Robust Optimization for Unit Commitment and Dispatch in Energy Markets
|
|
- Daniel Sherman
- 8 years ago
- Views:
Transcription
1 1/41 Robust Optimization for Unit Commitment and Dispatch in Energy Markets Marco Zugno, Juan Miguel Morales, Henrik Madsen (DTU Compute) and Antonio Conejo (Ohio State University) Modeling and Optimization of Heat and Power System DTU, 12th January 2015
2 2/41 Outline 1 Mathematical Framework 2 Energy & Reserve Dispatch 3 CHP Unit Commitment & Scheduling 4 Conclusion
3 3/41 Why Optimization Under Uncertainty Sources of uncertainty in energy systems Decision-making in energy markets Better decisions if optimization accounts for uncertainty
4 4/41 Deterministic Optimization Framework Unit commitment and dispatch formalized as optimization problems Min. x,y c x + q y s.t. Ax b, Tx + Wy h x, y represent decision variables: unit status, dispatch, storage level, network state, etc. inequality and equality constraints impose limits on dispatch: dispatch limits, ramping, supply/demand balance, etc.
5 5/41 Optimization Under Uncertainty Most often not all parameters are known in advance { } Min. c x + M ω q x,y ω ωy ω s.t. Ax b, Tx + Wy ω h ω, ω Ω Uncertain parameters depend on random variable ω Variables y ω are adjustable (recourse) Measure of recourse cost in the objective function (expectation, etc.) Constraints hold for different realizations of the uncertainty
6 6/41 Stochastic Programming Tractable version of stochastic problem by sampling the uncertainty Min. x,y ωs c x + S p ωs q ω s y ωs s=1 s.t. Ax b, Tx + Wy ωs h ωs, s = 1,..., S Probably the most popular method of optimization under uncertainty Only used as a comparison in this presentation
7 7/41 Adaptive Robust Optimization (ARO) Tractable version of stochastic problem based on notion of uncertainty set Min. x c x+ max s.t. Ax b, min q,h y q y Tx + Wy h, (q, h) U, M( ) replaced by worst-case realization of recourse cost Inner minimization problem guarantees feasibility of solution Three-level game against nature
8 8/41 ARO and Non-Anticipativity t 1 t 2 t 3 t 4 Worst-case trajectory of uncertainty revealed before making recourse decisions Recourse has perfect knowledge of future worst-case uncertainty
9 9/41 Affinely Adjustable Robust Optimization (AARO) Enforcing Linear Decision Rules Optimal recourse functions y(ω) approximated by linear rules: y = Yω Computational Simplification Linear coefficient Y is a first-stage variable: essentially a max-min problem y ω
10 10/41 Affinely Adjustable Robust Optimization (AARO) Problem Formulation { } Min. x,y c x + E ω ω Q Yω s.t. Ax b, Tx + WYω Hω, ω U { } Min. x,y c x + E ω ω Q Yω s.t. Ax b, max {Tx + (WY H) ω} 0 ω Ω Differences from ARO: Not bound to optimize worst-case cost: can be made less conservative min-max problem can be simplified via duality argument: single stage problem
11 11/41 Energy and Reserve Dispatch using Adaptive Robust Optimization
12 12/41 Electricity Market Setup Day-ahead market: 12- to 36-hour ahead Energy and reserve are co-optimized (first stage) Followed by a balancing (real-time) market (second stage) Network is represented as in US markets The right of starting-up and shutting-down own units is not confiscated as in EU markets
13 13/41 Adaptive Robust Optimization Approach Two-stage robust optimization turn into three-level optimization problems 1 Make dispatch decision x with prognosis of the future (minimize cost) 2 Uncertainty (wind power production w) unfolds (maximize cost) 3 Make operation (recourse) decision y (minimize cost)
14 14/41 The Robust Optimization Approach min x c x T x +max w s.t. Fx = d ŵ, Gx g. min y c y T y s.t. Py = w Qx, : λ, s.t. H w h, Ly l Mx N w, : µ, This is a complicated problem! There is no general solution technique for three-level optimization problems
15 15/41 Modeling the Uncertainty Set Uncertain stochastic power production at different nodes described by polyhedral uncertainty set W can be modeled via linear inequalities H w h can include budget constraints on the stochastic production can limit the variation of stochastic production among adjacent plants w 1 w 2 ŵ W
16 Column & Constraint Generation Approach In the linear case, for any feasible x the worst-case realization of w is a vertex of the polytope W the worst-case recourse cost is the maximum of a finite number of affine functions of x 1 Find optimal first-stage solution for approximation (lower bound) 2 Calculate worst-case recourse cost (upper bound) 3 Generate cut 16/41
17 Column & Constraint Generation Approach In the linear case, for any feasible x the worst-case realization of w is a vertex of the polytope W the worst-case recourse cost is the maximum of a finite number of affine functions of x Cost 1 Find optimal first-stage solution for approximation (lower bound) 2 Calculate worst-case recourse cost (upper bound) 3 Generate cut (p,r) 16/41
18 Column & Constraint Generation Approach In the linear case, for any feasible x the worst-case realization of w is a vertex of the polytope W the worst-case recourse cost is the maximum of a finite number of affine functions of x Cost 1 Find optimal first-stage solution for approximation (lower bound) 2 Calculate worst-case recourse cost (upper bound) 3 Generate cut (p,r) 16/41
19 Column & Constraint Generation Approach In the linear case, for any feasible x the worst-case realization of w is a vertex of the polytope W the worst-case recourse cost is the maximum of a finite number of affine functions of x Cost gap 1 Find optimal first-stage solution for approximation (lower bound) 2 Calculate worst-case recourse cost (upper bound) 3 Generate cut (p,r) 16/41
