•
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 .
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'.

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.
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.
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.
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
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)
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
TABLE 1: Objective Functions for Distillation Columns
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)
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.
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:
For case (1), maximum lies in (x2, b).
For case (2), maximum lies in (x1, x3).

•
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
•
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.
• 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.

• 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(x
1
, x
2
)
Assume f
1
 f f f
2
, f
3
)
•
Points 2,3, and 4 form a new equilateral triangle.
• The pattern search approach usually results in a zigzag pattern as it moves to the optimum (see figure 20.6-next 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 (OP) Problems: a) Quadratic objective function plus linear equality and inequality constraints. b) Computationally efficient methods are available.

•
Linear Programming (LP) Problems
Both objective function and constraints are linear.
Solutions are highly structured and can be rapidly obtained.
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:
P
 i n 

1 y i
P i
 
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: max f
 i n 

1 c x i i subject to: x i

0 i

1, 2, , n j n 

1 a x ij j
 b i i

1, 2, , n
•
Solution of LP Problems
- Simplex Method
- Examine only constraint boundaries
- Very efficient, even for large problems