Tight Piecewise Convex Relaxations for Global Optimization of … · – exploit the narrow bounds...

Tight Piecewise Convex Relaxations for Global Optimization of Optimal Power Flow

Harsha NagarajanLos Alamos National Laboratory

Mowen Lu & Russell BentDiscussions with Prof. J. Linderoth

Jan 10, 2019

LA-UR-19-20884

Optimal Power Flow• Introduced in 1962

• The basis for many of the economic decisions made by modern grid operators

• Rising interest in solving AC OPF• DOE ARPA-E Go Competition

https://gocompetition.energy.gov

• Generator dispatch, Unit commitment, Transmission switching, etc.

• AC OPF is NP hard (Bienstock, Verma - 2006)

https://gocompetition.energy.gov

Optimal Power Flow

Minimize cost of generation

Kirchhoff ’s law

Ohm’s law

Engineering limits

Convex relaxations for OPF

SDP relaxations– Bai, Wei, Fujisawa, Wang (2008)– R. Madani, S. Sojoudi, J. Lavaei (2014) SOCQC

SDPAC

Sgi ´ Sd

i “ÿ

pi,jqPEYER

Sij @i P N

Sij “ Y ˚ij ViV

˚i ´ Y ˚

ij ViV˚j pi, jq P E Y ER

SOC-based relaxations– R. Jabr (2006) – B. Kocuk, S. S. Dey, and X. A. Sun (2016)

Convex quadratic (QC) relaxations– H. Hijazi, C. Coffrin, and P. Van Hentenryck (2015)

Source: C. Coffrin, et. al. “The QC relaxation: A theoretical and computational study on optimal power flow”, 2016

And many others

QC-relaxation overviewWii = v2i i 2 N

<(Wij) = vivj cos(✓i � ✓j) 8(i, j) 2 E

=(Wij) = vivj sin(✓i � ✓j) 8(i, j) 2 E

(1)

‣ Key ideas– factorable-functions relaxation– exploit the narrow bounds in power systems– convexify transcendental functions (sin, cos)

‣ Resulting optimization model– quadratic and convex (computationally better)

H. Hijazi, C. Coffrin, P.V. Hentenryck, Convex Quadratic Relaxations for Mixed-Integer Nonlinear Programs in Power Systems, Math. Prog.-C, 2015




(1)

Recursive McCormick relaxation

Trilinear monomials

Wii = hv2i iT i 2 N

<(Wij) = hhvivjiM hcos(✓i � ✓j)iCiM 8(i, j) 2 E

=(Wij) = hhvivjiM hsin(✓i � ✓j)iSiM 8(i, j) 2 E




(1)

Trilinear monomials

H-representation

�10

1 �1 �0.5 0 0.5 1

�1

0

1

vivj

dv ivj

Convex Hull of Bilinear Function

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dvivj > vlivj + vljvi � vlivlj

dvivj > vui vj + vuj vi � vui vuj

dvivj 6 vlivj + vuj vi � vlivuj

dvivj 6 vui vj + vljvi � vui vlj

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(1)

Trilinear monomials

H-representation

Apply recursively on

May not capture it’s convex hull

(asymmetric bounds on voltage and phase-

angle variables)

dvivj > vlivj + vljvi � vlivlj

dvivj > vui vj + vuj vi � vui vuj

dvivj 6 vlivj + vuj vi � vlivuj

dvivj 6 vui vj + vljvi � vui vlj

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Recursive vs. Convex hull relaxationsSymmetric bounds:

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7hx1x2x3x4i - gap

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

hhx

1x

2ix

3ix

4-

gap

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7hx1x2x3x4i - gap

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

hx1x

2ih

x3x

4i-

gap

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7hx1x2x3x4i - gap

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x1hx

2hx

3x

4ii

-ga

p

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7hx1x2x3x4i - gap

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

hx1hx

2x

3ii

x4

-ga

p

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Recursive vs. Convex hull relaxationsAsymmetric bounds:

