1. Throughput maximization by improved bottleneck control
Elvira Marie B. Aske*&, Sigurd Skogestad* and Stig Strand&
*Department of Chemical Engineering, Norwegian University of Science and Technology, Trondheim, Norway
&Statoil R&D, Process Control, Trondheim, Norway
Throughput maximization by improved bottleneck control.
Work together with Sigurd Skogestad, professor at Norwegian University of Science and Technology and Stig Strand, researcher at Statoil R&D.
My background is working in oil industry, mainly with implementation of MPC.
In this case, the economic optimal operation is simplified to maximum throughput.

2. Outline Modes of optimal operation Maximum throughput
Throughput manipulator (TPM)
Max-flow min-cut theorem
Realizing maximum throughput
Single-loop
MPC
Back off
Conclusions
First, I will talk about modes of optimal operation, then I will talk about maximum throughput.
I will mentioned some background information about throughput manipulator and the max-flow min-cut theorem.
Then will I discuss how to realize maximum throughput, both single-loop and by using MPC, followed by discussions around back off before I will sum up.

3. Depending on marked conditions: Two main modes of optimal operation
Mode 1. Given throughput (“nominal case”)
Given feed or product rate
“Maximize efficiency”: Unconstrained optimum (“trade-off”)
Mode 2. Max/Optimum throughput
Throughput is a degree of freedom + good product prices
2a) Maximum throughput
Increase throughput until constraints give infeasible operation
Constrained optimum - identify active constraints (bottleneck!)
2b) Optimized throughput
Increase throughput until further increase is uneconomical
Unconstrained optimum
In operation/control there are two main modes of optimal operation:
Mode 1. Given throughput, feed or product rate is given. Optimal operation is where the operation is most efficient.
Mode 2. Max or optimum throughput. Here the throughput is a degree of freedom and the product prices is good compared to feed and utility costs.
We have two situations here. First, maximum throughput where it is optimal to increase the throughput until constraints give infeasible operation. This is a constrained optimum where the active constraints, that is, the bottlenecks should be identified.
Second, optimized throughput which is the unconstrained case, a further increase in throughput is uneconomical. For instance, a reactor conversion decreases rapidly (or purge streams increases sharply) at high feed rates.
Traditionally: Focus on mode I
But: Mode IIa is where we really can make “extra” money!

4. Mode 2a: Maximum throughput
Typical profit function:
Feed flows are set in proportion to F and assume constant efficiencies:
Leads to:
Maximize profit → Maximize throughput F
In many cases, and especially when the product prices are high, optimal operation of the plant is the same as maximizing throughput. To understand this, consider a typical profit function where Pj are the product flows, Fi the feed flows, Qk the utility duties (heating, cooling, power), and p denote the prices.
Then assume that all feed flows are set in proportion to F, Fi = kF,iF
and under the assumption of constant efficiency in the units (independent of throughput) and assuming that all intensive (property) variables are constant,
Substitute the above equations, this leads to constant operational profit per unit of feed F processed. Assume p>0 such that we have a meaningful case where the products are worth more than the feedstocks and utilities. Then it is clear that maximizing the profit (-J) is equivalent to maximizing the throughput F.
However, F cannot go to infinity, because the operational constraints (g ≤ 0) related to achieving feasible operation (indirectly) impose a maximum value for F. In practice, the gains kP,j and kQ,k and are not constant, because the efficiency of the plant changes. Usually, operation becomes less efficient and p decreases when F increases. Nevertheless, as long as p remains positive, d(−J)/dF = p > 0 is nonzero, and we have a constrained optimum with respect to the throughput F. From the profit function we see that p will remain positive and optimal operation is the same as maximum throughput if the feed is available and product prices pP,j are sufficiently high compared to the prices of feeds and utilities.
If the assumption of constant efficiency do not hold, then mode 2b should be used.
For instance if there is purge streams and there is no limit for the purge rate, then there is no bottleneck. Ref: Ammonia synthesis process, Antonio Carlos Brandao de Arajo – Studies on Plantwide Control, PhD Thesis, NTNU, Also submitted for publication in Computers and Chemical Engineering.

5. Throughput manipulator (TPM)
Before we continue with maximum throughput, I will point out some terms.
Buckley and Price, Lyman and Georgakis points out three basic schemes for inventory control. Inventory control deals with how the mass balance is maintained in the plant.
They defined throughput manipulator (TPM): The TPM is the degree of freedom used to set the throughput in the primary process path (from the major feed to the major products)
The three basic schemes are:
TPM at feed, inventory control in direction of flow
TPM at product, inventory control in direction opposite to flow
TPM inside plant, radiating inventory control
Direction of inventory control is due to self-consistency, which is that the flow is maintained through the plant by use of the inventory loops only.
The placement of the throughput manipulator must be taken with a plant-wide understanding.
Buckley (1964). Techniques of Process Control
Price, Lyman and Georgakis (1994). Throughput manipulation
in plantwide control structures. Ind. Eng. Chem. Res. 33, 1197–1207.

