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Instance powerflow0009p
Optimal Power Flow problem modeled using trigonometric functions (polar coordinates)
| Formatsⓘ | ams gms mod nl osil py |
| Primal Bounds (infeas ≤ 1e-08)ⓘ | |
| Other points (infeas > 1e-08)ⓘ | |
| Dual Boundsⓘ | 3560.23041600 (COUENNE) 5296.66880000 (GUROBI) 2357.82828700 (LINDO) 4574.81589600 (SCIP) |
| Referencesⓘ | Hijazi, H L, Coffrin, C, and Van Hentenryck, P, Convex Quadratic Relaxations of Nonlinear Programs in Power Systems, Tech. Rep. 2013-09, Optimization Online, 2013. |
| Applicationⓘ | Electricity Networks |
| Added to libraryⓘ | 18 Aug 2014 |
| Problem typeⓘ | NLP |
| #Variablesⓘ | 60 |
| #Binary Variablesⓘ | 0 |
| #Integer Variablesⓘ | 0 |
| #Nonlinear Variablesⓘ | 57 |
| #Nonlinear Binary Variablesⓘ | 0 |
| #Nonlinear Integer Variablesⓘ | 0 |
| Objective Senseⓘ | min |
| Objective typeⓘ | quadratic |
| Objective curvatureⓘ | convex |
| #Nonzeros in Objectiveⓘ | 3 |
| #Nonlinear Nonzeros in Objectiveⓘ | 3 |
| #Constraintsⓘ | 139 |
| #Linear Constraintsⓘ | 85 |
| #Quadratic Constraintsⓘ | 18 |
| #Polynomial Constraintsⓘ | 0 |
| #Signomial Constraintsⓘ | 0 |
| #General Nonlinear Constraintsⓘ | 36 |
| Operands in Gen. Nonlin. Functionsⓘ | cos mul sin sqr |
| Constraints curvatureⓘ | indefinite |
| #Nonzeros in Jacobianⓘ | 361 |
| #Nonlinear Nonzeros in Jacobianⓘ | 180 |
| #Nonzeros in (Upper-Left) Hessian of Lagrangianⓘ | 147 |
| #Nonzeros in Diagonal of Hessian of Lagrangianⓘ | 57 |
| #Blocks in Hessian of Lagrangianⓘ | 40 |
| Minimal blocksize in Hessian of Lagrangianⓘ | 1 |
| Maximal blocksize in Hessian of Lagrangianⓘ | 18 |
| Average blocksize in Hessian of Lagrangianⓘ | 1.425 |
| #Semicontinuitiesⓘ | 0 |
| #Nonlinear Semicontinuitiesⓘ | 0 |
| #SOS type 1ⓘ | 0 |
| #SOS type 2ⓘ | 0 |
| Minimal coefficientⓘ | 1.0000e+00 |
| Maximal coefficientⓘ | 1.2250e+03 |
| Infeasibility of initial pointⓘ | 1.25 |
| Sparsity Jacobianⓘ |  |
| Sparsity Hessian of Lagrangianⓘ |  |
$offlisting
*
* Equation counts
* Total E G L N X C B
* 140 56 33 51 0 0 0 0
*
* Variable counts
* x b i s1s s2s sc si
* Total cont binary integer sos1 sos2 scont sint
* 61 61 0 0 0 0 0 0
* FX 0
*
* Nonzero counts
* Total const NL DLL
* 365 182 183 0
*
* Solve m using NLP minimizing objvar;
Variables x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19
,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35,x36
,x37,x38,x39,x40,x41,x42,x43,x44,x45,x46,x47,x48,x49,x50,x51,x52,x53
