This example shows how to optimize a design when there are uncertain variables. You optimize the dimensions of a Continuously Stirred Tank Reactor (CSTR) to minimize product concentration variation and production cost in case of varying, or uncertain, feed stock.

Continuously Stirred Tank Reactors (CSTRs) are common in the process industry. The Simulink model, `sdoCSTR`

, models a jacketed diabatic (i.e., non-adiabatic) tank reactor described in [1]. The CSTR is assumed to be perfectly mixed, with a single first-order exothermic and irreversible reaction, . , the reactant, is converted to , the product.

In this example, you use the following two-state CSTR model, which uses basic accounting and energy conservation principles:

, and - Concentrations of A in the CSTR and in the feed [kgmol/m^3]

, , and - CSTR, feed, and coolant temperatures [K]

and - Volumetric flow rate [m^3/h] and the density of the material in the CSTR [1/m^3]

and - Height [m] and heated cross-sectional area [m^2] of the CSTR.

- Pre-exponential non-thermal factor for reaction [1/h]

and - Activation energy and heat of reaction for [kcal/kgmol]

- Boltzmann's gas constant [kcal/(kgmol * K)]

and - Heat capacity [kcal/K] and heat transfer coefficients [kcal/(m^2 * K * h)]

Open the Simulink model.

```
open_system('sdoCSTR');
```

The model includes a cascaded PID controller in the `Controller`

subsystem. The controller regulates the reactor temperature, , and reactor residual concentration, .

Assume that the CSTR is cylindrical, with the coolant applied to the base of the cylinder. Tune the CSTR cross-sectional area, , and CSTR height, , to meet the following design goals:

Minimize the variation in residual concentration, . Variations in the residual concentration negatively affect the quality of the CSTR product. Minimizing the variations also improves CSTR profit.

Minimize the mean coolant temperature . Heating or cooling the jacket coolant temperature is expensive. Minimizing the mean coolant temperature improves CSTR profit.

The design must allow for variations in the quality of supply feed concentration, , and feed temperature, . The CSTR is fed with feed from different suppliers. The quality of the feed differs from supplier to supplier and also varies within each supply batch.

Select the following model parameters as design variables for optimization:

Cylinder cross-sectional area

Cylinder height

p = sdo.getParameterFromModel('sdoCSTR',{'A','h'});

Limit the cross-sectional area to a range of [1 2] m^2.

p(1).Minimum = 1; p(1).Maximum = 2;

Limit the height to a range of [1 3] m.

p(2).Minimum = 1; p(2).Maximum = 3;

Select the feed concentration and feed temperature as uncertain variables. You evaluate the design using different values of feed temperature and concentration.

pUnc = sdo.getParameterFromModel('sdoCSTR',{'FeedCon0','FeedTemp0'});

Create a parameter space for the uncertain variables. Use normal distributions for both variables. Specify the mean as the current parameter value. Specify a variance of 5% of the mean for the feed concentration and 1% of the mean for the temperature.

uSpace = sdo.ParameterSpace(pUnc); uSpace = setDistribution(uSpace,'FeedCon0',makedist('normal',pUnc(1).Value,0.05*pUnc(1).Value)); uSpace = setDistribution(uSpace,'FeedTemp0',makedist('normal',pUnc(2).Value,0.01*pUnc(2).Value));

The feed concentration is inversely correlated with the feed temperature. Add this information to the parameter space.

```
%uSpace.RankCorrelation = [1 -0.6; -0.6 1];
```

The rank correlation matrix has a row and column for each parameter with the (i,j) entry specifying the correlation between the i and j parameters.

Sample the parameter space. The scatter plot shows the correlation between concentration and temperature.

rng('default'); %For reproducibility uSmpl = sdo.sample(uSpace,60); sdo.scatterPlot(uSmpl)

Ideally you want to evaluate the design for every combination of points in the design and uncertain spaces, which implies 30*60 = 1800 simulations. Each simulation takes around 0.5 sec. You can use parallel computing to speed up the evaluation. For this example you instead only use the samples that have maximum & minimum concentration and temperature values, reducing the evaluation time to around 1 min.

[~,iminC] = min(uSmpl.FeedCon0); [~,imaxC] = max(uSmpl.FeedCon0); [~,iminT] = min(uSmpl.FeedTemp0); [~,imaxT] = max(uSmpl.FeedTemp0); uSmpl = uSmpl(unique([iminC,imaxC,iminT,imaxT]) ,:);

The design requirements require logging model signals. During optimization, the model is simulated using the current value of the design variables. Logged signals are used to evaluate the design requirements.

