efficiency of of the the Hall-H´ Hall-H´eeroult roult process process in is 3 3 6 continuously controchemical reaction, carbon anodes are improving is the cell voltage. One of the major limiting factors in trochemical reaction, carbon anodes are continuously conthat usually only the line current and cell voltage are improving the efficiency of the Hall-H´ e roult process is (Al2 O3while ) dissolved in cryolite (Naaluminum During the elecsumed the produced liquid is accumulated 3 AlF 6 ).continuously trochemical reaction, carbon anodes are conthat usually only the line current and cell voltage are sumed while the produced liquid aluminum is accumulated that usually only the line current and cell voltage are improving the efficiency of the Hall-H´ e roult process is sumed while the produced liquid aluminum is accumulated measured continuously online. This means that the cell that usually only the line current and cell voltage are trochemical reaction, carbon anodes are continuously conat of the serving the sumed while the liquid aluminum is accumulated continuously online. This means that the cell at the the bottom bottom of produced the cell, cell, effectively effectively serving as as the cathode. cathode. measured measured continuously online. This means that the cell that usually only the line current and cell voltage are at the bottom of the cell, effectively serving as the cathode. control logics can only be designed based on these two measured continuously online. This means that the cell sumed while the produced liquid aluminum is accumulated The main chemical reaction is shown below: at themain bottom of thereaction cell, effectively serving as the cathode. control logics can only be designed based on these two The chemical is shown below: control logics can be designed based on two continuously online. This cell means thatthese theWith cell The chemical is below: variables that represent conditions. control logics can only only the be averaged designed based on these two at themain bottom of thereaction cell, effectively serving as the cathode. measured The main chemical reaction is→shown shown below: variables that represent the averaged cell conditions. With 2 Al O + 3 C − − 4 Al + 3 CO (1) variables that represent the averaged cell conditions. With 2 O3 + 3 C −−→ 4 Al + 3 CO2 control logics can only be designed based on these two larger cells containing more anodes being built to meet 2 Al (1) variables that represent the averaged cell conditions. With The main chemical is→shown below: 2 3 3reaction 2 2 + 3 C −− + 3 cells containing more anodes being built to meet larger cells containing more anodes being built to meet 2 Al Al22aO Oschematic →4 4 Al Al + Hall-H´ 3 CO CO22 eroult cell.(1) (1) larger variables that represent the averaged cell conditions. With 3 + 3 C −−plot the growing market demands, it has become more evident Figure 1 depicts of a larger cells containing more anodes being built to meet the growing market demands, it has become more evident Figure 1 depicts a schematic plot of a Hall-H´ e roult cell. 2 Al O + 3 C − − → 4 Al + 3 CO (1) 2a schematic 3 2 eroult cell. the growing market demands, it has become more evident Figure 1 depicts plot of a Hall-H´ larger cells containing more anodes being built to meet that there can be significant spatial variations in the cell the growing market demands, it has become more evident Figure 1 depicts a schematic plot of a Hall-H´eroult cell. that there can be significant spatial variations in the cell that there can be significant spatial variations in the cell the growing market demands, it has become more evident Figure 1 depicts a schematic plot of a Hall-H´eroult cell. (Haupin and Seger, 2001; Keniry et al., 2001; Keniry and that there can be significant spatial variations in the cell (Haupin and Seger, 2001; Keniry et al., 2001; Keniry and (Haupin and Seger, 2001; Keniry et al., 2001; Keniry and that there can be significant spatial variations in the and cell Shaidulin, 2008). Therefore, it has become increasingly (Haupin and Seger, 2001; Keniry et al., 2001; Keniry Shaidulin, 2008). Therefore, it has become increasingly Shaidulin, 2008). Therefore, it has become increasingly (Haupin and Seger, 2001; Keniry et al., 2001; Keniry and important for the localized cell conditions to be properly Shaidulin, 2008). Therefore, it has become increasingly important for the localized cell conditions to be properly important for the localized cell conditions to be Shaidulin, 2008). Therefore, it that, has become dealt with. In order to achieve