accelerating the convergence of stochastic unit commitment ... · commitment problems by using...

Accelerating the Convergence of Stochastic UnitCommitment Problems by Using Tight and Compact

MIP Formulations

Germán Morales-España†,

and Andrés Ramos∗

†Delft University of Technology, Delft, The Netherlands

∗Universidad Pontificia Comillas, Madrid, Spain

PES General Meeting

Denver-USA, July 2015

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Outline

1 Introduction

2 Good and Ideal MIP formulations

3 Tight & Compact (TC) UC Formulations

4 Case StudiesDeterministic Selft-UCStochastic UCs: Different SolversStochastic UCs: IEEE-118 Bus System

5 Conclusions

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Introduction

UC and MIP

Unit Commitment (UC): essential tool for day(week)-aheadplanning in the electricity sector

Decide on units’ physical operation (e.g., on-off) at minimum costUC is an integer computationally demanding problem

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Introduction

UC and MIP



Significant breakthroughs in Mixed-Integer Programming (MIP)

Solving MIP 100 million times faster than 20 years ago1

The time to solve UC is still a critical limitation

1T. Koch, T. Achterberg, E. Andersen, O. Bastert, T. Berthold, R. E. Bixby, E. Danna, G. Gamrath, A. M. Gleixner,S. Heinz, A. Lodi, H. Mittelmann, T. Ralphs, D. Salvagnin, D. E. Steffy, and K. Wolter, “MIPLIB 2010,” Mathematical

Programming Computation, vol. 3, no. 2, pp. 103–163, Jun. 2011

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Introduction

UC and MIP






How to reduce solving times?

Computer power (e.g., clusters)Solving algorithms (e.g., solvers, decomposition techniques)



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Introduction

UC and MIP






How to reduce solving times?

Computer power (e.g., clusters)Solving algorithms (e.g., solvers, decomposition techniques)Improving the MIP-Based UC formulation ⇒ ↓ solving times



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Good & Ideal MIP

Outline

1 Introduction




5 Conclusions

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Good & Ideal MIP

An MIP Has Infinite LP Formulations

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Good & Ideal MIP


LP1 and LP2 represent the same MIP problem

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Good & Ideal MIP


LP1, LP2 and CH represent the same MIP problem

which one to choose?

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Good & Ideal MIP

Solving MIP Through The Powerful LP

Shaping the linear feasible region to arrive from vertex ZLP to ZMIP

To prove optimality ZMIP must become a vertex by:

Branch and bound (divide and conquer)

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Good & Ideal MIP

Solving MIP Through The Powerful LP

Shaping the linear feasible region to arrive from vertex ZLP to ZMIP

To prove optimality ZMIP must become a vertex by:

Branch and bound (divide and conquer)

and/or by adding cuts

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Good & Ideal MIP

Convex Hull: The Tightest Formulation

Convex Hull (CH)

Smallest convex feasible regioncontaining all feasible integerpoints2

2L. Wolsey, Integer Programming. Wiley-Interscience, 19983H. P. Williams, Model Building in Mathematical Programming, 5th Edition. John Wiley & Sons Inc, Feb. 2013

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Good & Ideal MIP


Convex Hull (CH)


The convex hull problem solves an MIP as an LP

Each vertex satisfies the integrality constraintsSo an LP optimum is also an MIP optimum

Unfortunately,


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Good & Ideal MIP


Convex Hull (CH)


The convex hull problem solves an MIP as an LP

Each vertex satisfies the integrality constraintsSo an LP optimum is also an MIP optimum

Unfortunately, the convex hull is typically too difficult to obtain2,3

To solve an MIP is usually easier than trying to find its convex hull


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Good & Ideal MIP

Choosing The Best FormulationMeasuring The Tightness

Integrality Gap (IGap)

Relative distance betweenMIP and LP optima

IGapLP1 = ZMIP−ZLP1

ZMIP> IGapLP2 = ZMIP−ZLP2

ZMIP

⇓

As an MIP problem:LP2 is expected to be solved faster than LP1

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Good & Ideal MIP

Choosing The Best FormulationMeasuring The Tightness

Integrality Gap (IGap)

Relative distance betweenMIP and LP optima

IGapLP1 = ZMIP−ZLP1

ZMIP> IGapLP2 = ZMIP−ZLP2

ZMIP>

IGapCH = ZMIP−ZCH

ZMIP=0

⇓

As an MIP problem:LP2 is expected to be solved faster than LP1

CH will be solved way faster than LP2

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Good & Ideal MIP

Concepts: Tightness and Compactness

Tightness: defines the search space (relaxed feasible region) thatthe solver needs to explore to find the solution

Compactness (problem size): defines the searching speed (data toprocess) that the solver takes to find the solution

