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1 Click to edit Master title style APOPT and BPOPT: New NLP and MINLP Solvers for Differential Algebraic Systems with Benchmark Testing John D. Hedengren Jose L. Mojica Brigham Young University Kody M. Powell Wesley Cole Thomas F. Edgar The University of Teas at Austin

2 Presentation Overview Overview of Benchmarking Testing. NLP Benchmarking Hock-Schittkowski Dynamic optimization Biological models MINLP Benchmarking MacMINLP Case studies Gravity Drained tank (MIDAE control problem) Energy Storage for smart grid Future Developments Oct

3 Oct Overview of Benchmark Testing NLP Benchmark Testing APOPT 1 BPOPT 1 IPOPT 2 SNOPT 3 MINOS 3 Problem characteristics: Hock Schittkowski Dynamic Opt SBML Nonlinear Programming (NLP) Differential Algebraic Equations (DAEs) APMonitor Modeling Language MINLP Benchmark Testing APOPT 1 BPOPT 1 BONMIN 2 DICOPT 4 Problem characteristics: MacMINLP Industrial Test Set Mied Integer Nonlinear Programming (MINLP) Mied Integer Differential Algebraic Equations (MIDAEs) APMonitor & AMPL Modeling Language m n u y u y h u y g u y t f s t u y J ) ( ) (.. ) ( min 1 APS LLC 2 EPL 3 SBS Inc. 4-CMU m m n z u y z u y h z u y g z u y t f s t z u y J ) ( ) (.. ) ( min

4 NLP Benchmark Hock-Schittkowski (116) Percentage (%) APOPT+BPOPT APOPT 1. BPOPT 1. IPOPT 3.1 IPOPT 2.3 SNOPT 6.1 MINOS Not worse than 2 times slower than the best solver () Oct

5 NLP Benchmark - Dynamic Optimization (37) Percentage (%) APOPT+BPOPT APOPT 1. BPOPT 1. IPOPT 3.1 IPOPT 2.3 SNOPT 6.1 MINOS Not worse than 2 times slower than the best solver () Oct

6 NLP Benchmarking SBML (341) Percentage (%) # of Models APOPT+BPOPT APOPT 1. BPOPT 1. IPOPT 3.1 IPOPT 2.3 SNOPT # of Physical Entities MINOS Not worse than 2 times slower than the best solver () Oct

7 Eample: Click ErbB to Signal edit Master Transduction title style Cascade Large ErbB signalling model (~54 physical entities)* Parameter estimation (simulated annealing) took 24 hours on a 1-node cluster computer *Chen et al. Mol Syst Biol. 29;5:239.

8 NLP Benchmark Summary (494) 1 9 Percentage (%) APOPT+BPOPT APOPT 1. BPOPT 1. IPOPT 3.1 IPOPT 2.3 SNOPT 6.1 MINOS Not worse than 2 times slower than the best solver () Oct

9 NLP Benchmark Combo Solvers (494) APOPT+BPOPT APOPT+IPOPT 3 APOPT+IPOPT 2 7 APOPT+SNOPT APOPT+MINOS BPOPT+IPOPT 3 Percentage (%) BPOPT+IPOPT 2 BPOPT+SNOPT BPOPT+MINOS IPOPT 3 +IPOPT 2 IPOPT 3 +SNOPT IPOPT 3 +MINOS IPOPT 2 +SNOPT IPOPT 2 +MINOS SNOPT+MINOS APOPT 1. 3 BPOPT 1. IPOPT 3.1 IPOPT SNOPT 6.1 MINOS Not worse than 2 times slower than the best solver () Oct

10 MacMINLP Benchmark Summary Results Variables Integers Equations NLP Obj MINLP Obj Solution Time Status E E+2.31 Success an_integer_test E E+5.25 Success batch E E 7.16 Success geartrain E+ 1.3E+1.94 Success mittelman E 1 6.1E+.31 Success synthes E+ 7.3E+1.47 Success synthes E E+1.62 Success synthes E E 1.31 Success wind fac E+ 1.1E Success c reload 14a E+ 1.2E Success c reload 14b E E Failure c reload 14c E+ 1.3E Success c reload 14d E+ 1.3E Success c reload 14e E+ 1.1E Success c reload 14f Oct

11 Case Study- MPC with MINLP PCT4 is a basic process control unit produced by Armfield Main Devices: Large Process Vessel CSTR Hot water Tank Sensors: temperature pressure and level sensors Devices: valves heating coils and pumps How does this work for MINLP Problems? Objective: Keep the tank level at 1 mm Continuous: Proportional Solenoid Valve (PSV) Discrete: On/Off Valve (SOL1) Oct

12 MINLP Model Predictive Control MIDAE System Variables: 576 Integers: 16 Equations: 544 Height (mm) Target Height Actual Height Features Warm Start Solutions Discrete (16) and Continuous (16) Decisions CPU Time (sec) Total Time Solver Time CPU Increases with Setpoint Changes Number of Cycles Cycle 5 Total CPU Time Oct CPU Time (sec)

13 Results set point changes Set Point Changes Large (5) ON Medium(2) ON Small (5) ON Very Small OFF Set Point Set Change Point Change Needs another constraint to limit SOL1 use. Oct

14 Results-disturbances Disturbance Opened up an eit valve which is not part of the model Reduced Solenoid fluctuations but still a problem add cost to opening SOL 1 Oct

15 Smart Grid Energy Systems Oct

16 Thermal Energy Storage (TES) Oct

17 Chiller-TES System 53 F 85 F 95 F MINLP System Variables: 672 Integers: 24 Equations: 576 Features Cold Start Solution Discrete (24) and Continuous (24) Decisions 4 F 53 F Oct

18 Problem Formulation (MINLP) Subject to: i t f J c () t P () t dt elec i chiller i building i chiller i yl L TES P y f L WBT f is nonlinear chiller i i chiller i detes TES dt ETES 8 2 TES L 1312 y i 1 chiller i Oct

19 Problem Formulation (Relaed) Subject to: 24 min elec i chiller1 i chiller 2 i i1 J c P P L L L TES chiller1 i chiller 2 i building i i P f L WBT f is nonlinear chiller1 i chiller1 i i P f L WBT chiller 2 i chiller 2 i i E E TES TES i TES i1 i E 8 TES i 2 TES 2 i Solved using SQP algorithm Oct

20 Relaing Binary Constraints Mied integer problem doubles # variables Branch and bound technique significantly increases computation time Continuous formulation? Power Consumed (kw) Power Consumed/Load (kw/ton) Load (tons) Oct Load (tons) 2

21 Results BONMIN APOPT Ecel MATLAB Cost ($/day) $ $ $ 81.93* $ 82.13* CPU Time (sec) *.1* TES (tons) Ecel Solution APM Solution Time (hours) * = Relaed Formulation (NLP) Oct

22 Optimization of investment decision in energy systems Can we apply a similar MPC with MINLP approach to optimize investment decisions for net generation energy systems? Oct

23 Future Development Most fleible and powerful platform for DAEs Need collaborators to create a new interfaces. Development influenced by multiple domains: Computational Biology Aerospace Engineering (UAVs) Petrochemical Industry Smart Grid Modeling and Optimization Solver Development Mied Integer Nonlinear Programming for DAEs (MIDAEs) Combine strengths of Active Set (APOPT) and Interior Point (BPOPT) methods Oct