APMOD04Application of the Benders decomposition ot a … · Application of the Benders...

Application of the Benders decomposition to a stochastic mixed integer UC problem

Santiago Cerisola, Álvaro Baíllo, Andrés RamosInstituto Investigación Tecnológica Universidad Pontificia Comillas

Ralf GollmerInstitut für Mathematik University Duisburg-Essen

Presentación Tesis Santiago Cerisola IIT - 2

Index

• Motivation• The L-Shaped method• Extension and sequential cut refinement strategy• Numerical application: Stochastic Unit Commitment• Conclusions


Index



Motivation

• Real optimization models presents matrix structures that may be exploited in their numerical solution.– Economic Dispatch – Optimal Power Flow – Unit Commitment– Generalized Unit Commitment– Multiperiod problems– Stochastic programming


Problems structure:Multiperiod problems


Problems structure : Stochastic Programming

• Stochastic programming deals with optimization models in which some of the parameters are random variables.

• Different goals: Mean Value minimization– Continuous distributions: Present difficulties for the

solution. It relies on numerical integration – Discrete distributions: The stochastic problem can be

solved by formulating its deterministic equivalent problem• Large scale problems that make necessary the use of

decomposition techniques• Staircase structures that induce solution through

decomposition techniques


Problems structure : Stochastic programming


Problems structure : Stochastic programming


Index



The L-shaped method

• Usually applied to problems with a clear difference in the collection of decision variables.

• The two stage problem is equivalently formulated as a master problem (first stage decision) and a subproblem(second stage decisions)

( ) minx qyWy h Txy Y

θ == −

∈

min

,

cx qyTx Wy hx X y Y

++ =∈ ∈

min ( )cx xx X

θ+∈


The L-shaped method (II)

• The recourse function indicates the variation of thesubproblem optimal objective value as a function of the first stage variables that modify the RHS. These variables are denoted as tender variables

• Properties– Linear subproblem – Recourse function is convex

because of LP duality– This property induces in a natural way an iterative

algorithm that solves the problem


The L-shaped method (III)

• Evolution of the Benders algorithm

( )cx xθ+


The L-shaped method (IV): GBD

• The incorporation of integer variables in the subproblemcomplicates the resolution of two-stage problems. The recourse function becomes– Non convex– No continuous

• A possibility consists in the use of the LagrangeanRelaxation to solve the subproblem. This method provides a partial convexification of the recourse function

0

0

( ) minx qyWy h Txy Y

θ == −

∈

0max min ( )

y qy Wy h Tx

y Yλ λ+ − +

∈


Index



Extension

• A modification is introduced via a perturbation variable that represents the domain of definition of the recourse function

• The perturbation variable is limited to

{ }, 0max min ( )

y r qy r Tx

Wy h ry Y

λ λ λ+ −

− =∈

{ },Tx x X− ∈

{ }, 0max min ( )

,

y r qy Tx Tx

Tx Wy hx X y Y

λ λ λ− +

+ =∈ ∈


Extension: example

• Consider the solution of the following problem

min 0.3 1.50 5, 0, 0

3.75.2

,

x y zx y z

x yy zy Z z Z

− − −≤ ≤ ≥ ≥+ ≤+ ≤∈ ∈

Master Problem

min 0.3 ( )0 5

x xx

θ− +≤ ≤

Subproblem

min 1.53.7

5.20, 0 ,

y zy xy zy z y Z z Z

− −≤ −+ ≤≥ ≥ ∈ ∈


Example

LINEAR

LR

LR + P


Extension for nested situations

Stage 1

Stage 2

Stage n

Stage 3

Stage 1 primal value

Stage 2 primal value

.

.

.Stage n-1 primal value

.

.

.

Stage n dual value (LR)

Stage 3 dual value (LR)

Stage 2 dual value (LR)

.

.

.


Extension for nested situations

• The extension of the algorithm to a nested situation is carried out in a natural way– In forward passes the subproblems are solved via a

branch-and-bound method– In backward passes the subproblems are solved via

LR algorithm


Index



Sequential Cut Refinement Strategy

• Sequential cut refinement method:– General strategy for computing cheap computational cuts

prior to those more expensive computational cuts– It includes the following phases:

LRMIPPhase 5

MIP + Lagrangean Subproblem

MIPPhase 4LP + Lagrangean SubproblemMIPPhase 3

LPMIPPhase 2LPLPPhase 1

Backward ResolutionForward Resolution


Index



Application: Stochastic Unit Commitment

• Short term – Unit commitment model (UC)– Objective: Minimization of the total expected cost

Subject to:Demand and spinning reserve satisfactionMaximum and minimum outputReserve dynamics constraintsStart up and shut downs management constraints

• VEAG system (VD & VA) – Time horizon: 1 week, division into 168 hours.– Disaggregated units (32 thermal and 22 pumped

storage)– Aggregated units (14 thermal and 8 pumped

storage)– Stochastic instances (1,4, 7 and 12 scenarios)



• Matrix structure for the deterministic problem



• Scenarios tree’s skeletons for the 7 and 12 scenario situation


Application: Model size (1 scenario size)

25205712Binaries

44163104440Nonzero

1646640150Columns

1276929902Rows

VEAG aggregatedVEAG disaggregated

Model size


Implementation

• Algorithms coded in C using ILOG Concert Technology– Concert Technology 1.2 and Cplex 7.5

• A generic code that performs and implements the proposed algorithm

• Special design oriented to permit the interconnection of different models.

Model 1 Model 2 Model 3

Model 5Model 4

Benders Solution


Application: computational tests

• Algorithm tests were performed with a Pentium III, 512 MB RAM , 1.8 GHz

• Accuracies for different problems:– VEAG Disaggregated model 1,4 and 7 scenarios

• accuracy of 0.06% after phase 2– VEAG Aggregated model 1,4,7 and 12 scenarios

• accuracy of 0.05% after phase 2


Application: time comparisons (seconds)

VEAG VA

VEAG VD

Model

3051110468042288177329184012

13760¿?7394560000473040031

BendersDirectScenarios


Application: time comparisons II (seconds)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

x 104

0

200

400

600

800

1000

1200

1400

1600

1800

2000

Num ber of binary variables

Com

puta

tion

times

Direc t ResolutionBenders


Index



Conclusions

• An extension of the GBD method has been presented that reduces the final duality gap at the optimum found by the L-shaped method

• The sequential cut refinement method “naturally” computes the cheaper cuts prior to the more expensive cuts

• The method has been coded and tested over a stochastic unit commitment problem.

• The decomposition considers subproblems for individual nodes of the tree in a different manner as traditional UC decompositions, that consider individual problems for each generator

Aplication of the Benders decomposition to a stochastic mixed integer UC problem

Santiago Cerisola, Álvaro Baíllo, Andrés RamosInstituto Investigación Tecnológica Universidad Pontificia Comillas

Ralf GollmerInstitut für Mathematik University Duisburg-Essen

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