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Good & Ideal MIP

Concepts: Tightness and Compactness

Tightness: defines the search space (relaxed feasible region) thatthe solver needs to explore to find the solution

Compactness (problem size): defines the searching speed (data toprocess) that the solver takes to find the solution

Convex hull: The tightest formulation ⇒ MIP solved as LP

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Good & Ideal MIP

Tightening an MIP Formulation

The most common strategy is adding cuts

In fact, this is the most effective strategy of current MIP solvers4

4R. Bixby and E. Rothberg, “Progress in computational mixed integer programming—a look back from the other side ofthe tipping point,” Annals of Operations Research, vol. 149, no. 1, pp. 37–41, Jan. 2007

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Good & Ideal MIP




They should be added as cuts in the B&B5,6 ⇒ ↓ Timeand not directly to the model, huge number of inequalities ⇒ ↑ Time


5P. Damci-Kurt, S. Kucukyavuz, D. Rajan, and A. Atamturk, “A Polyhedral Study of Ramping in Unit Committment,”University of California-Berkeley, Research Report BCOL.13.02 IEOR, Oct. 2013

6J. Ostrowski, M. F Anjos, and A. Vannelli, “Tight mixed integer linear programming formulations for the unitcommitment problem,” IEEE Transactions on Power Systems, vol. 27, no. 1, pp. 39–46, Feb. 2012

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Good & Ideal MIP




They should be added as cuts in the B&B5,6 ⇒ ↓ Timeand not directly to the model, huge number of inequalities ⇒ ↑ TimeTrade-off: Tightness vs. Compactness




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Good & Ideal MIP




They should be added as cuts in the B&B5,6 ⇒ ↓ Timeand not directly to the model, huge number of inequalities ⇒ ↑ TimeTrade-off: Tightness vs. Compactness

Improving the MIP formulation

Provide the convex hull for some set of constraintsIf available, use the convex hull for some set of constraints




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Tight & Compact UCs

Outline

1 Introduction




5 Conclusions

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Tight & Compact UCs

Tight and Compact (TC) Formulation

Let’s focus on the core of UC formulations:

Min/max outputsSU & SD rampsMinimum up/down (T U/T D) times

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Tight & Compact UCs

Tight and Compact (TC) Formulation

Let’s focus on the core of UC formulations:

Min/max outputsSU & SD rampsMinimum up/down (T U/T D) times, convex hull already available7

The whole formulation can be found in the paper TC-UC8 and theconvex hull proof in gentile et al.9

7D. Rajan and S. Takriti, “Minimum up/down polytopes of the unit commitment problem with start-up costs,” IBM,Research Report RC23628, Jun. 2005

8G. Morales-Espana, J. M. Latorre, and A. Ramos, “Tight and compact MILP formulation for the thermal unitcommitment problem,” IEEE Transactions on Power Systems, vol. 28, no. 4, pp. 4897–4908, Nov. 2013

9C. Gentile, G. Morales-Espana, and A. Ramos, “A tight MIP formulation of the unit commitment problem with start-upand shut-down constraints,” Institute for Research in Technology (IIT), Technical Report IIT-14-040A, 2014

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Tight & Compact UCs

Formulation for a generating unit (I)

Generation limits taking into account:

pt ≤(

P − P)

ut −(

P − SD)

wt+1 − max (SD−SU, 0) vt ∀t

(1)

pt ≤(

P − P)

ut −(

P − SU)

vt − max (SU−SD, 0) wt+1 ∀t

(2)

Total generation = P · ut + pt.

Variables Parameters

pt Energy production above P P Minimum power output

ut Commitment status P Maximum power output

vt Startup status SU Startup ramp

wt Shutdown status SD Shutdown ramp

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Tight & Compact UCs

Formulation for a generating unit (II)

Logical relationship:

ut − ut−1 = vt − wt ∀t (3)

vt ≤ ut ∀t (4)

wt ≤ 1 − ut ∀t (5)

where (4) and (5) avoid the simultaneous startup and shutdown.

