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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 Texas at Austin 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 17, 2012 APMonitor.com | APOPT.com BYU – UT Austin Overview of Benchmark Testing
NLP Benchmark Testing 1 1 2 3 3 min J (x, y,u) APOPT , BPOPT , IPOPT , SNOPT , MINOS x Problem characteristics: s.t. 0 f , x, y,u t Hock Schittkowski, Dynamic Opt, SBML 0 g(x, y,u) Nonlinear Programming (NLP)
Differential Algebraic Equations (DAEs) 0 h(x, y,u) n m APMonitor Modeling Language x, y u
MINLP Benchmark Testing min J (x, y,u, z) 1 1 2 4 APOPT , BPOPT , BONMIN , DICOPT x s.t. 0 f , x, y,u, z Problem characteristics: t
MacMINLP, Industrial Test Set 0 g(x, y,u, z) Mixed Integer Nonlinear Programming (MINLP) 0 h(x, y,u, z) Mixed Integer Differential Algebraic Equations (MIDAEs) x, y n u m z m APMonitor & AMPL Modeling Language
1–APS, LLC 2–EPL, 3–SBS, Inc., 4-CMU Oct 17, 2012 APMonitor.com | APOPT.com BYU – UT Austin NLP Benchmark – Hock-Schittkowski (116)
100
90
80
70
60 APOPT+BPOPT APOPT 50 1.0 BPOPT 40 1.0 Percentage (%) IPOPT 3.10 30 IPOPT 2.3 SNOPT 20 6.1 MINOS 5.5 10
0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Not worse than 2 times slower than the best solver ()
Oct 17, 2012 APMonitor.com | APOPT.com BYU – UT Austin NLP Benchmark - Dynamic Optimization (37)
100
90
80
70
60 APOPT+BPOPT APOPT 1.0 50 BPOPT 1.0 IPOPT 40 3.10 Percentage (%) IPOPT 2.3 30 SNOPT 6.1 MINOS 20 5.5
10
0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Not worse than 2 times slower than the best solver ()
Oct 17, 2012 APMonitor.com | APOPT.com BYU – UT Austin NLP Benchmarking – SBML (341)
100
90
80
70 120 APOPT+BPOPT 100 APOPT 1.0 60 80 BPOPT
Models 60 1.0
of 40 IPOPT 50 # 3.10 20
Percentage (%) IPOPT 2.3 40 0 SNOPT 6.1 MINOS 30 # of Physical Entities 5.5
20
10 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Not worse than 2 times slower than the best solver ()
Oct 17, 2012 APMonitor.com | APOPT.com BYU – UT Austin Example:Click ErbB to Signaledit Master Transduction title style Cascade
Large ErbB signalling model (~504 physical entities)*
Parameter estimation (simulated annealing) took “24 hours on a 100-node cluster computer”
*Chen et al. Mol Syst Biol. 2009;5:239 . NLP Benchmark – Summary (494)
100
90
80 APOPT+BPOPT APOPT 70 1.0 BPOPT 1.0 60 IPOPT 3.10 IPOPT 50 2.3 SNOPT
Percentage (%) 6.1 40 MINOS 5.5
30
20
10 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Not worse than 2 times slower than the best solver ()
Oct 17, 2012 APMonitor.com | APOPT.com BYU – UT Austin NLP Benchmark – Combo Solvers (494)
100
90
APOPT+BPOPT 80 APOPT+IPOPT 3 APOPT+IPOPT 2 APOPT+SNOPT 70 APOPT+MINOS BPOPT+IPOPT 3 BPOPT+IPOPT 2 60 BPOPT+SNOPT BPOPT+MINOS IPOPT +IPOPT 3 2 IPOPT +SNOPT 50 3 Percentage (%) IPOPT +MINOS 3 IPOPT +SNOPT 2 IPOPT +MINOS 40 2 SNOPT+MINOS APOPT 1.0 BPOPT 30 1.0 IPOPT 3.10 IPOPT 2.3 SNOPT 20 6.1 MINOS 5.5
10 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Not worse than 2 times slower than the best solver ()
Oct 17, 2012 APMonitor.com | APOPT.com BYU – UT Austin MacMINLP Benchmark Summary Results
Variables Integers Equations NLP Obj MINLP Obj Solution Time Status 10 4 4 1.55E+02 1.74E+02 0.031 Success an_integer_test 107 24 73 2.59E+05 2.86E+05 0.25 Success batch 44 06.30E‐13 7.78E‐07 0.016 Success geartrain1 27 16 7 5.02E+00 1.30E+01 0.094 Success mittelman 12 3 6 7.59E‐01 6.01E+00 0.031 Success synthes1 23 5 14 4.18E+00 7.30E+01 0.047 Success synthes2 38 8 24 1.65E+01 7.57E+01 0.062 Success synthes3 63 51.18E‐01 2.54E‐01 0.031 Success wind‐fac 510 168 308 ‐1.01E+00 ‐1.01E+00 8.865 Success c‐reload‐14a 510 168 308 ‐1.02E+00 ‐1.02E+00 13.728 Success c‐reload‐14b 510 168 308 ‐9.91E‐01 ‐9.56E‐01 48.344 Failure c‐reload‐14c 510 168 308 ‐1.03E+00 ‐1.03E+00 6.724 Success c‐reload‐14d 510 168 308 ‐1.03E+00 ‐1.03E+00 4.852 Success c‐reload‐14e 510 168 308 ‐1.01E+00 ‐1.01E+00 6.614 Success c‐reload‐14f
