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Instance powerflow0009r
Optimal Power Flow problem modeled using quadratic functions (rectangular coordinates)
| Formatsⓘ | ams gms lp mod nl osil pip py |
| Primal Bounds (infeas ≤ 1e-08)ⓘ | |
| Other points (infeas > 1e-08)ⓘ | |
| Dual Boundsⓘ | 5296.15653500 (ANTIGONE) 5296.68458100 (BARON) 5296.65000000 (COUENNE) 5296.68280400 (GUROBI) 5154.91260300 (LINDO) 5296.68443000 (SCIP) 1188.75000000 (SHOT) |
| 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ⓘ | QCQP |
| #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ⓘ | 103 |
| #Linear Constraintsⓘ | 31 |
| #Quadratic Constraintsⓘ | 72 |
| #Polynomial Constraintsⓘ | 0 |
| #Signomial Constraintsⓘ | 0 |
| #General Nonlinear Constraintsⓘ | 0 |
| Operands in Gen. Nonlin. Functionsⓘ | |
| Constraints curvatureⓘ | indefinite |
| #Nonzeros in Jacobianⓘ | 307 |
| #Nonlinear Nonzeros in Jacobianⓘ | 216 |
| #Nonzeros in (Upper-Left) Hessian of Lagrangianⓘ | 129 |
| #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ⓘ | 5.7754e-01 |
| 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
* 104 56 15 33 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
* 311 92 219 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;
e1.. 1100*sqr(x55) + 500*x55 + 850*sqr(x56) + 120*x56 + 1225*sqr(x57) + 100*x57
- objvar =E= -1085;
e2.. 8.53242320819113*x42*x48 - 8.53242320819113*x39*x51 + 8.53242320819113*x48
*x42 - 8.53242320819113*x51*x39 + x1 =E= 0;
e3.. 8.53242320819113*x39*x51 - 8.53242320819113*x42*x48 - 8.53242320819113*x48
*x42 + 8.53242320819113*x51*x39 + x2 =E= 0;
e4.. 0.808561236623068*x43*x44 - 1.61712247324614*sqr(x43) - 6.84898929845422*
x43*x53 + 0.808561236623068*x44*x43 + 6.84898929845422*x44*x52 +
6.84898929845422*x52*x44 - 1.61712247324614*sqr(x52) + 0.808561236623068*
x52*x53 - 6.84898929845422*x53*x43 + 0.808561236623068*x53*x52 + x3 =E= 0;
e5.. 0.808561236623068*x43*x44 + 6.84898929845422*x43*x53 + 0.808561236623068*
x44*x43 - 1.61712247324614*sqr(x44) - 6.84898929845422*x44*x52 -
6.84898929845422*x52*x44 + 0.808561236623068*x52*x53 + 6.84898929845422*
x53*x43 + 0.808561236623068*x53*x52 - 1.61712247324614*sqr(x53) + x4 =E= 0
;
e6.. 0.641004569212057*x41*x42 - 1.28200913842411*sqr(x41) - 2.79412248118076*
x41*x51 + 0.641004569212057*x42*x41 + 2.79412248118076*x42*x50 +
2.79412248118076*x50*x42 - 1.28200913842411*sqr(x50) + 0.641004569212057*
x50*x51 - 2.79412248118076*x51*x41 + 0.641004569212057*x51*x50 + x5 =E= 0;
e7.. 0.641004569212057*x41*x42 + 2.79412248118076*x41*x51 + 0.641004569212057*
x42*x41 - 1.28200913842411*sqr(x42) - 2.79412248118076*x42*x50 -
2.79412248118076*x50*x42 + 0.641004569212057*x50*x51 + 2.79412248118076*
x51*x41 + 0.641004569212057*x51*x50 - 1.28200913842411*sqr(x51) + x6 =E= 0
;
e8.. 0.577543740445048*x42*x43 - 1.1550874808901*sqr(x42) - 4.89213521318159*
