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The importance of uncertainty covariance tuning for steady-state data reconciliation in mineral and metal processing
16th IFAC Symposium on Automation in Mining, Mineral and Metal Processing August 25-28, 2013. San Diego, California, USA
The importance of uncertainty covariance tuning for steady-state data reconciliation in mineral and metal processing Amir Vasebi*, Daniel Hodouin** and Éric Poulin* [email protected], [email protected], [email protected] LOOP – Process Observation and Optimization Laboratory * Department of Electrical and Computer Engineering ** Department of Mining, Metallurgical and Materials Engineering Université Laval, Québec (Québec), Canada Abstract: Data reconciliation is widely applied in metallurgical plants for steady-state estimation of process variables. Algorithms are mainly based on mass and energy conservation equations that could be mostly extracted from process flow diagrams. An important difficulty is to characterize uncertainties about the plant behavior. Usually, the Gaussian context is assumed and a Maximum-Likelihood estimator is retained. In practice, tuning the weighting matrix, i.e. selecting the variances of modeling and measurement errors, is a challenging task that is sometimes overlooked. The objective of the paper is to illustrate the impact of correctly choosing uncertainty covariance matrices. Effects of matrix structures are investigated to assess the sensitivity of data reconciliation to uncertainty. Analyses are supported by simulation of basic case-studies including a combustion chamber, a hydrocyclone, a flotation circuit and a separation unit. Results show that adjustment of the uncertainty covariance matrix has a significant impact on the precision of estimates. A correct selection is highly valuable for subsequent uses of reconciled data. Keywords: Data reconciliation, material balance, energy balance, measurement errors, model uncertainty, covariance matrices, steady-state systems
et al., 1997; Lachance et al., 2007) for tuning these matrices. They are established based on statistical analysis of plant data, analysis and quantification of the various sources of error in the measurement procedures, and evaluation of the uncertainties of the parameters involved in the empirical or phenomenological models. Since industrial applications of DR techniques most frequently assume a steady-state plant operation, the inherent deviations from this assumption requires a careful analysis of the uncertainties that it generates. The objective of this paper is to present the impact of correctly choosing the modeling and measurement uncertainties variance-covariance matrices on the performance of DR observers. The study focuses on steadystate DR that estimates the underlying SS process state values. The emphasis is put on the importance of correctly selecting the uncertainty covariance matrix structure, since most of the time diagonal matrices are assumed for practical applications, while many situations exist where the true uncertainties are correlated. Four case-studies taken from the mineral and metallurgical processing industries are simulated to illustrate the sensitivity of the DR performance to uncertainty variance tuning: a combustion chamber, a hydrocyclone, a flotation circuit, and a separation unit.
1. INTRODUCTION Performance of process control or optimization strategies vitally depends on the accuracy and availability of process data. In practical cases, measurements are always corrupted by different types of error, and also because of the technical and economic considerations, some important process variables are not measured. These problems can greatly reduce the process performance and profitability, and in the worst case, they can bring the plant to unsafe operating modes. Therefore, to maximize the profitability of the process and keep plants in safe operating regions, measured and unmeasured process variables must be accurately filtered and estimated. Steady-state (SS) data reconciliation (DR) observers (Kuehn & Davidson, 1961), based on mass and energy conservation equations, are widely applied in mineral and metal processing industries where the main applications are related to survey analysis and production accounting (Mirabedini & Hodouin, 1998) or process control and optimization (Vasebi et al., 2012b). Generally, SS data reconciliation observers are based on Maximum-Likelihood (ML) estimators in Gaussian environment, where modeling and measurement uncertainties are modeled by statistical properties. However, the tuning of uncertainty variance-covariance matrices is a challenging problem because these matrices, used in the weighting factors of the DR objective function (Vasebi et al., 2012a), have crucial effects on the observer performance. Various techniques have been proposed (Almasy & Mah, 1984; Chen 978-3-902823-42-7/2013 © IFAC
2. STEADY-STATE DATA RECONCILIATION A steady-state data reconciliation problem can be formulated using 3 sets of equations: a) model or constraint equations, b)
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IFAC MMM 2013 August 25-28, 2013. San Diego, USA
measurement equations, and c) an objective function to be minimized. The process model is written as:
widely applied in mineral processing industries where V de might not be zero (Bazin & Hodouin, 2001).
(1)
f ( x) m
For the purpose of illustration, the present study considers 1level linear and 2-level bilinear steady-state DR problems where the latter one is the most common situation met in plants. In 1-level DR, the state variables are total flowrates F or component flowrates, whereas in 2-level DR, the first level deals with total mass flowrate state variables, and the second level with the chemical and physical properties of flows z such as species mass fractions, densities, particle sizes, enthalpies, specific heats, etc. Therefore the state vector x for 2-level DR is defined as:
where f is the set of DR constraints, x the state vector that represents the SS underlying variables, and m the modeling uncertainties, assumed to obey a Gaussian statistical distribution: (2)
m ~ N ( 0,Vm )
Generally, Eq. 1 contains mass balance equations for the whole material, composition normalization equations, and energy balance equations (involving flowrates, concentrations and temperatures). The measurement equation for steady-state DR is:
F x z
(3)
y g ( x ) g ( xd x ) e
These 2-level state variables lead to linear (conservation of total mass and mass fractions normalization constraints) and bilinear conservation equations: conservation of minerals or chemical species, and conservation of physical properties such as particle size and density, temperature and enthalpy. Considering only total masses brings the problem to linear DR, while using chemical species and physical properties lead to bilinear DR where constraints contain cross products of total flowrates and species/properties concentrations. It is frequently proposed to linearize the 2-level problems by defining component flowrates as a new variable through a change of variable obtained by the cross product of total flowrates and mass fractions or temperatures. The impact of this variable change on the tuning of the uncertainty variances will be discussed in the fourth case-study of this paper.
where y is the vector of measured variables, g the process observation function, and x d the true dynamic state vector at the observation time. Measurement error e is assumed to obey the following statistical properties: (4)
e ~ N (0,Ve )
Eq. 3 separately describes two sources of uncertainty: the measurement error e and the process dynamic variations d g ( x d x ) around the underlying SS to be estimated. The process is locally assumed to operate in a stationary regime, therefore d has the following property: (5)
d ~ N ( 0 ,V d )
where V d
is the covariance-variance of the dynamic
3. FACTORS TO BE CONSIDERED FOR DR WEIGHTING MATRIX TUNING
variations around x . The ML estimator xˆ is then given, through the minimization of an objective function, by: T y g ( x ) y g ( x ) xˆ arg min W x f ( x ) f ( x )
Modeling errors in the constraints of a steady-state DR problem may contain biases. They may be related to forgotten streams in the network of mass and energy conservation (leakages or infiltrations) or to forgotten material transformation reactions, or to biases in the selection of the model parameters such as mass or heat transfer coefficients, equilibrium constants, or heats of reaction (Hodouin, 2010). These biases must be tracked and corrected prior to any uncertainty variance fine tuning, since they do not satisfy the assumptions as formulated in Eq. 6. But there are also unavoidably random modeling errors which are related to the inherent stochastic variations of the processed materials and of the operating plant conditions. These errors are plant specific, and necessarily require a careful analysis before selecting their variance which, most of the time, is simply set to zero in practice. At least, when this assumption is applied, it should be validated by evaluating that modeling uncertainties are small compared to the measurement errors.
