Part 3: Regulatory («stabilizing») control Outline Inventory (level) control structure Location of throughput manipulator Consistency and radiating rule Structure of regulatory control layer (PID) Selection of controlled variables (CV2) and pairing with manipulated variables (MV2) Main rule: Control drifting variables and "pair close" Summary: Sigurd’s rules for plantwide control  

Example flow control cascade Process: F = k(z)*sqrt(dp) Japanize slide. Linearize in two points. P-control Find closed-loop tf from Fs to F.

Procedure Skogestad procedure for control structure design I Top Down Step S1: Define operational objective (cost) and constraints Step S2: Identify degrees of freedom and optimize operation for disturbances Step S3: Implementation of optimal operation What to control ? (primary CV’s) Active constraints Self-optimizing variables for unconstrained, c=Hy Step S4: Where set the production rate? (Inventory control) II Bottom Up Step S5: Regulatory control: What more to control (secondary CV’s) ? Step S6: Supervisory control Step S7: Real-time optimization

Step S4. Where set production rate? Very important decision that determines the structure of the rest of the inventory control system! May also have important economic implications Link between Top-down (economics) and Bottom-up (stabilization) parts Inventory control is the most important part of stabilizing control “Throughput manipulator” (TPM) = MV for controlling throughput (production rate, network flow) Where set the production rate = Where locate the TPM? Traditionally: At the feed For maximum production (with small backoff): At the bottleneck

TPM and link to inventory control Liquid inventory: Level control (LC) Sometimes pressure control (PC) Gas inventory: Pressure control (PC) Component inventory: Composition control (CC, XC, AC)

Production rate set at inlet : Inventory control in direction of flow* TPM * Required to get “local-consistent” inventory control

Production rate set at outlet: Inventory control opposite flow* TPM * Required to get “local-consistent” inventory control

Production rate set inside process* TPM * Required to get “local-consistent” inventory control

General: “Need radiating inventory control around TPM” (Georgakis)

Consistency of inventory control Consistency (required property): An inventory control system is said to be consistent if the steady-state mass balances (total, components and phases) are satisfied for any part of the process, including the individual units and the overall plant. 11

QUIZ 1 CONSISTENT?

Local-consistency rule Rule 1. Local-consistency requires that 1. The total inventory (mass) of any part of the process must be locally regulated by its in- or outflows, which implies that at least one flow in or out of any part of the process must depend on the inventory inside that part of the process. 2. For systems with several components, the inventory of each component of any part of the process must be locally regulated by its in- or outflows or by chemical reaction. 3. For systems with several phases, the inventory of each phase of any part of the process must be locally regulated by its in- or outflows or by phase transition. Proof: Mass balances Note: Without the word “local” one gets the more general consistency rule

QUIZ 1 CONSISTENT? Rule (March 2017): Controlling pressure at inlet or outlet gives indirect flow control (because of pressure boundary condition)

Local concistency requirement -> “Radiation rule “(Georgakis)

Flow split: May give extra DOF TPM Split: Extra DOF (FC) Flash: No extra DOF

Example: Separator control (oil-gas separation offshore)

Example: Separator control Alternative TPM locations Compressor could be replaced by valve if p1>pG

Alt.1 Alt.2 Alt.4 Alt.3 Similar to original but NOT CONSISTENT (PC not direction of flow)

Example: Solid oxide fuel cell xCH4,s CC PC CH4 CH4 + H2O = CO + 3H2 CO + H2O = CO2 + H2 2H2 + O2- → 2H2O + 2e- H2O (in ratio with CH4 feed to reduce C and CO formation) Solid oxide electrolyte O2- e- TPM = current I [A] = disturbance O2 + 4e- → 2O2- Air (excess O2) PC TC Ts = 1070 K (active constraint)

LOCATION OF SENSORS Location flow sensor (before or after valve or pump): Does not matter from consistency point of view Locate to get best flow measurement Before pump: Beware of cavitation After pump: Beware of noisy measurement Location of pressure sensor (before or after valve, pump or compressor): Important from consistency point of view

Next slides: Exercises and solutions

For each of the five structures; Where is the TPM? Is it feasible? FC PC FC PC PC FC FC PC PC FC For each of the five structures; Where is the TPM? Is it feasible?

FC PC FC PC PC FC FC PC FT PT Same sensor location as case 3. Can you control both flow and temperature using valves 2 and 3, Valves 3 and 4? Some more. For each of the five structures; Where is the TPM? Is it feasible?