Log the following signals:

CSTR concentration, available at the second output port of the

`sdoCSTR/CSTR`

block

Conc = Simulink.SimulationData.SignalLoggingInfo; Conc.BlockPath = 'sdoCSTR/CSTR'; Conc.OutputPortIndex = 2; Conc.LoggingInfo.NameMode = 1; Conc.LoggingInfo.LoggingName = 'Concentration';

Coolant temperature, available at the first output of the

`sdoCSTR/Controller`

block

Coolant = Simulink.SimulationData.SignalLoggingInfo; Coolant.BlockPath = 'sdoCSTR/Controller'; Coolant.OutputPortIndex = 1; Coolant.LoggingInfo.NameMode = 1; Coolant.LoggingInfo.LoggingName = 'Coolant';

Create and configure a simulation test object to log the required signals.

```
simulator = sdo.SimulationTest('sdoCSTR');
simulator.LoggingInfo.Signals = [Conc,Coolant];
```

Create a function to evaluate the CSTR design. This function is called at each optimization iteration.

Use an anonymous function with one argument that calls the `sdoCSTR_design`

function.

evalDesign = @(p) sdoCSTR_design(p,simulator,pUnc,uSmpl);

The `evalDesign`

function:

Has one input argument that specifies the CSTR dimensions

Returns the optimization objective value

The `sdoCSTR_design`

function uses a `for`

loop that iterates through the sample values specified for the feed concentration. Within the loop, the function:

Simulates the model using the current iterate, feed concentration, and feed temperature values

Calculates the residual concentration variation and coolant temperature costs

To view the objective function, type `edit sdoCSTR_design`

.

Call the `evalDesign`

function with the initial CSTR dimensions.

dInit = evalDesign(p)

dInit = struct with fields: F: 11.3356 costConc: 6.4390 costCoolant: 4.8965

Plot the model response for the initial design. Simulate the model using the sample feed concentration values. The plot shows the variation in the residual concentration and coolant temperature.

sdoCSTR_plotModelResponse(p,simulator,pUnc,uSmpl);

The `sdoCSTR_plotModelResponse`

function plots the model response. To view this function, type `edit sdoCSTR_plotModelResponse`

.

Pass the objective function and initial CSTR dimensions to `sdo.optimize`

.

pOpt = sdo.optimize(evalDesign,p)

Optimization started 29-Feb-2020 03:32:07 max First-order Iter F-count f(x) constraint Step-size optimality 0 4 5.17535 0 1 8 3.81522 0 2.01 7.83 2 12 2.63696 0 0.57 3.01 3 16 2.53152 0 0.159 0.252 4 20 2.51758 0 0.0144 0.365 5 24 2.49208 0 0.105 0.814 6 29 2.4815 0 0.0506 0.248 7 33 2.47929 0 0.0118 0.0315 8 34 2.47929 0 0.000659 0.0183 Local minimum possible. Constraints satisfied. fmincon stopped because the size of the current step is less than the value of the step size tolerance and constraints are satisfied to within the value of the constraint tolerance. pOpt(1,1) = Name: 'A' Value: 2 Minimum: 1 Maximum: 2 Free: 1 Scale: 0.5000 Info: [1x1 struct] pOpt(2,1) = Name: 'h' Value: 2.2441 Minimum: 1 Maximum: 3 Free: 1 Scale: 2 Info: [1x1 struct] 2x1 param.Continuous

Call the `evalDesign`

function with the optimized CSTR dimensions.

dFinal = evalDesign(pOpt)

dFinal = struct with fields: F: 2.4793 costConc: 1.4506 costCoolant: 1.0287

Plot the model response for the optimized design. Simulate the model using the sample feed concentration values. The optimized design reduces the residual concentration variation and average coolant temperature for different feed stocks.

sdoCSTR_plotModelResponse(pOpt,simulator,pUnc,uSmpl);

To learn how to use sensitivity analysis to explore the CSTR design space and select an initial design for optimization, see "Design Optimization with Uncertain Variables (Code)".

[1] Bequette, B.W. *Process Dynamics: Modeling, Analysis and Simulation.* 1st ed. Upper Saddle River, NJ: Prentice Hall, 1998.

% Close the model bdclose('sdoCSTR')