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1 Optimal operation strategies for dynamic processes under uncertainty Public PhD Defence Candidate : Vinicius de Oliveira Supervisors: Sigurd Skogestad Johannes Jschke Department of Chemical Engineering, Faculty of Natural Sciences and

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Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty

Optimal operation strategies for dynamic processes under uncertainty

Public PhD Defence

Candidate: Vinicius de Oliveira

Supervisors: Sigurd Skogestad Johannes Jäschke

Department of Chemical Engineering, Faculty of Natural Sciences and Technology NTNU, Trondheim, Norway

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Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty

Main goal

Find implementation strategies for the optimal operation of processes during transients  Focus on cases where dynamic behavior is important in terms of economic performance We are not only interested in finding (numerical) optimal solutions  but specially in the practical implementation strategies using feedback control  Challenge: disturbances and uncertainties!!

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Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty

Main question

How to achieve acceptable performance in the face of unknown disturbances and uncertainties? By acceptable we mean:

  • Near-optimal economic cost
  • stable operation
  • minimum constraint violations

Our focus is to find simple policies to achieve this goal

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Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty

Presentation outline

Introduction Near-optimal operation of uncertain batch systems

 Chapters 7 and 8

Optimal operation of energy storage systems

 Chapters 2, 3 and 4

Optimal operation of dynamic systems at their stability limit: anti-slug control system for oil production optimization

 Chapters 5 and 6

Concluding remarks

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Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty

Presentation outline

Introduction Near-optimal operation of uncertain batch systems

 Chapters 7 and 8

Optimal operation of energy storage systems

 Chapters 2, 3 and 4

Optimal operation of dynamic systems at their stability limit: Application to anti-slug control

 Chapters 5 and 6

Concluding remarks

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Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty

Null-space method for optimal operation

  • f transient processes (Ch. 8)

We consider a dynamic optimization problem in the form

𝑦 ∈ ℛ𝑜𝑦 :=differential states 𝑣 ∈ ℛ𝑜𝑣 :=control inputs y ∈ ℛ𝑜𝑧 :=measurements d ∈ ℛ𝑜𝑒 :=uncertain parameters

Nominal solution:

  • 𝑒0, 𝑣0, 𝑦0, 𝑧0
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Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty

Main goal

Achieve near-optimal economic performance despite uncertainty/disturbances without the need for re-optimization*

(*) Solving dynamic optimization problems can be veeery time- consuming

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Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty

Self-optimizing control

Step 3) Be optimal without re-optimizing despite uncertainties in 𝑒

Step 1) Find a function of measurements 𝑑 ≔ ℎ(𝑧) whose

  • ptimal is invariant to changes in 𝑒

𝑑𝑝𝑞𝑢 𝑢, 𝑒0 = 𝑑𝑝𝑞𝑢 𝑢, 𝑒1 = ⋯ Step 2) Control c(t) to its reference 𝑑𝑡 = 𝑑𝑝𝑞𝑢(𝑢, 𝑒0) using your favorite controller

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Proposed method

Control a linear combination 𝑑 𝑢 = 𝐼 𝑢 𝑧 𝑢 , (𝐼 is a 𝑜𝑣 × 𝑜𝑧 matrix ) This is the (local) optimal choice if 𝐼(𝑢)𝐺(𝑢) = 0 𝐼(𝑢) must lie in the left nullspace of 𝐺 𝑢 ∗  Thus the name, ‘Nullspace method’

(*) Nullspace method for steady-state problems originally published in Alstad (2007).