Solution according to radiation rule FC PC Solution according to radiation rule TPM NO FC PC YES TPM PC FC TPM NO FC PC YES (it does not matter where the flow is measured; the valve determines the location of the TPM) TPM PC FC YES TPM Note: Can never control pressure at ends (upstream first control valve or downstream last valve)!

TPM NO NO. New Rule: Another controller cannot cross the TPM, FC PC TPM NO FC PC NO. New Rule: Another controller cannot cross the TPM, Not even with a PC TPM PC FC NO, cannot cross TPM TPM FC PC NO TPM FT PT No, but OK if we use valve 4 for PC Some more. For each of the five structures; Where is the TPM? Is it feasible?

Where should we place TPM? TPM = MV used to control throughput Traditionally: TPM = Main feed valve (or pump/compressor) Gives inventory control “in direction of flow” Consider moving TPM if: There is an important CV that could otherwise not be well controlled Dynamic reasons Special case: Max. production important: Locate TPM at process bottleneck* ! TPM can then be used to achieve tight bottleneck control (= achieve max. production) Economics: Max. production is very favorable in “sellers marked” If placing it at the feed may yield infeasible operation (“overfeeding”) If “snowballing” is a problem (accumulation in recycle loop), then consider placing TPM inside recycle loop BUT: Avoid a variable that may (optimally) saturate as TPM (unless it is at bottleneck) Reason: To keep controlling CV=throughput, we would need to reconfigure (move TPM)** *Bottleneck: Last constraint to become active as we increase throughput -> TPM must be used for bottleneck control **Sigurd’s general pairing rule (to reduce need for reassigning loops): “Pair MV that may (optimally) saturate with CV that may be given up”

Example TPM: Two distillation columns See end of part 1

Often optimal: Locate TPM at bottleneck! "A bottleneck is a unit where we reach a constraints which makes further increase in throughput infeasible" If feed is cheap and available: Located TPM at bottleneck (dynamic reasons) If the flow for some time is not at its maximum through the bottleneck, then this loss can never be recovered.

Single-loop alternatives for bottleneck control Want max flow here Traditional: Manual control of feed rate TPM Alt.1. Feedrate controls bottleneck flow (“long loop”…): FC Fmax TPM Alt. 2: Feedrate controls lost task (another “long loop”…): Fmax TPM Alt. 3: Reconfigure all upstream inventory loops: Fmax TPM

May move TPM to inside recycle loop to avoid snowballing Example: Eastman esterification process Alcohol recycle Reach max mass transfer rate: R increases sharply (“snowballing”) Ester product Alcohol + water + extractive agent (e)

First improvement: Located closer to bottleneck

Final improvement: Located “at” bottleneck + TPM is inside “snowballing” loop Follows Luyben’s law 1 to avoid snowballing(modified): “Avoid having all streams in a recycle system on inventory control”

CASE STUDY: Recycle plant (Luyben, Yu, etc.) Part 1 -3 Recycle of unreacted A (+ some B) 5 Feed of A 4 1 2 Assume constant reactor temperature. Given feedrate F0 and column pressure: 3 Dynamic DOFs: Nm = 5 Column levels: N0y = 2 Steady-state DOFs: N0 = 5 - 2 = 3 Product (98.5% B)

Recycle plant: Optimal operation Part 1: Economics (Given feed) Recycle plant: Optimal operation mT 1 remaining unconstrained degree of freedom, CV=?

J=V as a function of reflux L Optimum = Nominal point With fixed active constraints: Mr = 2800 kmol (max), xB= 1.5% A (max)

Control of recycle plant: Conventional structure (“Two-point”: CV=xD) LC TPM LC xD XC XC xB LC Control active constraints (Mr=max and xB=0.015) + xD

Luyben law no. 1 (to avoid snowballing): “Fix a stream in the recycle loop” (CV=F or D)

Luyben rule: CV=D (constant) LC LC XC LC

“Brute force” loss evaluation:Disturbance in F0 Luyben rule: Conventional Loss with nominally optimal setpoints for Mr, xB and c

Loss evaluation: Implementation error Luyben rule: Loss with nominally optimal setpoints for Mr, xB and c

Conclusion: Control of recycle plant Active constraint Mr = Mrmax Self-optimizing L/F constant: Easier than “two-point” control Assumption: Minimize energy (V) Active constraint xB = xBmin

Modified Luyben’s law to avoid snowballing Luyben law no. 1 (“Plantwide process control”, 1998, pp. 57): “A stream somewhere in all recycle loops must be flow controlled” Luyben rule is OK dynamically (short time scale), BUT economically (steady-state): Recycle should increase with throughput Modified Luyben’s law 1 (by Sigurd): “Consider moving the TPM inside the recycle loop”

NOTE: There are actually two recycles One through the reactor (D or F) One through the column (L) One flow inside both recycle loops: V Alt.6: TPM=V if we want to break both recycle loops! PC TC

TPM = V Alt. 6 PC LC LC L XC F TC TPM XC LC Simulations (to be done) confirm This is the best! L and F for composition control: OK!