the measurement of important for the localized cell conditions to increasingly be properly properly dealt with. In order to achieve that, the measurement of dealt with. In order to achieve the measurement of important for the localized cell that, conditions to be(Barnett, properly individual anode currents has been proposed dealt with. In order to achieve that, the measurement of individual anode anode currents currents has has been been proposed proposed (Barnett, (Barnett, individual dealt with. In and order to achieve that, the measurement of 1988; Evans Urata, 2012). The individual anode individual anode currents has been proposed (Barnett, 1988; Evans and Urata, 2012). The individual anode 1988; Evans and Urata, 2012). The individual anode individual anode currents has been proposed (Barnett, current is attractive because it reflects the cell conditions 1988; Evans and Urata, 2012). The individual anode current is attractive because it reflects the cell conditions current is because it cell conditions 1988; Evans and Urata, 2012). Theitthe individual anode in the vicinity of each anode and has found great current is attractive attractive because it reflects reflects the cell conditions in the vicinity of each anode and it has found great in the vicinity of each anode and it has found great current is attractive because it reflects the cell conditions potential in a number of proposed applications (Barber, in the vicinity of eachof anode andapplications it has found great potential in a number proposed (Barber, potential in a number of proposed applications (Barber, in theKeniry vicinity ofShaidulin, eachof anode and it ethas found great 1992; 2008; Rye al., 1998; Dion potential in aand number proposed applications (Barber, 1992; Keniry and Shaidulin, 2008; Rye et al., 1998; Dion 1992; Keniry and Shaidulin, 2008; Rye et al., 1998; Dion potential in aCheung number ofal., proposed applications (Barber, et al., 2015; et 2012, 2013; Yao et al., 2016, 1992; Keniry and Shaidulin, 2008; Rye et al., 1998; Dion et al., 2015; Cheung et al., 2012, 2013; Yao et al., 2016, et al., 2015; Cheung et al., 2012, 2013; Yao et al., 2016, 1992; Keniry and Shaidulin, 2008; Rye et al., 1998; Dion 2017). et al., 2015; Cheung et al., 2012, 2013; Yao et al., 2016, 2017). 2017). et al., 2015; Cheung et al., 2012, 2013; Yao et al., 2016, 2017). The use of state observers in the Hall-H´ eeroult process 2017). The use of state observers in the Hall-H´ process The use of state observers in the Hall-H´ eeroult roult process with the measurement of individual anode currents has The use of state observers in the Hall-H´ roult process with the the measurement measurement of of individual individual anode anode currents currents has with has The use of state observers in itthe Hall-H´ eroult process been proved to be beneficial as provides the estimation with the measurement of individual anode currents has been proved to be beneficial as it provides the estimation been proved to be beneficial as it provides the estimation with the measurement of individual anodethe currents has of localized state variables such as alumina concentration been proved to be beneficial as it provides estimation of localized state variables such as alumina concentration of localized such alumina concentration been provedstate to bevariables beneficial as itas provides the estimation and anode-cathode distance (ACD) (Jakobsen et al., 2001; of localized state variables such as alumina concentration and anode-cathode distance (ACD) (Jakobsen et al., 2001; and anode-cathode distance (ACD) (Jakobsen et al., of localized state variables such as alumina concentration Hestetun and Hovd, 2005; Hestutun and Hovd, Yao Fig. 1. Schematic plot of a Hall-H´ cell. and anode-cathode distance (ACD) (Jakobsen et2006; al., 2001; 2001; Hestetun and Hovd, 2005; Hestutun and Hovd, 2006; Yao Fig. 1. Schematic plot of aa Hall-H´ Hall-H´eeeroult roult cell. Hestetun and Hovd, 2005; Hestutun and Hovd, 2006; Yao Fig. 1. Schematic plot of roult cell. and anode-cathode (ACD) (Jakobsen et2006; al., 2001; et al., 2017). normal operation, the bath flow rates Hestetun andDuring Hovd,distance 2005; Hestutun and Hovd, Yao Fig. 1. Schematic plot of a Hall-H´eroult cell. et al., 2017). During normal operation, the bath flow rates et al., 2017). During normal operation, the bath flow rates Hestetun and Hovd, 2005; Hestutun and Hovd, 2006; Yao Fig. 1. Schematic plot of a Hall-H´ e roult cell. in a Hall-H´ e roult cell often change slowly, thus it is possible et al., 2017). During normal operation, the