(6)
where W is a weighting matrix which in Gaussian context has the optimal value of: W
*
d var m
e
1
(7)
For mineral and metallurgical applications, it is frequently assumed that mass conservation equations are exactly known, and therefore: (8)
m 0
Generally, this assumption could not be valid, when more complex models than basic conservation equations are used, especially for energy balance problems. However, under this assumption, Eq. 6 comes down to: xˆ arg min x
s .t .
where V de
y g ( x ) V
T
T
d
V e V de V de
y g ( x )
(10)
The first source of random uncertainty in the observation equations is related to the fact that no process can strictly operate in a perfectly steady-state regime. When an instantaneous or a time-averaged measurement is undertaken, it necessarily contains an error d due to the dynamic variations of the sampled variable. This contribution to the measurement equation error, named integration error, has
1
(9)
f ( x) 0
is the covariance between process dynamic
fluctuation and measurement errors. Normally, V de is zero for instantaneous sampling. However, composite sampling is 19
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been extensively studied by Gy (1979). When considering simultaneous measurements at different locations in a plant network, these errors are necessarily correlated through the dynamic behavior of the different units (Mirabedini & Hodouin, 1998).
4. SIMULATION RESULTS AND DISCUSSIONS The effects of uncertainty variance matrix tuning on SS data reconciliation performance are illustrated in this section using four simulated case-studies. To evaluate the steady-state DR observer performance, the following index is applied:
The second source of uncertainties is related to the measurement process itself and leads to the e term in Eq. 3. Several phenomena contribute to this error: the flow primary sampling, the sample extraction, the secondary sampling, the sample preparation, the analytical instrument precision which is sensitive to calibration and ambient conditions, the raw measurement processing, the data transmission process, etc. As underlined by Gy (1979), the sources are many and their variances should be added up. The V e covariance matrix is usually diagonal, if all measurement processes are independent (Narasimhan & Jordache, 2000). This is usually the case for total flowrate measurements. However, concentrations could be measured for various species by the same devices (e.g. same sampler, and same X-ray analyser), or physical properties such as particle size distributions can be measured by a single device (for instance a sieving column). These measuring techniques lead to correlation between the errors of different measured variables and come down to nondiagonal V e .
t
trace ( P )
(11)
trace (V y )
where P and V y are the estimation error and measurement covariance matrices (Poulin et al., 2010). This index measures the average improvement of the estimation of the plant underlying SS regime compared to the raw measurements. Smaller t indicates better observer performance. Here, it is assumed that all the states are measured. In the absence of dynamic fluctuations, the first example (a combustion chamber) illustrates the impact of using the covariance terms of the modeling error, while the second one (a hydrocyclone) applies same procedure for the measurement errors. The third example (a flotation plant) illustrates the role of correlation terms induced by the dynamic fluctuations around the SS to be estimated. For a single separation process, the fourth example shows the impact of correctly recalculating the measurement error variance when a change of variable is performed to linearize a 2-level DR procedure. 4.1 Case-study I: Combustion chamber
A study using a simulated flotation circuit has shown that taking into account the correlation between different measurements can improve the precision of a DR estimation process (Mirabedini & Hodouin, 1998). Bazin & Hodouin (2001) have investigated the importance of uncertainty matrix tuning for measurement of particle size using sieving columns. The particle mass fractions in various size classes are correlated due to the physical principles of sieving, and also to the final data processing step used for mass fractions calculation. The paper has shown that these correlations should be applied to data reconciliation problem, and they can improve the SS observer performance (Vasebi et al., 2012a).
In contrast to mass balance equations, energy conservation equations usually exhibit a larger model uncertainty due to model parameter inaccuracies. To illustrate SS data reconciliation with modeling errors in a context of simultaneous mass and energy balances, a gas burner unit is chosen. Combustion chambers are common parts of mineral and metallurgical processes, for instance in drying or conversion processes. Fig. 1 depicts the scheme of such a device. This plant has three input streams (fuel gas, pure oxygen, and humid air with respective mass flowrates Q g , and Q a ) and one output stream of flowrate Q . The fuel gas contains a mixture of butane and propane where the varying mass fraction of butane in Q g is expressed by c b . Qo
Even by knowing the structure of uncertainty matrices, it is still difficult to assign values to the different terms. It is particularly difficult to determine and separate the relative magnitude of modeling and measurement errors. Process modeling using data acquisition campaigns may greatly help to quantify parameter uncertainties. Although the best approach is to try to quantify separately dynamic fluctuations and measurement errors, it must be underlined that the strict separation between them is not compulsory for steady-state DR methods, since both contributions appear in the observation equation. Therefore, statistical analysis of time variations of plant data may give useful insight into the magnitude of total error in measurement equation. Obviously, this is not acceptable if the objective is to estimate the true values of the dynamic states for process control purposes, as in dynamic or stationary DR methods. Also, when the dynamic fluctuations are not stationary on a long term basis, it may be better to separate both contributions to timely adapt the weighting factor tuning ( Keller et al., 1992; Chen et al., 1997; Lachance et al., 2007).
The input air stream includes oxygen, nitrogen, and water vapor, where the varying fraction of water in Q a is expressed by c h . It is also assumed that the relative content of oxygen with respect to nitrogen is constant. In Fig. 1, T g , To , T a and T stand for input gas, pure oxygen, air, and output gas temperatures. In this case, complete burning reaction is assumed and so the oxygen Q o is in excess to the required amount for complete gas burning. Therefore, output stream contains water, oxygen, nitrogen and carbon dioxide.
Qg , Tg
Qo , To
Combustion Chamber
Qa , Ta
Fig. 1. Combustion chamber scheme.
