Dynamic Modeling of Chemical Processes

1
Dynamic Modeling of Chemical Processes
Statistical Challenges and Approaches

• Kim B. McAuley
• Department of Chemical Engineering
• Queens University
• Kingston, Canada

2
Fundamental models of chemical processes are
valuable. After the models are derived,
parameter estimation is the main challenge.
3
Fundamental models of chemical processes are
valuable. After the models are derived,
parameter estimation is the main challenge.

• How do chemical engineers derive differential
  equations?
• Why is parameter estimation so difficult?
• Statistical approaches to aid parameter
  estimation
• - Accounting for model imperfections and
  stochastic disturbances
• Selecting parameters to estimate using limited
  data

4
Fundamental Dynamic Models of Chemical Processes

• Model Equations
• Material balances on chemical species, energy
  balances
• Differential equations
• Algebraic equations that describe
• Rates of chemical reactions
• Movement of chemical species from one phase to
  another
• Conversion from one type of energy to another
• Heat liberated by chemical reactions
• Energy required to vaporize liquids and melt
  solids
• Energy required for mixing and pumping

5
Fundamental Dynamic Models of Chemical Processes

• Model Equations
• Material balances on chemical species, energy
  balances
• Differential equations
• Algebraic equations that describe
• Rates of chemical reactions
• Movement of chemical species from one phase to
  another
• Conversion from one type of energy to another
• Heat liberated by chemical reactions
• Energy required to vaporize liquids and melt
  solids
• Energy required for mixing and pumping

Chemical engineers spend years learning to
develop mathematical expressions to describe
physical and chemical phenomena. We derive sets
of nonlinear differential equations with many
unknown parameters
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Fundamental Dynamic Models of Chemical Processes

• Model Equations
• Material balances on chemical species, energy
  balances
• Equations that describe
• Rates of chemical reactions
• Movement of chemical species from one phase to
  another
• Conversion from one type of energy to another
• Example - Ethylene/hexene copolymerization model
  (BP Chemicals)
• 22 ordinary differential equations
• 45 parameters (mostly kinetic rate constants and
  activation energies)
• Model predicts gas composition, polymer
  production rate, copolymer composition, average
  molecular weight
• Model for scaling up from laboratory-scale to
  commercial reactors

7
Fundamental Dynamic Models of Chemical Processes

• Two Uses of Fundamental Models
• Developing understanding of new chemical
  processes
• Are our beliefs consistent with what really
  happens?
• Testing mythology and assumptions
• Gaining knowledge that we can use to guide
  experimentation and innovation.
• Example Nitroxide-mediated mini-emulsion
  polymerization (Xerox)
• We are happiest when model predictions match
  experiments.
• We learn more when they dont.

8
Fundamental Dynamic Models of Chemical Processes

• Two Uses of Fundamental Models
• Predicting behaviour of existing industrial
  processes
• How can we
• make different products using the same equipment?
• improve production rates and product quality?
• reduce off-specification product during start-up
  and grade changes?
• Examples Models of industrial reactors for
  nylon 66 and polyethylene.
• For these applications we need accurate model
  predictions.
• Main challenge Getting model parameters.

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Steps in Fundamental Model Development

• Derive equations based on knowledge and
  assumptions.
• Solve equations numerically using initial guesses
  for the parameters.
• Get values of some parameters from the
  literature.
• Use data from existing or new experiments to
  estimate remaining model parameters.
• Test model fit.
• Design and conduct more experiments?
• Revise model structure based on new knowledge?
• Validate model using additional experimental
  data.
• Use the model.
• Why is step 4 so difficult?

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Why is the Parameter Estimation Step Difficult?