20 Column & Constraint Generation Approach In the linear case, for any feasible x the worst-case realization of w is a vertex of the polytope W the worst-case recourse cost is the maximum of a finite number of affine functions of x Cost gap 1 Find optimal first-stage solution for approximation (lower bound) 2 Calculate worst-case recourse cost (upper bound) 3 Generate cut (p,r) 16/41
21 Column & Constraint Generation Approach In the linear case, for any feasible x the worst-case realization of w is a vertex of the polytope W the worst-case recourse cost is the maximum of a finite number of affine functions of x Cost gap 1 Find optimal first-stage solution for approximation (lower bound) 2 Calculate worst-case recourse cost (upper bound) 3 Generate cut (p,r) 16/41
22 Column & Constraint Generation Approach In the linear case, for any feasible x the worst-case realization of w is a vertex of the polytope W the worst-case recourse cost is the maximum of a finite number of affine functions of x 1 Find optimal first-stage solution for approximation (lower bound) 2 Calculate worst-case recourse cost (upper bound) 3 Generate cut Cost gap (p,r) 16/41
23 Column & Constraint Generation Approach In the linear case, for any feasible x the worst-case realization of w is a vertex of the polytope W the worst-case recourse cost is the maximum of a finite number of affine functions of x 1 Find optimal first-stage solution for approximation (lower bound) 2 Calculate worst-case recourse cost (upper bound) 3 Generate cut Cost gap (p,r) 16/41
24 Column & Constraint Generation Approach In the linear case, for any feasible x the worst-case realization of w is a vertex of the polytope W the worst-case recourse cost is the maximum of a finite number of affine functions of x Cost 1 Find optimal first-stage solution for approximation (lower bound) 2 Calculate worst-case recourse cost (upper bound) 3 Generate cut (p,r) 16/41
25 17/41 Modified IEEE 24-Node Reliability Test System Block offers by generators Flexible generators provide reserve Three lines with reduced capacity Wind farms at 6 nodes of the system
26 18/41 Wind Power Data Data from 6 locations in Spain Uncertainty set parameters estimated from historical data Historical data also used as scenarios
27 19/41 Uncertainty Set for Wind Power Production Budget of uncertainty: Intervals: Wn max Budget: w n W max n Φ N n w n W max n, n Φ N Γ One additional type of constraint to model geographical correlation ρ n1 n 2 w n 1 w n 2 ρ wn max 1 wn max n1 n 2, n 1, n 2 Φ N, n 1 n 2 2
28 Robust Optimization vs Stochastic Programming Risk-Neutral Case Cost Robust Optimization Stochastic Programming Energy disp Up reserve Down reserve Total day-ahead Mean CVaR 95% VaR 100% Mean CVaR 95% VaR 100% Energy redisp Load-shedding Total balancing Total aggregate /41
29 21/41 Robust Optimization vs Stochastic Programming Risk-Averse Case CVaR Expectation Cost [$] Cost [$] 2.3 x In-sample α 2.9 x SP RO SP RO Cost [$] Cost [$] 2.35 x Out-of-sample α 2.95 x SP RO SP RO α α
30 22/41 Robust Optimization vs Stochastic Programming Computational Properties Computational results with different IEEE power system models Computation time (s) 10 2 SP (24 bus) 10 1 RO (24 bus) SP (118 bus) RO (118 bus) SP (300 bus) 10 0 RO (300 bus) # scenarios
31 22/41 Unit Commitment and Scheduling for Heat & Power Systems using Affinely Adjustable Robust Optimization
32 23/41 Heat and Power System Example Heat plant District heating CHP (extraction) CHP (backpressure) Heat tank Electricity market
33 24/41 A Multi-Stage Uncertain Problem The problem is uncertain (heat demand, prices) and multi-stage: Day-ahead Determine unit commitment and schedule for the following day (periods 1,..., N T ) Oper. h 1 Uncertainty at time 1 unfolds and units are re-dispatched (based on realizations until time 1) Oper. h 2 Uncertainty at time 2 unfolds and units are re-dispatched (based on realizations until time 2). Oper. h N T Uncertainty at time N T unfolds and units are re-dispatched (based on all realizations)
34 24/41 A Multi-Stage Uncertain Problem The problem is uncertain (heat demand, prices) and multi-stage: Day-ahead Determine unit commitment and schedule for the following day (periods 1,..., N T ) Oper. h 1 Uncertainty at time 1 unfolds and units are re-dispatched (based on realizations until time 1) Oper. h 2 Uncertainty at time 2 unfolds and units are re-dispatched (based on realizations until time 2). Oper. h N T Uncertainty at time N T unfolds and units are re-dispatched (based on all realizations)
35 24/41 A Multi-Stage Uncertain Problem The problem is uncertain (heat demand, prices) and multi-stage: Day-ahead Determine unit commitment and schedule for the following day (periods 1,..., N T ) Oper. h 1 Uncertainty at time 1 unfolds and units are re-dispatched (based on realizations until time 1) Oper. h 2 Uncertainty at time 2 unfolds and units are re-dispatched (based on realizations until time 2). Oper. h N T Uncertainty at time N T unfolds and units are re-dispatched (based on all realizations)
36 24/41 A Multi-Stage Uncertain Problem The problem is uncertain (heat demand, prices) and multi-stage: Day-ahead Determine unit commitment and schedule for the following day (periods 1,..., N T ) Oper. h 1 Uncertainty at time 1 unfolds and units are re-dispatched (based on realizations until time 1) Oper. h 2 Uncertainty at time 2 unfolds and units are re-dispatched (based on realizations until time 2). Oper. h N T Uncertainty at time N T unfolds and units are re-dispatched (based on all realizations)
