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

2

Power Grid Operations Zavala, Constantinescu, Wang, and Botterud, 2009

3

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

7

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

9

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

18

Optimization Under Uncertainty

19

Deterministic Newton Methods (State-of-the-Art)

20

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

24

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

26

Network Compression

27

- 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

28

Fusion

Mira

Implementing in Toolkit for Advanced Optimization (TAO) & Leveraging PETSc Constructs

Coupled Infrastructure Systems

29

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