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Q, T
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The SS simulator calculates the output process variables (flowrates, temperatures and gas composition) for a given set of process inputs. Only two input variables c b and c h are able to vary; they are randomly changed around their nominal values according to a Gaussian distribution law. Their standard deviations are 10% and 30% of c b and c h nominal values, respectively. In this example, balance equations are nonlinear (not bilinear) because of the dependence of the gas compositions specific heats and temperatures. Twelve plant process states are then supposed to be measured with Gaussian measurement errors, which are added to the simulated values.
equations. In case 3, the full covariance matrix is used. Simulation results are presented in Table 1. Table 1. Case-study 1: simulation results Index t
and c h are set at their nominal values. With these approximations, the steady-state DR equations become linear and bilinear, and can be formulated as: (12)
Q Q g Qa Qo 0
(13)
c H 2 O c O 2 c N 2 c CO 2 1
(14)
c h Q a m1 Q g c H 2 O Q H 2 O
(15)
m 2 Q g c CO 2 Q CO 2
(16)
m 3Q a Q o m 4 Q g cO 2 Q O 2
(17)
m5 Qa c N 2Q N 2
(18)
Diag. elements (case 2) 1.092
Fully tuned (case 3) 0.700
Case 3 (optimal tuning) provides the best estimation of the state variables, as expected since the modeling uncertainty is adequately weighted in the ML estimator. Using modeling errors with the correct variances without consideration to their inherent correlation structure (case 2) shows that, on an average, DR deteriorates the estimation compared to the raw measurements, while completely neglecting the modeling errors is less detrimental (case 1) and produces better the reconciled values than the raw measurements. This is not a general conclusion, since it is specific to this case-study where the correlation between modeling errors is quite significant. However in complex industrial plants, the number of mass and energy equations is usually much larger than the number of fluctuation sources in process operating regimes, a situation that brings significant correlation in process variables.
The SS reconciliation is then performed on the measured data generated by the simulator. For that purpose, the heat and mass balance equations are written under linear and bilinear expressions using the following approximations: 1) the specific heats of the species are assumed to be independent of temperature by selecting averaged values in the nominal range of temperature variations; 2) the specific heats of the gas mixtures are assumed to be composition independent and tuned for the nominal operating regime; 3) the values of c b
e1 Q g e 2 Q g T g e3 Q g T e 4 Q g To E e 5 Q a T e 6 Q a T a e 7 Q o To e8 Q o T
No modeling error (case 1) 0.830
4.2 Case-study II: Hydrocyclone As a second example, a particle size separator (hydrocyclone HC), where mass fraction measurements are inherently correlated, is selected for illustrating the effect of measurement error correlation. The process has one feed stream and two output streams: the underflow and overflow products. It is assumed that it operates in a given constant SS regime and measurement noise is the only source of uncertainty. The total mass split factor d is assumed to be constant and perfectly known; therefore the DR constraints are linear. The HC behavior is characterized by the separation coefficients of five size classes (fraction of feed class directed to underflow stream). The three particle size distributions are measured by sieving. The objective of this example is to illustrate the role of the measurement noise variance tuning in DR problems. The feed particle size (PSD) and the separation coefficients (SC) are given in Table 2. The value of the split factor to the underflow is d 0 . 4354 .
where e 1 to e 8 and m 1 to m 5 stand for coefficients of energy balance equation (Eq. 12) and mass conservation equations, respectively. Also E , H 2 O , CO 2 , O 2 and N 2 represent modeling error in conservation equations. They are induced by approximations made for writing the DR equations. They are roughly correlated Gaussian variables, since the sources of variations of the simulated data have only two degrees of freedom ( c b and c h ). The measurement error standard deviation is set to 1% in order to make the magnitude of the residuals of the DR equations for the nominal operating states at the same order of the modeling error standard deviations.
Table 2. Case-study 2: feed PSD and SC Tyler Mesh 48 100 200 400 -400
Here, three different SS data reconciliation procedures are compared for illustrating the impact of the structure of the model uncertainty matrix. In case 1, the process modeling error is assumed to have zero values, in such a way that the measurement noise is the only uncertainty source. In case 2, modeling error is considered, but only the diagonal elements of its covariance matrix are used, thus neglecting the correlation between the modeling errors of the various DR
Feed PSD (gr) 6.25 25.66 32.71 24.23 45.70
SC 0.90 0.75 0.60 0.30 0.15
Repetitive measurements of the 15 state variables are simulated using the following procedure: a) extraction of three samples of random mass from the three streams; b) sieving of the three samples, with stochastic variations of material retained on the sieves at constant total mass; c) weighing of the masses retained on the four sieves and the pan with addition of a random error; d) calculation of the three PSDs; 21
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e) data reconciliation with diagonal and full weighting matrix, with and without normalization constraints of the mass fractions. The DR constraints are: (19) f i d u i (1 d ) o i 0 n
i 1
n
fi
u i 1
n
i
o
i
Rougher Feed
(20)
100
where f , u and o are the particle size mass fractions, i is the particle size index, n is the number of particle size classes. The normalization constraint (Eq. 20) will be applied or not in the following DR tests.
Concentration
Fig. 2. Flotation circuit flow sheet. Table 4. Case-study 3: simulation results
Table 3 gives the trace indices for the reconciled PSD. The quality of the reconciled estimates using the full matrix tuning is clearly better. When the full matrix weighting is used, the mass fraction normalization constraint (Eq. 20) is not compulsory since the weighting matrix structure warrants it is obeyed. However, as it is impossible to have the exact weighting matrix in practical cases, it must be always applied (Bazin & Hodouin, 2001).
Eq. 19 Eq. 19 Eqs. 19, 20 Eqs. 19, 20
Diagonal Full Diagonal Full
No process dyn. (case 1) 0.691
Index t
Diag. elements (case 2) 0.680
Fully tuned (case 3) 0.651
4.4 Case-study IV: Linearization by variable change (a single separation unit) In bilinear DR problem where the state variable is given by Eq. 6 and the observation equation function g ( x ) is linear and written as Cx ( C is the observation matrix), it is reasonable to transform the ML criterion minimization problem into a LQ problem which has a direct analytical solution by making the following change of variables:
Table 3. Case-study 2: simulation results Weighting
Reject
Cleaner
i 1
Constraints
Scavenger
PSD t 0.673 0.630 0.647 0.630
F X F
4.3 Case-study III: Flotation circuit
z
(21)
where is Hadamard product operator and for m species, F is defined as follow:
The objective of the third example is to illustrate the role of process dynamic variations on the structure of the error in measurement equation to estimate the underlying steady-state by SS data reconciliation. The flotation plant of Fig. 2 is simulated using an empirical dynamic model described by transfer functions and separation coefficients. The feed stream contains two mineral species, pyrite and chalcopyrite bearing Fe , Cu and gangue. The plant is assumed to operate in stationary conditions where ore feedrate and composition vary around nominal points. The variance of the measurement noise due to sampling and analysis is perfectly known and all flowrates and compositions are measured with a relative precision of 5 % of their nominal values. The plant feed dynamic fluctuations, responsible for an additional error to the observation equations, are generated using transfer functions driven by Gaussian white noises. As a consequence, the ore feedrate and compositions fluctuate with standard deviations equal to 7% and 13% of their nominal values.
