Addressing Uncertainty How to Model and Solve Energy Optimization Problems
- 1. Addressing Uncertainty: How to Model and Solve Energy Optimization Problems Alkis Vazacopoulos Vice President Optimization Direct, Inc. 1
- 2. Acknowledgments • Susara van den Heever, IBM • Jeremy Bloom, IBM • Charles (Chip) Wilkins, IBM • Robert Ashford (Optimization Direct, Inc.) 2
- 3. Agenda • Intro Optimization Direct • CPLEX Optimization Studio • Energy Applications • Unit Commitment • Uncertainty toolbox IBM 3
- 4. Optimization Direct • IBM Business Partner • More than 30 years of experience in developing and selling Optimization software • Experience in implementing optimization technology in all the verticals • Sold to end users – Fortune 500 companies • Train our customers to get the maximum out of the IBM software • Help the customers get a kick start and get the maximum from the software right from the start 4
- 5. Which software? • CPLEX Optimization Studio • CPLEX is the leader in optimization technology • CPLEX can handle large scale problems and solve them very fast 5
- 6. Why IBM? Why CPLEX? • Fast • Reliable • IBM software • Large scale • Gives you the ability to model develop and solve your decision problem • Complete solution 6
- 7. How can we help? • Benchmark your problems? • Help you with next steps for developing your solution! • Develop optimization prototypes using OPL 7
- 9. Energy Problems • Network Planning • Product Portfolio Planning • Capital Investment • Resource Planning • Unit Commitment/Economic Dispatch • Optimal Power flow / Security Constrained Dispatch 9
- 10. Unit Commitment Paradigm • June 1989, Electrical Power Energy, EPRI GS-6401: • “Mixed Integer Programming is a powerful Modeling tool, They are , However, theoretical complicated and computationally cumbersome” • California 7-day model: • Reported results 1989 – machine unknown • 2 day model: 8 hours, no progress • 7 day model: 1 hour only to solve the LP 10
- 11. CPLEX MIP Performance and the Unit Commitment Paradigm • California 7-day model • 1999 results on a desktop PC • CPLEX 6.5: 22 minutes, Optimal • 2007 results on a desktop PC • CPLEX 11.0: 71 seconds, optimal • What has happened? • CPLEX MIP has become the standard approach for UC applications • CPLEX MIP early adopters 11
- 12. CPLEX MIP Performance and the Unit Commitment Paradigm • What has happened? • CPLEX MIP has become the standard approach for UC applications • CPLEX MIP early adopters gained a competitive advantage • Applications have expanded and changed • 1000-2000 generation units simultaneously (Day Ahead Market) • Solution Cycles less than 5 minutes (Real Time Market) • Uncertainty – We start solving problems with • Scenario Generation • Stochastic • Robust 12
- 13. Why we can succeed with CPLEX MIP? • Computers are faster • Good model formulations – “good modeling” • Cutting Planes: Valid, redundant inequalities that tighten the linear relaxation • Heuristics: inexpensive methods for converting a relaxation solution into an integer feasible solution 13
- 14. Computers are faster • Parallel cores on commodity chips have become standard in recent years • CPLEX has the best • Parallel implementation for Barrier • Parallel NonDeterministic MIP • Parallel Deterministic MIP (Make the regulators Happy) • Parallelism is enabled by default 14
- 15. CPLEX – Cuts – Valid Inequalities • Reduce size of LP feasible region • Cut out parts where there are no integer solutions • (Usually) reduce number of integer infeasibilities • Improves branching • Improves performance of heuristics • Mostly added during root solve, some added in tree • Dramatic benefit in overall performance 15
- 16. CPLEX – Cuts – Valid Inequalities • 1335 models in IBM test set which take ≥ 10 secs and ≤ 10,000 secs (Cplex 12.5, Xeon E5430 12 cores 2.66GHZ) • Fail to solve 28% at all without cuts • Those that do solve take 6 X longer Achterberg and Wonderling, 2013 16
- 17. CPLEX - Heuristics • Attempt to find integer feasible solutions • (Relatively) quick • Work either by inspection or by solving a (possibly sequence of) small sub-models • Can reduce solution times by reduced-cost fixing, root termination during cutting and pruning search tree • On those 1335 test models • 11% fail to solve without heuristics • Those that do take 2 X longer 17