x42*x52 + 0.577543740445048*x43*x42 + 4.89213521318159*x43*x51 +
4.89213521318159*x51*x43 - 1.1550874808901*sqr(x51) + 0.577543740445048*
x51*x52 - 4.89213521318159*x52*x42 + 0.577543740445048*x52*x51 + x7 =E= 0;
e9.. 0.577543740445048*x42*x43 + 4.89213521318159*x42*x52 + 0.577543740445048*
x43*x42 - 1.1550874808901*sqr(x43) - 4.89213521318159*x43*x51 -
4.89213521318159*x51*x43 + 0.577543740445048*x51*x52 + 4.89213521318159*
x52*x42 + 0.577543740445048*x52*x51 - 1.1550874808901*sqr(x52) + x8 =E= 0;
e10.. 8*x38*x53 - 8*x44*x47 - 8*x47*x44 + 8*x53*x38 + x9 =E= 0;
e11.. 8*x44*x47 - 8*x38*x53 + 8*x47*x44 - 8*x53*x38 + x10 =E= 0;
e12.. 0.971095624357363*x40*x41 - 1.94219124871473*sqr(x40) - 5.25534102593397*
x40*x50 + 0.971095624357363*x41*x40 + 5.25534102593397*x41*x49 +
5.25534102593397*x49*x41 - 1.94219124871473*sqr(x49) + 0.971095624357363*
x49*x50 - 5.25534102593397*x50*x40 + 0.971095624357363*x50*x49 + x11
=E= 0;
e13.. 0.971095624357363*x40*x41 + 5.25534102593397*x40*x50 + 0.971095624357363*
x41*x40 - 1.94219124871473*sqr(x41) - 5.25534102593397*x41*x49 -
5.25534102593397*x49*x41 + 0.971095624357363*x49*x50 + 5.25534102593397*
x50*x40 + 0.971095624357363*x50*x49 - 1.94219124871473*sqr(x50) + x12
=E= 0;
e14.. 8.68055555555556*x40*x46 - 8.68055555555556*x37*x49 + 8.68055555555556*
x46*x40 - 8.68055555555556*x49*x37 + x13 =E= 0;
e15.. 8.68055555555556*x37*x49 - 8.68055555555556*x40*x46 - 8.68055555555556*
x46*x40 + 8.68055555555556*x49*x37 + x14 =E= 0;
e16.. 0.68259385665529*x40*x45 + 5.80204778156997*x40*x54 + 0.68259385665529*
x45*x40 - 1.36518771331058*sqr(x45) - 5.80204778156997*x45*x49 -
5.80204778156997*x49*x45 + 0.68259385665529*x49*x54 + 5.80204778156997*
x54*x40 + 0.68259385665529*x54*x49 - 1.36518771331058*sqr(x54) + x15
=E= 0;
e17.. 0.68259385665529*x40*x45 - 1.36518771331058*sqr(x40) - 5.80204778156997*
x40*x54 + 0.68259385665529*x45*x40 + 5.80204778156997*x45*x49 +
5.80204778156997*x49*x45 - 1.36518771331058*sqr(x49) + 0.68259385665529*
x49*x54 - 5.80204778156997*x54*x40 + 0.68259385665529*x54*x49 + x16 =E= 0
;
e18.. 0.593802189645574*x44*x45 - 1.18760437929115*sqr(x44) - 2.9875672666543*
x44*x54 + 0.593802189645574*x45*x44 + 2.9875672666543*x45*x53 +
2.9875672666543*x53*x45 - 1.18760437929115*sqr(x53) + 0.593802189645574*
x53*x54 - 2.9875672666543*x54*x44 + 0.593802189645574*x54*x53 + x17 =E= 0
;
e19.. 0.593802189645574*x44*x45 + 2.9875672666543*x44*x54 + 0.593802189645574*
x45*x44 - 1.18760437929115*sqr(x45) - 2.9875672666543*x45*x53 -
2.9875672666543*x53*x45 + 0.593802189645574*x53*x54 + 2.9875672666543*x54
*x44 + 0.593802189645574*x54*x53 - 1.18760437929115*sqr(x54) + x18 =E= 0;
e20.. 8.53242320819113*x39*x42 - 17.0648464163823*sqr(x39) + 8.53242320819113*
x42*x39 - 17.0648464163823*sqr(x48) + 8.53242320819113*x48*x51 +