T T T F F | F | | F m
T
(22)
Total mass and species conservation equations then become AX , where A is an incidence matrix. In SS data reconciliation context, is equal to zero. Moreover, applying this modification on measurement equation (Eq. 3) leads to (23)
Y CX C ( X d X ) E
where E represents measurement noise for new variable X . Under this change of variable, the analytical solution of Eq. 6 for X is:
Xˆ I W
1
T
A ( AW
1
T
A )
1
A Y
(24)
where I is the identity matrix with proper dimension. The weighting matrix W of the DR must be obtained through the calculation of the total measurement error variance of the new variable X . In Eq. 23, there are two error components: a) process dynamic fluctuations which are correlated because of the bilinear DR nature, and b) the measurement errors. Here, measurement errors are necessarily correlated because the measurement error of F appear in the product Fz and, obviously, in F itself. As a consequence, for calculating the measurement error of the new variable X , one has to calculate the variance of the cross-products Fz , but also the covariance between the species flowrates Fz and the flowrate F , and furthermore the covariance between the component flowrates, particularly on the same stream since they contain
A large number of simulation tests were performed followed by the application of a steady-state DR procedure. Three different DR weighting scenarios were used: a) assuming that process operates in truly SS and there is no process dynamics, the only uncertainty source being the measurement noise of variance V e (case one), b) assuming process dynamic fluctuations, but taking into account only of the diagonal elements of uncertainty covariance matrix (case two), and c) considering the full dynamic uncertainty covariance matrix. As expected, the performance index t has the smallest value for the fully tuned uncertainty matrix (Table 4). 22
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the same variable F . Those covariance terms have been neglected by some authors who used this linearization method (Narasimhan & Jordache, 2000).
REFERENCES Almasy, G.A. & Mah, R.S.H., 1984. Estimation of Measurement Error Variances from Process Data. Industrial & Engineering Chemistry Research, 23, pp.779–784.
For illustrating the importance of these covariance terms, a simple example of a single node separation plant is considered in the absence of dynamic fluctuations to focus only on the measurement error itself. The separation unit has one feed and two outputs (a reject and a concentrate), and only two different species are considered. Four SS data reconciliation methods are simulated to establish a strong comparison. The first one corresponds to the solving of the bilinear DR problem by using nonlinear technique (SQP); the second method is based on a linearization of the DR constraints using a first order Taylor development approximation (Romagnoli & Sanchez, 2000) and the last two ones use the constraints linearization by the proposed variable change with the difference that third method takes advantage of the full error variance matrix of X , while in the fourth technique only diagonal approximation of the error variance matrix is used (Narasimhan & Jordache, 2000). Table 5 presents estimation error trace indices for different DR methods, showing again that the correct use of the variance matrix including the covariance terms exhibits the best performance same as the true optimum of the bilinear DR problem.
Bazin, C. & Hodouin, D., 2001. Importance of covariance in mass balancing of particle size distribution data. Mineral Engineering, 14(8), pp.851–860. Chen, J., Bandoni, A. & Romagnoli, J., 1997. Robust estimation of measurements error variance/covariance from process sampling data. Computers & Chemical Engineering, 21, pp.593–600. Gy, P., 1979. Sampling of Particulate Materials Theory and Practice, Amsterdam: Elsevier. Hodouin, D., 2010. Process Observers and Data Reconciliation Using Mass and Energy Balance Equations. In D. Sbárbaro & R. Villar, eds. Advanced Control and Supervision of Mineral Processing Plants. Springer, pp. 15–83. Keller, J.Y., Zasadzinski, M. & Darouach, M., 1992. Analytical estimator of measurement error variances in data reconciliation. Computers & Chemical Engineering, 16(3), pp.185–188.
Table 5. Case-study 4: simulation results Index
Nonlinear
Const. linearization
Var. change (full)
Var. change (diagonal)
t
0.440
0.451
0.440
0.500
Kuehn, D.R. & Davidson, H., 1961. Computer Control II: Mathematics of Control. Chemical Engineering Progress, 57, pp.44–47. Lachance, L., Poulin, E., Hodouin, D. & Desbiens, A., 2007. Tuning stationary observers: Application to a flotation unit simulator. 12th IFAC Symposium on Automation in Mining, Mineral and Metal Processing. pp. 357–362.
5. CONCLUSION To maximize the precision of mass and energy balances in metallurgical processes, the statistical properties of the modeling and measurement uncertainties must be carefully tuned when applying steady-state DR techniques. The paper has presented the detrimental impact of neglecting the covariance terms of the uncertainties, as it is usual industrial practice. An example for a mass and energy balance of a combustion chamber has shown how taking into account of the correlations between the model parameter errors improves the accuracy of the state variable estimates. A second example involving particle size classification by a hydrocyclone has illustrated how important it is to take into account the correlations of the measurement errors produced by sieving columns for analyzing particle size distributions. A third example for a flotation plant has revealed that considering measurement errors variance and covariance induced by dynamic fluctuations improves the accuracy of the steadystate mass balance of the plant. The paper has shown that linearization of mass balance equations by generating pseudomeasurements of components flowrates is efficient, but it is important to properly calculate the variances of the pseudomeasurements without forgetting the correlations induced between the errors of the new variables. As a general conclusion, it must be emphasized that the usual practice of tuning the raw data variance only is detrimental for the accuracy of plant material and energy balances, and so a careful investigation of the modeling and measurement errors correlation is strongly recommended.
Mirabedini, A. & Hodouin, D., 1998. Calculation of variance and covariance of sampling errors in complex mineral processing systems, using state–space dynamic models. International Journal of Mineral Processing, 55(1), pp.1–20. Narasimhan, S. & Jordache, C., 2000. Data reconciliation & gross error detection: an intelligent use of process data, Gulf Publisher. Poulin, E., Hodouin, D. & Lachance, L., 2010. Impact of plant dynamics on the performance of steady-state data reconciliation. Computers & Chemical Engineering, 34(3), pp.354–360. Romagnoli, J. & Sanchez, M., 2000. Data Processing and Reconciliation for Chemical Process Operations, Academic Press. Vasebi, A., Poulin, É. & Hodouin, D., 2012a. Dynamic data reconciliation based on node imbalance autocovariance functions. Computers & Chemical Engineering, 43, pp.81–90. Vasebi, A., Poulin, É. & Hodouin, D., 2012b. Dynamic data reconciliation in mineral and metallurgical plants. Annual Reviews in Control, 36(2), pp.235–243.
