1 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 1 Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty
2 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!! 2 Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty
3 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 3 Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty
4 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 4 Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty
5 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 5 Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty
6 Null-space method for optimal operation of transient processes (Ch. 8) We consider a dynamic optimization problem in the form 𝑦 ∈ ℛ 𝑜 𝑦 :=differential states Nominal solution: 𝑣 ∈ ℛ 𝑜 𝑣 :=control inputs • 𝑒 0 , 𝑣 0 , 𝑦 0 , 𝑧 0 y ∈ ℛ 𝑜 𝑧 :=measurements d ∈ ℛ 𝑜 𝑒 :=uncertain parameters 6 Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty
7 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 7 Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty
8 Self-optimizing control Step 1) Find a function of measurements 𝑑 ≔ ℎ(𝑧) whose optimal is invariant to changes in 𝑒 𝑑 𝑝𝑞𝑢 𝑢, 𝑒 0 = 𝑑 𝑝𝑞𝑢 𝑢, 𝑒 1 = ⋯ Step 2) Control c(t) to its reference 𝑑 𝑡 = 𝑑 𝑝𝑞𝑢 (𝑢, 𝑒 0 ) using your favorite controller Step 3) Be optimal without re-optimizing despite uncertainties in 𝑒 8 Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty
9 Optimal sensitivities 𝐺(𝑢) = 𝜖𝑧 𝑝𝑞𝑢 (𝑢, 𝑒) Proposed method 𝜖𝑒 Control a linear combination 𝑑 𝑢 = 𝐼 𝑢 𝑧 𝑢 , (𝐼 is a 𝑜 𝑣 × 𝑜 𝑧 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). 9 Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty
10 Outline of the procedure • Define main uncertainties 𝑒 • Compute nominal solution 𝑒 0 , 𝑣 0 , 𝑦 0 , 𝑧 0 Offline steps • Compute sensitivities 𝐺(𝑢) and the matrix 𝐼(𝑢) • Compute the reference trajectory 𝑑 𝑡 𝑢 = 𝐼 𝑢 𝑧 0 𝑢 • Track references 𝑑 𝑡 using feedback control Online step • By doing so, we are near-optimal without the need for re-optimization, despite d 10 Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty
ሶ ሶ ሶ 11 Simulation example: fed-batch reactor We have two chemical reactions happening u 𝐵 + 𝐶 → 𝐷 and 𝐶 → 𝐸 Subject to the following dynamics 𝑑 𝐵 𝑣 𝑑 𝐵 = −𝑙 1 𝑑 𝐵 𝑑 𝐶 − 𝑊 𝑑 𝐶 −𝑑 𝐶,𝑗𝑜 𝑣 𝑑 𝐶 = −𝑙 1 𝑑 𝐵 𝑑 𝐶 − 2𝑙 2 𝑑 𝐶 − 𝑊 𝑊 = 𝑣 We want to compute to maximize C − 𝐸 Main uncertainties ( 𝑙 1 and 𝑙 2 ) 11 Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty
12 Nominal solution Next steps • Compute sensitivity matrix 𝐺(𝑢) and combination 𝐼(𝑢) • Obtain 𝑑 𝑡 (𝑢) = 𝐼 𝑢 𝑧 0 (𝑢) 12 Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty
13 Example of invariant trajectory 𝑑 𝐵 𝑑 𝐶 𝑑 = [ℎ 1 ℎ 2 ℎ 3 ] 𝑊 Control 𝑑 using a PI controller 13 Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty
14 Results with 20% error in 𝑙 𝟐 and 𝑙 𝟑 Cost comparison 𝐾 𝑝𝑞𝑢 𝐾 𝑞𝑠𝑝𝑞𝑝𝑡𝑓𝑒 𝐾 𝑝𝑞𝑓𝑜𝑚𝑝𝑝𝑞 Open-loop -0.1957 -0.1957 -0.1904 Proposed Optimal Near-optimal operation without re-optimization despite disturbances 14 Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty
16 What you should remember Step 1) Compute reference 𝑑 𝑡 (𝑢) ≔ 𝐼 𝑢 𝑧 0 (𝑢) whose optimal is invariant due to disturbances. We showed how to compute 𝐼(𝑢) . Step 2) Control 𝑑(𝑢) to its reference 𝑑 𝑡 = 𝑑 𝑝𝑞𝑢 (𝑢, 𝑒 0 ) using your favorite controller PID Step 3) Be (almost) optimal without re-optimizing despite uncertainties in 𝑒 16 Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty
17 How could you best use the approach? Combine with EMPC 17 Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty
18 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 18 Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty
19 Motivation Increase of use of renewable energy Strong dependence on weather conditions Energy production must cover demand at all times Influence demand by real-time pricing 19 Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty
20 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/ 20 Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty
21 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 21 Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty
22 Storage capacity is not enough Users are unlikely to Need automatic change their control and behavior optimization Main requirements: • Near-optimal results good savings without sacrifices • Low (computational) cost for widespread use 22 Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty
23 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) 23 Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty
24 Process model Control degrees of freedom ( 𝑣 ) • Electric power: 𝑅 • Inflows: 𝑟 𝑑𝑥 , and 𝑟 𝑗𝑜 Differential variables ( 𝑦 ) • Liquid temperature: 𝑈 • Liquid volume: 𝑊 Algebraic variables ( 𝑧 ) • Hot water temp. 𝑈 ℎ𝑥 • Tank outlet: 𝑟 𝑝𝑣𝑢 Disturbances ( 𝑒 ) • Hot water flow rate: 𝑟 ℎ𝑥 • Hot water temp. setpoint: 𝑈 ℎ𝑥,𝑡𝑞 • Electricity price: 𝑞 24 Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty
ሶ 25 Problem formulation ∞ 𝑞 𝑢 𝑅 𝑢 𝑒𝑢 𝐾 = (𝑓𝑜𝑓𝑠𝑧 𝑑𝑝𝑡𝑢) Minimize: 𝑢 0 subject to: 𝑊 𝑛𝑗𝑜 ≤ 𝑊 ≤ 𝑊 𝑛𝑏𝑦 Satisfy demand at all 𝑈 𝑛𝑗𝑜 ≤ 𝑈 ≤ 𝑈 𝑛𝑏𝑦 times 0 ≤ 𝑅 ≤ 𝑅 𝑛𝑏𝑦 𝑦 = 𝑔(𝑦, 𝑒, 𝑣) Most important constraint for optimization 𝑈 ≥ 𝑈 𝑛𝑗𝑜 = 𝑈 ℎ𝑥,𝑡𝑞 25 Vinicius de Oliveira | Optimal operation strategies for dynamic processes under uncertainty
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