,x54,x55,x56,x57,x58,x59,x60,objvar;
Equations e1,e2,e3,e4,e5,e6,e7,e8,e9,e10,e11,e12,e13,e14,e15,e16,e17,e18,e19
,e20,e21,e22,e23,e24,e25,e26,e27,e28,e29,e30,e31,e32,e33,e34,e35,e36
,e37,e38,e39,e40,e41,e42,e43,e44,e45,e46,e47,e48,e49,e50,e51,e52,e53
,e54,e55,e56,e57,e58,e59,e60,e61,e62,e63,e64,e65,e66,e67,e68,e69,e70
,e71,e72,e73,e74,e75,e76,e77,e78,e79,e80,e81,e82,e83,e84,e85,e86,e87
,e88,e89,e90,e91,e92,e93,e94,e95,e96,e97,e98,e99,e100,e101,e102,e103
,e104,e105,e106,e107,e108,e109,e110,e111,e112,e113,e114,e115,e116
,e117,e118,e119,e120,e121,e122,e123,e124,e125,e126,e127,e128,e129
,e130,e131,e132,e133,e134,e135,e136,e137,e138,e139,e140;
e1.. 1100*sqr(x55) + 500*x55 + 850*sqr(x56) + 120*x56 + 1225*sqr(x57) + 100*x57
- objvar =E= -1085;
e2.. -17.0648464163823*x3*x6*sin(x48 - x51) + x10 =E= 0;
e3.. -17.0648464163823*x6*x3*sin(x51 - x48) + x11 =E= 0;
e4.. 1.61712247324614*x7*x8*cos(x52 - x53) - 1.61712247324614*sqr(x7) -
13.6979785969084*x7*x8*sin(x52 - x53) + x12 =E= 0;
e5.. 1.61712247324614*x8*x7*cos(x53 - x52) - 1.61712247324614*sqr(x8) -
13.6979785969084*x8*x7*sin(x53 - x52) + x13 =E= 0;
e6.. 1.28200913842411*x5*x6*cos(x50 - x51) - 1.28200913842411*sqr(x5) -
5.58824496236153*x5*x6*sin(x50 - x51) + x14 =E= 0;
e7.. 1.28200913842411*x6*x5*cos(x51 - x50) - 1.28200913842411*sqr(x6) -
5.58824496236153*x6*x5*sin(x51 - x50) + x15 =E= 0;
e8.. 1.1550874808901*x6*x7*cos(x51 - x52) - 1.1550874808901*sqr(x6) -
9.78427042636317*x6*x7*sin(x51 - x52) + x16 =E= 0;
e9.. 1.1550874808901*x7*x6*cos(x52 - x51) - 1.1550874808901*sqr(x7) -
9.78427042636317*x7*x6*sin(x52 - x51) + x17 =E= 0;
e10.. -16*x8*x2*sin(x53 - x47) + x18 =E= 0;
e11.. -16*x2*x8*sin(x47 - x53) + x19 =E= 0;
e12.. 1.94219124871473*x4*x5*cos(x49 - x50) - 1.94219124871473*sqr(x4) -
10.5106820518679*x4*x5*sin(x49 - x50) + x20 =E= 0;
e13.. 1.94219124871473*x5*x4*cos(x50 - x49) - 1.94219124871473*sqr(x5) -
10.5106820518679*x5*x4*sin(x50 - x49) + x21 =E= 0;
e14.. -17.3611111111111*x1*x4*sin(x46 - x49) + x22 =E= 0;
e15.. -17.3611111111111*x4*x1*sin(x49 - x46) + x23 =E= 0;
e16.. 1.36518771331058*x9*x4*cos(x54 - x49) - 1.36518771331058*sqr(x9) -
11.6040955631399*x9*x4*sin(x54 - x49) + x24 =E= 0;
e17.. 1.36518771331058*x4*x9*cos(x49 - x54) - 1.36518771331058*sqr(x4) -
11.6040955631399*x4*x9*sin(x49 - x54) + x25 =E= 0;
e18.. 1.18760437929115*x8*x9*cos(x53 - x54) - 1.18760437929115*sqr(x8) -
5.97513453330859*x8*x9*sin(x53 - x54) + x26 =E= 0;
