Transcript Real-Time Optimization (RTO) • Chapter 19 • In previous chapters we have emphasized control system performance for load and set-point changes. Now we will be concerned.

Real-Time Optimization (RTO)
•
Chapter 19
•
In previous chapters we have emphasized control
system performance for load and set-point changes.
Now we will be concerned with how the set points are
specified.
In real-time optimization, a computer is used to optimize set
points for control loops.
Implementation of RTO/Supervisory Control:
1) Set point calculations are performed on a digital
computer (e.g., steady-state optimization).
2) Set points could be transmitted to either:
(a) analog controllers (before 1990)
(b) digital controllers (after 1990)
3) Case (a) is referred to as "analog supervisory control";
case (b) is "direct digital control'.
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Chapter 19
Chapter 19
Selection of Processes for RTO
Sources of information for the analysis:
1) Profit and loss statements for the plant
•Sales, prices
•Manufacturing costs etc.
2) Operating records
•Material and energy balances
•Unit efficiencies, production rate etc.
Categories of Interest:
1) Sales limited by production
•Increases in throughput desirable
•Incentives for improved operating conditions and schedules.
2) Sales limited by market
•Seek improvements in efficiency.
•Example: Reduction in manufacturing costs (utilities, feedstocks)
3) Large throughput units
•Small savings in production costs per unit are greatly magnified.
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4) Process Variability
•Excursions in process variables => offspec
products and a need for larger storage capacities.
•Reduction in variability allows set points to be
moved closer to a limiting constraint, e.g. product
quality.
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5) High Raw Material or Energy
Consumption
•e.g., Minimize energy consumption by optimal
allocation of fuel supplies and steam.
6) Losses of Valuable Components in waste
Streams
•Detect via chemical analysis plant exit streams
(e.g., air and water). Adjust air/fuel ratio in furnace to
minimize hydrocarbon consumption.
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Common Types of Optimization Problems
1. Operating Conditions
•Tower reflux ratio
•Reactor temperature
2. Allocation
•Fuel use
•Feedstock selection
3. Scheduling
•Cleaning (e.g., heat exchangers)
•Maintenance
•Batch processes
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Formulation and Solution of Optimization Problems
(see book by Edgar et al., “Optimization of Chemical
Processes”)
In order to perform on-line optimization, a series of
steps are required:
Step 1: Determine the process variables of interest.
Step 2: Definition of the Objective Function
•Difficult part of the problem!
•Example: Minimize the amount of offspec product from a
distillation column while avoiding a flooding condition.
- Relate off-spec product to costs of utilities and feedstocks.
Treat flooding condition as a constraint on vapor and
liquid flow rates.
•Specific objective functions will depend on plant configuration
and supply and demand.
(See TABLE 1 - next slide)
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DIFFERENT OBJECTIVE FUNCTIONS FOR DISTILLATION COLUMNS
1. Maximum yield of more valuable components from given feed, within purity
specifications
2. Maximize product purity at a given production rate from a given feed
3. Minimize energy consumption: reboilers, condensers, within purity specifications
4. Optimize energy consumption versus product recovery value, separation
5. Maximize distillate production, within specification
6. Optimize feed rate, tradeoff capacity versus recovery
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Step 3. Development of Process Models
•Process model is used in on-line optimization.
Consists of:
- Equality constraints which relate principal
process variables.
- Inequality constraints (e.g., physical limits on
pumps, compressors, metallurgical limits)
•Process model is usually a steady-state model.
•Some batch optimization methods use the process
as the model. (e.g., design of experiments)
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Step 4. Simplification of the Process Model
•Reduce the size of the optimization problem as much as
possible without losing the essence of the problem.
- Ignore process variables which have a negligible effect on
the objective function.
Step 5. Computation of the Optimum
•Choose a computational technique to determine optimum.
•Virtually all optimization techniques are iterative in nature –
linear programming, nonlinear programming.
•Good estimate of the optimum accelerates convergence.
- Experience on which inequality constraints might be active is
also useful.
Step 6. Sensitivity Studies
•Can be used to determine which variables and parameters
are most important in determining the optimum.
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Chapter 19
One Dimensional Search Techniques
Selection of a method involves a trade-off between
the number of objective function evaluations
(computer time) and complexity.
(1) "Brute Force" Approach
Small grid spacing (x) and evaluate f(x) at each
grid point  can get close to the optimum but
very inefficient.
(2) Equal Interval Search
• Example: Compare objective function values at
3 equally spaced points.
• Suppose we are interested in the region, a x
b . Thus initially, the "region of uncertainty, L" is
L0 = b - a .
