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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
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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)
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Optimal Power Flow
Minimize cost of generation
Kirchhoff ’s law
Ohm’s law
Engineering limits
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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
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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
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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)
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
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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)
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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Recursive McCormick relaxation
![Page 8: Tight Piecewise Convex Relaxations for Global Optimization of … · – exploit the narrow bounds in power systems – convexify transcendental functions (sin, cos) ‣ Resulting](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a71b55fcd3410ad1ae9d1/html5/thumbnails/8.jpg)
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)
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 McCormick relaxation
![Page 9: Tight Piecewise Convex Relaxations for Global Optimization of … · – exploit the narrow bounds in power systems – convexify transcendental functions (sin, cos) ‣ Resulting](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a71b55fcd3410ad1ae9d1/html5/thumbnails/9.jpg)
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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![Page 10: Tight Piecewise Convex Relaxations for Global Optimization of … · – exploit the narrow bounds in power systems – convexify transcendental functions (sin, cos) ‣ Resulting](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a71b55fcd3410ad1ae9d1/html5/thumbnails/10.jpg)
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
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![Page 11: Tight Piecewise Convex Relaxations for Global Optimization of … · – exploit the narrow bounds in power systems – convexify transcendental functions (sin, cos) ‣ Resulting](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a71b55fcd3410ad1ae9d1/html5/thumbnails/11.jpg)
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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k
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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
![Page 12: Tight Piecewise Convex Relaxations for Global Optimization of … · – exploit the narrow bounds in power systems – convexify transcendental functions (sin, cos) ‣ Resulting](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a71b55fcd3410ad1ae9d1/html5/thumbnails/12.jpg)
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
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Comparison of trilinear envelopes on OPF relaxations
MediumIPOPT
LargeIPOPT
MediumGUROBI
LargeGUROBI
MediumCPLEX
LargeCPLEX
101
102
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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.
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2.000
4.000
6.000
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10.000
12.000
14.000
16.000
18.000
20.000
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case240_wecc
case30_fsr
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case73_iee
e_rts_ap
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case89_pegase
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case118_ieee_api
case189_edin_api
case29_edin_sad
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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
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Local-solver (Ipopt)
Global optimum - 80% of hard instances
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Global optimization: Tight piecewise relaxations
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2
Bilinear Function
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![Page 16: Tight Piecewise Convex Relaxations for Global Optimization of … · – exploit the narrow bounds in power systems – convexify transcendental functions (sin, cos) ‣ Resulting](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a71b55fcd3410ad1ae9d1/html5/thumbnails/16.jpg)
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=">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</latexit>
![Page 17: Tight Piecewise Convex Relaxations for Global Optimization of … · – exploit the narrow bounds in power systems – convexify transcendental functions (sin, cos) ‣ Resulting](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a71b55fcd3410ad1ae9d1/html5/thumbnails/17.jpg)
Tight piecewise formulations
�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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Tight piecewise formulations
SOS-2 type constraints: Extreme points of the lifted-variable polytope are integral
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Tight piecewise formulations
Uniformly spaced partitions induce too many binaries. Hence, formulating tractable mixed-integer convex programs are crucial
![Page 20: Tight Piecewise Convex Relaxations for Global Optimization of … · – exploit the narrow bounds in power systems – convexify transcendental functions (sin, cos) ‣ Resulting](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a71b55fcd3410ad1ae9d1/html5/thumbnails/20.jpg)
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
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x
⇤ Active partition
x
⇤ Active partition
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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
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Adaptive variable partitioning
Was found useful in many applications
![Page 22: Tight Piecewise Convex Relaxations for Global Optimization of … · – exploit the narrow bounds in power systems – convexify transcendental functions (sin, cos) ‣ Resulting](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a71b55fcd3410ad1ae9d1/html5/thumbnails/22.jpg)
POD.jl: An open-source global MINLP solverhttps://github.com/lanl-ansi/POD.jl
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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
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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%)
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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
case73 ieee rts api 10.97 9.64
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
case73 ieee rts sad 2.53 2.38
case118 ieee sad 4.61 4.14
case179 goc api 7.18 7.21
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Unexpected
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Revisiting extreme-point formulation
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Extreme point captures the convex hull locally on trilinear
monomials
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Recursive McCormick shares the lifted variable across the trilinear
monomials
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But neither captures the convex hull of
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Tying constraint
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Computational results on bound tightening with improved relaxationshttps://arxiv.org/abs/1809.04565
Revisiting extreme-point formulation
![Page 28: Tight Piecewise Convex Relaxations for Global Optimization of … · – exploit the narrow bounds in power systems – convexify transcendental functions (sin, cos) ‣ Resulting](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a71b55fcd3410ad1ae9d1/html5/thumbnails/28.jpg)
Revisiting extreme-point formulation
Instances 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
case73 ieee rts api 10.97 9.64
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
case73 ieee rts sad 2.53 2.38
case118 ieee sad 4.61 4.14
case179 goc api 7.18 7.21
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QC-RMC
QC-LMAC
QC-TLM
https://arxiv.org/abs/1809.04565
![Page 29: Tight Piecewise Convex Relaxations for Global Optimization of … · – exploit the narrow bounds in power systems – convexify transcendental functions (sin, cos) ‣ Resulting](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a71b55fcd3410ad1ae9d1/html5/thumbnails/29.jpg)
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?
![Page 30: Tight Piecewise Convex Relaxations for Global Optimization of … · – exploit the narrow bounds in power systems – convexify transcendental functions (sin, cos) ‣ Resulting](https://reader034.vdocuments.us/reader034/viewer/2022050301/5f6a71b55fcd3410ad1ae9d1/html5/thumbnails/30.jpg)
Thank you!
Questions?