Real-time Optimal Resource Allocation using Online Primal Decomposition
Explore the concept of real-time optimal resource allocation through online primal decomposition in the context of control, optimization, and automation in mining, mineral, and metal processing. This study discusses challenges, examples, and the application of dual decomposition for decentralized optimization. Learn about techniques like Lagrange multipliers and feedback control to address constraints and enhance operational efficiency.
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Real Real- -time Optimal Resource time Optimal Resource Allocation using using Online Primal Online Primal Decomposition Allocation Decomposition Risvan Dirza, Md. Rizwan, Sigurd Skogestad, Dinesh Krishnamoorthy 19th IFAC Symposium on Control, Optimization and Automation in Mining, Mineral and Metal Processing (IFAC MMM 2022) August 15-17, 2022 Montreal, Canada
Background Background Shared Shared resources resources: : Plant Plant E E Shared Shared facility facility Plant Plant E E Steam Electric power Waste handling Plant Plant D D Plant Plant A A Plant Plant A A Challenge: Challenge: Plant Plant B B Plant Plant C C Optimal Allocation when Resource is limited Plant Plant B B 2
Example Example. . Independent Independent oil oil wells wells Optimization problem: Maximize oil revenue while minimizing gas lift injection cost. ????:? = ?????,?? = ??,????,?+ ??,????,? ? ? ? ? Shared resource : Maximum produced gas handling ??? 0 ? = ???,? ??? ? ? Each well operated locally 3
Real Real- -time time optimization optimization (RTO) problem: Optimal Optimal operation operation under under changing (RTO) problem: changing constraints constraints Original problem (constrained): Equivalent Lagrange problem (unconstrained): ?,? ?,?,? = ???,? + ??? ?,? min (2) min ? ???,? = ?????,?? (1?) ? ? ? : primal (manipulated) variables (MVs) ? 0: Lagrange multipliers (dual variables; shadow prices) s.t. ? ?,? = ????,?? 0 (1?) ? ? Necessary Conditions of Optimality (KKT-conditions): ? = ????,? + ?T ?? ?,? = 0 ? ?,? 0 (3?) Stationary Condition Primal Feasibility Dual Feasibility Complementary (3?) ? 0 ?T? ?,? = 0 (3?) (3?) 4
Challenge RTO Centralized optimizer, slow Want to decompose the problem Use feedback control if possible 5
work: Dual : Dual decomposition decomposition[1,2,3] Previous Previous work Central Central Constraint Constraint controller (Lagrange (Lagrange multiplier multiplier update controller update) ) Master Problem ? 0 MAX ? (Dual Var.) ???, ?? Local Local gradient gradient controller ( (primal primal MVs MVs update ????,? + ?T ?? ?,? = 0 controller update) ) Local Problem Local Local ( (dynamic dynamic) ) Estimator Estimator ? (primal MVs) Process Process [1] Krishnamoorthy, D., 2021. A distributed feedback-based online process optimization framework for optimal resource sharing. Journal of Process Control,97, pp.72-83. [2] Dirza, R., et. al. 2021. Optimal Resource Allocation using Distributed Feedback-based Real-time Optimization. IFAC-PapersOnLine54 (3), 706-711. [3] Dirza, R., et. al. 2022. Experimental validation of distributed feedback-based real-time optimization in a gas-lifted oil well rig. Control Engineering Practice, 126, pp. 105253 6
Example Example. Dual . Dual decomposition decomposition Constraint control (master problem): Simple but on SLOW time scale 0 MAX 8
Decomposition Decomposition methods Decomposition methods Dual Dual Decomposition Primal Primal Decomposition Decomposition ?= ????? Price Price ( ?) controllers ) controllers with normal, and normal, and compensator subsystem subsystem Complex with virtual virtual, , compensator Central Central Constraint Constraint controller (Lagrange (Lagrange multiplier multiplier update controller update) ) SLOW Master Problem ? ???? 0 MAX ?? (Aux. Primal Var.) ?? ? (Dual Var.) ???, ?? ?? Local Local constraint constraint controllers ( (primal primal MVs MVs update ????,?? ?? controllers update) ) ??= 0 Local Local gradient gradient controller ( (primal primal MVs MVs update ????,? + ?T ?? ?,? = 0 Complex controller update) ) Local Problem Local Local ( (dynamic dynamic) ) Estimator Estimator Local Local ( (dynamic dynamic) ) Estimator Estimator Local Local ( (dynamic dynamic) ) Estimator Estimator ? (primal MVs) ? (primal MVs) Process Process Process Process 9