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The importance of uncertainty covariance tuning for steady-state data reconciliation in mineral and metal processing Amir Vasebi*, Daniel Hodouin** and Éric Poulin* [email protected], [email protected], [email protected] LOOP – Process Observation and Optimization Laboratory * Department of Electrical and Computer Engineering ** Department of Mining, Metallurgical and Materials Engineering Université Laval, Québec (Québec), Canada Abstract: Data reconciliation is widely applied in metallurgical plants for steady-state estimation of process variables. Algorithms are mainly based on mass and energy conservation equations that could be mostly extracted from process flow diagrams. An important difficulty is to characterize uncertainties about the plant behavior. Usually, the Gaussian context is assumed and a Maximum-Likelihood estimator is retained. In practice, tuning the weighting matrix, i.e. selecting the variances of modeling and measurement errors, is a challenging task that is sometimes overlooked. The objective of the paper is to illustrate the impact of correctly choosing uncertainty covariance matrices. Effects of matrix structures are investigated to assess the sensitivity of data reconciliation to uncertainty. Analyses are supported by simulation of basic case-studies including a combustion chamber, a hydrocyclone, a flotation circuit and a separation unit. Results show that adjustment of the uncertainty covariance matrix has a significant impact on the precision of estimates. A correct selection is highly valuable for subsequent uses of reconciled data. Keywords: Data reconciliation, material balance, energy balance, measurement errors, model uncertainty, covariance matrices, steady-state systems
et al., 1997; Lachance et al., 2007) for tuning these matrices. They are established based on statistical analysis of plant data, analysis and quantification of the various sources of error in the measurement procedures, and evaluation of the uncertainties of the parameters involved in the empirical or phenomenological models. Since industrial applications of DR techniques most frequently assume a steady-state plant operation, the inherent deviations from this assumption requires a careful analysis of the uncertainties that it generates. The objective of this paper is to present the impact of correctly choosing the modeling and measurement uncertainties variance-covariance matrices on the performance of DR observers. The study focuses on steadystate DR that estimates the underlying SS process state values. The emphasis is put on the importance of correctly selecting the uncertainty covariance matrix structure, since most of the time diagonal matrices are assumed for practical applications, while many situations exist where the true uncertainties are correlated. Four case-studies taken from the mineral and metallurgical processing industries are simulated to illustrate the sensitivity of the DR performance to uncertainty variance tuning: a combustion chamber, a hydrocyclone, a flotation circuit, and a separation unit.
1. INTRODUCTION Performance of process control or optimization strategies vitally depends on the accuracy and availability of process data. In practical cases, measurements are always corrupted by different types of error, and also because of the technical and economic considerations, some important process variables are not measured. These problems can greatly reduce the process performance and profitability, and in the worst case, they can bring the plant to unsafe operating modes. Therefore, to maximize the profitability of the process and keep plants in safe operating regions, measured and unmeasured process variables must be accurately filtered and estimated. Steady-state (SS) data reconciliation (DR) observers (Kuehn & Davidson, 1961), based on mass and energy conservation equations, are widely applied in mineral and metal processing industries where the main applications are related to survey analysis and production accounting (Mirabedini & Hodouin, 1998) or process control and optimization (Vasebi et al., 2012b). Generally, SS data reconciliation observers are based on Maximum-Likelihood (ML) estimators in Gaussian environment, where modeling and measurement uncertainties are modeled by statistical properties. However, the tuning of uncertainty variance-covariance matrices is a challenging problem because these matrices, used in the weighting factors of the DR objective function (Vasebi et al., 2012a), have crucial effects on the observer performance. Various techniques have been proposed (Almasy & Mah, 1984; Chen 978-3-902823-42-7/2013 © IFAC
2. STEADY-STATE DATA RECONCILIATION A steady-state data reconciliation problem can be formulated using 3 sets of equations: a) model or constraint equations, b)
18
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measurement equations, and c) an objective function to be minimized. The process model is written as:
widely applied in mineral processing industries where V de might not be zero (Bazin & Hodouin, 2001).
(1)
f ( x) m
For the purpose of illustration, the present study considers 1level linear and 2-level bilinear steady-state DR problems where the latter one is the most common situation met in plants. In 1-level DR, the state variables are total flowrates F or component flowrates, whereas in 2-level DR, the first level deals with total mass flowrate state variables, and the second level with the chemical and physical properties of flows z such as species mass fractions, densities, particle sizes, enthalpies, specific heats, etc. Therefore the state vector x for 2-level DR is defined as:
where f is the set of DR constraints, x the state vector that represents the SS underlying variables, and m the modeling uncertainties, assumed to obey a Gaussian statistical distribution: (2)
m ~ N ( 0,Vm )
Generally, Eq. 1 contains mass balance equations for the whole material, composition normalization equations, and energy balance equations (involving flowrates, concentrations and temperatures). The measurement equation for steady-state DR is:
F x z
(3)
y g ( x ) g ( xd x ) e
These 2-level state variables lead to linear (conservation of total mass and mass fractions normalization constraints) and bilinear conservation equations: conservation of minerals or chemical species, and conservation of physical properties such as particle size and density, temperature and enthalpy. Considering only total masses brings the problem to linear DR, while using chemical species and physical properties lead to bilinear DR where constraints contain cross products of total flowrates and species/properties concentrations. It is frequently proposed to linearize the 2-level problems by defining component flowrates as a new variable through a change of variable obtained by the cross product of total flowrates and mass fractions or temperatures. The impact of this variable change on the tuning of the uncertainty variances will be discussed in the fourth case-study of this paper.
where y is the vector of measured variables, g the process observation function, and x d the true dynamic state vector at the observation time. Measurement error e is assumed to obey the following statistical properties: (4)
e ~ N (0,Ve )
Eq. 3 separately describes two sources of uncertainty: the measurement error e and the process dynamic variations d g ( x d x ) around the underlying SS to be estimated. The process is locally assumed to operate in a stationary regime, therefore d has the following property: (5)
d ~ N ( 0 ,V d )
where V d
is the covariance-variance of the dynamic
3. FACTORS TO BE CONSIDERED FOR DR WEIGHTING MATRIX TUNING
variations around x . The ML estimator xˆ is then given, through the minimization of an objective function, by: T y g ( x ) y g ( x ) xˆ arg min W x f ( x ) f ( x )
Modeling errors in the constraints of a steady-state DR problem may contain biases. They may be related to forgotten streams in the network of mass and energy conservation (leakages or infiltrations) or to forgotten material transformation reactions, or to biases in the selection of the model parameters such as mass or heat transfer coefficients, equilibrium constants, or heats of reaction (Hodouin, 2010). These biases must be tracked and corrected prior to any uncertainty variance fine tuning, since they do not satisfy the assumptions as formulated in Eq. 6. But there are also unavoidably random modeling errors which are related to the inherent stochastic variations of the processed materials and of the operating plant conditions. These errors are plant specific, and necessarily require a careful analysis before selecting their variance which, most of the time, is simply set to zero in practice. At least, when this assumption is applied, it should be validated by evaluating that modeling uncertainties are small compared to the measurement errors.