Oct 17, 2012 APMonitor.com | APOPT.com BYU – UT Austin Case Study- MPC with MINLP
PCT40 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 100 mm
Continuous: Proportional Solenoid Valve (PSV) Discrete: On/Off Valve (SOL1)
Oct 17, 2012 APMonitor.com | APOPT.com BYU – UT Austin MINLP Model Predictive Control
250 Target Height Actual Height MIDAE System 200 Variables: 576 150
Integers: 16 100 Height (mm) Height Equations: 544 50
0 0 50 100 150 200 250 300 350 400 450 500 Features
1 Warm Start Solutions 10 Total Time Solver Time 0 Discrete (16) and 10 Continuous (16) -1 10
Decisions CPU Time(sec)
-2 10 0 50 100 150 200 250 300 350 400 450 500 CPU Increases with Cycle Setpoint Changes 500 Total CPU Time 400
300
200
Number of Cycles 100
0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 CPU Time (sec) Oct 17, 2012 APMonitor.com | APOPT.com BYU – UT Austin Results –set point changes
Set Point Changes
Large (50) – ON
Medium(20) – ON
Small (5) – ON
Very Small – OFF
Set PointSet Change Point Change Needs another constraint to limit SOL1 use.
Oct 17, 2012 APMonitor.com | APOPT.com BYU – UT Austin Results-disturbances
Disturbance – Opened up an exit valve which is not part of the model
Reduced Solenoid fluctuations, but still a problem, add cost to opening SOL 1
Oct 17, 2012 APMonitor.com | APOPT.com BYU – UT Austin Smart Grid Energy Systems
Oct 17, 2012 APMonitor.com | APOPT.com BYU – UT Austin Thermal Energy Storage (TES)
Oct 17, 2012 APMonitor.com | APOPT.com BYU – UT Austin Chiller-TES System
85°F MINLP System Variables: 672 95°F Integers: 24 Equations: 576 53°F Features Cold Start Solution
Discrete (24) and Continuous (24) Decisions
40°F 53°F
Oct 17, 2012 APMonitor.com | APOPT.com BYU – UT Austin Problem Formulation (MINLP)
t J f ct() P () tdt 0 elec chiller, i i Subject to: yL L TES i i chiller, i building
PyfLWBTfchiller,, i i chiller i , is nonlinear dE TES TES dt
08000ETES 2000TES 2000
175Lchiller, i 1312
yi 0,1
Oct 17, 2012 APMonitor.com | APOPT.com BYU – UT Austin Problem Formulation (Relaxed)
24 JcPPmin elec,1,2, i chiller i chiller i i1 Subject to:
LLchiller1, i chiller 2, i LTES building , i i
PfLWBTfchiller1, i chiller 1, i, i is nonlinear PfLWBTchiller2, i chiller 2, i, i
EETES,,1 i TES i TES i
08000ETES, i
2000TESi 2000 Solved using SQP algorithm
Oct 17, 2012 APMonitor.com | APOPT.com BYU – UT Austin Relaxing Binary Constraints
4000 Mixed integer 3500 problem doubles 3000 # variables 2500 2000 Branch and 1500 bound technique 1000 500
significantly (kW) Consumed Power increases 0 0 1000 2000 3000 4000 5000 6000 7000 computation time Load (tons)
Continuous 5 formulation? 4 3 2 Power Power (kW/ton) 1 Consumed/Load 0 0 1000 2000 3000 4000 5000 6000 7000
Load (tons)
Oct 17, 2012 APMonitor.com | APOPT.com BYU – UT Austin20 Results
BONMIN APOPT Excel MATLAB
Cost ($/day) $ 807.53 $ 804.65 $ 801.93* $ 802.13* CPU Time (sec) 11.6 45.0 58.4* 0.1* * = Relaxed Formulation (NLP) 2500 2000 Excel Solution 1500 APM Solution 1000 500 0 -500 TES (tons) -1000 -1500 -2000 -2500 0 5 10 15 20 25 Time (hours) Oct 17, 2012 APMonitor.com | APOPT.com BYU – UT Austin Optimization of investment decision in energy systems
Can we apply a similar MPC with MINLP approach to optimize investment decisions for next generation energy systems?
Oct 17, 2012 APMonitor.com | APOPT.com BYU – UT Austin Future Development
Most flexible 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
Mixed Integer Nonlinear Programming for DAEs (MIDAEs)
Combine strengths of Active Set (APOPT) and Interior Point (BPOPT) methods
Oct 17, 2012 APMonitor.com | APOPT.com BYU – UT Austin