264887 Uncertainties and Stochastic Optimal Control in Batch Crystallization for Different Types of Objective Functions
Uncertainties and Stochastic Optimal Control in Batch Crystallization for different types of Objective Functions
Kirti M. Yenkie^{1,2 }and Urmila Diwekar ^{1,2}
^{1}Department of Bio Engineering, University of Illinois, Chicago, IL 60607  USA
^{2}Center for uncertain Systems: Tools for Optimization & Management (CUSTOM),
Vishwamitra Research Institute, Clarendon Hills, IL 60514 – USA
Abstract:
Batch crystallization is an industrial separation and purification process. It is dependent on parameters like temperature, supersaturation, agitation, solution behavior, etc. The fundamental process parameters like solubility and crystal structure can have physical uncertainties, while the process can have uncertainties in engineering design parameters and operating conditions. These uncertainties were generally ignored while predicting operating policies and hence, the predicted outcomes were not fully in accordance with the actual outcomes. To overcome these shifts in the desired results uncertainties are taken into account while predicting the crystallization operating policies.
Uncertainties in the system are characterized as a special class of stochastic processes, known as Ito processess if they follow certain properties. These uncertainties on propagation into the process model yields dynamic uncertainties within the process states. Using the theory of stochastic modeling and Ito's calculus, these uncertainties in the process states are represented in the mathematical form. The modified process model involves the state equations with the stochastic term representing uncertainty. The objective functions in batch crystallization are dependent on the desired outcome and can be classified into two categories late growth and early growth functions. Thus, we consider several objectives to see the effect of uncertainty on both the categories of functions. This results into stochastic optimal control problems and they are solved using Ito's lemma, stochastic calculus and stochastic optimal control theory.
The aim is to study all possible sources of uncertainty in batch crystallization and model them as Ito processes based on their behavior. Observe the effects of uncertainty inclusion into the process model and their effects on the operating policles for the early and late growth categories of objective functions. Thus, providing operating policies which shall mimic the actual behavior of the system and yield results with better predictive value.
Keywords: uncertainty, operating policy, stochastic processes, Ito's calculus, optimal control
Case study 1: Optimal control while considering kinetic parameter uncertainty
1. Kinetic parameter uncertainty:
The kinetic parameters are generally empirical constants determined by fitting experimental data to the model, and hence are a source of uncertainty within the system. In batch crystallization kinetics, the growth and nucleation expressions have empirical constants shown in table 3, they can be assumed to follow a Gaussian distribution, with the fitted values from previous experiments to be the mean. The values are assumed to deviate around ±5% about the mean.
Table 1: Kinetic parameter uncertainty in batch crystallization model
Uncertainty
 Kinetic constants
 Value from experiments/ model fitting
 Range of Values

G
 k_{g}
 1.44 x 10^{8} µm s^{1}
 1.368 x 10^{8} – 1.512 x 10^{8}

E_{g}/R
 4859 K
 4616.05 – 5101.95
 
g
 1.5
 1.425 – 1.575
 
B
 k_{b}
 285 (s µm^{3})^{1}
 270.75 – 299.25

E_{b}/R
 7517 K
 7141.15 – 7892.85
 
b
 1.45
 1.3775 – 1.5225

We consider a 95 % confidence interval and hence, the kinetic parameters lie within two standard deviations of their values (μ±2σ ). Thus, we evaluate the standard deviation for all of them using the extreme deviations as minimum and maximum values.
ü The sampling operation for multivariable uncertain parameter domain is performed using the Monte Carlo sampling technique
ü 100 sample values for each of the kinetic parameter are generated using inverse transformation over cumulative probability distribution.
ü After generating 100 samples for the kinetic parameter data, the model is simulated using each set.
ü The resulting dynamic uncertainty in the state variables due to static uncertainty in the kinetic parameters is observed in the plots of the dynamic uncertainty as shown in figure 1.
Figure 1 The dynamic uncertainties in state variables C, μ1s, μ2s, μ3s
2. Modeling uncertainties as Ito processes
By studying the dynamic uncertainty plots of the process variables and their correlation to Ito processes, it has been observed that the uncertainties can be best modeled with a simple Ito process known as Brownian motion with drift.
It is defined as: dy=ay,tdt+by,tdz (1)
where dz is the increment of the Wiener process equal to ε_{t}√Δt, and a(y,t) and b(y,t) are known functions. The random value ε_{t} has a unit normal distribution with zero mean and standard deviation of one. In this paper, a simplification of the above equation is used to represent the time dependent uncertainties in the concentration, seed and nucleation moments:
Figure 2. State variables modeled as Ito Processes
3. Stochastic Maximum Principle:
We use the stochastic maximum principle similar to the method illustrated in Ramirez and Diwekar. The objective is to maximize the expected value of mass of seeded crystals and minimize the expected value of mass of fines, considering the uncertainties associated with the concentration and moments of batch crystallization, finding the best operating temperature profile for the process.
The objective function for the stochastic formulation can be written as equation 3a or 3b where E is the expected value.
MaximizeT L=E0tdμ3sdt dμ3ndtdt (3a)
MaximizeT L=Eμ3s(tf)μ3n(tf) (3b)
Subject to seeded batch crystallization state variables modeled as Ito processes, initial conditions, constraints for supersaturation condition maintenance. The state equations in general can be represented as shown in eq 4.
where, yi=[C μ0s μ1s μ2s μ3s μ0n μ1n μ2n μ3n ]
Figure 3. Flowchart for optimal temperature profile evaluation using stochastic maximum principle (active constraint strategy)
4. Results
Figure 4 summarizes the optimal temperature profiles evaluated by the deterministic method and stochastic method. The results for the linear cooling profile, which is most commonly used is also compared against the optimal trajectories. Figure 5 compares the final size distributions predicted from the moment values, and it can be seen that the stochastic case yields a much narrower probability density function for the distribution in case of seed as well as nucleated crystals.
Table 2: Results of Comparison between linear, deterministic and stochastic cases
Case
 Objective function
 % Increment in stochastic case

Linear
 8.53 x 10^{9}
 5.88

Deterministic
 8.79 x 10^{9}
 2.73

Stochastic
 9.03 x 10^{9}
 N.A.

5. Conclusion
The static uncertainty in the kinetics result in dynamic uncertainties within the state variables. These uncertainties could be modeled as Ito processes, since they followed certain properties and had a behavioral pattern. The most important aspect was to solve the stochastic optimal control problem involving Ito processes and application of stochastic calculus, Ito's Lemma and stochastic maximum principle.
6. Future work
We would like to consider several kinds of objective functions used by researchers for predicting the best optimal policy. The uncertainties associated with these functions would be incorporated in the study and a similar strategy to study and compare the determinstic and stochastic cases will be adopted. Other sources of uncertainty along with kinetic parameter uncertainty would also be taken into account to make the operating policies more robust.
Figure 4. Comparison of Temperature profiles
(Linear, Deterministic and Stochastic case)
Figure 5. Predicted particle size distributions (PSD) from moment values for 3 cases.
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