(6)
where W is a weighting matrix which in Gaussian context has the optimal value of: W
*
d var m
e
1
(7)
For mineral and metallurgical applications, it is frequently assumed that mass conservation equations are exactly known, and therefore: (8)
m 0
Generally, this assumption could not be valid, when more complex models than basic conservation equations are used, especially for energy balance problems. However, under this assumption, Eq. 6 comes down to: xˆ arg min x
s .t .
where V de
y g ( x ) V
T
T
d
V e V de V de
y g ( x )
(10)
The first source of random uncertainty in the observation equations is related to the fact that no process can strictly operate in a perfectly steady-state regime. When an instantaneous or a time-averaged measurement is undertaken, it necessarily contains an error d due to the dynamic variations of the sampled variable. This contribution to the measurement equation error, named integration error, has
1
(9)
f ( x) 0
is the covariance between process dynamic
fluctuation and measurement errors. Normally, V de is zero for instantaneous sampling. However, composite sampling is 19
IFAC MMM 2013 August 25-28, 2013. San Diego, USA
been extensively studied by Gy (1979). When considering simultaneous measurements at different locations in a plant network, these errors are necessarily correlated through the dynamic behavior of the different units (Mirabedini & Hodouin, 1998).
4. SIMULATION RESULTS AND DISCUSSIONS The effects of uncertainty variance matrix tuning on SS data reconciliation performance are illustrated in this section using four simulated case-studies. To evaluate the steady-state DR observer performance, the following index is applied:
The second source of uncertainties is related to the measurement process itself and leads to the e term in Eq. 3. Several phenomena contribute to this error: the flow primary sampling, the sample extraction, the secondary sampling, the sample preparation, the analytical instrument precision which is sensitive to calibration and ambient conditions, the raw measurement processing, the data transmission process, etc. As underlined by Gy (1979), the sources are many and their variances should be added up. The V e covariance matrix is usually diagonal, if all measurement processes are independent (Narasimhan & Jordache, 2000). This is usually the case for total flowrate measurements. However, concentrations could be measured for various species by the same devices (e.g. same sampler, and same X-ray analyser), or physical properties such as particle size distributions can be measured by a single device (for instance a sieving column). These measuring techniques lead to correlation between the errors of different measured variables and come down to nondiagonal V e .
t
trace ( P )
(11)
trace (V y )
where P and V y are the estimation error and measurement covariance matrices (Poulin et al., 2010). This index measures the average improvement of the estimation of the plant underlying SS regime compared to the raw measurements. Smaller t indicates better observer performance. Here, it is assumed that all the states are measured. In the absence of dynamic fluctuations, the first example (a combustion chamber) illustrates the impact of using the covariance terms of the modeling error, while the second one (a hydrocyclone) applies same procedure for the measurement errors. The third example (a flotation plant) illustrates the role of correlation terms induced by the dynamic fluctuations around the SS to be estimated. For a single separation process, the fourth example shows the impact of correctly recalculating the measurement error variance when a change of variable is performed to linearize a 2-level DR procedure. 4.1 Case-study I: Combustion chamber
A study using a simulated flotation circuit has shown that taking into account the correlation between different measurements can improve the precision of a DR estimation process (Mirabedini & Hodouin, 1998). Bazin & Hodouin (2001) have investigated the importance of uncertainty matrix tuning for measurement of particle size using sieving columns. The particle mass fractions in various size classes are correlated due to the physical principles of sieving, and also to the final data processing step used for mass fractions calculation. The paper has shown that these correlations should be applied to data reconciliation problem, and they can improve the SS observer performance (Vasebi et al., 2012a).
In contrast to mass balance equations, energy conservation equations usually exhibit a larger model uncertainty due to model parameter inaccuracies. To illustrate SS data reconciliation with modeling errors in a context of simultaneous mass and energy balances, a gas burner unit is chosen. Combustion chambers are common parts of mineral and metallurgical processes, for instance in drying or conversion processes. Fig. 1 depicts the scheme of such a device. This plant has three input streams (fuel gas, pure oxygen, and humid air with respective mass flowrates Q g , and Q a ) and one output stream of flowrate Q . The fuel gas contains a mixture of butane and propane where the varying mass fraction of butane in Q g is expressed by c b . Qo
Even by knowing the structure of uncertainty matrices, it is still difficult to assign values to the different terms. It is particularly difficult to determine and separate the relative magnitude of modeling and measurement errors. Process modeling using data acquisition campaigns may greatly help to quantify parameter uncertainties. Although the best approach is to try to quantify separately dynamic fluctuations and measurement errors, it must be underlined that the strict separation between them is not compulsory for steady-state DR methods, since both contributions appear in the observation equation. Therefore, statistical analysis of time variations of plant data may give useful insight into the magnitude of total error in measurement equation. Obviously, this is not acceptable if the objective is to estimate the true values of the dynamic states for process control purposes, as in dynamic or stationary DR methods. Also, when the dynamic fluctuations are not stationary on a long term basis, it may be better to separate both contributions to timely adapt the weighting factor tuning ( Keller et al., 1992; Chen et al., 1997; Lachance et al., 2007).
The input air stream includes oxygen, nitrogen, and water vapor, where the varying fraction of water in Q a is expressed by c h . It is also assumed that the relative content of oxygen with respect to nitrogen is constant. In Fig. 1, T g , To , T a and T stand for input gas, pure oxygen, air, and output gas temperatures. In this case, complete burning reaction is assumed and so the oxygen Q o is in excess to the required amount for complete gas burning. Therefore, output stream contains water, oxygen, nitrogen and carbon dioxide.
Qg , Tg
Qo , To
Combustion Chamber
Qa , Ta
Fig. 1. Combustion chamber scheme.