Variable bounds

pt ≥ 0 ∀t (6)

0 ≤ ut, vt, wt ≤ 1 ∀t (7)

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Tight & Compact UCs

Tightness of the Formulation

Let’s study the polytope (1)-(7) using PORTA10:

PORTA enumerates all vertices of a convex feasible region

10T. Christof and A. Löbel, “PORTA: POlyhedron representation transformation algorithm, version 1.4.1,”Konrad-Zuse-Zentrum für Informationstechnik Berlin, Germany, 2009

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Tight & Compact UCs

Tightness of the Formulation

Let’s study the polytope (1)-(7) using PORTA10:

PORTA enumerates all vertices of a convex feasible region

Example: 3 periods and P = 200, P = SU = SD = 100 for:

Case 1: T U = T D = 1Case 2: T U = T D = 2

10T. Christof and A. Löbel, “PORTA: POlyhedron representation transformation algorithm, version 1.4.1,”Konrad-Zuse-Zentrum für Informationstechnik Berlin, Germany, 2009

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Tight & Compact UCs

Case 1: Providing The Convex HullFormulation:

pt ≤(

P − P)

ut −(

P − SD)

wt+1

− max (SD−SU, 0) vt (1)

pt ≤(

P − P)

ut −(

P − SU)

vt

− max (SU−SD, 0) wt+1 (2)

ut − ut−1 = vt − wt (3)

vt ≤ ut (4)

wt ≤ 1 − ut (5)

PORTA results for (T U =T D =1)

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Tight & Compact UCs


pt ≤(

P − P)

ut −(

P − SD)

wt+1

− max (SD−SU, 0) vt (1)

pt ≤(

P − P)

ut −(

P − SU)

vt

− max (SU−SD, 0) wt+1 (2)

ut − ut−1 = vt − wt (3)

vt ≤ ut (4)

wt ≤ 1 − ut (5)

PORTA results for (T U =T D =1)u1, u2, u3, v2, v3, w2, w3, p1, p2, p3:

DIM = 10

CONV_SECTION( 1) 0 0 0 0 0 0 0 0 0 0( 2) 0 0 1 0 1 0 0 0 0 0( 3) 1 0 0 0 0 1 0 0 0 0( 4) 0 1 0 1 0 0 1 0 0 0( 5) 0 1 1 1 0 0 0 0 0 0( 6) 0 1 1 1 0 0 0 0 0 100( 7) 1 1 0 0 0 0 1 0 0 0( 8) 1 1 0 0 0 0 1 100 0 0( 9) 1 1 1 0 0 0 0 0 0 0( 10) 1 1 1 0 0 0 0 0 0 100( 11) 1 1 1 0 0 0 0 0 100 0( 12) 1 1 1 0 0 0 0 0 100 100( 13) 1 1 1 0 0 0 0 100 0 0( 14) 1 1 1 0 0 0 0 100 0 100( 15) 1 1 1 0 0 0 0 100 100 0( 16) 1 1 1 0 0 0 0 100 100 100( 17) 1 0 1 0 1 1 0 0 0 0

END

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Tight & Compact UCs


pt ≤(

P − P)

ut −(

P − SD)

wt+1

− max (SD−SU, 0) vt (1)

pt ≤(

P − P)

ut −(

P − SU)

vt

− max (SU−SD, 0) wt+1 (2)

ut − ut−1 = vt − wt (3)

vt ≤ ut (4)

wt ≤ 1 − ut (5)

All vertices are integer⇓

Convex Hull


DIM = 10

CONV_SECTION( 1) 0 0 0 0 0 0 0 0 0 0( 2) 0 0 1 0 1 0 0 0 0 0( 3) 1 0 0 0 0 1 0 0 0 0( 4) 0 1 0 1 0 0 1 0 0 0( 5) 0 1 1 1 0 0 0 0 0 0( 6) 0 1 1 1 0 0 0 0 0 100( 7) 1 1 0 0 0 0 1 0 0 0( 8) 1 1 0 0 0 0 1 100 0 0( 9) 1 1 1 0 0 0 0 0 0 0( 10) 1 1 1 0 0 0 0 0 0 100( 11) 1 1 1 0 0 0 0 0 100 0( 12) 1 1 1 0 0 0 0 0 100 100( 13) 1 1 1 0 0 0 0 100 0 0( 14) 1 1 1 0 0 0 0 100 0 100( 15) 1 1 1 0 0 0 0 100 100 0( 16) 1 1 1 0 0 0 0 100 100 100( 17) 1 0 1 0 1 1 0 0 0 0

END

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Tight & Compact UCs

Case 2: Providing and Using Convex Hulls (I)Formulation + T U/T D Convex hull:

pt ≤(

P − P)

ut −(

P − SD)

wt+1

− max (SD−SU, 0) vt (1)

pt ≤(

P − P)

ut −(

P − SU)

vt

− max (SU−SD, 0) wt+1 (2)

ut − ut−1 = vt − wt (3)

t∑

i=t−T U+1

vi≤ ut (4)

t∑

i=t−T D+1

wi≤ 1 − ut (5)

PORTA results for (T U =T D =2)

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Tight & Compact UCs

Case 2: Providing and Using Convex Hulls (I)Formulation + T U/T D Convex hull:

pt ≤(

P − P)

ut −(

P − SD)

wt+1

− max (SD−SU, 0) vt (1)

pt ≤(

P − P)

ut −(

P − SU)

vt

− max (SU−SD, 0) wt+1 (2)

ut − ut−1 = vt − wt (3)

t∑

i=t−T U+1

vi≤ ut (4)

t∑

i=t−T D+1

wi≤ 1 − ut (5)

How to remove the fractionalvertices?