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014hx1x2x3x4i - gap

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

hhx

1x

2ix

3ix

4-

gap

0.000 0.002 0.004 0.006 0.008 0.010hx1x2x3x4i - gap

0.0

0.1

0.2

0.3

0.4

0.5

hx1x

2ih

x3x

4i-

gap

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014hx1x2x3x4i - gap

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

x1hx

2hx

3x

4ii

-ga

p

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014hx1x2x3x4i - gap

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08hx

1hx

2x

3ii

x4

-ga

p

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Term-wise convex hull representation

2666664x1

x2

y

3777775= �1

2666664`1

`2

`1`2

3777775+ �2

2666664`1

u2

`1u2

3777775+ �3

2666664u1

`2

u1`2

3777775+ �4

2666664u1

u2

u1u2

3777775�1 + �2 + �3 + �4 = 1 and �

i

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V-representation (Bilinear)

�10

1 �1 �0.5 0 0.5 1

�1

0

1

x1x2

x

1x

2

Convex Hull of Bilinear Function

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V-representation (Trilinear)

’k=1..8

�

k

= 1,

�

k

> 0, [k = 1, . . . , 8,

bx =’k=1..8

�

k

�(⇠k),

x

i

=’k=1..8

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Meyer, C.A. and Floudas, C.A. “Trilinear monomials with mixed sign domains: Facets of the convex and concave envelopes” Journal of Global Optimization - 2004H-representation

(Trilinear)

Convex combination of extreme points

Improved QC-relaxation gaps

Without Bound Tightening

Instances QCrmc (�) QCconv (�)

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Instances: C. Coffrin et. al, “NESTA, the NICTA energy system test archive,” 2014

Comparison of trilinear envelopes on OPF relaxations

MediumIPOPT

LargeIPOPT

MediumGUROBI

LargeGUROBI

MediumCPLEX

LargeCPLEX

101

102

103

Com

puta

tion

tim

e(s

econ

ds)

Recursive MC

Extreme point

Meyer-Floudas

Narimani, M.R, Molzahn, D.K, Nagarajan, H, Crow, M.L. “Comparison of Various Trilinear Monomial Envelopes for Convex Relaxations of Optimal Power Flow Problems”. IEEE Global Conference on Signal and Information Processing (GlobalSIP). IEEE, 2018.

0.000

2.000

4.000

6.000

8.000

10.000

12.000

14.000

16.000

18.000

20.000

case5_pjm

case240_wecc

case30_fsr

_api

case73_iee

e_rts_ap

i

case89_pegase

_api

case118_ieee_api

case189_edin_api

case29_edin_sad

case9_na_cao

_nco

case9_nb_cao_nco

case14_s_cao_nco

Without Obj upper bnd With Obj upper bnd (OBBT)

With Optimization-based Bound Tightening (OBBT)

Improved QC-relaxation gaps

X

i2Gc2i(R(Sg

i )2) + c1iR(Sg

i ) + c0i 6 UBfeas

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Local-solver (Ipopt)

Global optimum - 80% of hard instances

Global optimization: Tight piecewise relaxations

�10

1 �1 �0.5 0 0.5 1

�1

0

1

x1x2

x

1x

2

Bilinear Function

<latexit sha1_base64="LQM/oczlqDU3l4eb8elSolpN8Ag=">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</latexit><latexit sha1_base64="LQM/oczlqDU3l4eb8elSolpN8Ag=">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</latexit><latexit sha1_base64="LQM/oczlqDU3l4eb8elSolpN8Ag=">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</latexit>