• Models are nonlinear in the parameters
• multiple optima, initial parameter guesses
• numerical solution of differential equations
  required at each iteration
• Experiments not well-designed. Additional
  experiments expensive.
• How to design experiments for building these
  dynamic models?
• Traditional least-squares assumptions not valid
• Imperfect model structure
• Random errors not independent
• Heteroskedastic responses
• Some parameters have little influence on model
  predictions.
• Effects of some parameters are highly correlated
  with effects of others
• Multiple sets of parameters give nearly the same
  predictions
• Poorly conditioned estimation problem

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Invalid Least-Squares Assumptions
Four Replicate Experiments
Polymerization Rate

• Strong evidence of heteroskedasticity - Use
  transformed responses
• Model will give auto-correlated residuals.
  - We should account for lack of independence
  when fitting parameters and judging models

12
Why is the Parameter Estimation Step Difficult?

• Models are nonlinear in the parameters
• multiple optima, initial parameter guesses
• numerical solution of differential equations
  required at each iteration
• Experiments not well-designed. Additional
  experiments expensive.
• How to design experiments for building these
  dynamic models?
• Traditional least-squares assumptions not valid
• Imperfect model structure
• Random errors not independent
• Heteroskedastic responses
• Some parameters have little influence on model
  predictions.
• Effects of some parameters are highly correlated
  with effects of others
• Multiple sets of parameters give nearly the same
  predictions
• Poorly conditioned estimation problem

13
Parameter Estimation for Stochastic Dynamic
Models

• Saeed Varziri, K.B. McAuley and P.J. McLellan

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Traditional Parameter Estimation in Differential
Equations

• Estimate the model parameters ?, given noisy
  observations y and known system inputs u.

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Traditional Parameter Estimation in Differential
Equations

• Estimate the model parameters ?, given noisy
  observations y and known system inputs u.
• But, models are simplifications of reality
• f has minor structural imperfections
• Less-important inputs may be left out (or
  unknown)
• Current deviations in f influence future values
  of x and y
• residuals will be correlated

16
Parameter Estimation in Stochastic Differential
Equations
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Parameter Estimation in Stochastic Differential
Equations

• Two noise sources
• Measurement noise
• Stochastic process disturbances
• Uncertainties in u
• Unknown or unmeasured inputs
• Minor structural imperfections
• We can also incorporate nonstationary
  disturbances

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Parameter Estimation in Stochastic Differential
Equations

• Our approach
• Assume that the solution to the differential
  equations can be represented using B-splines or
  other basis functions

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Parameter Estimation in Stochastic Differential
Equations

• Our approach
• Assume that the solution to the differential
  equations can be represented using B-splines or
  other basis functions
• is used to convert our ODEs into
  algebraic equations

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Approximate Maximum Likelihood Estimation

• Assume the solution of the dynamic system can be
  well approximated by B-splines with unknown
  coefficients ?
• We want to estimate the fundamental model
  parameters ? and we need to estimate the unknown
  spline coefficients ?
• Select and to minimize

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Approximate Maximum-Likelihood Estimation

• Assume the solution of the dynamic system can be
  well approximated by B-splines with unknown
  coefficients ?
• We want to estimate the fundamental model
  parameters ? and we need to estimate the unknown
  spline coefficients ?
• Select and to minimize

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What Weighting to Use?

• Heuristically
• A large ? is appropriate when
• Model is accurate and data are noisy
• A small ? is appropriate when
• Data are good and model is inaccurate

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What Weighting to Use?

• Heuristically
• A large ? is appropriate when
• Model is accurate and data are noisy
• A small ? is appropriate when
• Data are good and model is inaccurate

Very large ? corresponds to traditional
least-squares parameter estimation, which assumes
a perfect model and no disturbances
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Objective Function for a Multivariate Example
with Known Variances
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Recent Progress

• Solving optimization problem using IP-OPT
• Confidence intervals for parameters
• Parameter estimation when ?m2 or Q is unknown
• Whats Next?
• Testing on larger-scale chemical engineering
  problems

26
Features of Proposed Estimation Method

• Readily handles systems with
• Unknown or uncertain initial conditions
• Irregular sampling
• Unmeasured states
• Meandering (nonstationary) disturbances
• No need for repeated numerical solution of ODEs
• Collocation method
• ODEs are satisfied (or not) using soft
  constraints in the objective function

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Why is the Parameter Estimation Difficult?