37 25/41 Robust Optimization Approach Our approach based on Robust Optimization (RO) } Minimize c x + E ω {q ωy(ω) x,y( ) where s.t. Ax b, Tx + Wy(ω) h ω, ω U (3) x are first-stage decision (unit commitment, schedule) y( ) are recourse functions of the uncertainty ω (unit operation) The solution is optimal in expectation (1) The solution is feasible for any ω in an uncertainty set U (3) (1) (2)
38 26/41 Linear Decision Rules Optimal recourse decision approximated via linear decision rules y t y it (ω) = ŷ it + Y itτ ω τ, τ=1 i, t ω Recourse decision depends on history of uncertainty up to now (t) Independence on future uncertainty realizations (τ > t): non-anticipative
39 27/41 Final Linear Reformulation We make the following assumptions: linear dependence of parameters to uncertainty: q ω = Qω, h ω = Hω budget uncertainty set defined by Dω d By duality arguments, the problem can be reformulated linearly: { Min. x,y,λ c x + tr Q Y (Σ ω + E{ω}E{ω} )} s.t. Ax b, Λ d Tx, D Λ = (WY H), Λ R R L 2 0
40 28/41 Case-Study Setup Three production facilities with storage Extraction CHP (AMV3, AVV1) Back-pressure CHP (AMV1) heat-only plant Historical heat load from VEKS RO: budget uncertainty set (tunable interval size and budget) SP: Gaussian scenarios Power prices from NordPool SP: Gaussian scenarios correlated with heat demand Figure from HOFOR.dk
41 29/41 Back-pressure vs Extraction CHP Units P Back-pressure CHP unit constant heat/power ratio (r b ) rb H Extraction CHP unit more flexible operation in heat/power space feasible region approximated by polyhedron P rv rb H
42 Parameters of the System Extraction Back-pressure Heat-only Power/heat ratio r b Electricity loss for heat prod. r v Fuel per electricity unit ϕ p Fuel per heat unit ϕ q Minimum fuel input f Maximum fuel input f Ramp-up/-down for fuel input r/ r Minimum heat output q Maximum heat output q Fuel marginal cost c No-load cost c Start-up/shut-down cost c SU /c SD Minimum up-time T U Minimum down-time T D Provides real-time flexibility yes no yes 30/41
43 31/41 Scheduling Example Results from the Deterministic Model Time period (h) 600 Extraction Backpressure Storage Demand Schedule (MWh) Time period (h)
44 32/41 Scheduling Example Results from the Robust Optimization Model Time period (h) 600 Extraction Backpressure Storage Demand Schedule (MWh) Time period (h)
45 Tuning the Need for Flexibility The extraction unit is the main provider of flexibility Unit Γ Total heat dispatch MWh Periods online h /41
46 34/41 Piecewise Linear Decision Rules We can define different decision rules for partitions of the uncertainty set 2 different rules for each dimension of the uncertainty partitioning demand deviation about 0 straightforward choice different response to deficit/surplus heat demand y LDR PLDR ω
47 Linear vs Piecewise Linear Decision Rules Dispatch Improvement Dispatch for extraction unit during peak-demand day Decision rule Unit Γ Linear MWh Piecewise-linear MWh /41
48 Linear vs Piecewise Linear Decision Rules In-Sample vs Out-of-Sample Results Theoretical improvement Empirical improvement % Γ Γ % Interval size (# sd) Interval size (# sd) decision rules not binding in practice anticipative out-of-sample evaluation (stochastic programming) 36/41
49 Price of Robustness Sensitivity analysis can help tune the RO parameters (cost vs robustness) Average profit Worst-case load-not-served (MWh) Γ Γ Interval size (# sd) Interval size (# sd) 37/41
50 RO vs Alternative Methods Out-of-Sample Analysis Model Interval Load not served (MWh) Γ Average profit (e) radius (# sd) largest expected RO-PLDR , DET SP Load-not-served not penalized in the objective function Out-of-sample analysis based on SP 38/41
51 39/41 Conclusion Robust optimization addresses some of the shortcomings of stochastic programming intractability (large scenario sets) violation of non-anticipativity in multi-stage problems Simpler models of the uncertainty are required uncertainty sets low-order moments (mean, standard deviation, etc.) Trade-off between optimality and conservativeness can be tuned via uncertainty set parameters similar results in terms of CVaR for risk-averse SP (ARO model for energy and reserve) tunable conservativeness in AARO model for CHP units Promising results in terms of scalability
52 40/41 References M. Zugno, A.J. Conejo (2014). A Robust Optimization Approach to Energy and Reserve Dispatch in Electricity Markets, (under review). M. Zugno, J.M. Morales, H. Madsen (2014). Commitment and Dispatch of Heat and Power Units via Affinely Adjustable Robust Optimization (working paper, available soon). M. Zugno, J.M. Morales, H. Madsen (2014). Robust Management of Combined Heat and Power Systems via Linear Decision Rules, 2014 IEEE Energy Conference (ENERGYCON) Proceedings,
53 41/41 Thank you for your attention! Q&A Session
54 42/41 Inner Problem Reformulation via Duality From min-max-min to min-max-max: min x c x T x+max w s.t. Fx = d ŵ, Gx g. max λ,µ ( w Qx)T λ + (l Mx N w) T µ s.t. P T λ + L T µ = c y, µ 0, s.t. H w h, Inner max-max problem can be merged into a single bilinear maximization problem
55 43/41 Solution via Cutting-Plane Method min x,β c x T x + β s.t. β w T k λ k + (l N w k ) T µ k (λ k Q + µ k M) x, k, Fx = d ŵ, Gx g. 1 Start with a lower bound β for the recourse cost 2 Find optimal first-stage solution x for the approximation (lower bound) 3 Calculate worst-case recourse cost (upper bound) 4 Generate cut for the vertex, go back to 2