Optimal sensitivities 𝐺(𝑢) = 𝜖𝑧𝑝𝑞𝑢(𝑢, 𝑒) 𝜖𝑒

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Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty

Outline of the procedure

Offline steps

  • Define main uncertainties 𝑒
  • Compute nominal solution 𝑒0, 𝑣0, 𝑦0, 𝑧0
  • Compute sensitivities 𝐺(𝑢) and the matrix 𝐼(𝑢)
  • Compute the reference trajectory 𝑑𝑡 𝑢 = 𝐼 𝑢 𝑧0 𝑢

Online step

  • Track references 𝑑𝑡 using feedback control
  • By doing so, we are near-optimal without the need

for re-optimization, despite d

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Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty

Simulation example: fed-batch reactor

u

We have two chemical reactions happening 𝐵 + 𝐶 → 𝐷 and 𝐶 → 𝐸 Subject to the following dynamics ሶ 𝑑𝐵 = −𝑙1𝑑𝐵𝑑𝐶 −

𝑑𝐵𝑣 𝑊

ሶ 𝑑𝐶 = −𝑙1𝑑𝐵𝑑𝐶 − 2𝑙2𝑑𝐶 −

𝑑𝐶−𝑑𝐶,𝑗𝑜 𝑣 𝑊

ሶ 𝑊 = 𝑣

We want to compute to maximize C − 𝐸 Main uncertainties (𝑙1and 𝑙2)

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Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty

Nominal solution

Next steps

  • Compute sensitivity matrix

𝐺(𝑢) and combination 𝐼(𝑢)

  • Obtain 𝑑𝑡(𝑢) = 𝐼 𝑢 𝑧0(𝑢)
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Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty

Example of invariant trajectory

𝑑 = [ℎ1 ℎ2 ℎ3 ] 𝑑𝐵 𝑑𝐶 𝑊 Control 𝑑 using a PI controller

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Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty

Results with 20% error in 𝑙𝟐 and 𝑙𝟑

Open-loop Proposed Optimal Cost comparison 𝐾𝑝𝑞𝑢 𝐾𝑞𝑠𝑝𝑞𝑝𝑡𝑓𝑒 𝐾𝑝𝑞𝑓𝑜𝑚𝑝𝑝𝑞

  • 0.1957
  • 0.1957
  • 0.1904

Near-optimal operation without re-optimization despite disturbances

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What you should remember

Step 3) Be (almost) optimal without re-optimizing despite uncertainties in 𝑒

Step 1) Compute reference 𝑑𝑡(𝑢) ≔ 𝐼 𝑢 𝑧0(𝑢) whose optimal is invariant due to disturbances. We showed how to compute 𝐼(𝑢).

Step 2) Control 𝑑(𝑢) to its reference 𝑑𝑡 = 𝑑𝑝𝑞𝑢(𝑢, 𝑒0) using your favorite controller

PID

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How could you best use the approach? Combine with EMPC

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Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty

Presentation outline

Introduction Near-optimal operation of uncertain batch systems

 Chapters 7 and 8

Optimal operation of energy storage systems

 Chapters 2, 3 and 4

Optimal operation of dynamic systems at their stability limit: anti- slug control system for oil production optimization

 Chapters 5 and 6

Concluding remarks

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Motivation

Increase of use of renewable energy Strong dependence

  • n weather

conditions Energy production must cover demand at all times

Influence demand by real-time pricing

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Example of electricity price in Norway*

Future trend: use of smart meters Consumer charged in a hourly basis Electricity price available in real-time

(*) http://www.nordpoolspot.com/

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How can end-user take advantage of this scenario?

Key requirement: energy storage

  • Allows us to move the consumption to more

favorable periods → flexible consumption

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Storage capacity is not enough

Main requirements:

  • Near-optimal results good savings without sacrifices
  • Low (computational) cost for widespread use

Users are unlikely to change their behavior Need automatic control and

  • ptimization
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Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty

Some examples of energy storage

  • Batteries
  • Ice banks
  • Building's mass (Topic of Ch. 4)
  • Compressed air storage
  • Hot-water tanks (Topic of Ch. 2 and 3)
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Process model

Control degrees of freedom (𝑣)

  • Electric power: 𝑅
  • Inflows: 𝑟𝑑𝑥, and 𝑟𝑗𝑜

Differential variables (𝑦)

  • Liquid temperature: 𝑈
  • Liquid volume: 𝑊

Algebraic variables (𝑧)

  • Hot water temp. 𝑈

ℎ𝑥

  • Tank outlet: 𝑟𝑝𝑣𝑢

Disturbances (𝑒)