What about keeping V constant? (in addition to having another TPM) Alt. 7 What about keeping V constant? (in addition to having another TPM) PC TC LC TPM L LC F0 F V LC XC With feedrate F0 fixed (TPM) L for compostion control in bottom (xB) Top composition floating NO! Never control cost J=V

Bottleneck: max. vapor rate in column   Reactor-recycle process: Want to maximize feedrate: reach bottleneck in column Bottleneck: max. vapor rate in column TPM

Bottleneck: max. vapor rate in column Reactor-recycle process with max. feedrate Alt.A: Feedrate controls bottleneck flow Bottleneck: max. vapor rate in column TPM Vs FC Vmax V Vmax-Vs=Back-off = Loss Get “long loop”: Need back-off in V

Bottleneck: max. vapor rate in column Reactor-recycle process with max. feedrate: Alt. B Move TPM to bottleneck (MAX). Use feedrate for lost task (xB) Bottleneck: max. vapor rate in column MAX TPM =Alt.6 TPM Get “long loop”: May need back-off in xB instead…

Reactor-recycle process with max. feedrate: Alt Reactor-recycle process with max. feedrate: Alt. C: Best economically: Move TPM to bottleneck (MAX) + Reconfigure upstream loops MAX LC TPM OK, but reconfiguration undesirable… =Alt.6 TPM

Reactor-recycle process: Alt.C’: Move TPM + reconfigure (permanently!) LC CC TPM F0s =Alt.6 TPM For cases with given feedrate: Get “long loop” but no associated loss

Alt.4: Multivariable control (MPC) Can reduce loss BUT: Is generally placed on top of the regulatory control system (including level loops), so it still important where the production rate is set! One approach: Put MPC on top that coordinates flows through plant By manipulating feed rate and other ”unused” degrees of freedom (including level setpoints): E.M.B. Aske, S. Strand and S. Skogestad, ``Coordinator MPC for maximizing plant throughput'', Computers and Chemical Engineering, 32, 195-204 (2008).

Conclusion TPM (production rate manipulator) Think carefully about where to place it! Difficult to undo later

Session 5: Design of regulatory control layer

Outline Skogestad procedure for control structure design I Top Down Step S1: Define operational objective (cost) and constraints Step S2: Identify degrees of freedom and optimize operation for disturbances Step S3: Implementation of optimal operation What to control ? (primary CV’s) (self-optimizing control) Step S4: Where set the production rate? (Inventory control) II Bottom Up Step S5: Regulatory control: What more to control (secondary CV’s) ? Distillation example Step S6: Supervisory control Step S7: Real-time optimization

Step 5. Regulatory control layer Regulatory layer Step 5. Regulatory control layer Purpose: “Stabilize” the plant using a simple control configuration (usually: local SISO PID controllers + simple cascades) Enable manual operation (by operators) Main structural decisions: What more should we control? (secondary cv’s, CV2, use of extra measurements) Pairing with manipulated variables (mv’s u2) CV1 CV2 = ?

Structure of regulatory control layer (PID) Main decisions: Selection of controlled variables (CV2) Pairing with manipulated variables (MV2) Main rules: Control drifting variables «Pair close"   CV2 ↕ MV2

Regulatory layer Stabilizing control: Use inputs MV2=u2 to control “drifting” variables CV2 Primary CV CV1 CV2s K u2 G CV2 Secondary CV (control for dynamic reasons) Key decision: Choice of CV2 (controlled variable) Also important: Choice of MV2=u2 (“pairing”) Process control: Typical «drifting» variables (CV2) are Liquid inventories (level) Vapor inventories (pressure) Some temperatures (reactor, distillation column profile)

Degrees of freedom unchanged Regulatory layer Degrees of freedom unchanged No degrees of freedom lost as setpoints y2s replace inputs u2 as new degrees of freedom for control of y1 Cascade control: G K CV2s u2 CV2 CV1 MV2=Original DOF CV2s=New DOF