56 43/41 Solution via Cutting-Plane Method min x,β c x T x + β s.t. β w T k λ k + (l N w k ) T µ k (λ k Q + µ k M) x, k, Fx = d ŵ, Gx g. While (upper bound - lower bound > ɛ) 1 Start with a lower bound β for the recourse cost 2 Find optimal first-stage solution x for the approximation (lower bound) 3 Calculate worst-case recourse cost (upper bound) 4 Generate cut for the vertex, go back to 2
57 43/41 Solution via Cutting-Plane Method min x,β c x T x + β s.t. β w T k λ k + (l N w k ) T µ k (λ k Q + µ k M) x, k, Fx = d ŵ, Gx g. While (upper bound - lower bound > ɛ) 1 Start with a lower bound β for the recourse cost 2 Find optimal first-stage solution x for the approximation (lower bound) 3 Calculate worst-case recourse cost (upper bound) 4 Generate cut for the vertex, go back to 2
58 43/41 Solution via Cutting-Plane Method min x,β c x T x + β s.t. β w T k λ k + (l N w k ) T µ k (λ k Q + µ k M) x, k, Fx = d ŵ, Gx g. While (upper bound - lower bound > ɛ) 1 Start with a lower bound β for the recourse cost 2 Find optimal first-stage solution x for the approximation (lower bound) 3 Calculate worst-case recourse cost (upper bound) 4 Generate cut for the vertex, go back to 2
59 43/41 Solution via Cutting-Plane Method min x,β c x T x + β s.t. β w T k λ k + (l N w k ) T µ k (λ k Q + µ k M) x, k, Fx = d ŵ, Gx g. While (upper bound - lower bound > ɛ) 1 Start with a lower bound β for the recourse cost 2 Find optimal first-stage solution x for the approximation (lower bound) 3 Calculate worst-case recourse cost (upper bound) 4 Generate cut for the vertex, go back to 2
60 So far only easy-to-enumerate polyhedral sets have been considered, or heuristic techniques 44/41 Subproblem Choice vs Uncertainty Set Point 3 in the cutting-plane method requires the solution of a bilinear program Simple, easy-to-enumerate uncertainty sets (budget) Heuristic methods (suboptimal) We want to be able to solve the problem exactly for any polyhedral uncertainty set w 2 ŵ W w 1
61 45/41 Reformulation of the Problem (1) Swapping Inner Problems The inner problems can be swapped: min x c x T x+max λ,µ s.t. Fx = d ŵ, Gx g, (Qx)T λ + (l Mx) T µ+max w s.t. P T λ + L T µ = c y, µ 0, ( λ T + µ T N ) w s.t. H w h, : ξ,
62 46/41 Reformulation of the Problem (2) Using Strong Duality The inner problem is linear strong duality holds: ( λ T + µ T N ) w = h T ξ } {{ } objective function We can exchange a bilinear term with a linear one, but the innermost problem becomes a minimization problem We can enforce KKT conditions to guarantee optimality and get rid of a level
63 indicates that the product of the operands is 0 this can be linearized using binary variables (MILP) 47/41 Reformulation of the Problem (3) Final Formulation min x c x T x+ max λ,µ, w,ξ (Qx)T λ + (l Mx) T µ + h T ξ s.t. Fx = d ŵ, Gx g. s.t. 0 ξ h H w 0, H T ξ = λ N T µ, P T λ + L T µ = c y, µ 0,
64 RO vs Alternative Methods Out-of-Sample Analysis (Spring) Model Interval radius (# sd) Γ Average profit (e) RO-PLDR Heat load not served (MWh) largest expected DET SP /41
65 RO vs Alternative Methods Out-of-Sample Analysis (Summer) Model Interval radius (# sd) Γ Average profit (e) Heat load not served (MWh) largest expected RO-PLDR DET SP /41
66 RO vs Alternative Methods Out-of-Sample Analysis (Autumn) Model Interval radius (# sd) Γ Average profit (e) RO-PLDR Heat load not served (MWh) largest expected DET SP /41
Two-Stage Stochastic Linear Programs
Two-Stage Stochastic Linear Programs Operations Research Anthony Papavasiliou 1 / 27 Two-Stage Stochastic Linear Programs 1 Short Reviews Probability Spaces and Random Variables Convex Analysis 2 Deterministic
More informationA Log-Robust Optimization Approach to Portfolio Management
A Log-Robust Optimization Approach to Portfolio Management Dr. Aurélie Thiele Lehigh University Joint work with Ban Kawas Research partially supported by the National Science Foundation Grant CMMI-0757983
More informationRandomization Approaches for Network Revenue Management with Customer Choice Behavior
Randomization Approaches for Network Revenue Management with Customer Choice Behavior Sumit Kunnumkal Indian School of Business, Gachibowli, Hyderabad, 500032, India sumit kunnumkal@isb.edu March 9, 2011
More informationAdvanced Lecture on Mathematical Science and Information Science I. Optimization in Finance
Advanced Lecture on Mathematical Science and Information Science I Optimization in Finance Reha H. Tütüncü Visiting Associate Professor Dept. of Mathematical and Computing Sciences Tokyo Institute of Technology
More informationRobust and Data-Driven Optimization: Modern Decision-Making Under Uncertainty
Robust and Data-Driven Optimization: Modern Decision-Making Under Uncertainty Dimtris Bertsimas Aurélie Thiele March 2006 Abstract Traditional models of decision-making under uncertainty assume perfect
More informationComputing the Electricity Market Equilibrium: Uses of market equilibrium models
Computing the Electricity Market Equilibrium: Uses of market equilibrium models Ross Baldick Department of Electrical and Computer Engineering The University of Texas at Austin April 2007 Abstract We discuss
More informationNan Kong, Andrew J. Schaefer. Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA
A Factor 1 2 Approximation Algorithm for Two-Stage Stochastic Matching Problems Nan Kong, Andrew J. Schaefer Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA Abstract We introduce
More informationA joint control framework for supply chain planning
17 th European Symposium on Computer Aided Process Engineering ESCAPE17 V. Plesu and P.S. Agachi (Editors) 2007 Elsevier B.V. All rights reserved. 1 A joint control framework for supply chain planning
More informationARTICLE IN PRESS. European Journal of Operational Research xxx (2004) xxx xxx. Discrete Optimization. Nan Kong, Andrew J.