bath flow rates This work is supported by Australian Research Council Discovery in aa Hall-H´ eeroult cell often change slowly, thus it is possible in Hall-H´ roult cell often change slowly, thus it is possible et al., 2017). During normal operation, the bath flow rates work is supported by Australian Research Council Discovery This to estimate the bath flow rates as process parameters along in a Hall-H´ e roult cell often change slowly, thus it is possible This work is supported by Australian Research Council Discovery to estimate the bath flow rates as process parameters along This work Project DP160101810. is supported by Australian Research Council Discovery to bath as along Project DP160101810. in Hall-H´ethe roult cellflow oftenrates change slowly, parameters thus it is possible to aestimate estimate the bath flow rates as process process parameters along Project DP160101810. This work is supported by Australian Research Council Discovery Project DP160101810. to estimate the bath flow rates as process parameters along
Project DP160101810. 2405-8963 © 2018 2018, IFAC IFAC (International Federation of Automatic Control) Copyright © 36 Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 36 Peer review© responsibility of International Federation of Automatic Copyright © under 2018 IFAC IFAC 36 Control. Copyright © 2018 IFAC 36 10.1016/j.ifacol.2018.09.389 Copyright © 2018 IFAC 36
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with the state variables. The dynamics of the Hall-H´eroult cell has very strong output nonlinearity, which should be effectively dealt with by using an Iterated Extended Kalman Filter (IEKF) to achieve better accuracy and robustness to ill-set initial estimation. In this paper, a HallH´eroult cell is discretized into a number of subsystems and an IEKF is developed to estimate the above local state variables and process parameters. This paper is organized as follows. The process model is briefly introduced in the next section, followed by the description of IEKF. The state and parameter estimation results are presented and compared with conventional EKF. A discussion and conclusion is provided at the end of the paper.
37
Fig. 2. Discretization of a Hall-H´eroult cell where accum(Iline ) is the height accumulation rate of molten aluminum at the bottom of the cell, consum(Iline ) is the rate of the anode height reduction through the carbon consumption, D is the ACD and BM is the distance of the busbar movement.
2. PROCESS MODELING
2.3 Cell Voltage
The process model used in this paper is similar to the one described in Yao et al. (2017) and is briefly discussed in this section.
The cell voltage is a nonlinear function of cell conditions, including alumina concentration and ACD, and a number of process parameters that change slowly (Haupin, 1998). The general form of the voltage equation is (6) Vcell = Erev + Eover + Iline Rpath + Vexternal , where Erev and Eover are the reversible and over-potential, Rcell is the ohmic resistance of the cell and Vexternal is the external voltage drop. In this paper, the cell voltage is represented as (7) Vcell (k) = h(Iline (k), D(k), Cd (k), θ), where θ includes cell design and operating parameters.
2.1 Alumina Concentration It is assumed that the dissolution of alumina in the bath are governed by first-order rate equations with two rate constants (Biedler, 2003). The faster dissolution rate constant is 0.099 s−1 , which is equivalent to a residence time of 50 seconds. In order to simplify the model, one minute is chosen as the time step so that the dynamics of the alumina with faster dissolution rate constant can be ignored. The discrete-time equation for alumina dissolution is shown in equation (2). g(k)r Cun (k + 1) = Cun (k) − kdiss Cun (k) + , (2) m where Cun is the concentration of the undissolved alumina, kdiss is the respective dissolution rate constant, g is the amount of alumina fed, r is the weight ratio between the fast and slow dissolving alumina and m is the mass of the bath. It follows that, with alumina consumption governed by Faraday’s Law, the equation for dissolved alumina is g(k)(1 − r) Cd (k + 1) = Cd (k) + kdiss Cun (k) + m F a(Iline ,Al2 O3 (k)) , − m (3) where Cd is the concentration of the dissolved alumina, F a(Iline ,Al2 O3 ) is the alumina consumption rate based on the Faraday’s Law, as shown in equation (4). Iline × MAl2 O3 × η , (4) F a(Iline ,Al2 O3 ) = F ×z where MAl2 O3 is the molar mass of alumina, η is the current efficiency, F is the Faraday constant and z is the number of electrons transferred.
