Post on 14-Jan-2016
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Optimization Under Uncertainty: Structure-Exploiting Algorithms
Victor M. ZavalaAssistant Computational MathematicianMathematics and Computer Science Division Argonne National LaboratoryFellowComputation InstituteUniversity of Chicago
March, 2013
Outline
Background
Project Objectives and Progress
On-Going Work
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Power Grid Operations Zavala, Constantinescu, Wang, and Botterud, 2009
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Grid Operated with Expected Values of Demands, Renewables, and Topology
Robustness Embedded in “Reserves”
Prices at Illinois Hub, 2009
Grid Time Volatility
Volatility Reflects System Instabilities and Uneven Distributions of Welfare
Uncertainties Not Properly Anticipated/Factored In Decisions
Grid Spatial Volatility
WindRamps
Wind Power Adoption
Newton’s Method
Solve Sequence of BPs with
NLP Barrier Problem
KKT Matrix
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Scalable Optimization: Interior Point Solvers
Huge Advances in Convergence Theory and Scalability- Available Implementations: IPOPT, OOQP, KNITRO, LOQO, Gurobi, CPLEX
Key Advantages:- Superlinear Convergence and Polynomial Complexity- Enables Sparse and Structured Linear Algebra- “Easy” Extensions to Nonlinear Problems
Scalable Stochastic OptimizationNeed to Make Decision Now While Anticipating Future Scenarios
Typically: Scenarios Sampled a-priori From Given Distribution (e.g., Weather)
Problem Induces Arrow-Head Structure in KKT System
Key Bottlenecks: - Number and Size of Scenarios and First-Stage Variables - Decomposition Based on Schur Complement : Dense Sequential Step - Hard To Get Good Preconditioners (Inequality Constraints, Unstructured Grids)
Illinois System Zavala, Constantinescu, Wang, and Botterud, 2009, Lubin, Petra,
Anitescu, Zavala 2011
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1900 Buses 261 Generators 24 Hours
• O(104-105) Scenarios Needed to Cover High-Dimensional Spatio-Temporal Space (Wind Fields)
• 6 Billion Variables Solved in Less than an Hour on Intrepid (128,000 Cores)
• O(103) First-Stage Variables
• Strong Scaling on Intrepid – 128,000 Cores
• O(105) First-Stage Enabled with Parallel Dense Solvers
PIPS Petra, Lubin, Anitescu and Zavala 2011
Based on OOQP Gertz & Wright, Schur Complement-Based, Hybrid MPI/OpenMPIncite Award Granting Access to BlueGene/P (Intrepid)
Scalability Results Interior-Point Solver
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Reducing Grid Volatility (Zavala, Anitescu, Birge 2012)
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Distribution of Social Welfare (Zavala, Anitescu, Birge 2012)
Mean Price Field - Deterministic
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Mean Price Field - Stochastic
Distribution of Social Welfare (Zavala, Anitescu, Birge 2012)
Exploring Asymptotic Statistical Behavior with HPC Zavala, et.al. 2012
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Analysis Requires Problems with O(109) Complexity
Ambiguity : Weather Forecasting Ambiguity : Weather Forecasting Constantinescu, Zavala, Anitescu, 2010
Demand
Thermal
Wind
- WRF Forecasts are -In General- Accurate with Tight Uncertainty Bounds
- Excursions Occur: Probability Distribution of 3rd Day is Inaccurate! Resolution? Frequency Data Assimilation? Missing Physics? 100m Sensors?
Major Advances in Meteorological Models (WRF) Highly Detailed Phenomena High Complexity 4-D Fields (106- 108 State Variables)
Model Reconciled to Measurements From Meteo Stations
Data Assimilation -Every 6-12 hours-: 3-D Var Courtier, et.al. 1998 4-D Var (MHE) Navon et.al., 2007 Extended and Ensemble Kalman Filter Eversen, et.al. 1998
Ambiguity : Weather Forecasting Ambiguity : Weather Forecasting Constantinescu, Zavala, Anitescu, 2010
Current Time
Data Assimilation (Least-Squares) Forecast (Sampling)
Forecast Distribution Function of PDE Resolution
Need to Embed Distributional Error Bounds in Stochastic Optimization
Dealing with Ambiguity in Decision Can Relax Resolution Needs (Need Integration with UQ)
Forecast 24 hr in One Hour
Ambiguity – Weather Forecasting Ambiguity – Weather Forecasting Constantinescu, Zavala, Anitescu, 2010
Outline
Background
Project Objectives and Progress
On-Going Work
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Optimization Under Uncertainty
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Deterministic Newton Methods (State-of-the-Art)
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Implementations: PIPS (Petra, Anitescu), OOPS (Gondzio, Grothey)
Bottleneck in HPC: Limited Algorithmic Flexibility 1. How To Construct Steps From Smaller Sample Sets? Need to Allow for Inexactness 2. Progress and Termination Is Deterministic Not Probabilistic Need to Relax Criteria – Probabilistic Metrics 3. Inefficient Management of Redundancies
Stochastic Newton Methods
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Scenario Compression Zavala, 2013
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Residual Characterization: - Cluster Based on Effect on First-Stage Direction
- Clustering Techniques: Hierarchical, k-Means, etc…
Network Expansion Network Expansion Zavala, 2013Zavala, 2013
- Number of Iterations as Function of Compression Rates – 100 Total Scenarios
Sparse Multi-Level Preconditioning Zavala(b), 2013
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Numerical Tests Numerical Tests Zavala, 2013
- Test Effectiveness of Preconditioner Using Scenario Clustering
- Compare Against Scenario Elimination and No Preconditioning
Observations:- Clustering 2-3 Times More Effective Than Elimination
- Compression Rates of 70% Achievable - Multilevel Enables Rates > 80%
Outline
Background
Project Objectives and Progress
On-Going Work
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Network Compression
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- Compression Possible in Networks- Enables Multi-Level- KKT System Structure Becomes Nested
Observations: -If Link is Not Congested, Nodes Can be Clustered -Use Link Lagrange Multiplier as Weight
Scalable Linear Algebra & HPC
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Fusion
Mira
Implementing in Toolkit for Advanced Optimization (TAO) & Leveraging PETSc Constructs
Coupled Infrastructure Systems
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Natural Gas Electricity
Urban Energy Systems
Optimization Under Uncertainty: Structure-Exploiting Algorithms
Victor M. ZavalaAssistant Computational MathematicianMathematics and Computer Science Division Argonne National LaboratoryFellowComputation InstituteUniversity of Chicago
March, 2013