A factor 1 European Journal of Operational Research xxx (00) xxx xxx Discrete Optimization approximation algorithm for two-stage stochastic matching problems Nan Kong, Andrew J. Schaefer * Department of
More informationDistributionally Robust Optimization with ROME (part 2)
Distributionally Robust Optimization with ROME (part 2) Joel Goh Melvyn Sim Department of Decision Sciences NUS Business School, Singapore 18 Jun 2009 NUS Business School Guest Lecture J. Goh, M. Sim (NUS)
More informationMoral Hazard. Itay Goldstein. Wharton School, University of Pennsylvania
Moral Hazard Itay Goldstein Wharton School, University of Pennsylvania 1 Principal-Agent Problem Basic problem in corporate finance: separation of ownership and control: o The owners of the firm are typically
More informationDuality in General Programs. Ryan Tibshirani Convex Optimization 10-725/36-725
Duality in General Programs Ryan Tibshirani Convex Optimization 10-725/36-725 1 Last time: duality in linear programs Given c R n, A R m n, b R m, G R r n, h R r : min x R n c T x max u R m, v R r b T
More informationEconometrics Simple Linear Regression
Econometrics Simple Linear Regression Burcu Eke UC3M Linear equations with one variable Recall what a linear equation is: y = b 0 + b 1 x is a linear equation with one variable, or equivalently, a straight
More informationA joint chance-constrained programming approach for call center workforce scheduling under uncertain call arrival forecasts
*Manuscript Click here to view linked References A joint chance-constrained programming approach for call center workforce scheduling under uncertain call arrival forecasts Abstract We consider a workforce
More informationManaging Financial Risk in Planning under Uncertainty
PROCESS SYSTEMS ENGINEERING Managing Financial Risk in Planning under Uncertainty Andres Barbaro and Miguel J. Bagajewicz School of Chemical Engineering and Materials Science, University of Oklahoma, Norman,
More information5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1
5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 General Integer Linear Program: (ILP) min c T x Ax b x 0 integer Assumption: A, b integer The integrality condition
More informationA Log-Robust Optimization Approach to Portfolio Management
A Log-Robust Optimization Approach to Portfolio Management Ban Kawas Aurélie Thiele December 2007, revised August 2008, December 2008 Abstract We present a robust optimization approach to portfolio management
More informationSTRATEGIC CAPACITY PLANNING USING STOCK CONTROL MODEL
Session 6. Applications of Mathematical Methods to Logistics and Business Proceedings of the 9th International Conference Reliability and Statistics in Transportation and Communication (RelStat 09), 21
More informationChapter 9 Short-Term Trading for Electricity Producers
Chapter 9 Short-Term Trading for Electricity Producers Chefi Triki, Antonio J. Conejo, and Lina P. Garcés Abstract This chapter considers a price-taker power producer that trades in an electricity pool
More informationVI. Real Business Cycles Models
VI. Real Business Cycles Models Introduction Business cycle research studies the causes and consequences of the recurrent expansions and contractions in aggregate economic activity that occur in most industrialized
More informationStatistical Machine Learning
Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes
More informationVariance Reduction. Pricing American Options. Monte Carlo Option Pricing. Delta and Common Random Numbers
Variance Reduction The statistical efficiency of Monte Carlo simulation can be measured by the variance of its output If this variance can be lowered without changing the expected value, fewer replications
More informationA Robust Formulation of the Uncertain Set Covering Problem
A Robust Formulation of the Uncertain Set Covering Problem Dirk Degel Pascal Lutter Chair of Management, especially Operations Research Ruhr-University Bochum Universitaetsstrasse 150, 44801 Bochum, Germany
More informationRobust Geometric Programming is co-np hard
Robust Geometric Programming is co-np hard André Chassein and Marc Goerigk Fachbereich Mathematik, Technische Universität Kaiserslautern, Germany Abstract Geometric Programming is a useful tool with a
More informationRobust Optimization Strategies for Total Cost Control in Project Management. Joel Goh. Nicholas G. Hall. Melvyn Sim
Robust Optimization Strategies for Total Cost Control in Project Management Joel Goh Nicholas G. Hall Melvyn Sim Graduate School of Business, Stanford University Department of Management Sciences, The
More informationSupport Vector Machine (SVM)
Support Vector Machine (SVM) CE-725: Statistical Pattern Recognition Sharif University of Technology Spring 2013 Soleymani Outline Margin concept Hard-Margin SVM Soft-Margin SVM Dual Problems of Hard-Margin
More informationINTEGER PROGRAMMING. Integer Programming. Prototype example. BIP model. BIP models
Integer Programming INTEGER PROGRAMMING In many problems the decision variables must have integer values. Example: assign people, machines, and vehicles to activities in integer quantities. If this is
More informationDiscrete Optimization
Discrete Optimization [Chen, Batson, Dang: Applied integer Programming] Chapter 3 and 4.1-4.3 by Johan Högdahl and Victoria Svedberg Seminar 2, 2015-03-31 Todays presentation Chapter 3 Transforms using
More informationNonlinear Optimization: Algorithms 3: Interior-point methods
Nonlinear Optimization: Algorithms 3: Interior-point methods INSEAD, Spring 2006 Jean-Philippe Vert Ecole des Mines de Paris Jean-Philippe.Vert@mines.org Nonlinear optimization c 2006 Jean-Philippe Vert,
More informationTwo Topics in Parametric Integration Applied to Stochastic Simulation in Industrial Engineering
Two Topics in Parametric Integration Applied to Stochastic Simulation in Industrial Engineering Department of Industrial Engineering and Management Sciences Northwestern University September 15th, 2014
More informationLecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method
Lecture 3 3B1B Optimization Michaelmas 2015 A. Zisserman Linear Programming Extreme solutions Simplex method Interior point method Integer programming and relaxation The Optimization Tree Linear Programming