- 18. CPLEX- Heuristics • Useful in their own right if don’t require proof of optimality • Essential for many large models where never get a solution from branching • Cplex heuristics include • local branching • RINS • feasibility pump • (genetic) solution polishing 18
- 19. CPLEX – Model Structure • CPLEX Optimization Studio • Write models quickly • Test • Debug • And start deploying 19
- 20. Unit Commitment and the future • GOAL 1: FERC Meeting (June 2014): most of the new problems involve • Stochastic • Robust • Scenario Based • Monte Carlo Simulation and run many problems • Goal 2: Solve many problems faster • Take advantage of the architecture • Do better and faster modeling 20
- 21. IBM – Toolbox for Uncertainty Optimization • Joint Program between IBM Research and Decision Optimization • If you want more info please contact Optimization Direct and we can organize a more detailed Webinar 21
- 22. Design & long-term planning Tactical planning Operations planning Planning levels Real-time Hours Days Weeks Months Years Time horizon Decision aggregation Operations scheduling Real-time control 22
- 23. Examples of decisions Real-time Hours Days Weeks Months Years Time horizon Decision aggregation Design & longterm planning Tactical planning Operations planning Operations scheduling Real-time control Plant & network design, capacity expansion Mid-term production targets Production, maintenance plans Equipment scheduling Equipment set points 23
- 24. Impact of uncertainty Real-time Hours Days Weeks Months Years Time horizon Decision aggregation Design & longterm planning Tactical planning Operations planning Operations scheduling Real-time control Plant & network design, capacity expansion Mid-term production targets Production, maintenance plans Equipment scheduling Equipment set points Population growth Long-term demand patterns Prices, demand, supplier reliability Rainfall, renewable energy sources, instrumentation error 24
- 25. Uncertainty Toolkit goals • 2013 Joint Program between IBM Research and Decision Optimization • Goals • Increase customer solution resilience, reliability, and stability • Improve trust & understanding of optimization technology • Our approach • Leverage Decision Optimization & mathematical optimization to hedge against uncertainty (e.g. uncertain demand, task durations, prices, resource availability) • A user-friendly toolkit as plug-in to Decision Optimization Center • 5 steps to resilient decisions in the face of uncertainty 1. Define decision model 2. Characterize uncertainty 3. Generate uncertain model 4. Generate decisions 5. Analyze trade-offs
- 26. Stable decisions, stable profits • Test examples • Supply chain planning for a motorcycle vendor 2% increase in profits vs. deterministic optimization • Inventory optimization for IBM Microelectronics Division Greater than 7x increase in feasibility vs. deterministic optimization • Case studies • Energy cost minimization for Cork County Council Estimated 30% value-add in cost reduction vs. deterministic optimization • Leakage reduction for Dublin City Council Estimated 10 times increased stability vs. deterministic optimization • Other benefits • Automated toolkit reduces dependence on PhD-level experts & statistical data • Visualize trade-off between multiple KPIs across multiple scenarios and plans 26
- 27. Effect of data uncertainty on decision resilience re·sil·ient1 adjective ri-ˈzil-yənt a : capable of withstanding shock without permanent deformation or rupture b : tending to recover from or adjust easily to misfortune or change ve·rac·i·ty1 noun və-ˈra-sə-tē : truth or accuracy un·cer·tain1 adjective ˌən-ˈsər-tən : not exactly known or decided : not definite or fixed : not sure : having some doubt about something “Resilient” how decisions should be “Veracity” the data quality decision makers and decision software often assume “Uncertain” the actual data quality 27 Assuming data veracity in the face of uncertainty leads to decision instability, as well as distrust in decision optimization technology.