• Models are nonlinear in the parameters
• multiple optima, initial parameter guesses
• numerical solution of differential equations
  required at each iteration
• Experiments not well-designed. Additional
  experiments expensive.
• How to design experiments for building these
  dynamic models?
• Traditional least-squares assumptions not valid
• Imperfect model structure
• Random errors not independent
• Heteroskedastic responses
• Some parameters have little influence on model
  predictions.
• Effects of some parameters are highly correlated
  with effects of others
• Multiple sets of parameters give nearly the same
  predictions
• Poorly conditioned estimation problem

28
Why is the Parameter Estimation Difficult?

• Models are nonlinear in the parameters
• multiple optima, initial parameter guesses
• numerical solution of differential equations
  required at each iteration
• Experiments not well-designed. Additional
  experiments expensive.
• How to design experiments for building these
  dynamic models?
• Traditional least-squares assumptions not valid
• Imperfect model structure
• Random errors not independent
• Heteroskedastic responses
• Some parameters have little influence on model
  predictions.
• Effects of some parameters are highly correlated
  with effects of others
• Multiple sets of parameters give nearly the same
  predictions
• Poorly conditioned estimation problem

29
Estimability Analysis

• Addresses problems of
• Too many parameters for available data
• Parameters with little influence
• Correlated effects of parameters

30
Estimability Analysis

• Addresses problems of
• Too many parameters for available data
• Parameters with little influence
• Correlated effects of parameters
• The idea
• With insufficient information to estimate all
  parameters, select only the most important ones
  and adjust them so model predictions match data.
• Leave unimportant parameters at nominal values
  or simplify model to get rid of them.

31
Estimability Analysis

• Addresses problems of
• Too many parameters for available data
• Parameters with little influence
• Correlated effects of parameters
• The idea
• With insufficient information to estimate all
  parameters, select only the most important ones
  and adjust them so model predictions match data
• Leave unimportant parameters at nominal values
  or simplify model to get rid of them
• How many parameters can we estimate?
• Which ones?
• Answers depend on available data, model
  structure, and how much we believe initial
  guesses for various parameters.
• - Use a sensitivity-based approach

32
Estimability Analysis

• The Approach
• 1. Construct sensitivity matrix containing
  derivatives of model predictions with respect to
  the parameters
• Each column of Z contains sensitivity
  coefficients for a particular parameter
• Each row contains sensitivity coefficients for a
  particular response at a particular time
• Sensitivities from multiple experimental runs
  stacked vertically, so Z has many rows
• Rows deleted when some responses not available
  at some times
• Initial guesses required

33
Estimability Analysis

• 2. Scale elements of Z to permit effective
  comparisons
• Scale using information about
  reproducibility of yr and uncertainty in initial
  guess for??p
• Calculate magnitude of each column of Z. Select
  parameter whose column has the largest magnitude
  as most estimable parameter.

34
Estimability Analysis

• 2. Scale elements of Z to permit effective
  comparisons
• Scale using information about
  reproducibility of yr and uncertainty in initial
  guess for??p
• Calculate magnitude of each column of Z. Select
  parameter whose column has the largest magnitude
  as most estimable parameter.Columns with large
  sensitivity coefficients correspond to parameters
  with large influence on model predictions of
  experimental data.
• How do we pick the 2nd most estimable
  parameter?

35
Estimability Analysis

• Use selected column, X1, to calculate
  least-squares prediction of the full sensitivity
  matrix, Z, using
• Columns in are multiples of X1
• Columns in will be nearly the same as
  columns in Zfor parameters whose effects are
  correlated with the most estimable parameter

36
Estimability Analysis

• Use selected column, X1, to calculate
  least-squares prediction of the full sensitivity
  matrix, Z, using
• Calculate residual matrix
• Column of R1 with the largest magnitude is the
  next most estimable parameter. Augment matrix X
  with new column.
• Obtain least-squares estimate of Z using
  augmented X.
• Continue until parameters ranked from most
  estimable to least.