2.4 Cell Discretization Equations (2)-(7) can be slightly modified to describe the subsystems in a discretized Hall-H´eroult cell shown in Figure 2. The cell is discretized into six subsystems according to the location of the feeders. As can be seen, each of the subsystems 2 to 5 contains one feeder and the alumina in subsystems 1 and 6 is supplied by the electrolyte flows from subsystems 2 and 5. It is assumed that the mass transfer is induced by electrolyte flows (bath flows), and the flow rates across any boundary between two subsystems are the same (for example, the bath flow from subsystem 1 to subsystem 2 is the same as the one from subsystem 2 to subsystem 1). Therefore, the process model of, for example subsystem 2, can be written as g2 (k)r (8) Cun,2 (k + 1) = Cun,2 (k) − kdiss Cun,2 (k) + m2 Cd,2 (k + 1) = Cd,2 (k) + kdiss Cun,2 (k) F a(Isum,Al2 O3 ,2 (k)) g2 (k)(1 − r) − + m2 m2 (9) M T1,2 (−Cd,2 (k) + Cd,1 (k)) + m2 M T2,3 (−Cd,2 (k) + Cd,3 (k)) + m2 D2 (k + 1) = D2 (k) − accum(Isum,2 (k)) (10) + consum(Isum,2 (k)) + BM (k) (11) Vcell,2 (k) = h(Isum,2 (k), D2 (k), Cd,2 (k), θ2 ), where M T represents mass transfer rate of bath between two neighboring subsystems and Isum,2 represents the sum of anode currents in subsystem 2, which can be obtained by solving the following system of equations
2.2 Anode-cathode Distance The dynamics of ACD include the accumulation of the liquid aluminum at the bottom of the cell, which gradually reduces the ACD, and the consumption of carbon anodes, which increases the ACD. Thus, the dynamic equation for the ACD is D(k + 1) = D(k) − accum(Iline (k)) (5) + consum(Iline (k)) + BM (k), 37
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Vcell,1 = Vcell,2 Vcell,2 = Vcell,3 ··· Vcell,N −1 = Vcell,N N
∂h |xˆ(k|k)i (19) ∂x ∂h |xˆ(k|k)i (20) M (k)i = ∂v T K(k)i = P (k|k − 1)(H(k))i (H(k)i P (k|k − 1)(H(k)i )T
H(k)i = (12)
+ M (k)i R(k)(M (k)i )T )−1
Isum,j = Iline ,
(21) (22) (23)
j=1
where N is the number of subsystems or anodes.
x ˆ(k|k)i+1 = x ˆ(k|k − 1) + K(k)i (yk − h(ˆ x(k|k)i ) x(k|k − 1) − x ˆ(k|k)i )), − H(k)i (ˆ until ˆ(k|k)i || ≤ , (24) ||ˆ x(k|k)i+1 − x where is a pre-defined threshold. Therefore, the posterior state estimation can be give as (25) x ˆ(k|k) = x ˆ(k|k)i+1 P (k|k) = (I − K(k)i H(k)i )P (k|k − 1). (26) It is easy to see that the IEKF becomes EKF when i = 0 and the additional iterations provide refining steps to reduce the linearization error.
3. ITERATED EXTENDED KALMAN FILTER The IEKF is a higher order state observer for highly nonlinear system that reduces the linerization error in the EKF (Simon, 2006). Its procedure is outlined in this section. The standard procedure for parameter estimation is used, which treats the bath flow rates M T as state variables that is corrupted by a white noise. M T (k + 1) = M T (k) + ω(k). (13) Equations (7)-(10) can be written in the following form: xj (k + 1) = f (xj (k), uj (k), ωj (k)) (14) Vcell,j (k) = h(xj (k), uj (k), vj (k)).