- 28. Example use cases for the Uncertainty Toolkit Industry Typical company Problem type Government Government agencies Project portfolio management Tourism Hotel operators, Airlines Revenue management Transport Railroads Railroad locomotive planning Transport Supermarket chain, cement Delivery / pick-up truck routing Utilities Electricity company Production planning Utilities Water company Tactical reservoir planning Utilities Water company Water distribution network configuration Utilities Electricity company Unit commitment Utilities Water network operators Pump scheduling Utilities Water network operators Pressure management Utilities Electricity company Energy trading Oil and gas Oil company Vessel scheduling Manufacturing Manufacturer Operational project scheduling Manufacturing Car manufacturer Manufacturing line load balancing Manufacturing Aircraft manufacturer Plant assembly Manufacturing Car manufacturer Sales and operations planning Supply chain Manufacturer Contractor to transport leg assignment Supply chain Manufacturer Product to store allocation Supply chain Manufacturer, oil&gas Inventory optimization Supply chain Manufacturer, oil&gas Supply chain network configuration Supply chain Manufacturer Procurement planning Supply chain Manufacturer Emergency operations planning Commercial Banks, insurance, TV Marketing campaign optimization Finance Banks Collateral allocation 28
- 29. 5 steps to resilience with the Uncertainty Toolkit 1. Define decision model 2. Characterize uncertainty 3. Generate uncertain model 4. Generate decisions 5. Analyze trade-offs 29 • Create optimization model with IBM CPLEX Studio • Some modeling skill required, or existing assets • Embed in IBM Decision Optimization Center “Steve” the IT expert, & “Keith” the OR consultant • OR consultant’s “wizard”: 7 screens • Defines uncertainty, scenario generation, risk measures • Built-in automated reformulation, based on steps 1 and 2 • No modeling knowledge required • “Robustification” (make the original model robust to change) • Business user’s “wizard” • Automated solution generation • Automated scenario comparison “Anne” the business user • Built-in visual analytics • Analyze KPI trade-offs across multiple plans & scenarios
- 30. Uncertainty Toolkit: automated reformulations Robust / Stochastic approach Applicabl e model types Resulting model types Uncertainty characterizatio n Restrictions Single-stage penalty approach (Mulvey et al., 1995) LP LP (or QP) Scenarios (finite) No uncertain data in MILP MILP (or objective function MIQP) Two-stage penalty approach (Mulvey et al., 1995) LP LP (or QP) Scenarios (finite) No uncertain data in MILP MILP (or objective function MIQP) Multistage Stochastic (e.g. King & Wallace, 2012) LP LP Scenarios (finite) None MILP MILP Safety margin approach with ellipsoidal uncertainty sets (Ben-Tal & Nemirovski, 1999) LP QCP Range No uncertain data in standalone parameters or equality constraints MILP MIQCP Safety margin approach with polyhedral uncertainty sets (Bertsimas & Sim, 2004) LP LP Range No uncertain data in standalone parameters or equality constraints MILP MILP Extreme Scenario approach (Lee, 2014) LP LP Range No uncertain data in MILP MILP variable coefficients Distributionally robust reformulation (Mevissen et al., 2013) LP LP Scenarios Uncertainty in standalone parameters handled as penalty term in objective MILP MILP 30
- 31. Uncertainty Toolkit: automated reformulations Robust / Stochastic approach Applicab le model types Resulting model types Uncertainty characterizati on Restrictions Single-stage penalty approach (Mulvey et al., 1995) LP LP (or QP) Scenarios (finite) No uncertain data in MILP MILP (or objective function MIQP) Two-stage penalty approach (Mulvey et al., 1995) LP LP (or QP) Scenarios (finite) No uncertain data in MILP MILP (or objective function Q: How do I know which of these methods to use? MIQP) Multistage Stochastic (e.g. King & Wallace, 2012) LP LP Scenarios A: The Uncertainty Toolkit will decide automatically based on your (finite) None MILP MILP input into the Consultant’s Wizard Safety margin approach with ellipsoidal uncertainty sets (Ben-Tal & Nemirovski, 1999) LP QCP Range No uncertain data in standalone parameters or equality constraints MILP MIQCP Safety margin approach with polyhedral uncertainty sets (Bertsimas & Sim, 2004) LP LP Range No uncertain data in standalone parameters or equality constraints MILP MILP Extreme Scenario approach (Lee, 2014) LP LP Range No uncertain data in MILP MILP variable coefficients Distributionally robust reformulation (Mevissen et al., 2013) LP LP Scenarios Uncertainty in standalone parameters handled as penalty term MILP MILP 31