Tight piecewise formulations

�10

1 �1 �0.5 0 0.5 1

�1

0

1

x1x2

x

1x

2

Piecewise Mccormick Envelopes

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z1 = 1) �1 6 x2 6 0

2666664x1

x2

y

3777775= �1

2666664�1�11

3777775+ �2

2666664�100

3777775+ �3

26666641�1�1

3777775+ �4

2666664100

3777775�1 + �2 + �3 + �4 = 1 and �5 + �6 = 0

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z2 = 1) 0 6 x2 6 1

2666664x1

x2

y

3777775= �2

2666664�100

3777775+ �5

2666664�11�1

3777775+ �4

2666664100

3777775+ �6

2666664111

3777775�2 + �4 + �5 + �6 = 1 and �1 + �3 = 0

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z1 + z2 = 1, z1, z2 2 {0, 1}, and �i > 0 i 2 {1, . . . , 6}<latexit sha1_base64="NPki55L2RfWY/GM1G/XGve+0lds=">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</latexit><latexit sha1_base64="NPki55L2RfWY/GM1G/XGve+0lds=">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</latexit><latexit sha1_base64="NPki55L2RfWY/GM1G/XGve+0lds=">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</latexit><latexit sha1_base64="NPki55L2RfWY/GM1G/XGve+0lds=">AAADrnicdVLbbtNAEN3WXEq4NIVHXlYkSEhYwY4EFKRIlRCIxyIaUqm2rPF6HK+yXrveNdSx/HV8BZ/AK/wA69QInMJIlsbn7JmdnTlhLrjSjvNtZ9e6dv3Gzb1bg9t37t7bHx7c/6SysmA4Z5nIitMQFAouca65FniaFwhpKHARrt60/OIzFopn8kRXOfopLCWPOQNtoGDoj9eBS5/SdTClM+qObdoC9ubf45J6tWO7XmNwkBEde8KUjiDg1FviuRIgNXWod15CRPnledf2okwr+4VRBcORM3E2Qa8mbpeMSBfHwcHua6NmZYpSMwFKnblOrv0aCs2ZwGbglQpzYCtY4plJJaSo/Hozh4Y+NkhE46wwn2lsg/6tqCFVqkpDczIFnahtrgX/yYVp/+YT16/bK1CyrZZ0fOjXXOblhqM9UsURxlAKbSsGAqPZ5PC5X7/jBXwEqXqFaolf9IWKm/ZJcSmozmi7PhrxApkWlUmAFdxMhbIECmDaLLlXIV/GuTCL6NcFscyMLEn5Vuf+H2ran3Ot+Wrd/K/271ShNmtIc9Az0zsq3Sk6WvCwgKKqDc+STtsXDyhF49SKwgVXzyaQ52gcp3Ql0Diz1glnq8Y2D15tgXYooKUa4zZ321tXk/l08mrifpiOjt52ttsjD8kj8oS45CU5Iu/JMZkTRr6S7+QH+Wm51sLyreDy6O5Op3lAemElvwDFXzVB</latexit>


�10

1 �1 �0.5 0 0.5 1

�1

0

1

x1x2

x

1x

2

Bilinear Function

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�10

1 �1 �0.5 0 0.5 1

�1

0

1

x1x2

x

1x

2

Piecewise Mccormick Envelopes

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Bivariate partitioning

�(x̂s

) = x1,i · x2,j

x1

x2

x̂1x̂2 x̂3

x̂4

x̂5x̂6 x̂7

x̂8

x̂9x̂10 x̂11

x̂12

x̂13x̂14 x̂15

x̂16

x1,1 x1,2 x1,3 x1,4

x2,1

x2,2

x2,3

x2,4

z

11 z

21 z

31

z

12

z

22

z

32

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SOS-2 type constraints: Extreme points of the lifted-variable polytope are integral


Uniformly spaced partitions induce too many binaries. Hence, formulating tractable mixed-integer convex programs are crucial

Adaptive variable partitioning: Tightening gaps using mixed-integer convex programsLocal solvers (IPOPT) are amazing on ACOPF

�3 �2 �1 0 1 2 3

�2.5 6 x 6 2.5

Iterations

�3 �2 �1 0 1 2 3

x

⇤ Active partition

x

⇤ Active partition

�3 �2 �1 0 1 2 3

Non-uniform, dynamically added partitions guided by local and lower-bounding solutions