More informationRe-Solving Stochastic Programming Models for Airline Revenue Management
Re-Solving Stochastic Programming Models for Airline Revenue Management Lijian Chen Department of Industrial, Welding and Systems Engineering The Ohio State University Columbus, OH 43210 chen.855@osu.edu
More informationDistributed Control in Transportation and Supply Networks
Distributed Control in Transportation and Supply Networks Marco Laumanns, Institute for Operations Research, ETH Zurich Joint work with Harold Tiemessen, Stefan Wörner (IBM Research Zurich) Apostolos Fertis,
More informationIntroduction to General and Generalized Linear Models
Introduction to General and Generalized Linear Models General Linear Models - part I Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby
More information24. The Branch and Bound Method
24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NP-complete. Then one can conclude according to the present state of science that no
More informationStochastic control of HVAC systems: a learning-based approach. Damiano Varagnolo
Stochastic control of HVAC systems: a learning-based approach Damiano Varagnolo Something about me 2 Something about me Post-Doc at KTH Post-Doc at U. Padova Visiting Scholar at UC Berkeley Ph.D. Student
More informationA stochastic programming approach for supply chain network design under uncertainty
A stochastic programming approach for supply chain network design under uncertainty Tjendera Santoso, Shabbir Ahmed, Marc Goetschalckx, Alexander Shapiro School of Industrial & Systems Engineering, Georgia
More informationSpatial Decomposition/Coordination Methods for Stochastic Optimal Control Problems. Practical aspects and theoretical questions
Spatial Decomposition/Coordination Methods for Stochastic Optimal Control Problems Practical aspects and theoretical questions P. Carpentier, J-Ph. Chancelier, M. De Lara, V. Leclère École des Ponts ParisTech
More informationA General Approach to Variance Estimation under Imputation for Missing Survey Data
A General Approach to Variance Estimation under Imputation for Missing Survey Data J.N.K. Rao Carleton University Ottawa, Canada 1 2 1 Joint work with J.K. Kim at Iowa State University. 2 Workshop on Survey
More informationThe Advantages and Disadvantages of Online Linear Optimization
LINEAR PROGRAMMING WITH ONLINE LEARNING TATSIANA LEVINA, YURI LEVIN, JEFF MCGILL, AND MIKHAIL NEDIAK SCHOOL OF BUSINESS, QUEEN S UNIVERSITY, 143 UNION ST., KINGSTON, ON, K7L 3N6, CANADA E-MAIL:{TLEVIN,YLEVIN,JMCGILL,MNEDIAK}@BUSINESS.QUEENSU.CA
More informationA Robust Optimization Approach to Supply Chain Management
A Robust Optimization Approach to Supply Chain Management Dimitris Bertsimas and Aurélie Thiele Massachusetts Institute of Technology, Cambridge MA 0139, dbertsim@mit.edu, aurelie@mit.edu Abstract. We
More informationInvestment and Risk Management in Electrical Infrastructure
Investment and Risk Management in Electrical Infrastructure Oliver Wright Supervisor: Dr. Daniel Kuhn Second Marker: Professor Berç Rustem Department of Computing, Imperial College London, United Kingdom,
More informationModern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh
Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem
More informationA simulation based optimization approach to supply chain management under demand uncertainty
Computers and Chemical Engineering 28 (2004) 2087 2106 A simulation based optimization approach to supply chain management under demand uncertainty June Young Jung a, Gary Blau a, Joseph F. Pekny a, Gintaras
More informationSpatial Statistics Chapter 3 Basics of areal data and areal data modeling
Spatial Statistics Chapter 3 Basics of areal data and areal data modeling Recall areal data also known as lattice data are data Y (s), s D where D is a discrete index set. This usually corresponds to data
More informationInteger Programming Approaches for Appointment Scheduling with Random No-shows and Service Durations
Vol. 00, No. 0, Xxxxx 0000, pp. 000 000 issn 0030-364X eissn 1526-5463 00 0000 0001 Integer Programming Approaches for Appointment Scheduling with Random No-shows and Service Durations Ruiwei Jiang Department
More informationEquilibrium computation: Part 1
Equilibrium computation: Part 1 Nicola Gatti 1 Troels Bjerre Sorensen 2 1 Politecnico di Milano, Italy 2 Duke University, USA Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium
More informationIntroduction to Support Vector Machines. Colin Campbell, Bristol University
Introduction to Support Vector Machines Colin Campbell, Bristol University 1 Outline of talk. Part 1. An Introduction to SVMs 1.1. SVMs for binary classification. 1.2. Soft margins and multi-class classification.
More informationNumerical methods for American options
Lecture 9 Numerical methods for American options Lecture Notes by Andrzej Palczewski Computational Finance p. 1 American options The holder of an American option has the right to exercise it at any moment
More informationMathematical finance and linear programming (optimization)
Mathematical finance and linear programming (optimization) Geir Dahl September 15, 2009 1 Introduction The purpose of this short note is to explain how linear programming (LP) (=linear optimization) may
More informationASSET-LIABILITY MANAGEMENT FOR CZECH PENSION FUNDS USING STOCHASTIC PROGRAMMING
ASSET-LIABILITY MANAGEMENT FOR CZECH PENSION FUNDS USING STOCHASTIC PROGRAMMING Jitka Dupačová and Jan Polívka, Department of Probability and Mathematical Statistics, Charles University Prague, Sokolovská
More informationUSING SPECTRAL RADIUS RATIO FOR NODE DEGREE TO ANALYZE THE EVOLUTION OF SCALE- FREE NETWORKS AND SMALL-WORLD NETWORKS
USING SPECTRAL RADIUS RATIO FOR NODE DEGREE TO ANALYZE THE EVOLUTION OF SCALE- FREE NETWORKS AND SMALL-WORLD NETWORKS Natarajan Meghanathan Jackson State University, 1400 Lynch St, Jackson, MS, USA natarajan.meghanathan@jsums.edu
More informationECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE
ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE YUAN TIAN This synopsis is designed merely for keep a record of the materials covered in lectures. Please refer to your own lecture notes for all proofs.