- 32. Questions • What is the right model? • Can we automate the selection of the model? 32
- 33. Uncertainty Toolkit Decision Tree (automated) Select the uncertain data item(s) Is the uncertainty represented as (1) a set of scenarios, or (2) a data range? (1) (2) Can some decisions change when you know the outcome of the uncertain data? Yes Select a risk measure to optimize: (1) Expected value (2) Worst-case performance (3) Conditional Value at Risk No Single-stage scenario-based • Single-stage penalty approach • Distributionally robust optimization Can some constraints be violated? No Multi-stage scenario-based approaches • Stochastic Constraint Programming • Stochastic Mathematical Programming Yes Extreme scenario approach Is the uncertain data item a variable coefficient? For these constraints, do you want to use: (1) Chance constraints (i.e. chance of violation < 5%) (2) A violation penalrty? (1) (2) No Two-stage penalty approach Yes Do you have correlated uncertain data items? Yes Safety margin with ellipsoidal uncertainty Do you want to work with a budget of uncertainty? Yes Safety margin with polyhedral uncertainty 33
- 34. Uncertainty Toolkit Decision Tree (automated) Select the uncertain data item(s) Is the uncertainty represented as (1) a set of scenarios, or (2) a data range? (1) (2) Can some decisions change when you know the outcome of the uncertain data? Yes Select a risk measure to optimize: (1) Expected value (2) Worst-case performance (3) Conditional Value at Risk No Single-stage scenario-based • Single-stage penalty approach • Distributionally robust optimization Can some constraints be violated? No Multi-stage scenario-based approaches • Stochastic Constraint Programming • Stochastic Mathematical Programming Yes Extreme scenario approach Is the uncertain data item a variable coefficient? For these constraints, do you want to use: (1) Chance constraints (i.e. chance of violation < 5%) (2) A violation penalrty? (1) (2) No Two-stage penalty approach Yes Do you have correlated uncertain data items? Yes Safety margin with ellipsoidal uncertainty Do you want to work with a budget of uncertainty? Yes Safety margin with polyhedral uncertainty 34
- 35. Question: • What is the architecture? 35
- 36. Uncertainty Toolkit architecture Business User OR Consultant 36
- 37. Question • Can I automate the reformulation from the deterministic model? 37
- 38. Example: Automated model reformulation for stochastic CP Input: Deterministic model Automated model reformulation Output: Stochastic model IBM Confidential 38
- 39. Question • What is the right Software Platform? 39
- 40. Decision Optimization Center (DOC) is about Decision Support • DOC for OR (Operations Research) experts: Eclipse-based development environment to create optimization solutions • CPLEX Studio embedded for OR needs • Data modeling & connections • Visualization • Custom Java extensions • DOC for business users: Supports decision making leveraging optimization • Scenario-based analysis • Manual planning in addition to optimization • Alternative business goals • Business rules • Tradeoff visualization 40 • “Freeze” partial solution and solve again
- 41. Decision Optimization Center IDE OPL Model Development 41
- 42. Decision Optimization Center IDE Configuration of built-in visualization 42
- 43. Decision Optimization Center IDE Custom Java extensions 43
- 44. 44 Displays using Simple Tables and Charts – out of the box
- 45. Pivot Tables and Scenario Comparison – out of the box 45