Objectives regulatory control layer Regulatory layer Objectives regulatory control layer Allow for manual operation Simple decentralized (local) PID controllers that can be tuned on-line Take care of “fast” control Track setpoint changes from the layer above Local disturbance rejection Stabilization (mathematical sense) Avoid “drift” (due to disturbances) so system stays in “linear region” “stabilization” (practical sense) Allow for “slow” control in layer above (supervisory control) Make control problem easy as seen from layer above Use “easy” and “robust” measurements (pressure, temperature) Simple structure Contribute to overall economic objective (“indirect” control) Should not need to be changed during operation

Example: Exothermic reactor (unstable) Regulatory layer Example: Exothermic reactor (unstable) Active constraints (economics): Product composition c + level (max) u = cooling flow (q) CV1 = composition (c) CV2 = temperature (T) feed CC CV1=c CV1s Ls=max TC CV2=T CV2s LC product u cooling

Optimizer (RTO) PROCESS H2 H y ny d Stabilized process u1 u2 CV1s Supervisory controller (MPC) Regulatory controller (PID) H2 H y1=CV1 CV2s y ny d Stabilized process CV1s y2=CV2 Physical inputs (valves) Optimally constant valves Always active constraints u1 u2 Degrees of freedom for optimization (usually steady-state DOFs), MVopt = CV1s Degrees of freedom for supervisory control, MV1=CV2s + unused valves Physical degrees of freedom for stabilizing control, MV2 = valves (dynamic process inputs)

Details Step 5 (Structure regulatory control layer) Regulatory layer (a) What to control (CV2)? Control CV2 that “stabilizes the plant” (stops drifting) Select CV2 which is easy to control (favorable dynamics) In addition, active constraints (CV1) that require tight control (small backoff) may be assigned to the regulatory layer.* *Comment: usually not necessary with tight control of unconstrained CVs because optimum is usually relatively flat

Regulatory layer “Control CV2 that stabilizes the plant (stops drifting)” In practice, control: Levels (inventory liquid) Pressures (inventory gas/vapor) (note: some pressures may be left floating) Inventories of components that may accumulate/deplete inside plant E.g., amine/water depletes in recycle loop in CO2 capture plant E.g., butanol accumulates in methanol-water distillation column E.g., inert N2 accumulates in ammonia reactor recycle Reactor temperature Distillation column profile (one temperature inside column) Stripper/absorber profile does not generally need to be stabilized

Identify MVs (u2) to control CV2, taking into account: Details Step 5b…. Regulatory layer (b) Identify pairings = Identify MVs (u2) to control CV2, taking into account: Want “local consistency” for the inventory control Implies radiating inventory control around given flow Avoid selecting as MVs in the regulatory layer, variables that may optimally saturate at steady-state (active constraint on some region), because this would require either reassigning the regulatory loop (complication penalty), or requiring back-off for the MV variable (economic penalty) Want tight control of important active constraints (to avoid back-off) General rule: ”pair close” (see next slide)

Step 5b…. Main rule: “Pair close” Regulatory layer Step 5b…. Main rule: “Pair close” The response (from input to output) should be fast, large and in one direction. Avoid dead time and inverse responses!

Cascade control in regulatory layer May be helpful to reduce nonlinearity and improve disturbance rejection Controller (“master”) gives setpoint to another controller (“slave”) Without cascade: “Master” controller directly adjusts u (input, MV) to control y With cascade: Local “slave” controller uses u to control “extra”/fast measurement (y’). “Master” controller adjusts setpoint y’s. Example: Flow controller on valve (very common!) y = level H in tank (or could be temperature etc.) u = valve position (z) y’ = flowrate q through valve WITHOUT CASCADE WITH CASCADE flow in Hs LC H Hs flow in flow out MV=qs FC q z master slave H LC MV=z valve position measured flow flow out

What are the benefits of adding a flow controller (inner cascade)? qs Extra measurement y’ = q q z f(z) Counteracts nonlinearity in valve, f(z) With fast flow control we can assume q = qs Eliminates effect of disturbances in p1 and p2 (FC reacts faster than outer level loop) 1 linear valve z (valve opening) 1

Counteracting nonlinearity using cascade control: Process gain variation -> Time constant variation Proof: Slave controller with u = z (valve position) and y=q (flow) Nonlinear valve with varying gain k: G = k / (τs+1) PI-controller with gain Kc and integral time τI= τ. With slave (flow) controller: Transfer function from ys to y (for master loop): T = L/(1+L) = 1/(τCL s + 1) where τCL = τ/(k Kc) So variation in k translates into variation in τCL In practise this gives a variation in the effective time delay in the master loop Low gain k for valve gives large effective time delay (bad)