  • Hot water flow rate: 𝑟ℎ𝑥
  • Hot water temp. setpoint: 𝑈

ℎ𝑥,𝑡𝑞

  • Electricity price: 𝑞
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Problem formulation

Minimize: 𝐾 = ׬

𝑢0 ∞ 𝑞 𝑢 𝑅 𝑢 𝑒𝑢

(𝑓𝑜𝑓𝑠𝑕𝑧 𝑑𝑝𝑡𝑢) subject to: 𝑊

𝑛𝑗𝑜 ≤ 𝑊 ≤ 𝑊 𝑛𝑏𝑦

𝑈𝑛𝑗𝑜 ≤ 𝑈 ≤ 𝑈

𝑛𝑏𝑦

0 ≤ 𝑅 ≤ 𝑅𝑛𝑏𝑦 ሶ 𝑦 = 𝑔(𝑦, 𝑒, 𝑣) Satisfy demand at all times Most important constraint for optimization 𝑈 ≥ 𝑈𝑛𝑗𝑜 = 𝑈ℎ𝑥,𝑡𝑞

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Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty

Main complications:

  • Time-varying electricity price 𝑞(𝑢)
  • Time-varying and highly uncertain hot water demand 𝑟ℎ𝑥
  • Nonlinear dynamics

Demand varies in a fast time-scale (s-min)  need fast sampling time Economics evolve in a slower pace (hours- days) need long horizon

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Popular at the moment: Economic Model Predictive Control (EMPC)

Combine optimization and control in one big layer Computational cost very high!

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Proposed hierarchical control structure

Great simplification of the problem is achieved with this structure

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Great simplification of the problem by

Right choice of DoF for the optimization. Use of time-scale separation Make use of periodic behavior of problem

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Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty

Optimization layer problem formulation

Right choice of decision variables

We use the concept of energy storage 𝐹 = 𝜍𝑑𝑞𝑊(𝑈 − 𝑈

𝑑𝑥)

Because of the choice of reference temp (𝑈0 = 𝑈

𝑑𝑥), 𝑟𝑗𝑜 does

not affect 𝐹  Reduction of # of degrees of freedom Using 𝐹(𝑢) as our decision variable  problem becomes linear

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Optimization layer problem formulation

time-scale separation

Disturbances can be split into two frequency components 𝑒 = 𝑒𝑡𝑚𝑝𝑥 + ∆𝑒𝑔𝑏𝑡𝑢 Assume 𝑒𝑡𝑚𝑝𝑥 is more important for the economics  e.g. electricity price (hours)

  • Optimize 𝐹 according to 𝑒𝑡𝑚𝑝𝑥 time-scale
  • Use feedback control to reject fast variations

∆𝑒𝑔𝑏𝑡𝑢

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Optimization layer problem formulation

Take advantage of the periodicity of the problem

Add a final constraint 𝐹 𝑢𝑔 = 𝐹𝑛𝑏𝑦 This constraint the

  • ptimization problem
  • f two consecutive

days

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Proposed formulation (in terms of energy storage)

Linear program (LP) + Small number of decision variables

Very low computational cost

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Controlled variable selection

Optimization DoF:

𝐹 = 𝜍𝑑𝑞𝑊(𝑈 − 𝑈

𝑑𝑥)

Obvious CV candidates:

  • Liquid volume 𝑊
  • Liquid temperature 𝑈
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Case study

Compare our approach with:

  • Maximum storage policy

(full tank all the time)

  • Ideal case (assume

perfect knowledge of the future)

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Alternative strategy:

Maximum storage policy:

  • 𝑈

𝑡 = 𝑈 𝑛𝑏𝑦

  • 𝑊

𝑡 = 𝑊 𝑛𝑏𝑦

Safest policy in terms of constraint violations

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Ideal case

EMPC with perfect knowledge about the future Not achievable in practice

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Results

Considerable savings at low computational cost

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Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty

Results

Considerable savings at low computational cost

Increased performance by increasing

  • ptimization frequency
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Great simplification of the problem by

Right choice of DoF for the

  • ptimization. Use

process insight.