- 46. Case study: Water treatment/distribution energy cost reduction • Big picture: Cork County Council must reduce energy consumption by 20% by 2020 • 95% of this utility’s water-related energy costs due to pump operations • New dynamic energy pricing schemes leverage renewables (wind energy) • Trade-off: Cleaner energy at lower prices, but uncertainty in price due to • Wind uncertainty • Network outages • Other weather conditions • Goal: Schedule pumps leveraging dynamic prices, while hedging against uncertainty in price prediction 46
- 47. Simplified network (illustration purposes) Goal: Optimize pump schedules to minimize (uncertain) energy costs while meeting demand and respecting plant and network constraints* Pumphouse Water source Treatment plant Pumphouse * Based on Cork County Council’s Inniscarra network 47
- 48. Uncertainty in price prediction • Forecasted (D-1) post ante price from supplier • Considers forecasted demand based on weather, special events, wind, etc. • Actual (D+4) price charged 4 days after the event • Forecasted (D-1) and Settled price (D+4) can differ due to changes in predicted wind energy availability, weather, and unpredicted grid events 48 SMP Energy Pricing Event 8th May 2011 80 70 60 50 40 30 20 10 0 8/5:16 8/5:17 8/5:17 8/5:18 8/5:18 8/5:19 8/5:19 8/5:20 8/5:20 8/5:21 8/5:21 8/5:22 8/5:22 8/5:23 8/5:23 9/5:0 9/5:0 Date:Hour €/MW D+4 Actual Price D-1 Forecast Price Question: Should utility switch to a dynamic pricing scheme? Step 1: Prove dynamic pricing benefits Step 2: Prove optimization benefits Step 3: Deal with uncertainty
- 49. Step 1: Define decision model • Define objective, decisions, constraints (mathematical modeling skill required) • Objective: minimize energy costs from pump operations • Decisions: when to switch pumps on/off (decided every 30 minutes for 24 hours in advance) • Constraints: satisfy tank levels, pump operation rules, customer demand, network constraints • Model using CPLEX Studio, assuming certain data (“deterministic” model) 49 Note: When data is fairly certain, deterministic models are sufficient to provide significant benefit
- 50. Decision model in Decision Optimization Center prototype: Predicted energy cost savings from pump optimization and (known) dynamic energy prices Example: Steps 1 & 2 – benefit of dynamic pricing combined with optimization Energy cost 7.5% 5.5% 6% • Savings from dynamic pricing: 5.5% • Savings from optimization: 13.5% • Total 19% cost reduction Optimized schedule; dynamic prices Existing schedule; dynamic prices Existing schedule; day/night prices Optimizing day/night prices Switching to dynamic prices Optimizing dynamic prices Optimized schedule; day/night prices Next…deal with price uncertainty 50
- 51. Step 2: Characterize uncertainty • Price scenarios, with likelihoods: • From energy provider • From IBM Research forecasts Price (euro/kWh) Time of day (30 min increments) 51
- 52. Step 2: Uncertainty Toolkit wizard for consultant input (1 of 2) 1. Select uncertain data 2. Select data scenarios vs. ranges 4. Select risk measures 3. Define decision stages 52
- 53. Step 2: Uncertainty Toolkit wizard for consultant input (2 of 2) 6. Define KPIs 5. Specify constraint satisfaction 7. Define correlation (optional) 8. Save your recipe for later use 53
- 54. Step 3: Generate uncertain model • Uncertainty Toolkit automatically generates the uncertain model(s) depending on choices in Steps 1 and 2 • Uncertain models are typically classified as • “Robust”: hedging against worst case outcome(s) • “Stochastic”: optimizing for expected outcome(s) • If choice unclear, use both & visualize trade-offs Step 4: Generate plans Business user’s wizard Uncertainty Toolkit generates multiple solutions (deterministic, robust, stochastic) Uncertainty Toolkit automatically does solution-scenario cross-comparison – What is the impact of change on each plan Robust optimization Stochastic optimization Scenario/solution cross-comparison 54