• Consider two cases:
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• For case (1), maximum lies in (x2, b)
• For case (2), maximum lies in (x1, x3)
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• Polynomial fitting technique
• Fit quadratic polynomial to three data
points. Find analytical optimum, then
calculate f(x*). Discard worst value of f, then
repeat the process.
Multivariable Optimization
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•Computational efficiency is important.
•"Brute force" techniques are not practical for problems with
more than 3 or 4 variables to be optimized.
•Typical Approach: Reduce the multivariable optimization
problem to a series of one dimensional problems:
(1) From a given starting point, specify a search direction.
(2) Find the optimum along the search direction, i.e., a
one-dimensional search.
(3) Determine a new search direction.
(4) Repeat steps (2) and (3) until the optimum is located
•Two general categories for MV optimization techniques:
(1) Methods requiring derivatives of the objective function.
(2) Methods that do not require derivatives.
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• Nonderivative methods are attractive for real-time optimization
applications where a process measurement is used in place
of the objective function.
• Two nonderivative methods which have been used in industrial
applications:
(1) Evolutionary Operation (EVOP)
(2) Simplex or Pattern Search Method.
Evolutionary Operation (EVOP)
•Procedure:
(1) Choose a base point.
(2) Evaluate objective function at base point and a set of
regularly spaced points around the base point.
• For 2 variables, form a square around the base point.
• For 3 variables, form a cube.
(3) Move the process to the point that has the largest value
of the objective function.
(4) Use this point as the new base point.
(5) Repeat steps (2) and (4) until no further improvement
occurs.
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•Disadvantage of EVOP:
- It is slow due to the large number of steadystate points
which must be evaluated at each iteration.
Sequential Simplex Method [8]
•Uses a regular geometric figure (a simplex) as a basis.
- Two variable problem: use an equilateral triangle.
- Three variable problem: use a regular tetrahedron.
•Objective function is evaluated at the vertices of the simplex.
•General direction of search is projected away form the worst
vertex.
•Thus search direction can change and only one new operating
point must be evaluated at each iteration.
•Example: The two variable problem, max f(x1, x2)
Assume
f1  min f (f1 , f 2 , f3 )
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•Points 2,3, and 4 form a new equilateral triangle.
•The pattern search approach usually results in a zig-zag
pattern as it moves to the optimum (see figure 20.6next slide).
Constrained Optimization
•Optimization problems commonly involve equality and
inequality constraints.
•Nonlinear Programming (NLP) Problems:
a) Involve nonlinear objective function (and possible
nonlinear constraints).
b) Efficient off-line optimization methods are available (e.g.
conjugate gradient, variable metric).
c) On-line use? May be limited by computer time and
storage requirements.
•Quadratic Programming (QP) Problems:
a) Quadratic objective function plus linear equality and
inequality constraints.
b) Computationally efficient methods are available.
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•Linear Programming (LP) Problems
Both objective function and constraints are linear.
Solutions are highly structured and can be rapidly obtained.
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Linear Programming (LP)
•Has gained widespread industrial acceptance for on-line
optimization, blending etc.
•Linear constraints can arise due to:
1. Production limitation e.g. equipment limitations, storage
limits, market constraints.
2. Raw material limitation
3. Safety restrictions, e.g. allowable operating ranges for
temperature and pressures.
4. Physical property specifications e.g. product quality
constraints when a blend property can be calculated as
an average of pure component properties:
n
P   yi Pi  
i 1
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5. Material and Energy Balances
- Tend to yield equality constraints.
- Constraints can change frequently, e.g. daily or hourly.
•Effect of Inequality Constraints
- Consider the linear and quadratic objective functions on
the next page.
- Note that for the LP problem, the optimum must lie on one
or more constraints.
•General Statement of the LP Problem:
n
max f   ci x i
subject to:
i 1
xi  0
n
a
j 1
ij
x j  bi
i  1, 2,..., n
i  1, 2,..., n
•Solution of LP Problems
- Simplex Method
- Examine only constraint boundaries
- Very efficient, even for large problems
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The effect of an inequality constraint
on the maximum of quadratic function,
f ( x)  a0  a1 x  a2 x 2 (The arrows
indicate the allowable values of x.)
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Most LP applications involve more than two variables
and can involve 1000s of variables.
So we need a more general computational approach,
based on the Simplex method. There are many
variations of the Simplex method.
One that is readily available is the Excel Solver.
Basic features of LP:
•Linear objective function
•Linear equality/inequality constraints
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Nonlinear Programming (NLP) example
-nonlinear objective function
-nonlinear constraints
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