20
Q, T
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The SS simulator calculates the output process variables (flowrates, temperatures and gas composition) for a given set of process inputs. Only two input variables c b and c h are able to vary; they are randomly changed around their nominal values according to a Gaussian distribution law. Their standard deviations are 10% and 30% of c b and c h nominal values, respectively. In this example, balance equations are nonlinear (not bilinear) because of the dependence of the gas compositions specific heats and temperatures. Twelve plant process states are then supposed to be measured with Gaussian measurement errors, which are added to the simulated values.
equations. In case 3, the full covariance matrix is used. Simulation results are presented in Table 1. Table 1. Case-study 1: simulation results Index t
and c h are set at their nominal values. With these approximations, the steady-state DR equations become linear and bilinear, and can be formulated as: (12)
Q Q g Qa Qo 0
(13)
c H 2 O c O 2 c N 2 c CO 2 1
(14)
c h Q a m1 Q g c H 2 O Q H 2 O
(15)
m 2 Q g c CO 2 Q CO 2
(16)
m 3Q a Q o m 4 Q g cO 2 Q O 2
(17)
m5 Qa c N 2Q N 2
(18)
Diag. elements (case 2) 1.092
Fully tuned (case 3) 0.700
Case 3 (optimal tuning) provides the best estimation of the state variables, as expected since the modeling uncertainty is adequately weighted in the ML estimator. Using modeling errors with the correct variances without consideration to their inherent correlation structure (case 2) shows that, on an average, DR deteriorates the estimation compared to the raw measurements, while completely neglecting the modeling errors is less detrimental (case 1) and produces better the reconciled values than the raw measurements. This is not a general conclusion, since it is specific to this case-study where the correlation between modeling errors is quite significant. However in complex industrial plants, the number of mass and energy equations is usually much larger than the number of fluctuation sources in process operating regimes, a situation that brings significant correlation in process variables.
The SS reconciliation is then performed on the measured data generated by the simulator. For that purpose, the heat and mass balance equations are written under linear and bilinear expressions using the following approximations: 1) the specific heats of the species are assumed to be independent of temperature by selecting averaged values in the nominal range of temperature variations; 2) the specific heats of the gas mixtures are assumed to be composition independent and tuned for the nominal operating regime; 3) the values of c b
e1 Q g e 2 Q g T g e3 Q g T e 4 Q g To E e 5 Q a T e 6 Q a T a e 7 Q o To e8 Q o T
No modeling error (case 1) 0.830
4.2 Case-study II: Hydrocyclone As a second example, a particle size separator (hydrocyclone HC), where mass fraction measurements are inherently correlated, is selected for illustrating the effect of measurement error correlation. The process has one feed stream and two output streams: the underflow and overflow products. It is assumed that it operates in a given constant SS regime and measurement noise is the only source of uncertainty. The total mass split factor d is assumed to be constant and perfectly known; therefore the DR constraints are linear. The HC behavior is characterized by the separation coefficients of five size classes (fraction of feed class directed to underflow stream). The three particle size distributions are measured by sieving. The objective of this example is to illustrate the role of the measurement noise variance tuning in DR problems. The feed particle size (PSD) and the separation coefficients (SC) are given in Table 2. The value of the split factor to the underflow is d 0 . 4354 .
where e 1 to e 8 and m 1 to m 5 stand for coefficients of energy balance equation (Eq. 12) and mass conservation equations, respectively. Also E , H 2 O , CO 2 , O 2 and N 2 represent modeling error in conservation equations. They are induced by approximations made for writing the DR equations. They are roughly correlated Gaussian variables, since the sources of variations of the simulated data have only two degrees of freedom ( c b and c h ). The measurement error standard deviation is set to 1% in order to make the magnitude of the residuals of the DR equations for the nominal operating states at the same order of the modeling error standard deviations.
Table 2. Case-study 2: feed PSD and SC Tyler Mesh 48 100 200 400 -400
Here, three different SS data reconciliation procedures are compared for illustrating the impact of the structure of the model uncertainty matrix. In case 1, the process modeling error is assumed to have zero values, in such a way that the measurement noise is the only uncertainty source. In case 2, modeling error is considered, but only the diagonal elements of its covariance matrix are used, thus neglecting the correlation between the modeling errors of the various DR
Feed PSD (gr) 6.25 25.66 32.71 24.23 45.70
SC 0.90 0.75 0.60 0.30 0.15
Repetitive measurements of the 15 state variables are simulated using the following procedure: a) extraction of three samples of random mass from the three streams; b) sieving of the three samples, with stochastic variations of material retained on the sieves at constant total mass; c) weighing of the masses retained on the four sieves and the pan with addition of a random error; d) calculation of the three PSDs; 21
IFAC MMM 2013 August 25-28, 2013. San Diego, USA
e) data reconciliation with diagonal and full weighting matrix, with and without normalization constraints of the mass fractions. The DR constraints are: (19) f i d u i (1 d ) o i 0 n
i 1
n
fi
u i 1
n
i
o
i
Rougher Feed
(20)
100
where f , u and o are the particle size mass fractions, i is the particle size index, n is the number of particle size classes. The normalization constraint (Eq. 20) will be applied or not in the following DR tests.
Concentration
Fig. 2. Flotation circuit flow sheet. Table 4. Case-study 3: simulation results
Table 3 gives the trace indices for the reconciled PSD. The quality of the reconciled estimates using the full matrix tuning is clearly better. When the full matrix weighting is used, the mass fraction normalization constraint (Eq. 20) is not compulsory since the weighting matrix structure warrants it is obeyed. However, as it is impossible to have the exact weighting matrix in practical cases, it must be always applied (Bazin & Hodouin, 2001).
Eq. 19 Eq. 19 Eqs. 19, 20 Eqs. 19, 20
Diagonal Full Diagonal Full
No process dyn. (case 1) 0.691
Index t
Diag. elements (case 2) 0.680
Fully tuned (case 3) 0.651
4.4 Case-study IV: Linearization by variable change (a single separation unit) In bilinear DR problem where the state variable is given by Eq. 6 and the observation equation function g ( x ) is linear and written as Cx ( C is the observation matrix), it is reasonable to transform the ML criterion minimization problem into a LQ problem which has a direct analytical solution by making the following change of variables:
Table 3. Case-study 2: simulation results Weighting
Reject
Cleaner
i 1
Constraints
Scavenger
PSD t 0.673 0.630 0.647 0.630
F X F
4.3 Case-study III: Flotation circuit
z
(21)
where is Hadamard product operator and for m species, F is defined as follow:
The objective of the third example is to illustrate the role of process dynamic variations on the structure of the error in measurement equation to estimate the underlying steady-state by SS data reconciliation. The flotation plant of Fig. 2 is simulated using an empirical dynamic model described by transfer functions and separation coefficients. The feed stream contains two mineral species, pyrite and chalcopyrite bearing Fe , Cu and gangue. The plant is assumed to operate in stationary conditions where ore feedrate and composition vary around nominal points. The variance of the measurement noise due to sampling and analysis is perfectly known and all flowrates and compositions are measured with a relative precision of 5 % of their nominal values. The plant feed dynamic fluctuations, responsible for an additional error to the observation equations, are generated using transfer functions driven by Gaussian white noises. As a consequence, the ore feedrate and compositions fluctuate with standard deviations equal to 7% and 13% of their nominal values.