- 55. Automated plan generation for varying scenarios Reduced risk due to stochastic / robust solutions Industry Solutions Joint Program Step 5: Analyze trade-offs Plan comparison across scenarios Scenarios Energy cost (euro/d) Scenario drill-down analysis (e.g. best and worst case) Trade-off analysis across scenarios Best average performance (stochastic model) Best worst-case performance (robust model) Pump scheduling use case: Value-add from Uncertainty Toolkit ~ 30% improvement in energy cost reduction 55
- 56. Benefits of Uncertainty Toolkit – pressure management use case Water network demand scenarios Greater area covered = better Blue (robust) plan wins! Stability of leakage reduction Worst-case infeasibility Stability of hydraulic feasibility Robust plans = most stable (~ 10 times more stable than deterministic (current state)) Worst-case leakage reduction Average leakage reduction Average infeasibility Robust plan = best performance in worst case Robust plan = best performance in average case RobustPlanA RobustPlanJ DeterministicPlan_2 UT = Uncertainty Toolkit plug-in to ODM Studio Pressure management use case: 56 Water network operational decisions 10 times more stable than current state, continue to perform well when data changes (i.e. “robust” plans)
- 57. Benefits of Uncertainty Toolkit – pressure management use case Better << Objective values >> Worse Visualization of trade-off: robustness vs. cost Robust << % infeasible scenarios >> Fragile Low cost and robust Robust, but high cost Low cost, but fragile 57
- 58. Benefits of Uncertainty Toolkit – pressure management use case Deterministic plan(s) quickly infeasible for alternative scenarios, high cost per additional 58 Number of feasible scenarios for each plan Objective value (cost) Robust plan(s) feasible for most scenarios, at minimal cost increase per additional scenario scenario Effect on feasibility (robustness) by increasing cost
- 59. Example: Unit Commitment Uncertain demand 59
- 60. Unit Commitment Problem Given Power generation units with Costs (start-up, fuel, CO2) Operational properties (capacity, ramp) Demand over several periods find generation plan Which units to use (unit commitment) How much to produce (dispatch) such that Demand is satisfied Operational constraints are satisfied Total cost is minimized 60 Industry Solutions Joint Program 60
- 61. Unit Commitment Problem – Stochastic Version Problem: How to deal with uncertain loads? Question: Is the dispatch plan still feasible under a slight perturbation of the load? Stochastic Programming Approach Separate decisions into stages to be able to “react” to uncertainty Decision Stages Stage 1: unit commitment “Here-and-now” decisions Stage 2: dispatch “Wait-and-see” decisions 61 Industry Solutions Joint Program 61
- 62. Step 6a: Inspecting the results: Table View • Stochastic Plan is feasible for all scenarios • Deterministic plans are only feasible for “their” scenario 62
- 63. Step 6b: Solution-Scenario Cross- Comparison 63
- 64. Step 6c: Cross-Comparison: Spinning Capacity 64
- 65. Summary • Energy applications can benefit from Optimization • Cplex Optimization Studio can speedup solving your problems and Deployment • MIP is becoming standard for solving Energy Optimization Problems 65
- 66. References & contact info • References • A. J. King and S. W. Wallace. Modeling with Stochastic Programming. Springer, 2012. • J. M. Mulvey, R. J. Vanderbei, and S. A. Zenios, “Robust Optimization of Large- Scale Systems,” Operations Research, vol. 43, no. 2, pp. 264-281, 1995. • A. Ben-Tal and A. Nemirovski, “Robust solutions of uncertain linear programs,” Operations research letters, vol. 25, no. 1, pp. 1-13, 1999. • D. Bertsimas and M. Sim, “The price of robustness,” Operations Research, vol. 52, no. 1, pp. 35-53, 2004. • C. Lee, “Extreme scenario approach,” forthcoming paper, to be presented at the International Federation of Operational Research Societies (IFORS) conference, Barcelona, 2014. • M. Mevissen, E. Ragnoli, and Y. Y. Jia, “Data-driven Distributionally Robust Polynomial Optimization,” in Advances in Neural Information Processing Systems, Nevada, United States, pp. 37-45, 2013. • Contacts • Alkis Vazacopoulos • 201 256 7323 • alkis@optimizationdirect.com • www.optimizationdirect.com • Susara van den Heever • svdheever@fr.ibm.com 66