6. From network theory: Max-flow min-cut theorem
Maximum flow through the network is equal to the capacity of the minimal cut (Ford and Fulkerson, 1962)
From network theory we have the max-flow min-cut theorem about maximal flows in flow networks.
In layman terms, the theorem states that the maximum flow in a network is dictated by its bottleneck.
Lets illustrate the max-flow min-cut theorem with bottlenecks and minimum cuts:
Consider a plant with parallel trains, with two feeds, a feed split and a crossover.
The maximum throughput is decided by the bottlenecks in the plant.
1) It can be the two last units in each train
2) It can be the first and the last unit (note that the crossover is one-directional)
3) A third possibility is reaching the maximum capacity in the parallel units here, however, the other train feed can be increase, until a unit has reached its maximum capacity, however, there is still possibility to route flow through the bottleneck and the feed can be increased until a forth bottleneck is reached.

7.  Maximum throughput requires tight control of the bottleneck unit
Maximum throughput achieved by maximizing the flow through the bottleneck
If the flow for some time is not at its maximum through the bottleneck, then this loss can never be recovered
 Maximum throughput requires tight control of the bottleneck unit
The term bottleneck is, as far I have noticed, a term mostly used in network theory and design. Now we look at the term bottleneck in a dynamic perspective. That is, the bottleneck may move under presence of disturbances or the maximum flow through the bottleneck (“size”) may vary.
Maximum flow is the maximum feed rate that the unit can accept subject to achieving feasible operation.
From the bottleneck definition the unit has no available capacity left when the unit its a bottleneck.
In some cases the bottleneck can not be located to a specific unit, but rather to a system of units (“system bottleneck”)
Example system bottleneck: Reactor-separator-recycle case.
If the flow its not at its maximum for some time in the bottleneck, then this loss can never be recovered.
This require tight control of the bottleneck unit!

8. Rules for achieving maximum throughput
Maximize flow F through bottleneck at all times
Use TPM for control of bottleneck unit
Locate TPM to achieve tight control at bottleneck
Back off: usually needed to ensure feasibility dynamically
Fmax F
How to achieve maximum throughput? Tight bottleneck control.
Maximize flow thought the bottleneck at all times.
Use TPM for control of the bottleneck unit. This follows because TPM is a DOF for throughput, which should be maximized.
However, the location of TPM is important at should be places so tight control at the bottleneck is possible.
However, some back off is needed to ensure feasibility dynamically due to disturbances etc. Back off gives loss, and therefore important to reduce the back off
Fset point
Back off
Time

9. Realize maximum throughput
Best result (minimize back-off) if TPM permanently is moved to bottleneck unit
Max = bottleneck
If the bottleneck is fixed at a unit, then single-loop control sufficient.
Use throughput manipulator to control the bottleneck unit.
Best result is achieved (minimized back off) if throughput manipulator is permanently moved to the bottleneck unit.
This has been discussed in literature, e.g. Skogestad “Control structure design for complete chemical plants”
Skogestad (2004) Control structure design for complete chemical plants
Comp. Chem. Eng 28 p

10. Realize maximum throughput in more complex cases
Bottleneck moves
Multiple feeds and crossovers
Proposed solution: Coordinator MPC*
Estimate of remaining capacity in each unit is obtained from local MPCs
Coordinator MPC manipulate TPMs (+ crossovers) to maximize flow through bottlenecks
In more complex cases where the bottleneck moves or if we have multiple feeds and crossovers in the chemical plant.
It the bottleneck moves to another unit, then reassignment of level loops is probably unavoidable (ensure self-consistency) and hence single-loop control is not attractive.
A better approach is to use a multivariable controller (for example MPC).
A proposed solution to maximize throughput with moving bottlenecks is coordinator MPC.
The idea here is to use the models in the local MPCs to estimate the remaining feed capacity in each unit, whereas the coordinator MPC is used to manipulate on the TPMs (usually feeds) and crossovers inside the plant.
“Coordinator” the term because the MPC takes decisions involves all the unit (or at least several), like throughput manipulator and crossovers. No set point coordination in the local MPCs. The coordinator MPC can be viewed as decentralized controller.
*Aske et al. (2007) Coordinator MPC for maximizing plant throughput Submitted to Comp. Chem. Eng