e19.. 1.18760437929115*x9*x8*cos(x54 - x53) - 1.18760437929115*sqr(x9) -
5.97513453330859*x9*x8*sin(x54 - x53) + x27 =E= 0;
e20.. 17.0648464163823*x3*x6*cos(x48 - x51) - 17.0648464163823*sqr(x3) + x28
=E= 0;
e21.. 17.0648464163823*x6*x3*cos(x51 - x48) - 17.0648464163823*sqr(x6) + x29
=E= 0;
e22.. 13.6979785969084*x7*x8*cos(x52 - x53) - 13.6234785969084*sqr(x7) +
1.61712247324614*x7*x8*sin(x52 - x53) + x30 =E= 0;
e23.. 13.6979785969084*x8*x7*cos(x53 - x52) - 13.6234785969084*sqr(x8) +
1.61712247324614*x8*x7*sin(x53 - x52) + x31 =E= 0;
e24.. 5.58824496236153*x5*x6*cos(x50 - x51) - 5.40924496236153*sqr(x5) +
1.28200913842411*x5*x6*sin(x50 - x51) + x32 =E= 0;
e25.. 5.58824496236153*x6*x5*cos(x51 - x50) - 5.40924496236153*sqr(x6) +
1.28200913842411*x6*x5*sin(x51 - x50) + x33 =E= 0;
e26.. 9.78427042636317*x6*x7*cos(x51 - x52) - 9.67977042636317*sqr(x6) +
1.1550874808901*x6*x7*sin(x51 - x52) + x34 =E= 0;
e27.. 9.78427042636317*x7*x6*cos(x52 - x51) - 9.67977042636317*sqr(x7) +
1.1550874808901*x7*x6*sin(x52 - x51) + x35 =E= 0;
e28.. 16*x8*x2*cos(x53 - x47) - 16*sqr(x8) + x36 =E= 0;
e29.. 16*x2*x8*cos(x47 - x53) - 16*sqr(x2) + x37 =E= 0;
e30.. 10.5106820518679*x4*x5*cos(x49 - x50) - 10.4316820518679*sqr(x4) +
1.94219124871473*x4*x5*sin(x49 - x50) + x38 =E= 0;
e31.. 10.5106820518679*x5*x4*cos(x50 - x49) - 10.4316820518679*sqr(x5) +
1.94219124871473*x5*x4*sin(x50 - x49) + x39 =E= 0;
e32.. 17.3611111111111*x1*x4*cos(x46 - x49) - 17.3611111111111*sqr(x1) + x40
=E= 0;
e33.. 17.3611111111111*x4*x1*cos(x49 - x46) - 17.3611111111111*sqr(x4) + x41
=E= 0;
e34.. 11.6040955631399*x9*x4*cos(x54 - x49) - 11.5160955631399*sqr(x9) +
1.36518771331058*x9*x4*sin(x54 - x49) + x42 =E= 0;
e35.. 11.6040955631399*x4*x9*cos(x49 - x54) - 11.5160955631399*sqr(x4) +
1.36518771331058*x4*x9*sin(x49 - x54) + x43 =E= 0;
e36.. 5.97513453330859*x8*x9*cos(x53 - x54) - 5.82213453330859*sqr(x8) +
1.18760437929115*x8*x9*sin(x53 - x54) + x44 =E= 0;
e37.. 5.97513453330859*x9*x8*cos(x54 - x53) - 5.82213453330859*sqr(x9) +
1.18760437929115*x9*x8*sin(x54 - x53) + x45 =E= 0;
e38.. sqr(x10) + sqr(x28) =L= 9;
e39.. sqr(x11) + sqr(x29) =L= 9;
e40.. sqr(x12) + sqr(x30) =L= 6.25;
e41.. sqr(x13) + sqr(x31) =L= 6.25;
e42.. sqr(x14) + sqr(x32) =L= 2.25;
e43.. sqr(x15) + sqr(x33) =L= 2.25;
e44.. sqr(x16) + sqr(x34) =L= 2.25;
e45.. sqr(x17) + sqr(x35) =L= 2.25;
e46.. sqr(x18) + sqr(x36) =L= 6.25;
e47.. sqr(x19) + sqr(x37) =L= 6.25;
e48.. sqr(x20) + sqr(x38) =L= 6.25;
e49.. sqr(x21) + sqr(x39) =L= 6.25;
e50.. sqr(x22) + sqr(x40) =L= 6.25;