T T T F F | F | | F m
T
(22)
Total mass and species conservation equations then become AX , where A is an incidence matrix. In SS data reconciliation context, is equal to zero. Moreover, applying this modification on measurement equation (Eq. 3) leads to (23)
Y CX C ( X d X ) E
where E represents measurement noise for new variable X . Under this change of variable, the analytical solution of Eq. 6 for X is:
Xˆ I W
1
T
A ( AW
1
T
A )
1
A Y
(24)
where I is the identity matrix with proper dimension. The weighting matrix W of the DR must be obtained through the calculation of the total measurement error variance of the new variable X . In Eq. 23, there are two error components: a) process dynamic fluctuations which are correlated because of the bilinear DR nature, and b) the measurement errors. Here, measurement errors are necessarily correlated because the measurement error of F appear in the product Fz and, obviously, in F itself. As a consequence, for calculating the measurement error of the new variable X , one has to calculate the variance of the cross-products Fz , but also the covariance between the species flowrates Fz and the flowrate F , and furthermore the covariance between the component flowrates, particularly on the same stream since they contain
A large number of simulation tests were performed followed by the application of a steady-state DR procedure. Three different DR weighting scenarios were used: a) assuming that process operates in truly SS and there is no process dynamics, the only uncertainty source being the measurement noise of variance V e (case one), b) assuming process dynamic fluctuations, but taking into account only of the diagonal elements of uncertainty covariance matrix (case two), and c) considering the full dynamic uncertainty covariance matrix. As expected, the performance index t has the smallest value for the fully tuned uncertainty matrix (Table 4). 22
IFAC MMM 2013 August 25-28, 2013. San Diego, USA
the same variable F . Those covariance terms have been neglected by some authors who used this linearization method (Narasimhan & Jordache, 2000).
REFERENCES Almasy, G.A. & Mah, R.S.H., 1984. Estimation of Measurement Error Variances from Process Data. Industrial & Engineering Chemistry Research, 23, pp.779–784.
For illustrating the importance of these covariance terms, a simple example of a single node separation plant is considered in the absence of dynamic fluctuations to focus only on the measurement error itself. The separation unit has one feed and two outputs (a reject and a concentrate), and only two different species are considered. Four SS data reconciliation methods are simulated to establish a strong comparison. The first one corresponds to the solving of the bilinear DR problem by using nonlinear technique (SQP); the second method is based on a linearization of the DR constraints using a first order Taylor development approximation (Romagnoli & Sanchez, 2000) and the last two ones use the constraints linearization by the proposed variable change with the difference that third method takes advantage of the full error variance matrix of X , while in the fourth technique only diagonal approximation of the error variance matrix is used (Narasimhan & Jordache, 2000). Table 5 presents estimation error trace indices for different DR methods, showing again that the correct use of the variance matrix including the covariance terms exhibits the best performance same as the true optimum of the bilinear DR problem.
Bazin, C. & Hodouin, D., 2001. Importance of covariance in mass balancing of particle size distribution data. Mineral Engineering, 14(8), pp.851–860. Chen, J., Bandoni, A. & Romagnoli, J., 1997. Robust estimation of measurements error variance/covariance from process sampling data. Computers & Chemical Engineering, 21, pp.593–600. Gy, P., 1979. Sampling of Particulate Materials Theory and Practice, Amsterdam: Elsevier. Hodouin, D., 2010. Process Observers and Data Reconciliation Using Mass and Energy Balance Equations. In D. Sbárbaro & R. Villar, eds. Advanced Control and Supervision of Mineral Processing Plants. Springer, pp. 15–83. Keller, J.Y., Zasadzinski, M. & Darouach, M., 1992. Analytical estimator of measurement error variances in data reconciliation. Computers & Chemical Engineering, 16(3), pp.185–188.
Table 5. Case-study 4: simulation results Index
Nonlinear
Const. linearization
Var. change (full)
Var. change (diagonal)
t
0.440
0.451
0.440
0.500
Kuehn, D.R. & Davidson, H., 1961. Computer Control II: Mathematics of Control. Chemical Engineering Progress, 57, pp.44–47. Lachance, L., Poulin, E., Hodouin, D. & Desbiens, A., 2007. Tuning stationary observers: Application to a flotation unit simulator. 12th IFAC Symposium on Automation in Mining, Mineral and Metal Processing. pp. 357–362.
5. CONCLUSION To maximize the precision of mass and energy balances in metallurgical processes, the statistical properties of the modeling and measurement uncertainties must be carefully tuned when applying steady-state DR techniques. The paper has presented the detrimental impact of neglecting the covariance terms of the uncertainties, as it is usual industrial practice. An example for a mass and energy balance of a combustion chamber has shown how taking into account of the correlations between the model parameter errors improves the accuracy of the state variable estimates. A second example involving particle size classification by a hydrocyclone has illustrated how important it is to take into account the correlations of the measurement errors produced by sieving columns for analyzing particle size distributions. A third example for a flotation plant has revealed that considering measurement errors variance and covariance induced by dynamic fluctuations improves the accuracy of the steadystate mass balance of the plant. The paper has shown that linearization of mass balance equations by generating pseudomeasurements of components flowrates is efficient, but it is important to properly calculate the variances of the pseudomeasurements without forgetting the correlations induced between the errors of the new variables. As a general conclusion, it must be emphasized that the usual practice of tuning the raw data variance only is detrimental for the accuracy of plant material and energy balances, and so a careful investigation of the modeling and measurement errors correlation is strongly recommended.
Mirabedini, A. & Hodouin, D., 1998. Calculation of variance and covariance of sampling errors in complex mineral processing systems, using state–space dynamic models. International Journal of Mineral Processing, 55(1), pp.1–20. Narasimhan, S. & Jordache, C., 2000. Data reconciliation & gross error detection: an intelligent use of process data, Gulf Publisher. Poulin, E., Hodouin, D. & Lachance, L., 2010. Impact of plant dynamics on the performance of steady-state data reconciliation. Computers & Chemical Engineering, 34(3), pp.354–360. Romagnoli, J. & Sanchez, M., 2000. Data Processing and Reconciliation for Chemical Process Operations, Academic Press. Vasebi, A., Poulin, É. & Hodouin, D., 2012a. Dynamic data reconciliation based on node imbalance autocovariance functions. Computers & Chemical Engineering, 43, pp.81–90. Vasebi, A., Poulin, É. & Hodouin, D., 2012b. Dynamic data reconciliation in mineral and metallurgical plants. Annual Reviews in Control, 36(2), pp.235–243.
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