More information2007/26. A tighter continuous time formulation for the cyclic scheduling of a mixed plant
CORE DISCUSSION PAPER 2007/26 A tighter continuous time formulation for the cyclic scheduling of a mixed plant Yves Pochet 1, François Warichet 2 March 2007 Abstract In this paper, based on the cyclic
More informationNonparametric adaptive age replacement with a one-cycle criterion
Nonparametric adaptive age replacement with a one-cycle criterion P. Coolen-Schrijner, F.P.A. Coolen Department of Mathematical Sciences University of Durham, Durham, DH1 3LE, UK e-mail: Pauline.Schrijner@durham.ac.uk
More informationLocating and sizing bank-branches by opening, closing or maintaining facilities
Locating and sizing bank-branches by opening, closing or maintaining facilities Marta S. Rodrigues Monteiro 1,2 and Dalila B. M. M. Fontes 2 1 DMCT - Universidade do Minho Campus de Azurém, 4800 Guimarães,
More informationSome representability and duality results for convex mixed-integer programs.
Some representability and duality results for convex mixed-integer programs. Santanu S. Dey Joint work with Diego Morán and Juan Pablo Vielma December 17, 2012. Introduction About Motivation Mixed integer
More informationα α λ α = = λ λ α ψ = = α α α λ λ ψ α = + β = > θ θ β > β β θ θ θ β θ β γ θ β = γ θ > β > γ θ β γ = θ β = θ β = θ β = β θ = β β θ = = = β β θ = + α α α α α = = λ λ λ λ λ λ λ = λ λ α α α α λ ψ + α =
More informationDynamic Thermal Ratings in Real-time Dispatch Market Systems
Dynamic Thermal Ratings in Real-time Dispatch Market Systems Kwok W. Cheung, Jun Wu GE Grid Solutions Imagination at work. FERC 2016 Technical Conference June 27-29, 2016 Washington DC 1 Outline Introduction
More informationPortfolio Distribution Modelling and Computation. Harry Zheng Department of Mathematics Imperial College h.zheng@imperial.ac.uk
Portfolio Distribution Modelling and Computation Harry Zheng Department of Mathematics Imperial College h.zheng@imperial.ac.uk Workshop on Fast Financial Algorithms Tanaka Business School Imperial College
More informationPricing Transmission
1 / 37 Pricing Transmission Quantitative Energy Economics Anthony Papavasiliou 2 / 37 Pricing Transmission 1 Equilibrium Model of Power Flow Locational Marginal Pricing Congestion Rent and Congestion Cost
More informationProbabilistic Models for Big Data. Alex Davies and Roger Frigola University of Cambridge 13th February 2014
Probabilistic Models for Big Data Alex Davies and Roger Frigola University of Cambridge 13th February 2014 The State of Big Data Why probabilistic models for Big Data? 1. If you don t have to worry about
More information1 Teaching notes on GMM 1.
Bent E. Sørensen January 23, 2007 1 Teaching notes on GMM 1. Generalized Method of Moment (GMM) estimation is one of two developments in econometrics in the 80ies that revolutionized empirical work in
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationMulti-Faceted Solution for Managing Flexibility with High Penetration of Renewable Resources
Multi-Faceted Solution for Managing Flexibility with High Penetration of Renewable Resources FERC Technical Conference Increasing RT & DA Market Efficiency Through Improved Software June 24 26, 2013 Nivad
More informationAn Internal Model for Operational Risk Computation
An Internal Model for Operational Risk Computation Seminarios de Matemática Financiera Instituto MEFF-RiskLab, Madrid http://www.risklab-madrid.uam.es/ Nicolas Baud, Antoine Frachot & Thierry Roncalli
More informationLecture 3: Linear methods for classification
Lecture 3: Linear methods for classification Rafael A. Irizarry and Hector Corrada Bravo February, 2010 Today we describe four specific algorithms useful for classification problems: linear regression,
More informationWorst-Case Conditional Value-at-Risk with Application to Robust Portfolio Management
Worst-Case Conditional Value-at-Risk with Application to Robust Portfolio Management Shu-Shang Zhu Department of Management Science, School of Management, Fudan University, Shanghai 200433, China, sszhu@fudan.edu.cn
More informationUSBR PLEXOS Demo November 8, 2012
PLEXOS for Power Systems Electricity Market Simulation USBR PLEXOS Demo November 8, 2012 Who We Are PLEXOS Solutions is founded in 2005 Acquired by Energy Exemplar in 2011 Goal People To solve the challenge
More informationCyber-Security Analysis of State Estimators in Power Systems
Cyber-Security Analysis of State Estimators in Electric Power Systems André Teixeira 1, Saurabh Amin 2, Henrik Sandberg 1, Karl H. Johansson 1, and Shankar Sastry 2 ACCESS Linnaeus Centre, KTH-Royal Institute
More informationOptimization Methods in Finance
Optimization Methods in Finance Gerard Cornuejols Reha Tütüncü Carnegie Mellon University, Pittsburgh, PA 15213 USA January 2006 2 Foreword Optimization models play an increasingly important role in financial
More informationModelling framework for power systems. Juha Kiviluoma Erkka Rinne Niina Helistö Miguel Azevedo
S VISIONS SCIENCE TECHNOLOGY RESEARCH HIGHLIGHT 196 Modelling framework for power systems Juha Kiviluoma Erkka Rinne Niina Helistö Miguel Azevedo VTT TECHNOLOGY 196 Modelling framework for power systems
More informationIncreasing Supply Chain Robustness through Process Flexibility and Inventory
Increasing Supply Chain Robustness through Process Flexibility and Inventory David Simchi-Levi He Wang Yehua Wei This Version: January 1, 2015 Abstract We study a hybrid strategy that uses both process
More informationAn Hybrid Optimization Method For Risk Measure Reduction Of A Credit Portfolio Under Constraints
An Hybrid Optimization Method For Risk Measure Reduction Of A Credit Portfolio Under Constraints Ivorra Benjamin & Mohammadi Bijan Montpellier 2 University Quibel Guillaume, Tehraoui Rim & Delcourt Sebastien
More informationOperations and Supply Chain Management Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras
Operations and Supply Chain Management Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Lecture - 36 Location Problems In this lecture, we continue the discussion
More information3.1 Solving Systems Using Tables and Graphs
Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system
More informationRevenue Management for Transportation Problems
Revenue Management for Transportation Problems Francesca Guerriero Giovanna Miglionico Filomena Olivito Department of Electronic Informatics and Systems, University of Calabria Via P. Bucci, 87036 Rende
More informationProximal mapping via network optimization
L. Vandenberghe EE236C (Spring 23-4) Proximal mapping via network optimization minimum cut and maximum flow problems parametric minimum cut problem application to proximal mapping Introduction this lecture:
More informationRisk-limiting dispatch for the smart grid: some research problems
Risk-limiting dispatch for the smart grid: some research problems Janusz Bialek, University of Edinburgh Pravin Varaiya, University of California, Berkeley Felix Wu, University of Hong Kong 1 Outline 1.