  • Introduction of energy storage 𝐹 allows a linear

formulation

  • Fewer decision variables since water refilling 𝑟𝑗𝑜 has

no effect in 𝐹 Use of time-scale separation

  • Energy storage 𝐹 varies in a slower time-scale

compared to heat input 𝑅

  • Control layer takes of fast varying disturbances and

handle constraints Make use of periodic behavior

  • f problem
  • We add a constraint 𝐹 = 𝐹𝑛𝑏𝑦 late in the night.
  • Decouples the optimization problem of two

consecutive days

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Main benefits

  • Optimal operation
  • Minimum modeling efforts
  • Very low computational cost suitable for embedded

hardware

Low-cost solutions Enable widespread usage of energy storage

Ease integration of renewable energy sources into the grid

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Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty

Presentation outline

Introduction Near-optimal operation of uncertain batch systems

 Chapters 7 and 8

Optimal operation of energy storage systems

 Chapters 2, 3 and 4

Optimal operation of dynamic systems at their stability limit: anti- slug control system for oil production optimization

 Chapters 5 and 6

Concluding remarks

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The big picture

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The slug cycle

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The slug cycle (video)

Experiments performed by the Multiphase Laboratory, NTNU

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p1 p2 z

Slug cycle (stable limit cycle)

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Problems caused by severe slugging

  • Large disturbances in the separators

– Causing poor separation performance – Can cause total plant shutdown  production losses! – Increase flaring.

  • Large and rapid variation in compressor load
  • Limits production capacity (increase pressure in pipeline)
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p1 p2 z

Avoid slugging: Close valve (but increases pressure)

Problematic for aging fields  increased pressure limits production No slugging when valve is closed

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Avoid slugging: ”Active” feedback control

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Anti slug control: Full-scale offshore experiments at Hod-Vallhall field (Havre,1999)

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Problems with current anti-slug control systems

  • Tend to become unstable (oscillating) after some time

– Inflow conditions change – Require frequent retuning by an expert  costly

  • Ideal operating point (pressure set-point) is unknown

– If pressure setpoint is too high  production is reduced – If pressure setpoint is too low  system may become unstable

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Motivation

We want to increase valve

  • pening

But larger

  • penings = worse

controllability

  • The lager the valve opening the more difficult it is to stabilize the system

– Controller gets more sensitive to uncertainties – Process gain is reduced

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Our proposed autonomous control system

Setpoint change is key for the adaptation to work well

Robust adaptive control Plant Autonomous supervisor

  • Periodically checks the stability of the system
  • Reduces setpoint if control loop is working fine

𝑄 𝑎 𝑄

𝑡𝑞

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How does it work?

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Adaptive control based on adaptive augmentation

Relies on state-of-the-art output feedback adaptive control techniques

 Very successful in the aerospace industry

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Adaptive control design

Open-loop system dynamics

ሶ x = 𝐵𝑦 + 𝐶Λ 𝑣 + Θ𝑈Φ 𝑦 𝑧𝑛𝑓𝑏𝑡 = 𝐷𝑦

Uncertainty model Λ → control effectiveness uncertainty. Affects the process gain Θ𝑈Φ 𝑦 → state-dependent nonlinear uncertainty. Affects poles and zeros Θ → matrix of unknown coefficients Φ 𝑦 → vector of Lipschitz basis functions

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Adaptive control design

Define reference model ሶ ො 𝑦 = 𝐵𝑠𝑓𝑔 ො 𝑦 + 𝐶𝑠𝑓𝑔𝑠 + 𝑴𝒘(𝒛 − ෝ 𝒛) Output Feedback Adaptive Laws

  • ሶ

෡ Θ = ΓΘProj(෡ Θ, Φ ො 𝑦, 𝑣𝑐𝑚 𝑧 − ො 𝑧 𝑈)

  • ሶ

෡ 𝐿𝑣 = ΓuProj(𝐿𝑣, 𝑣𝑐𝑚 𝑧 − ො 𝑧 𝑈)