37
Estimability Analysis

• Deflation algorithm ranks parameters according
  to
• Large influence on model predictions (for
  available experiments)
• Lack of correlation with previously selected
  parameters
• Uncertainty in initial parameter guesses
• How many parameters should we select for
  estimation?
• Until now, trial and error
• Try estimating top 5 or top 10 using simulated
  experiments
• Do the estimates converge? Are the model
  predictions reasonable?
• What happens to the objective function as we move
  down the list?
• What if we know that some parameters are
  important and they arent near the top of the
  list?
• Do what the estimability analysis says?
• Fudge the scaling to move them up the list?
• Select important parameters by hand and put them
  in X at start
• Algorithm ranks remaining parameters

38
Estimability Analysis

• What have we used it for?
• How did it work?
• Polyisobutylene modeling
• Dynamic modeling of lab-scale polyethylene
  reactor
• Polyacrylamide gel dosimeter modeling

39
Polyisobutylene Modeling

• Living carbocationic polymerization at low
  temperatures
• Judit Puskas (U. of Akron) developed a kinetic
  mechanism and performed experiments
• Mathematical model developed to assess if
  mechanism is consistent with polymerization rate,
  initiator concentration and molecular weight data
• Judits group tried estimating parameters, but
  estimation failed
• Estimability analysis showed not all kinetic
  parameters can be estimated together from
  available data
• Effects of some parameters are highly correlated
  with others
• We determined which parameters cant be estimated
  together
• We tested whether parameters would be estimable
  if additional measurements were available

40
Dynamic PE Reactor Model

• Gas phase ethylene/hexene polymerization
• lab-scale reactor
• Dynamic experiments using a variety of
  temperatures and gas compositions
• Polymerization rate and gas composition measured
  during each run
• Polymer properties measured at the end of each
  run
• 16 experimental runs (4 saved for validation)
• 45 parameters
• 22 differential equations

41
Dynamic PE Reactor Model

• Estimability Analysis and Results
• One row in Z matrix for each measured response in
  each run
• Measurements at irregular times are easy to
  handle
• Estimated 30 parameters using data
• Parameter values changed during estimation, so
  sensitivities changed
• Updated estimability analysis during nonlinear
  regression
• Estimability analysis guided model simplification

42
Polymer Gel Dosimeter Model

• Polymer gel dosimeters verify 3-D radiation doses
  in cancer radiotherapy equipment
• Adrian Fuxman developed a kinetic scheme for
  radiation-induced copolymerization of
  acrylamide/bisacrylamide with phase change (27
  parameters)
• Adrian adjusted some parameters by hand to match
  available data.
• S. Babic did new experiments. Adrians model
  gave poor predictions of new data.
• Model predicts mass of cross-linked polymer
  formed, temperature rise, concentrations of
  species in each phase.
• 15 parameters available from free-radical
  polymerization literature
• 12 parameters are poorly known (guesses)
• S. Daneshvar ranked poorly-known parameters using
  estimability analysis and estimated 7 most
  estimable parameters using new data

43
Implementing Estimability Analysis

• Get sensitivity coefficients for dynamic models
  by solving sensitivity equations along with ODEs.
  Program the estimability algorithm in Matlab, C,
  Fortran.
• OR

44
Implementing Estimability Analysis

• Get sensitivity coefficients for dynamic models
  by solving sensitivity equations along with ODEs.
  Program the estimability algorithm in Matlab, C,
  Fortran.
• OR
• For any model on any platform, perturb the
  parameters and run simulations to get
• for each parameter, for each measurement, in
  each run
• Collect sensitivity coefficients in a spreadsheet
• Calculate sum of squared column entries
• Transpose, multiply and invert matrices

45
Other Uses of Estimability Analysis

• Tuning models for commercial processes
• Start with model and parameters from lab-scale
  studies
• Obtain new data at commercial scale
• Use estimability analysis to decide which
  parameters to adjust so model matches data from
  commercial process
• Sequential experimental design
• Do estimability analysis by adding rows to Z for
  different proposed experiments
• Which experiments will yield the most estimable
  parameters and smallest approximate joint
  confidence regions for parameters that will be
  estimated?
• Properly scaled ZTZ is the Fisher Information
  matrix

46
Why is the Parameter Estimation Difficult?

• Models are nonlinear in the parameters
• multiple optima, initial parameter guesses
• numerical solution of differential equations
  required at each iteration
• Experiments not well-designed. Additional
  experiments expensive.
• How to design experiments for building these
  dynamic models?
• Traditional least-squares assumptions not valid
• Imperfect model structure
• Random errors not independent
• Heteroskedastic responses
• Some parameters have little influence on model
  predictions.
• Effects of some parameters are highly correlated
  with effects of others
• Multiple sets of parameters give nearly the same
  predictions
• Poorly conditioned estimation problem