DIM = 10

CONV_SECTION( 1) 0 0 0 0 0 0 0 0 0 0( 2) 1/2 1 1/2 1/2 0 0 1/2 0 50 0( 3) 1/2 1 1/2 1/2 0 0 1/2 0 50 50( 4) 1/2 1 1/2 1/2 0 0 1/2 50 50 0( 5) 1/2 1 1/2 1/2 0 0 1/2 50 50 50( 6) 0 0 1 0 1 0 0 0 0 0( 7) 1 0 0 0 0 1 0 0 0 0( 8) 0 1 1 1 0 0 0 0 0 0( 9) 0 1 1 1 0 0 0 0 0 100( 10) 1 1 0 0 0 0 1 0 0 0( 11) 1 1 0 0 0 0 1 100 0 0( 12) 1 1 1 0 0 0 0 0 0 0( 13) 1 1 1 0 0 0 0 0 0 100( 14) 1 1 1 0 0 0 0 0 100 0( 15) 1 1 1 0 0 0 0 0 100 100( 16) 1 1 1 0 0 0 0 100 0 0( 17) 1 1 1 0 0 0 0 100 0 100( 18) 1 1 1 0 0 0 0 100 100 0( 19) 1 1 1 0 0 0 0 100 100 100

END

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Tight & Compact UCs

Case 2: Providing and Using Convex Hulls (II)Reformulating (1) and (2) for T U ≥ 2:

pt ≤(

P − P)

ut −(

P − SD)

wt+1

− max (SD−SU, 0) vt (1)

pt ≤(

P − P)

ut −(

P − SU)

vt

− max (SU−SD, 0) wt+1 (2)

pt ≤(

P − P)

ut −(

P − SU)

vt

−(

P − SD)

wt+1 (8)

ut − ut−1 = vt − wt (3)

t∑

i=t−T U+1

vi ≤ ut (4)

t∑

i=t−T D+1

wi ≤ 1 − ut (5)


DIM = 10

CONV_SECTION( 1) 0 0 0 0 0 0 0 0 0 0( 2) 0 0 1 0 1 0 0 0 0 0( 3) 1 0 0 0 0 1 0 0 0 0( 4) 0 1 1 1 0 0 0 0 0 0( 5) 0 1 1 1 0 0 0 0 0 100( 6) 1 1 0 0 0 0 1 0 0 0( 7) 1 1 0 0 0 0 1 100 0 0( 8) 1 1 1 0 0 0 0 0 0 0( 9) 1 1 1 0 0 0 0 0 0 100( 10) 1 1 1 0 0 0 0 0 100 0( 11) 1 1 1 0 0 0 0 0 100 100( 12) 1 1 1 0 0 0 0 100 0 0( 13) 1 1 1 0 0 0 0 100 0 100( 14) 1 1 1 0 0 0 0 100 100 0( 15) 1 1 1 0 0 0 0 100 100 100

END

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Tight & Compact UCs

Case 2: Providing and Using Convex Hulls (II)Reformulating (1) and (2) for T U ≥ 2:

pt ≤(

P − P)

ut −(

P − SD)

wt+1

− max (SD−SU, 0) vt (1)

pt ≤(

P − P)

ut −(

P − SU)

vt

− max (SU−SD, 0) wt+1 (2)

pt ≤(

P − P)

ut −(

P − SU)

vt

−(

P − SD)

wt+1 (8)

ut − ut−1 = vt − wt (3)

t∑

i=t−T U+1

vi ≤ ut (4)

t∑

i=t−T D+1

wi ≤ 1 − ut (5)


DIM = 10

CONV_SECTION( 1) 0 0 0 0 0 0 0 0 0 0( 2) 0 0 1 0 1 0 0 0 0 0( 3) 1 0 0 0 0 1 0 0 0 0( 4) 0 1 1 1 0 0 0 0 0 0( 5) 0 1 1 1 0 0 0 0 0 100( 6) 1 1 0 0 0 0 1 0 0 0( 7) 1 1 0 0 0 0 1 100 0 0( 8) 1 1 1 0 0 0 0 0 0 0( 9) 1 1 1 0 0 0 0 0 0 100( 10) 1 1 1 0 0 0 0 0 100 0( 11) 1 1 1 0 0 0 0 0 100 100( 12) 1 1 1 0 0 0 0 100 0 0( 13) 1 1 1 0 0 0 0 100 0 100( 14) 1 1 1 0 0 0 0 100 100 0( 15) 1 1 1 0 0 0 0 100 100 100