4. STATE AND PARAMETER ESTIMATION RESULTS 4.1 Simulated operating data
The state vector x ∈ Rn is denoted as [Cun , Cd , D, M T ]T , where n = 4j − 1 and j is the number of subsystems. The vector u = [g, Isum , BM ]T represents the input, and ω and v are the noises in the process model with noise covariance Q and R. The column vector function f (x, u) contains the right-hand side of relevant state equations in equations (7)-(10). Therefore, the problem can be regarded as the estimation of state x with the measurement of u and cell voltage Vcell .
The Hall-H´eroult process is simulated according to equations (7)-(12). In this model, there are 20 anodes in the cell, which is discretized into six subsystems. The feed rates of alumina is taken from the operation log of a real-time application and Figure 3 and 4 show the average anode current, ACD, cell voltage and alumina concentration in subsystem 2 and 6.
For each subsystem j, the priori estimate of state variables, x ˆ(k|k − 1), at the k th step can be obtained from (15)
+ L(k − 1)Q(k − 1)(L(k − 1)) , x ˆ(k|k − 1) = f (ˆ x(k − 1|k − 1), u(k − 1), 0), (16) where F and L,are the partial derivative matrices of state equations with respect to states and noises: ∂f |xˆ(k−1|k−1) F (k − 1) = ∂x (17) ∂f |xˆ(k−1|k−1) ; L(k − 1) = ∂ω
3.7
ACD(cm)
T
3.8
1.045 1.04 1.035 1.03
3.6 3.5
0
200
400
3.4 0
600
Time(min)
Cell voltage(V)
4.05
4
3.95
3.9 0
In the conventional EKF, the output equation h is approximate by the Taylor expansion around the prior estimation x ˆ(k|k − 1). However, the cell voltage equation for the HallH´eroult process is highly nonlinear and a simple Taylor approximation may not provide good accuracy. Therefore, in order to deal with the high nonlinearity in the cell voltage equation, this work proposes the use of the IEKF, in which the linerization error is reduced by re-approximating the output equation around new estimated states, in a iterative manner.
200
400
Time(min)
200
400
600
Time(min)
600
Alumina concentration,Cd (wt%)
T
Current(Amp)
P (k|k − 1) = F (k − 1)P (k − 1|k − 1)(F (k − 1))
10 4
1.05
3.6 3.4 3.2 3 2.8 0
200
400
600
Time(min)
Fig. 3. Simulation results of subsystem 2 The state and parameter estimation is performed based on the simulated data and Figure 5 and 6 show the comparison of the estimated local alumina concentration and ACD with the actual ones. Furthermore, the estimated bath flow rates are shown in Figure 7. It can be seen from Figures 5 to 7 that the proposed IEKF can effectively estimate the state variables and process parameters with fast converging rate and minimum error. In order to illustrate the differences between the IEKF and EKF, the estimation results from the EKF for subsystems 2 and 6
The measurement update of the states are realized by iterations with the following initial conditions: ˆ(k|k − 1) (18) x ˆ(k|k)0 = x For i = 0, 1, 2, · · · , evaluate the following 38
IFAC MMM 2018 Shanghai, China, August 23-25, 2018
10 4
1.025
3.8
1.02 1.015
3.7 3.6
1.01 0
200
400
3.5 0
600
200
Cell voltage(V)
4.05
4
3.95
3.9 0
200
400
600
400
600
Time(min)
3.6 3.4 3.2 3 2.8 2.6 0
200
400
600
160
Actual
3.5
3
300
400
100
200
500
600
120 100
0
100
200
300
400
500
600
4 Estimated Actual
3.5
3
0
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200
300
400
500
600
400
500
600
Time(min)
ACD(cm)
3.6
3.6 3.5
100
200
300
400
500
3.4 0
600
100
200
Fig. 8. Comparison of estimated and actual state variables of subsystem 2 using EKF
Fig. 5. Comparison of estimated and actual states of subsystem 2
IEKF. To summarize the differences in the performance of the two state estimation methods, Table 1 illustrates different normalized mean-squared errors (NMSE) of the states and parameter estimation, where the differences in initiation condition refer to the NMSE of the initial conditions.