More informationCoordinated Price-Maker Operation of Large Energy Storage Units in Nodal Energy Markets
Coordinated Price-Maker Operation of Large Energy torage Units in Nodal Energy Markets Hamed Mohsenian-Rad, enior Member, IEEE Abstract In this paper, a new optimization framework is proposed to coordinate
More informationTheory and applications of Robust Optimization
Theory and applications of Robust Optimization Dimitris Bertsimas, David B. Brown, Constantine Caramanis July 6, 2007 Abstract In this paper we survey the primary research, both theoretical and applied,
More informationStock Price Dynamics, Dividends and Option Prices with Volatility Feedback
Stock Price Dynamics, Dividends and Option Prices with Volatility Feedback Juho Kanniainen Tampere University of Technology New Thinking in Finance 12 Feb. 2014, London Based on J. Kanniainen and R. Piche,
More informationSummary of specified general model for CHP system
Fakulteta za Elektrotehniko Eva Thorin, Heike Brand, Christoph Weber Summary of specified general model for CHP system OSCOGEN Deliverable D1.4 Contract No. ENK5-CT-2000-00094 Project co-funded by the
More informationA Programme Implementation of Several Inventory Control Algorithms
BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume, No Sofia 20 A Programme Implementation of Several Inventory Control Algorithms Vladimir Monov, Tasho Tashev Institute of Information
More informationMarkups and Firm-Level Export Status: Appendix
Markups and Firm-Level Export Status: Appendix De Loecker Jan - Warzynski Frederic Princeton University, NBER and CEPR - Aarhus School of Business Forthcoming American Economic Review Abstract This is
More informationOptimal Haul Truck Allocation in the Syncrude Mine
University of Alberta Optimal Haul Truck Allocation in the Syncrude Mine by Chung Huu Ta A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for
More informationDepartment of Mathematics, Indian Institute of Technology, Kharagpur Assignment 2-3, Probability and Statistics, March 2015. Due:-March 25, 2015.
Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment -3, Probability and Statistics, March 05. Due:-March 5, 05.. Show that the function 0 for x < x+ F (x) = 4 for x < for x
More informationBilevel Models of Transmission Line and Generating Unit Maintenance Scheduling
Bilevel Models of Transmission Line and Generating Unit Maintenance Scheduling Hrvoje Pandžić July 3, 2012 Contents 1. Introduction 2. Transmission Line Maintenance Scheduling 3. Generating Unit Maintenance
More informationImproving Market Clearing Software Performance to Meet Existing and Future Challenges MISO s Perspective
Improving Market Clearing Software Performance to Meet Existing and Future Challenges MISO s Perspective FERC Technical Conference on Increasing Real-Time and Day-Ahead Market Efficiency through Improved
More informationAverage Redistributional Effects. IFAI/IZA Conference on Labor Market Policy Evaluation
Average Redistributional Effects IFAI/IZA Conference on Labor Market Policy Evaluation Geert Ridder, Department of Economics, University of Southern California. October 10, 2006 1 Motivation Most papers
More informationRecovery of primal solutions from dual subgradient methods for mixed binary linear programming; a branch-and-bound approach
MASTER S THESIS Recovery of primal solutions from dual subgradient methods for mixed binary linear programming; a branch-and-bound approach PAULINE ALDENVIK MIRJAM SCHIERSCHER Department of Mathematical
More informationLinear Programming Notes V Problem Transformations
Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material
More information5. Linear Regression
5. Linear Regression Outline.................................................................... 2 Simple linear regression 3 Linear model............................................................. 4
More informationLinear Programming: Chapter 11 Game Theory
Linear Programming: Chapter 11 Game Theory Robert J. Vanderbei October 17, 2007 Operations Research and Financial Engineering Princeton University Princeton, NJ 08544 http://www.princeton.edu/ rvdb Rock-Paper-Scissors
More informationThe Heat Equation. Lectures INF2320 p. 1/88
The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)
More informationPoint Biserial Correlation Tests
Chapter 807 Point Biserial Correlation Tests Introduction The point biserial correlation coefficient (ρ in this chapter) is the product-moment correlation calculated between a continuous random variable
More informationCredit Risk Models: An Overview
Credit Risk Models: An Overview Paul Embrechts, Rüdiger Frey, Alexander McNeil ETH Zürich c 2003 (Embrechts, Frey, McNeil) A. Multivariate Models for Portfolio Credit Risk 1. Modelling Dependent Defaults:
More informationA Note on the Bertsimas & Sim Algorithm for Robust Combinatorial Optimization Problems
myjournal manuscript No. (will be inserted by the editor) A Note on the Bertsimas & Sim Algorithm for Robust Combinatorial Optimization Problems Eduardo Álvarez-Miranda Ivana Ljubić Paolo Toth Received:
More informationUNCERTAINTY IN THE ELECTRIC POWER INDUSTRY Methods and Models for Decision Support
UNCERTAINTY IN THE ELECTRIC POWER INDUSTRY Methods and Models for Decision Support CHRISTOPH WEBER University of Stuttgart, Institute for Energy Economics and Rational of Use of Energy fyj. Springer Contents
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More information