Primal Primal decomposition decomposition Consider two (2) subsystems Introduce unused resource, ?0, and auxiliary primal variables ?0 ?? ,?1 ??,?2 ?? Reformulate Resource sharing problem (1): min u1 ,u2 ???,? = ??1u1,d1 + ??2u2,d2 +??0 s.t. ?1u1,d1 ?1 ?2u2,d2 ?2 (3?) ??= 0, ??= 0, (3?) (3?) ??+ ?1 ??+ ?2 ?? ????= 0 ?0 (3?) 10
Local Local constraint constraint problem problem ?? and ?2 ?? Given ?1 Local Problem Subsystem ? (for ? = 1,2 ): ??= 0 ??u?,d? ?? Local Setpoint Controller (for ? = 1,2 ): Discrete I-controller, But in simulation we use PI ??,?+1= u? ??,?+ ??,??? ??,? ? ?? u? Remark This controller is not necessary if you have input shared constraint 11
Estimate Estimate local multipliers multipliers ?? Given ?1 For what value of ??is the local problem optimal? local Lagrange Lagrange ?? and ?2 ?? ?? ???u?,d? ?? ? u??? = 0 ?= ???u?,d? + ?? Local Lagrange function: Local optimality condition: u? ?= u???+ ?? Estimate Local Lagrange Multiplier (for ? = 1,2 ) [4] : 1 ??= u??? u??? [4] Dirza, R., et. al. 2021. Real-Time Optimal Resource Allocation and Constraint Negotiation Applied to A Subsea Oil Production Network. SPE ATCE)2021. Dubai, United Arab Emirates.. 12
Master problem Master problem All Lagrange multipliers ??should be equal Discrete I-controller Master Controller 1 (normal Master Controller 1 (normal subsystem subsystem): ): ??,?+1= ?1 ??,?+ ??,1 ?1 ?+ ?2 ? ?1 Master Controller N=2 ( Master Controller N=2 (compensator compensator subsystem ?2 subsystem) : ) : ??,?+1= ???? ?0 ??,? ?1 ??,? Master Controller 0 ( Master Controller 0 (virtual virtual subsystem subsystem for ?0 for unused unused resource ??,?+ ??,0 ?2 resource): ): ?] ??,?+1= max[0,?0 Unconstrained case: Want ?? = 0 to get u???= 0 Constrained case: Unused resources ?0 ?? = 0, and ?? 0 13
Summary Summary Primal Primal Decomposition Decomposition Master Controller Master Controller ??,?+1= ?1 ??,?+ ??,1 ?1 ?+ ?? ? ?1 ??,?+1= max[0,?0 ??,?+ ??,0 ?? ?] ?0 ? 1 ??,?+1= ???? ??,? ?? ?? ?=0 ?? 1 ?? 1 ?1 ??= u??? u??? ?? ?1= u1?1 u1?1 Setpoint controller SS 1 Setpoint controller SS N ?1 ?2 EKF + Gradient Estimator EKF + Gradient Estimator u1 uN yN y1 dN d1 Subsystem (SS) 1 Subsystem (SS) N 14
Time Time scale scale separation separation Cascade structure: Master controllers provide setpoints for Local controllers. Time scale separation is required [3,6,7] ???,master ???,local ; ??? is closed-loop time constant. Simulations: ???,local = 75 s = 1.25 min ???,master= 600 ? = 10 ??? 75 600 = 0.125 [3,6,7] ? = ? = [3] Dirza, R., et. al. 2022. Experimental validation of distributed feedback-based real-time optimization in a gas-lifted oil well rig. Control Engineering Practice, 126, pp. 105253 [6] Baldea, M., and Daoutidis, P., 2007. Control of integrated process networks A multi-time scale perspective. Computers & Chemical Engineering, 31 (5) (2007), pp. 426-444 [7] Skogestad., S., and Postlethwaite., I., 2005. Multivariable feedback control, John Wiley and Sons, New York, NY, USA (2005) 15
Simulation Simulation Example Example Case Case study study: : Optimal resource/constraint sharing problem: w??,? , ? ? ??= min ??,????,?+ ??,????,? (1?) ? ? ??? ??? ??? 0 s.t. ? = ???,? ??? (1?) ?? ? ? GOR Disturbances Gas-oil-ratio (GOR) Manifold Pressure Maximum produced gas treatment capacity GOR ?? Primal decomposition with PI-contollers ( ) ??? ??? 17
Primal Primal Dual Dual h Time scale separation: ? = 0.125 18
Primal Primal Dual Dual with with Constraint Constraint Override Override Time scale separation: ? = 0.125 19
Dual with constraint override g Primal-dual Feedback-optimizing Control with Direct Constraint Control Risvan Dirza, Dinesh Krishnamoorthy, and Sigurd Skogestad PSE-2021+, Kyoto (2022) 20
Conclusion Conclusion RTO Problem: Optimal operation under changing constraints Want simpler, decomposed implementation of RTO Dual decomposition works well but controls the constraint on the slow time scale Alternative for shared resource constraint (this talk): Primal decomposition Yes, primal gives better constraint control But otherwise poorer performance than dual And much less general: Need one MV in each subsystem for each constraint Proposed solution: Dual decomposition with constraint override Similar to primal for constraint satisfaction Applies to any number of constraints 23