11. Coordinator MPC CV CV CV CV MV MV MV CV MV CV CV CV MV CV MV CV
- maximize throughput (CV with high, unreachable set point with lower priority)
- TPMs as MVs
- keep columns within their capacity (CV constraints)
- disturbances moves the bottlenecks CV CV CV CV MV MV MV CV MV
To explain the coordinator MPC I will illustrate with an example.
Consider a plant with two processing trains consisting of distillation columns. This is actually part of a gas processing plant in Norway.
There is two plant feeds, throughput manipulators,
then crossovers: a feed split here with different composition than the train feeds, a crossover between the trains and a feed split between the stabilizers.
The task is to maximize throughput, which is implemented as a CV with a high unreachable set point with lower priority.
The MVs are the TPMs marked, that is valves which routes the gas through the plant.
The CV constraints are the capacity in each unit, which is calculated in each unit using the local MPCs.
Disturbances like feed composition changes in the feed may move the bottlenecks in the plant. CV CV CV MV CV MV CV

12. Back off = loss (in throughput)
Back off can be reduced by
Improved control (to some extent)
Limited by network dynamics from TPM to bottleneck
Obtain TPM closer to bottleneck
Move TPM (Change in base control)
Add buffer tanks to get dynamic TPMs (Design change)
or use existing buffer volumes
Estimate back off to find economic incentive:
Worst case amplification:
Back off is the distance to the active constraint needed to avoid dynamic infeasibility in the presence of disturbances, model errors, delay and other sources for imperfect control (Narraway and Perkins, 1993; Govatsmark and Skogestad, 2005).
This distance to the active constraint implies a loss, in throughput (or profit).
Therefore, we want the back off to be as small as possible.
The throughput can be maximized with improved bottleneck control.
Reduce back off by improved control. However, this is limited due to the network dynamics from the throughput manipulator to the bottleneck, may be a “long loop” or large disturbances in the bottleneck unit.
This “long loop” is reduced by moving the TPM closer to the bottleneck, hence improved control can be obtained.
In addition, existing buffer volumes or adding buffer tanks gives dynamic throughput manipulators. That is, the hold-up volumes can be used to obtain a fast increase or decrease in the throughput for a shorter period, before the throughput manipulator affects the throughput in the bottleneck. This is most attractive in the case with moving bottlenecks, since the TPM will not be located at the bottleneck at all times (since reassigning is undesirable).
We want to estimate back off to find the economic incentive for moving TPM compared to the cost.
How large loss represent a larger back compared to the cost for moving TPM? This is actually a design question (process design including regulatory control).
Can we do any estimation of back off needed?
Assume feed back control and a linearized system, then the transfer function from y to d is given by SGd, where S is the sensitivity function and Gd is the disturbance model.
y max represents the worst case disturbances over all disturbances and all directions and therefore represent the minimum back off

13. Example – estimation of back off
Bottleneck
Bottleneck
Compare TPM at feed and at bottleneck
Feedback controller K tuned by Skogestad’s tuning rules, τc=3θeff
Disturbance rejection as function of frequency
Lets illustrate the estimation of back off with a scalar example.
Consider 5 tanks in series
Each tank represented with first order plus dead time model.
Each disturbance model is of first order
Feedback controller tuned using Skogestads tuning rules, tau c = 3 times effective dead time

14. Back off as a function of frequency
Peak unavoidable
Effect of disturbances reduced
If we plot the SGd as a function of frequency for the two cases, where red lines are SGd for disturbance 0 to 5 when TPM is located at the feed whereas the blue lines are for the case where TPM is placed at the bottleneck. The green lines are the first order disturbance models.
We see that
the peak of SGd0 is unavoidable, that is, the disturbance d0 which enters at the bottleneck. However the peak frequency changes
the effect of disturbances d1 to d5 is reduced. In worst case the disturbances may occur at the same time, leading to a larger total back off in the case where the TPM is located at the feed.

15. Conclusions Tighter bottleneck control can reduce back off
TPM should be used for control of the bottleneck unit to obtain maximum flow
Bottleneck fixed →single-loop control sufficient
Bottleneck moves → multivariable control
Consider moving/adding TPM if back off is large
To sum up how to maximize throughput by improved bottleneck control:
We have demonstrated that under the assumptions of feed, product and utility cost and constant efficiency maximum profit is the same as maximum throughput. By tighter bottleneck control we can improve maximum throughput because of less back off is required.
First, TPM should be used for control of the bottleneck unit.
Then we have to cases.
Bottleneck is fixed, then single-loop control is sufficient. The selection of TPM requires reassigning the inventory control to ensure self-consistency.
Bottleneck moves, then multivariable control is the obvious choice.
Especially in the latter case, “long loop” from the TPM to the bottleneck may still exist, so the use of crossovers (if there exist) and exploiting the hold-up volumes (dynamic TPMs) may be attractively.
Moving or/and adding TPM is especially attractive if the back of is large.