47
Why is the Parameter Estimation Difficult?

• Models are nonlinear in the parameters
• multiple optima, initial parameter guesses
• numerical solution of differential equations
  required at each iteration
• Experiments not well-designed. Additional
  experiments expensive.
• How to design experiments for building these
  dynamic models?
• Traditional least-squares assumptions not valid
• Imperfect model structure
• Random errors not independent
• Heteroskedastic responses
• Some parameters have little influence on model
  predictions.
• Effects of some parameters are highly correlated
  with effects of others
• Multiple sets of parameters give nearly the same
  predictions
• Poorly conditioned estimation problem

48
Consequences of Model Simplification or
Estimating only a few Parameters

• Roy Wu, K.B. McAuley and T. J. Harris

49
Consequences of Simplifying the Modelor
Estimating only a Subset of Parameters

• If we simplify a dynamic model so all parameters
  are estimable, but the simplified model structure
  is imperfect
• Parameter estimates are biased
• Model predictions are biased
• If we keep the full model structure and estimate
  some parameters with others fixed at poor initial
  guesses
• Parameter estimates are biased
• Model predictions are biased

50
Consequences of Simplifying the Modelor
Estimating only a Subset of Parameters

• If we simplify a dynamic model so all parameters
  are estimable, but the simplified model structure
  is imperfect
• Parameter estimates are biased
• Model predictions are biased
• If we keep the full model structure and estimate
  some parameters with others fixed at poor initial
  guesses
• Parameter estimates are biased
• Model predictions are biased
• BUT sometimes we achieve better predictions than
  if we estimated all parameters in the full model

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What do we mean by better?

• Smaller mean-square error
• Error comes from two parts
• Variance
• Bias

Which is more important?
MSE Squared Bias Variance
52
Linear Regression Example
Full Model
Simple Model
53
Linear Regression Example
Full Model
Simplified Model

• When is it better to use the simple model for
  predictions?
• Removing ?2 from the full model is analogous to
  leaving parameters at initial guesses in
  nonlinear regression problems

54
When is it Better to Use the Simplified Model
for Predictions?
when

• This condition holds when the data are very
  noisy and the input variables are strongly
  correlated and have limited range, especially
  when the true value of is small.

55
Recent and Ongoing Work

• Devising hypothesis tests and confidence
  intervals to determine whether the simplified
  model or full model will give better predictions
• Dynamic models as test problems
• Assessing how many parameters we should estimate
  (from the ranked estimability analysis list) to
  obtain the best predictions

56
Comforting Conclusions for Modelers

• Sometimes simpler models give better predictions
  than complex models, even when the simple model
  is structurally imperfect.
• If you have bad data
• correlated designs
• limited range of inputs
• noisy measurements
• dont try to estimate too many parameters.
• Dont be afraid to fix some parameters if you
  have prior information.
• Estimability analysis can help you decide which
  parameters to estimate and which to fix at
  initial guesses
• based on model structure, experimental design,
  reproducibility of data and feelings about
  initial parameter uncertainty

57
Acknowledgments

• Colleagues
• Jim McLellan, Jim Ramsay, Tom Harris, David Bacon
  
• Larry Biegler
• Jim Hsu, Judit Puskas, John Schreiner, Michael
  Cunningham, Keith Marchildon
• Dan Norman, Norman Rice
• Graduate Students and Postdocs
• Roy Wu, Saeed Varziri, Shahab Daneshvar, Adrian
  Fuxman, Bo Kou, Kevin Yao, Ben Shaw, Bo Kou
• Funding
• MITACS, Cybernetica, SAS, DuPont, PREA,
  Xerox,NSERC, MMO, BP Chemicals, Nova, Exxon

58
Summary

• Chemical engineers use fundamental models to
  improve operation of industrial processes
• Physical understanding ? many parameters
• Parameter estimation is the main challenge
• New techniques for
• Estimating parameters in imperfect dynamic models
• Selecting parameters to estimate when we have too
  many for the available data
• Assessing whether model simplification will lead
  to better predictions and parameter estimates