e51.. sqr(x23) + sqr(x41) =L= 6.25;
e52.. sqr(x24) + sqr(x42) =L= 6.25;
e53.. sqr(x25) + sqr(x43) =L= 6.25;
e54.. sqr(x26) + sqr(x44) =L= 6.25;
e55.. sqr(x27) + sqr(x45) =L= 6.25;
e56.. x55 =L= 2.5;
e57.. x56 =L= 3;
e58.. x57 =L= 2.7;
e59.. x55 =G= 0.1;
e60.. x56 =G= 0.1;
e61.. x57 =G= 0.1;
e62.. x58 =L= 3;
e63.. x59 =L= 3;
e64.. x60 =L= 3;
e65.. x58 =G= -3;
e66.. x59 =G= -3;
e67.. x60 =G= -3;
e68.. x1 =L= 1.1;
e69.. x2 =L= 1.1;
e70.. x3 =L= 1.1;
e71.. x4 =L= 1.1;
e72.. x5 =L= 1.1;
e73.. x6 =L= 1.1;
e74.. x7 =L= 1.1;
e75.. x8 =L= 1.1;
e76.. x9 =L= 1.1;
e77.. x1 =G= 0.9;
e78.. x2 =G= 0.9;
e79.. x3 =G= 0.9;
e80.. x4 =G= 0.9;
e81.. x5 =G= 0.9;
e82.. x6 =G= 0.9;
e83.. x7 =G= 0.9;
e84.. x8 =G= 0.9;
e85.. x9 =G= 0.9;
e86.. x48 - x51 =G= -0.26;
e87.. - x48 + x51 =G= -0.26;
e88.. x52 - x53 =G= -0.26;
e89.. - x52 + x53 =G= -0.26;
e90.. x50 - x51 =G= -0.26;
e91.. - x50 + x51 =G= -0.26;
e92.. x51 - x52 =G= -0.26;
e93.. - x51 + x52 =G= -0.26;
e94.. - x47 + x53 =G= -0.26;
e95.. x47 - x53 =G= -0.26;
e96.. x49 - x50 =G= -0.26;
e97.. - x49 + x50 =G= -0.26;
e98.. x46 - x49 =G= -0.26;
e99.. - x46 + x49 =G= -0.26;
e100.. - x49 + x54 =G= -0.26;
e101.. x49 - x54 =G= -0.26;
e102.. x53 - x54 =G= -0.26;
e103.. - x53 + x54 =G= -0.26;
e104.. x48 - x51 =L= 0.26;
e105.. - x48 + x51 =L= 0.26;
e106.. x52 - x53 =L= 0.26;
e107.. - x52 + x53 =L= 0.26;
e108.. x50 - x51 =L= 0.26;
e109.. - x50 + x51 =L= 0.26;
e110.. x51 - x52 =L= 0.26;
e111.. - x51 + x52 =L= 0.26;
e112.. - x47 + x53 =L= 0.26;
e113.. x47 - x53 =L= 0.26;
e114.. x49 - x50 =L= 0.26;
e115.. - x49 + x50 =L= 0.26;
e116.. x46 - x49 =L= 0.26;
e117.. - x46 + x49 =L= 0.26;
e118.. - x49 + x54 =L= 0.26;
e119.. x49 - x54 =L= 0.26;
e120.. x53 - x54 =L= 0.26;
e121.. - x53 + x54 =L= 0.26;
e122.. x46 =E= 0;
e123.. x22 - x55 =E= 0;
e124.. x19 - x56 =E= 0;
e125.. x10 - x57 =E= 0;
e126.. x40 - x58 =E= 0;
e127.. x37 - x59 =E= 0;
e128.. x28 - x60 =E= 0;
e129.. x20 + x23 + x25 =E= 0;
e130.. x14 + x21 =E= -0.9;
e131.. x11 + x15 + x16 =E= 0;
e132.. x12 + x17 =E= -1;
e133.. x13 + x18 + x26 =E= 0;
e134.. x24 + x27 =E= -1.25;
e135.. x38 + x41 + x43 =E= 0;
e136.. x32 + x39 =E= -0.3;
e137.. x29 + x33 + x34 =E= 0;
e138.. x30 + x35 =E= -0.35;
e139.. x31 + x36 + x44 =E= 0;
e140.. x42 + x45 =E= -0.5;
Model m / all /;
m.limrow=0; m.limcol=0;
m.tolproj=0.0;
$if NOT '%gams.u1%' == '' $include '%gams.u1%'
$if not set NLP $set NLP NLP
Solve m using %NLP% minimizing objvar;
Last updated: 2025-08-07 Git hash: e62cedfc