Regulatory layer Sigurd’s pairing rule for regulatory layer: “Avoid using MVs that may optimally saturate (at steady state) to control CV2s” Main reason: Minimizes need for reassigning loops Important: Always feasible (and optimal) to give up a CV when optimal MV saturation occurs. Proof (DOF analysis): When one MV disappears (saturates), then we have one less optimal CV. Failing to follow this rule: Need some “fix” when MV saturates to remain optimal, like reconfiguration (logic) backoff (loss of optimality) BUT: Rule may be in conflict with other criteria Dynamics (“pair close” rule) Interactions (“avoid negative steady-state RGA” rule) If conflict: Use reconfiguration (logic) or go for multivariable constraint control (MPC which may provide “built-in” logic) LV TC T s . loop TC TS (a) Normal: Control T using V (b) If V may saturate: Use L

Regulatory layer Why simplified configurations? Why control layers? Why not one “big” multivariable controller? Fundamental: Save on modelling effort Other: easy to understand easy to tune and retune insensitive to model uncertainty possible to design for failure tolerance fewer links reduced computation load

Hierarchical/cascade control: Time scale separation With a “reasonable” time scale separation between the layers (typically by a factor 5 or more in terms of closed-loop response time) we have the following advantages: The stability and performance of the lower (faster) layer (involving y2) is not much influenced by the presence of the upper (slow) layers (involving y1) Reason: The frequency of the “disturbance” from the upper layer is well inside the bandwidth of the lower layers With the lower (faster) layer in place, the stability and performance of the upper (slower) layers do not depend much on the specific controller settings used in the lower layers Reason: The lower layers only effect frequencies outside the bandwidth of the upper layers

Cascade control distillation ys y Ts T Ls L z With flow loop + T-loop in top y XC Ts T TC Ls L FC z XC

QUIZ: What are the benefits of adding a flow controller (inner cascade)? qs Extra measurement y2 = q q z Counteracts nonlinearity in valve, f(z) With fast flow control we can assume q = qs Eliminates effect of disturbances in p1 and p2

Summary: Rules for plantwide control Here we present a set of simple rules for economic plantwide control to facilitate a close-to-optimal control structure design in cases where the optimization of the plant model is not possible. the rules may be conflicting in some cases and in such cases, human reasoning is strongly advised.

Rule 1: Control the active constraints. Rules for Step S3: Selection of primary (economic) controlled variables, CV1 Rule 1: Control the active constraints. In general, process optimization is required to determine the active constraints, but in many cases these can be identified based on a good process knowledge and engineering insight. Here is one useful rule: Rule 1A: The purity constraint of the valuable product is always active and should be controlled. This follows, because we want to maximize the amount of valuable product and avoid product “give away” (Jacobsen and Skogestad, 2011). Thus, we should always control the purity of the valuable product at its specification. For “cheap” products we may want to overpurify (purity constraint may not be active) because this may reduce the loss of a more valuable component. In other cases, we must rely on our process knowledge and engineering insight. For reactors with simple kinetics, we usually find that, the reaction and conversion rates are maximized by operating at maximum temperature and maximum volume (liquid phase reactor). For gas phase reactor, high pressure may increase the reaction rate, but this must be balanced against the compression costs.

Rule 2: (for remaining unconstrained steady-state degrees of freedom, if any): Control the “self-optimizing” variables. This choice is usually not obvious, as there may be several alternatives, so this rule is in itself not very helpful. The ideal self-optimizing variable, at least, if it can be measured accurately, is the gradient of the cost function. Ju, which should be zero for any disturbance. Unfortunately, it is rarely possible to measure this variable directly and the “self-optimizing” variable may be viewed as an estimate of the gradient Ju The two main properties of a good “self-optimizing” (CV1=c=Hy) variable are: Its optimal value is insensitive to disturbances (such that the optimal sensitivity dcopt/dd =Fc HF = is small) It is sensitive to the plant inputs (so the process gain dc/du = G = HGy is large). The following rule shows how to combine the two desired properties: Rule 2A: Select the set CV1=c such that the ratio G-1Fc is minimized. This rule is often called the “maximum scaled gain rule”.