END

⇒ Convex Hull

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Case Studies Deterministic Selft-UC

Outline

1 Introduction




5 Conclusions

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Self-UC case Study

Case Study: Self-UC for 10-units, for 32-512 days time span

Basic constraints: max/min, SU/SD and TU/TD

All results are expressed as percentages of 1bin results

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Self-UC case Study

Case Study: Self-UC for 10-units, for 32-512 days time span

Basic constraints: max/min, SU/SD and TU/TD

Formulations tested –modeling the same MIP problem:

TC11: Proposed Tight & Compact1bin12: 1-binary variable (u)3binTUTD13: 3-binary variable version (u,v,w) + T U/T D convex

hull

All results are expressed as percentages of 1bin results


12M. Carrion and J. Arroyo, “A computationally efficient mixed-integer linear formulation for the thermal unitcommitment problem,” IEEE Transactions on Power Systems, vol. 21, no. 3, pp. 1371–1378, 2006


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Case Study: Self-UC (I)

Results presented as percentages of 1bin:

3binTUTD (%) TC (%)

Constraints <78 <48Nonzeros 89 72Real Vars 33.3 33.3Bin Vars =300 =300

⇓

TC is more Compact

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3binTUTD (%) TC (%)


Integrality Gap 34 =0

⇓

TC is Tighter and Simultaneously more Compact

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3binTUTD (%) TC (%)



MIP runtime (speedup) 4.9 (20x) 0.107 (995x)⇓


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Case Study: Self-UC (II)


3binTUTD (%) TC (%)



MIP runtime 4.9 0.107LP runtime 80 49.8

⇓

The TC formulation describe the convex hull

then solving MIP (non-convex) as LP (convex)

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Case Studies Stochastic UCs: Different Solvers

Outline

1 Introduction




5 Conclusions

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Stochastic UC: Case Study

10 generating units for a time span of 2 days

10 to 200 scenarios in demand

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Comparing TC, 1bin, 3binTUTD and Two additional formulationsSh14 and 3bin15

14T. Li and M. Shahidehpour, “Price-based unit commitment: a case of lagrangian relaxation versus mixed integerprogramming,” IEEE Transactions on Power Systems, vol. 20, no. 4, pp. 2015–2025, Nov. 2005

15J. Arroyo and A. Conejo, “Optimal response of a thermal unit to an electricity spot market,” Power Systems, IEEE

Transactions on, vol. 15, no. 3, pp. 1098–1104, 2000

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Comparing TC, 1bin, 3binTUTD and Two additional formulationsSh14 and 3bin15

Different Solvers

Cplex 12.6.0Gurobi 5.6.2XPRESS 25.01.07

Stop criteria:

Time limit: 5 hours orOptimality tolerance: 0.1 %

14T. Li and M. Shahidehpour, “Price-based unit commitment: a case of lagrangian relaxation versus mixed integerprogramming,” IEEE Transactions on Power Systems, vol. 20, no. 4, pp. 2015–2025, Nov. 2005

15J. Arroyo and A. Conejo, “Optimal response of a thermal unit to an electricity spot market,” Power Systems, IEEE

Transactions on, vol. 15, no. 3, pp. 1098–1104, 2000

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Stochastic: Cplex

0 20 40 60 80 100 120 140 160 180 20010

0

101

102

103

104

Demand Scenarios

Tim

e [s

]

TC3binTUTD3binSh1bin

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Stochastic: Cplex

0 20 40 60 80 100 120 140 160 180 20010

0

101

102

103

104

Demand Scenarios

Tim

e [s

]


TC deals with 200 scenarios within the time that others deal with 40

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Stochastic: Gurobi

0 20 40 60 80 100 120 140 160 180 20010

0

101

102

103

Demand Scenarios

Tim

e [s

]



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Stochastic: XPRESS

0 20 40 60 80 100 120 140 160 180 20010

−1

100

101

102

103

104

Demand Scenarios

Tim

e [s

]



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Case Studies Stochastic UCs: IEEE-118 Bus System

Outline

1 Introduction




5 Conclusions

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IEEE-118 Bus System

54 thermal units; 118 buses; 186 transmission lines; 91 loads

24 hours time span3 wind farms, 20 wind power scenariosStop Criteria in Cplex 12.6.0

0.05% opt. tolerance or 24h time limit

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UC performance comparisons (I)

TraditionalEnergy-Block Scheduling

3binTUTD16 TC

o.f. [k$] 829.04 829.02

opt.tol [%] 0.224 0.023

IntGap [%] 1.27 0.58

Compared with 3binTUTD, TC:

lowered IntGap by 53.3%

16FERC, “RTO unit commitment test system,” Federal Energy and Regulatory Commission, Washington DC, USA,Tech. Rep., Jul. 2012, p. 55