  • 𝑣𝑏𝑒𝑏𝑞𝑢𝑗𝑤𝑓 = −෡

𝐿𝑣𝑣𝑐𝑏𝑡𝑓𝑚𝑗𝑜𝑓 − ෡ Θ𝑈Φ(𝑦) Robust baseline + adaptive output feedback

𝑣 = 𝑣𝑐𝑏𝑡𝑓𝑚𝑗𝑜𝑓 + 𝑣𝑏𝑒𝑏𝑞𝑢𝑗𝑤𝑓

  • 𝑣𝑐𝑏𝑡𝑓𝑚𝑗𝑜𝑓 computed using your favorite method (PID, 𝐼∞, LQG/LTR, …)

Feedback term to improve transient dynamics

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How does it perform in practice?

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2009-2013: Esmaeil Jahanshahi, PhD-work supported by Siemens

Experimental mini-rig

Pump Buffer Tank Water Reservoir Seperator Air to atm. Mixing Point safety valve P1 Pipeline Riser Subsea Valve Top-side Valve Water Recycle FT water FT air P3 P4 P2

3m

  • its dynamical behavior is

quite similar to that of much larger rigs

water+air mixture

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Experimental Results

  • Baseline controller tuned for Z=30%
  • Linearized mechanistic or simple empirical models

can be used

Note: our models agree very well with experiments

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Autonomous supervisor and adaptive LTR controller Safely operates at very large valve openings

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Autonomous supervisor and adaptive LTR controller

Adaptation gains

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Oops, Big disturbance! Emulates a ‘gas-to-oil’ ratio change

  • ver 60%
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Large change in the operating conditions

Supervisor quickly detects major disturbance Moves to safer

  • perating point

Adaptive control stabilizes under new

  • perating conditions
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What happens if the baseline controller is poorly tuned?

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Poorly tuned PI control as baseline: Adaptation is OFF

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Poorly tuned PI control as baseline: Adaptation is ON

  • 1. Supervisor quickly detects major disturbance

Desired closed-loop performance is recovered!

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Comparison

Case Mean valve

  • pening

ISE Bad baseline + adaptation OFF 38,45 % 6,2 Bad baseline + adaptation ON 50,42% 0,76 Good baseline + adaptation ON 53,23% 0,64

Large is good

Small is good

𝐽𝑇𝐹 = ׬ 𝑓2𝑒𝑢

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Take home message

  • Our 2-layered anti-slug control system works very well in

practice

  • The interaction between the two layers create a very nice

synergy:

Setpoint changes triggered by the supervisor makes the adaptation work well A well functioning adaptive control makes it possible to safely operate at large valve openings, thus maximizing production

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Take home message Expected benefits

  • Stable and safe operation in a wide range of

conditions

  • Reduced need for control tuning
  • Reduced workload on operators
  • Increased production
  • This work resulted in a patent application
  • Cooperation agreement with industrial partner on the

way

  • Industrial pilot project (hopefully) coming soon
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Presentation outline

Introduction Near-optimal operation of uncertain batch systems

 Chapters 7 and 8

Optimal operation of energy storage systems

 Chapters 2, 3 and 4

Optimal operation of dynamic systems at their stability limit: anti-slug control system for oil production optimization

 Chapters 5 and 6

Concluding remarks

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Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty

Concluding remarks

We have seen different strategies for near-optimal operation under uncertainty:

  • Null-space method for batch processes
  • Simplified optimization scheme of energy storage

systems based on a hierarchical control structure

  • Intelligent adaptive anti-slug control system for oil

production maximization

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Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty

Thank you for your attention

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Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty

Not included in the presentation

  • Ch. 4: Dynamic online optimization of a house heating system

in a fluctuating energy price scenario.

  • Ch. 6: A comparison between Internal Model Control, optimal

PIDF and robust controllers for unstable flow in risers.

  • Ch. 7: Neighbouring-Extremal Control for Steady-State

Optimization Using Noisy Measurements.