4 Estimated Actual
3.5
3
100
300
Time(min)
Time(min)
Alumina concentration, Cd(wt%)
600
3.7
3.5
200
300
400
500
600
4.2 Real operating data
600
In order to show that the proposed IEKF can be applied to real-time applications, Figure 10 shows the estimated local alumina concentration, ACD and bath flow rate for one of the subsystems from the data collected from an operating cell. The figure shows that the estimation is converging and the estimated states and parameter are in the reasonable operating range.
Time(min) 3.9 3.8
ACD(cm)
500
3.8
3.7
0
400
140
Time(min)
3.4 0
300
Time(min) Bath flow rate between subsystem 5 and 6
160
3.8
ACD(cm)
0
Fig. 7. Comparison of estimated and actual bath flow rates
Estimated
200
Estimated Actual
180
Alumina concentration, Cd (wt%)
Alumina concentration, Cd (wt%)
180
Time(min)
4
100
200
Time(min)
Fig. 4. Simulation results of subsystem 6
0
39
Bath flow rate between subsystem 1 and 2
220
Time(min) Alumina concentration, Cd (wt%)
Time(min)
Bath flow rate(kg/min)
1.005
Bath flow rate(kg/min)
3.9
ACD(cm)
Current(Amp)
1.03
Yuchen Yao et al. / IFAC PapersOnLine 51-21 (2018) 36–41
3.7 3.6 3.5 0
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200
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500
Time(min)
Table 1. Comparison of NMSE of state and parameter estimation at different initiation conditions
Fig. 6. Comparison of estimated and actual states of subsystem 6
Difference in initiation 0.2173 0.1228 0.0576 0.0106
are shown in Figures 8 and 9. It should be noted that the compared methods have exactly the same initiation conditions and tuning parameters Q and R, while the estimation results are significantly worse than that of the 39
EKF 2.1613 1.0929 0.8316 0.1948
IEKF 0.0316 0.0256 0.0110 0.0021
Alumina concentration, Cd (wt%)
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dissolved alumina, which contributes to a significant factor in determining the efficiency of the process.
4 Estimated Actual
3.5
The further work of this paper will include the verification of the proposed IEKF by analysing bath samples taking from the operating cell. In addition, the correlation between the anode currents and bath flow rates should be explored so that the state and parameter estimation can provide reasonable results even when the bath flow rates vary rapidly.
3
0
100
200
300
400
500
600
Time(min) 3.9
ACD(cm)
3.8 3.7
REFERENCES
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Barber, G. (1992). The Impact of Anode-related Process Dynamics of Cell Behaviour During Aluminium Electrolysis. Ph.D. thesis, Department of Chemical and Materials Engineering, School of Engineering, The University of Auckland, New Zealand. Barnett, W. (1988). Measuring current distribution in an aluminium reduction cell, U.S. Patent: 4786379. Biedler, P. (2003). Modeling of an Aluminium Reudction Cell for the Development of a State Estimator. Ph.D. thesis, Department of Mechanical and Aerospace Engineering, West Virginia University. Cheung, C., Menictas, C., Bao, J., Skyllas-Kazacos, M., and Welch, B.J. (2013). Characterization of individual anode current signals in aluminum reduction cells. Industrial and Engineering Chemistry Research, 52(28), 9632 – 9644. Cheung, C., Menictas, C., Bao, J., Skyllas-Kazacos, M., and Welch, B. (2012). Spatial temperature profiles in an aluminium reduction cell under different anode current distributions. AIChE Journal, 59(5), 1544–1556. Dion, L., Lagace, C., Evans, J.W., Victor, R., and Kiss, L.I. (2015). On-line monitoring of individual anode currents to understand and improve the process control at Alouette. In Proceedings of TMS Light Metals, Orlando, FL., volume 2015, 723 – 728. Evans, J. and Urata, N. (2012). Wireless and noncontacting measurement of individual anode currents in Hall-H´eroult pots; experience and benefits. In Proceedings of TMS Light Metals, Orlando, FL., 939 – 942. Grjotheim, K. and Kvande, H. (eds.) (1993). Introduction to Aluminium Electrolysis: Understanding the HallH´eroult Process. D¨ usseldorf: Aluminium-Verlag. Grjotheim, K. and Welch, B.J. (1980). Aluminium Smelter Technology: a Pure and Applied Approach. D¨ usseldorf : Aluminium-Verlag. Haupin, W. and Seger, E. (2001). Aiming for zero anode effects. In Proceedings of TMS Light Metals, New Orleans, LA., 329 – 336. Haupin, W. (1998). Interpreting the components of cell voltage. In Proceedings of TMS Light Metals, San Antonio, TX., 531 – 537. Hestetun, K. and Hovd, M. (2005). Detecting abnormal feed rate in aluminium electrolysis using extended Kalman filter. In Proceedings of the 16th IFAC World Congress, Prague, Czech republic, volume 16, 85 – 90. Hestutun, K. and Hovd, M. (2006). Detection of abnormal alumina feed rate in aluminium electrolysis cells using state and parameter estimation. In 16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering, Garmisch-Partenkirchen, Germany.
3.5 3.4 0
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Fig. 9. Comparison of estimated and actual state variables of subsystem 6 using EKF 3.5 3 2.5 2
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Time(min)
Fig. 10. Estimation of local states and parameters based on real operation data 5. DISCUSSIONS AND CONCLUSIONS This paper proposes the IEKF as the state observer for the estimation of local alumina concentration, ACD and bath flow rate. It has been shown in the comparisons between the estimated variables and the simulated actual ones that the IEKF is effective in estimating the localized process variables and process parameters with small estimation error and fast converging rate. Comparing the IEKF with EKF, it is evident that IEKF is superior to the EKF in that it provides robustness to ill-set initiation conditions. In an extreme case, the MSE of the state and parameter estimation can be 10 times smaller than that of the EKF. This is because the iteration steps in the IEKF can greatly reduce the linearization error. The robustness to different initiation conditions is rather important as the initiation conditions can be crucial in state estimation and it can be difficult to set as the states are normally not measurable. This work is different from Yao et al. (2017) in that the bath flow rate is directly estimated as a process parameter, under the premise that it does not change significantly during the process. The flow rate is a crucial factor in the Hall-H´eroult cells as it is related to the uniformity of 40
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Jakobsen, S., Hestetun, K., Hovd, M., and Solberg, I. (2001). Estimating alumina concentration distribution in aluminium electrolysis cells. In Automation in Mining, Mineral, and Metal Processing: a proceedings volume from the 10th IFAC symposium, Tokyo, Japan. Keniry, J. and Shaidulin, E. (2008). Anode signal analysis: the next generation in reduction cell control. In Proceedings of TMS Light Metals, Warrendale, PA., 287 – 92. Keniry, J., Barber, G.C., Taylor, M.P., and Welch, B.J. (2001). Digital processing of anode current signals: An opportunity for improved cell diagnosis and control. In Proceedings of TMS Light Metals, New Orleans, LA., 1225 – 1232. Rye, K., K¨ onigsson, M., and Solberg, I. (1998). Current redistribution among individual anode carbons in a Hall-H´eroult prebake cell at low alumina concentrations. In Proceedings of TMS Light Metals, San Antonio,TX., 241 – 246. Simon, D. (2006). Optimal state estimation. John Wiley & Sons. Yao, Y., Cheung, C., Bao, J., Skyllas-Kazacos, M., Welch, B., and Akhmetov, S. (2016). Detection of local cell condition based on individual anode current measurements. In Proceedings of TMS Light Metals, Nashville,TN. Yao, Y., Cheung, C.Y., Bao, J., Skyllas-Kazacos, M., Welch, B.J., and Akhmetov, S. (2017). Estimation of spatial alumina concentration in an aluminum reduction cell using a multilevel state observer. AIChE Journal, 63(7), 2806–2818.
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