Rule 3: (for remaining unconstrained steady-state degrees of freedom, if any): Never try to control the cost function J (or any other variable that reaches a maximum or minimum at the optimum) u J J>Jmin Jmin J<Jmin ? First, the cost function J has no sensitivity to the plant inputs at the optimal point and so G = 0 which violates Rule 2A. Second, if we specify J lower than its optimal value, then clearly, the operation will be infeasible Also, specifying J higher than its optimal value is problematic, as we have multiplicity of solutions. As mentioned above, rather controlling the cost J, we should control its gradient, Ju.

Rule 4: Locate the TPM close to the process bottleneck Rules for Step S4: Location of throughput manipulator (TPM) Rule 4: Locate the TPM close to the process bottleneck . The justification for this rule is to take advantage of the large economic benefits of maximizing production in times when product prices are high relative to feed and energy costs (Mode 2). To maximize the production rate, one needs to achieve tight control of the active constraints, in particular, of the bottleneck, which is defined as the last constraint to become active when increasing the throughput rate (Jagtap et al., 2013).

Rule 5: (for processes with recycle) Locate the TPM inside the recycle loop. The point is to avoid “overfeeding” the recycle loop which may easily occur if we operate close to the throughput where “snowballing” in the recycle loop occurs. This is a restatement of Luyben’s rule “Fix a Flow in Every Recycle Loop” (Luyben et al., 1997). From this perspective, snowballing can be thought of as the dynamic consequence of operating close to a bottleneck which is within a recycle system. In many cases, the process bottleneck is located inside the recycle loop and Rules 4 and 5 give the same result.

Rules for Step S5: Structure of regulatory control layer. Rule 6: Arrange the inventory control loops (for level, pressures, etc.) around the TPM location according to the radiation rule. The radiation rule (Price et al., 1994), says that, the inventory loops upstream of the TPM location must be arranged opposite of flow direction. For flow downstream of TPM location it must be arranged in the same direction. This ensures “local consistency” i.e. all inventories are controlled by their local in or outflows.

Rule 7: Select “sensitive/drifting” variables as controlled variables CV2 for regulatory control. This will generally include inventories (levels and pressures), plus certain other drifting (integrating) variables, for example, a reactor temperature a sensitive temperature/composition in a distillation column. This ensures “stable operation, as seen from an operator’s point of view. Some component inventories may also need to be controlled, especially for recycle systems. For example, according to “Down’s drill” one must make sure that all component inventories are “self-regulated” by flows out of the system or by removal by reactions, otherwise their composition may need to be controlled (Luyben, 1999).

Rule 8: Economically important active constraints (a subset of CV1), should be selected as controlled variables CV2 in the regulatory layer. Economic variables, CV1, are generally controlled in the supervisory layer. Moving them to the faster regulatory layer may ensure tighter control with a smaller backoff. Backoff: difference between the actual average value (setpoint) and the optimal value (constraint).

Rule 9: “Pair-close” rule: The pairings should be selected such that, effective delays and loop interactions are minimal.

Rule 10: : Avoid using MVs that may optimally saturate (at steady state) to control CVs in the regulatory layer (CV2) The reason is that we want to avoid re-configuring the regulatory control layer. To follow this rule, one needs to consider also other regions of operation than the nominal, for example, operating at maximum capacity (Mode 2) where we usually have more active constraints.

Rules for Step S6: Structure of supervisory control layer. Rule 11: MVs that may optimally saturate (at steady state) should be paired with the subset of CV1 that may be given up. This rule applies for cases when we use decentralized control in the supervisory layer and we want to avoid reconfiguration of loops. The rule follows because when a MV optimally saturates, then there will be one less degree of freedom, so there will be a CV1 which may be given up without any economic loss. The rule should be considered together with rule 10. Example: Gives correct answer for the process where we want to control flow and have p>pmin: Pair the valve (MV) with CV1 (p) which may be given up.

Plantwide control. Main references The following paper summarizes the procedure: S. Skogestad, ``Control structure design for complete chemical plants'', Computers and Chemical Engineering, 28 (1-2), 219-234 (2004). There are many approaches to plantwide control as discussed in the following review paper: T. Larsson and S. Skogestad, ``Plantwide control: A review and a new design procedure'' Modeling, Identification and Control, 21, 209-240 (2000). The following paper updates the procedure: S. Skogestad, ``Economic plantwide control’’, Book chapter in V. Kariwala and V.P. Rangaiah (Eds), Plant-Wide Control: Recent Developments and Applications”, Wiley (2012). More information: http://www.nt.ntnu.no/users/skoge/plantwide All papers available at: http://www.nt.ntnu.no/users/skoge/