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UC performance comparisons (I)


3binTUTD16 TC

o.f. [k$] 829.04 829.02

opt.tol [%] 0.224 0.023

IntGap [%] 1.27 0.58

MIP runtime [s] 86400 206.5

Compared with 3binTUTD, TC:

lowered IntGap by 53.3%is more than 420x faster


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UC performance comparisons (II)


3binTUTD17 TC

o.f. [k$] 829.04 829.02

opt.tol [%] 0.224 0.023

IntGap [%] 1.27 0.58


LP runtime [s] 246.76 22.03

TC solved the MIP before 3binTUTD solved the LP

within the required opt. tolerance (0.05%)


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UC performance comparisons (III)

Traditional Power-BasedEnergy-based UC UC

3binTUTD TC P-TC

o.f. [k$] 829.04 829.02 818.13

opt.tol [%] 0.224 0.023 0.049

IntGap [%] 1.27 0.58


LP runtime [s] 246.76 22.03

P-TC18 has a more detailed and accurate UC representation

18G. Morales-Espana, A. Ramos, and J. Garcia-Gonzalez, “An MIP formulation for joint market-clearing of energy andreserves based on ramp scheduling,” IEEE Transactions on Power Systems, vol. 29, no. 1, pp. 476–488, 2014

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UC performance comparisons (III)

Traditional Power-BasedEnergy-based UC UC

3binTUTD TC P-TC

o.f. [k$] 829.04 829.02 818.13

opt.tol [%] 0.224 0.023 0.049

IntGap [%] 1.27 0.58 0.74

MIP runtime [s] 86400 206.5 867.9

LP runtime [s] 246.76 22.03 38.1

P-TC18 has a more detailed and accurate UC representation

it solved 100x faster than 3binTUTD

its UC core is also a convex hull19


19G. Morales-España, C. Gentile, and A. Ramos, “Tight MIP formulations of the power-based unit commitmentproblem,” en, OR Spectrum, pp. 1–22, May 2015

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Conclusions

Outline

1 Introduction




5 Conclusions

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Conclusions

Conclusions (I)

Beware of what matters in good MIP formulations

Tightness & Compactness

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Conclusions

Conclusions (I)


Tightness & Compactness↑ Binaries ⇒ ↑ Solving time False myth

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Conclusions

Conclusions (I)



Use the convex hull of some set of constraints

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Conclusions

Conclusions (I)



Use the convex hull of some set of constraints

Minimum up/down times20

Unit operation in Energy-based UC21

Unit operation in Power-based UC22

⇒ ↓ solving time by simultaneously T&Cing the final UCs23,24

20D. Rajan and S. Takriti, “Minimum up/down polytopes of the unit commitment problem with start-up costs,” IBM,Research Report RC23628, Jun. 2005

21C. Gentile, G. Morales-Espana, and A. Ramos, “A tight MIP formulation of the unit commitment problem with start-upand shut-down constraints,” Institute for Research in Technology (IIT), Technical Report IIT-14-040A, 2014

22G. Morales-España, C. Gentile, and A. Ramos, “Tight MIP formulations of the power-based unit commitmentproblem,” en, OR Spectrum, pp. 1–22, May 2015


24G. Morales-Espana, J. M. Latorre, and A. Ramos, “Tight and compact MILP formulation of start-up and shut-downramping in unit commitment,” IEEE Transactions on Power Systems, vol. 28, no. 2, pp. 1288–1296, 2013

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Conclusions

Conclusions (II)

Better UC core in stochastic UCs ⇒

Critical solving time reductionseven when the UC is modeled in more detail, i.e., P-UC

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Conclusions

Conclusions (II)



If convex hulls are not available ⇒

Create simultaneously tight and compact models

by reformulating the problem, e.g., CCGTs25

Key hint: start removing all big-M parameters

25G. Morales-Espana, C. Correa-Posada, and A. Ramos, “Tight and Compact MIP Formulation of Configuration-BasedCombined-Cycle Units,” IEEE Transactions on Power Systems, vol. PP, no. 99, pp. 1–10, 2015

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Conclusions

Conclusions (II)



If convex hulls are not available ⇒

Create simultaneously tight and compact models

by reformulating the problem, e.g., CCGTs25

Key hint: start removing all big-M parameters

Create tight cuts

Don’t add them directly to the modelUse them as cuts in the B&B algorithm26,27 ⇒ ↓ time

25G. Morales-Espana, C. Correa-Posada, and A. Ramos, “Tight and Compact MIP Formulation of Configuration-BasedCombined-Cycle Units,” IEEE Transactions on Power Systems, vol. PP, no. 99, pp. 1–10, 2015



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Conclusions

Questions

Thank you for your attention

Contact Information:[email protected]

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Conclusions For Further Reading

For Further Reading

J. Arroyo and A. Conejo, “Optimal response of a thermal unit to an electricity spotmarket,” Power Systems, IEEE Transactions on, vol. 15, no. 3, pp. 1098–1104, 2000.