H. Nagarajan, M. Lu, E. Yamangil, R. Bent, “Tightening McCormick Relaxations for Nonlinear Programs via Dynamic Multivariate Partitioning,” Constraint Programming, 2016

H. Nagarajan, M. Lu, S. Wang, R. Bent, K. Sundar, “An Adaptive, Multivariate Partitioning Algorithm for Global Optimization of Nonconvex Programs,” Journal of Global Optimization, 2018

Adaptive variable partitioning

Was found useful in many applications

POD.jl: An open-source global MINLP solverhttps://github.com/lanl-ansi/POD.jl

https://github.com/lanl-ansi/POD.jl

POD.jl: An open-source global MINLP solver

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Numerical results on ACOPF

3 hours time limit for piecewise, dynamic

partitioning algorithm

0.000

2.000

4.000

6.000

8.000

10.000

12.000

14.000

16.000

18.000

20.000

case5_pjm

case240_wecc

case30_fsr

_api

case73_iee

e_rts_ap

i

case89_pegase

_api

case118_ieee_api

case189_edin_api

case29_edin_sad

case9_na_cao

_nco

case9_nb_cao_nco

case14_s_cao_nco

Without Obj upper bnd With Obj upper bnd (OBBT)

Global optimum - 80% of hard instances

Global optimum - 95% of hard instances (N <= 300)

Challenging Instances

case89_pegase_api (4.5%)case118_ieee_api (1.1%)case9_bgm_nco (3.5%)case39_1_bgm_nco (3.3%)

Revisiting extreme-point formulationInstances QC

rmc

(%) QC

conv

(%)

case3 lmbd 1.21 0.96

case30 ieee 15.64 15.20

case3 lmbd api 1.79 1.59

case24 ieee rts api 11.88 8.78


case3 lmbd sad 1.42 1.37

case4 gs sad 1.53 0.96

case5 pjm sad 0.99 0.77

case24 ieee rts sad 2.93 2.77


case118 ieee sad 4.61 4.14

case179 goc api 7.18 7.21

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Unexpected

Revisiting extreme-point formulation

´gijvivj qcsij ´ bijvivj|snij<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Extreme point captures the convex hull locally on trilinear

monomials


Recursive McCormick shares the lifted variable across the trilinear

monomials


But neither captures the convex hull of

Tying constraint

'pvi, vj , qcsij , |snijq “ ´gij vivj qcsijlooomooon

�c

´bij vivj|snijlooomooon

�s<latexit sha1_base64="Hdggllwa4zy3Y4qG1H2U9Rk6UMk=">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</latexit><latexit sha1_base64="Hdggllwa4zy3Y4qG1H2U9Rk6UMk=">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</latexit><latexit sha1_base64="Hdggllwa4zy3Y4qG1H2U9Rk6UMk=">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</latexit><latexit sha1_base64="Hdggllwa4zy3Y4qG1H2U9Rk6UMk=">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</latexit>

Computational results on bound tightening with improved relaxationshttps://arxiv.org/abs/1809.04565


https://arxiv.org/abs/1809.04565


Instances QC

rmc

(%) QC

conv

(%)

case3 lmbd 1.21 0.96

case30 ieee 15.64 15.20

case3 lmbd api 1.79 1.59



case3 lmbd sad 1.42 1.37

case4 gs sad 1.53 0.96

case5 pjm sad 0.99 0.77



case118 ieee sad 4.61 4.14

case179 goc api 7.18 7.21

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

6.98%<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

QC-RMC

QC-LMAC

QC-TLM



Future directions

✦ Extension of tying constraints to the partitioned case

✦ Other OPF formulations (Rectangular, Current-Voltage-based, Tangent)

✦ Incorporating SDP-based cuts by exploiting graph sparsity

✦ Scaling to large-scale networks?

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Tight Piecewise Convex Relaxations for Global Optimization of … · – exploit the narrow bounds...

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