Optimization Under Uncertainty: Structure-Exploiting Algorithms Victor M. Zavala Assistant Computational Mathematician Mathematics and Computer Science Division Argonne National Laboratory Fellow Computation Institute University of Chicago March, 2013

Outline  Background  Project Objectives and Progress  On-Going Work 2

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

Wind Ramps Wind Power Adoption

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 Optimization Need 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 Buses 261 Generators 24 Hours

O( ) 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(10 3 ) First-Stage Variables Strong Scaling on Intrepid – 128,000 Cores O(10 5 ) First-Stage Enabled with Parallel Dense Solvers PIPS Petra, Lubin, Anitescu and Zavala 2011 Based on OOQP Gertz & Wright, Schur Complement-Based, Hybrid MPI/OpenMP Incite Award Granting Access to BlueGene/P (Intrepid) Scalability Results Interior-Point Solver

11 Reducing Grid Volatility (Zavala, Anitescu, Birge 2012)

12 Distribution of Social Welfare (Zavala, Anitescu, Birge 2012) Mean Price Field - Deterministic

13 Mean Price Field - Stochastic Distribution of Social Welfare (Zavala, Anitescu, Birge 2012)

Exploring Asymptotic Statistical Behavior with HPC Zavala, et.al Analysis Requires Problems with O(10 9 ) 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 3 rd 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 ( State Variables) Model Reconciled to Measurements From Meteo Stations Data Assimilation -Every 6-12 hours-: 3-D Var Courtier, et.al D Var (MHE) Navon et.al., 2007 Extended and Ensemble Kalman Filter Eversen, et.al 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 21

Scenario Compression Zavala, Residual Characterization: - Cluster Based on Effect on First-Stage Direction - Clustering Techniques: Hierarchical, k-Means, etc…

Network Expansion Zavala, Number of Iterations as Function of Compression Rates – 100 Total Scenarios

Sparse Multi-Level Preconditioning Zavala(b),

Numerical Tests Numerical Tests Zavala, 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. Zavala Assistant Computational Mathematician Mathematics and Computer Science Division Argonne National Laboratory Fellow Computation Institute University of Chicago March, 2013