R. Bixby and E. Rothberg, “Progress in computational mixed integer programming—alook back from the other side of the tipping point,” Annals of Operations Research,vol. 149, no. 1, pp. 37–41, Jan. 2007.

M. Carrion and J. Arroyo, “A computationally efficient mixed-integer linearformulation for the thermal unit commitment problem,” IEEE Transactions on Power

Systems, vol. 21, no. 3, pp. 1371–1378, 2006.

T. Christof and A. Löbel, “PORTA: POlyhedron representation transformationalgorithm, version 1.4.1,” Konrad-Zuse-Zentrum für Informationstechnik Berlin,

Germany, 2009.

P. Damci-Kurt, S. Kucukyavuz, D. Rajan, and A. Atamturk, “A Polyhedral Study ofRamping in Unit Committment,” University of California-Berkeley, Research ReportBCOL.13.02 IEOR, Oct. 2013.

FERC, “RTO unit commitment test system,” Federal Energy and RegulatoryCommission, Washington DC, USA, Tech. Rep., Jul. 2012, p. 55.

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For Further Reading (cont.)

C. Gentile, G. Morales-Espana, and A. Ramos, “A tight MIP formulation of the unitcommitment problem with start-up and shut-down constraints,” Institute forResearch in Technology (IIT), Technical Report IIT-14-040A, 2014.

X. Guan, F. Gao, and A. Svoboda, “Energy delivery capacity and generationscheduling in the deregulated electric power market,” IEEE Transactions on Power

Systems, vol. 15, no. 4, pp. 1275–1280, Nov. 2000.

T. Koch, T. Achterberg, E. Andersen, O. Bastert, T. Berthold, R. E. Bixby,E. Danna, G. Gamrath, A. M. Gleixner, S. Heinz, A. Lodi, H. Mittelmann, T. Ralphs,D. Salvagnin, D. E. Steffy, and K. Wolter, “MIPLIB 2010,” Mathematical

Programming Computation, vol. 3, no. 2, pp. 103–163, Jun. 2011.

T. Li and M. Shahidehpour, “Price-based unit commitment: a case of lagrangianrelaxation versus mixed integer programming,” IEEE Transactions on Power Systems,vol. 20, no. 4, pp. 2015–2025, Nov. 2005.

G. Morales-Espana, C. Correa-Posada, and A. Ramos, “Tight and Compact MIPFormulation of Configuration-Based Combined-Cycle Units,” IEEE Transactions on

Power Systems, vol. PP, no. 99, pp. 1–10, 2015.

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For Further Reading (cont.)G. Morales-Espana, C. Gentile, and A. Ramos, “Tight MIP formulations of thepower-based unit commitment problem,” Institute for Research in Technology (IIT),Technical Report, 2014.

G. Morales-Espana, J. M. Latorre, and A. Ramos, “Tight and compact MILPformulation for the thermal unit commitment problem,” IEEE Transactions on Power

Systems, vol. 28, no. 4, pp. 4897–4908, Nov. 2013.

G. Morales-Espana, A. Ramos, and J. Garcia-Gonzalez, “An MIP formulation for jointmarket-clearing of energy and reserves based on ramp scheduling,” IEEE

Transactions on Power Systems, vol. 29, no. 1, pp. 476–488, 2014.

G. Morales-España, C. Gentile, and A. Ramos, “Tight MIP formulations of thepower-based unit commitment problem,” en, OR Spectrum, pp. 1–22, May 2015.

G. Morales-Espana, J. M. Latorre, and A. Ramos, “Tight and compact MILPformulation of start-up and shut-down ramping in unit commitment,” IEEE

Transactions on Power Systems, vol. 28, no. 2, pp. 1288–1296, 2013.

J. Ostrowski, M. F Anjos, and A. Vannelli, “Tight mixed integer linear programmingformulations for the unit commitment problem,” IEEE Transactions on Power

Systems, vol. 27, no. 1, pp. 39–46, Feb. 2012.

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For Further Reading (cont.)

D. Rajan and S. Takriti, “Minimum up/down polytopes of the unit commitmentproblem with start-up costs,” IBM, Research Report RC23628, Jun. 2005.

H. P. Williams, Model Building in Mathematical Programming, 5th Edition. JohnWiley & Sons Inc, Feb. 2013.

L. Wolsey, Integer Programming. Wiley-Interscience, 1998.

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Power-Based UC

Tow main features are included:

Schedules Power instead of Energy (for feasibility). To avoid:

Infeasible energy delivery28

Overestimated ramp availability29

28X. Guan, F. Gao, and A. Svoboda, “Energy delivery capacity and generation scheduling in the deregulated electricpower market,” IEEE Transactions on Power Systems, vol. 15, no. 4, pp. 1275–1280, Nov. 2000


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Power-Based UC





Includes startup (SU) and shutdown (SD) power trajectories

SU & SD ramps are deterministic events in day-ahead UCsIgnoring them change commitment decisions and increase costs21,30




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Power-Based UC





Includes startup (SU) and shutdown (SD) power trajectories

SU & SD ramps are deterministic events in day-ahead UCsIgnoring them change commitment decisions and increase costs21,30

The convex hull of the Power-Based UC for basic operatingconstraints is provided31.




31G. Morales-Espana, C. Gentile, and A. Ramos, “Tight MIP formulations of the power-based unit commitmentproblem,” Institute for Research in Technology (IIT), Technical Report, 2014

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Performance of the Energy-Based formulations:

3binTUTD (%) TC (%) R-TC (%)

Constraints <78 <48 <56Nonzeros 89 72 94Real Vars 33.3 33.3 66.7Bin Vars =300 =300 =300

Integrality Gap 34 =0 =0

MIP runtime 4.9 0.107 0.104MIP run (best-worst) 0.46 – 92 0.011 – 4.4 0.012 – 3.6

LP runtime 80 49.8 45.9

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Performance of the Energy-Based formulations:

3binTUTD (%) TC (%) R-TC (%)

Constraints <78 <48 <56Nonzeros 89 72 94Real Vars 33.3 33.3 66.7Bin Vars =300 =300 =300

Integrality Gap 34 =0 =0

MIP runtime 4.9 0.107 0.104MIP run (best-worst) 0.46 – 92 0.011 – 4.4 0.012 – 3.6

LP runtime 80 49.8 45.9⇓

The TC formulations describe the convex hull

then solving MIP (non-convex) as LP (convex)

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Conclusions UC for 40 power system mixes

Outline

UC for 40 power system mixes

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UC for 40 power system mixes

Case Study B: UC for 40 power system mixes, from 28 to 1870units32

Including demand, ramps, reserves, variable SU costs


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Integrality Gap: Small Cases 1-10280-540 units x 1 day, OptTol: 0.001

1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140P

ropo

rtio

n [%

]

Case [#]

0.6e−3

0.6e−3

0.6e−3

0.4e−3

0.6e−3

1.3e−3

0.4e−3

0.6e−3

0.6e−3

0.5e−3

0h0’

1bin3binTUTDTC

Geometric Averages: 3bin 64%; TC 35%

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CPU Time: Small Cases 1-10280-540 units x 1 day, OptTol: 0.001

1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140P

ropo

rtio

n [%

]

Case [#]

3’60’’

15’21’’

8’19’’

8’19’’

17’12’’

5’46’’

16’28’’

7’23’’

49’44’’

13’5’’

0h0’

1bin3binTUTDTC


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Integrality Gap: Large Cases 11-201320-1870 units x 1 day, OptTol: 0.01

11 12 13 14 15 16 17 18 19 200

20

40

60

80

100

120

140P

ropo

rtio

n [%

]

Case [#]

0.1e−3

0.1e−3

0.2e−3

0.2e−3

0.2e−3

0.1e−3

0.1e−3

0.1e−3

0.1e−3

0.1e−3

0h0’

1bin3binTUTDTC


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CPU Time: Large Cases 11-201320-1870 units x 1 day, OptTol: 0.01

11 12 13 14 15 16 17 18 19 200

20

40

60

80

100

120

140P

ropo

rtio

n [%

]

Case [#]

1h15’

2h2’

1h49’

7h24’

8h45’

4h2’

7h27’

8h58’

4h3’

10h0’

0h0’

1bin3binTUTDTC


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UC for 40 power system mixesResults presented as percentages of 1bin:

3binTUTD (%) TC (%)

Constraints <99 <40Nonzeros ~100 <35Real Vars 75 50Bin Vars =300 <500

Integrality Gap 72 40

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3binTUTD (%) TC (%)



⇓


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3binTUTD (%) TC (%)



Total Average runtime 71 7runtime (best-worst) 11 – 269 2 – 57

⇓


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3binTUTD (%) TC (%)



Total Average runtime 71 7runtime (best-worst) 11 – 269 2 – 57

Runtime Small Cases 67 11Runtime Large Cases 77 4.5

⇓


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