8.53242320819113*x51*x48 + x19 =E= 0;
e21.. 8.53242320819113*x39*x42 + 8.53242320819113*x42*x39 - 17.0648464163823*
sqr(x42) + 8.53242320819113*x48*x51 + 8.53242320819113*x51*x48 -
17.0648464163823*sqr(x51) + x20 =E= 0;
e22.. 6.84898929845422*x43*x44 - 13.6234785969084*sqr(x43) + 0.808561236623068*
x43*x53 + 6.84898929845422*x44*x43 - 0.808561236623068*x44*x52 -
0.808561236623068*x52*x44 - 13.6234785969084*sqr(x52) + 6.84898929845422*
x52*x53 + 0.808561236623068*x53*x43 + 6.84898929845422*x53*x52 + x21
=E= 0;
e23.. 6.84898929845422*x43*x44 - 0.808561236623068*x43*x53 + 6.84898929845422*
x44*x43 - 13.6234785969084*sqr(x44) + 0.808561236623068*x44*x52 +
0.808561236623068*x52*x44 + 6.84898929845422*x52*x53 - 0.808561236623068*
x53*x43 + 6.84898929845422*x53*x52 - 13.6234785969084*sqr(x53) + x22
=E= 0;
e24.. 2.79412248118076*x41*x42 - 5.40924496236153*sqr(x41) + 0.641004569212057*
x41*x51 + 2.79412248118076*x42*x41 - 0.641004569212057*x42*x50 -
0.641004569212057*x50*x42 - 5.40924496236153*sqr(x50) + 2.79412248118076*
x50*x51 + 0.641004569212057*x51*x41 + 2.79412248118076*x51*x50 + x23
=E= 0;
e25.. 2.79412248118076*x41*x42 - 0.641004569212057*x41*x51 + 2.79412248118076*
x42*x41 - 5.40924496236153*sqr(x42) + 0.641004569212057*x42*x50 +
0.641004569212057*x50*x42 + 2.79412248118076*x50*x51 - 0.641004569212057*
x51*x41 + 2.79412248118076*x51*x50 - 5.40924496236153*sqr(x51) + x24
=E= 0;
e26.. 4.89213521318159*x42*x43 - 9.67977042636317*sqr(x42) + 0.577543740445048*
x42*x52 + 4.89213521318159*x43*x42 - 0.577543740445048*x43*x51 -
0.577543740445048*x51*x43 - 9.67977042636317*sqr(x51) + 4.89213521318159*
x51*x52 + 0.577543740445048*x52*x42 + 4.89213521318159*x52*x51 + x25
=E= 0;
e27.. 4.89213521318159*x42*x43 - 0.577543740445048*x42*x52 + 4.89213521318159*
x43*x42 - 9.67977042636317*sqr(x43) + 0.577543740445048*x43*x51 +
0.577543740445048*x51*x43 + 4.89213521318159*x51*x52 - 0.577543740445048*
x52*x42 + 4.89213521318159*x52*x51 - 9.67977042636317*sqr(x52) + x26
=E= 0;
e28.. 8*x38*x44 + 8*x44*x38 - 16*sqr(x44) + 8*x47*x53 + 8*x53*x47 - 16*sqr(x53)
+ x27 =E= 0;
e29.. 8*x38*x44 - 16*sqr(x38) + 8*x44*x38 - 16*sqr(x47) + 8*x47*x53 + 8*x53*x47
+ x28 =E= 0;
e30.. 5.25534102593397*x40*x41 - 10.4316820518679*sqr(x40) + 0.971095624357363*
x40*x50 + 5.25534102593397*x41*x40 - 0.971095624357363*x41*x49 -
0.971095624357363*x49*x41 - 10.4316820518679*sqr(x49) + 5.25534102593397*
x49*x50 + 0.971095624357363*x50*x40 + 5.25534102593397*x50*x49 + x29
=E= 0;
e31.. 5.25534102593397*x40*x41 - 0.971095624357363*x40*x50 + 5.25534102593397*
x41*x40 - 10.4316820518679*sqr(x41) + 0.971095624357363*x41*x49 +
0.971095624357363*x49*x41 + 5.25534102593397*x49*x50 - 0.971095624357363*
x50*x40 + 5.25534102593397*x50*x49 - 10.4316820518679*sqr(x50) + x30
=E= 0;
e32.. 8.68055555555556*x37*x40 - 17.3611111111111*sqr(x37) + 8.68055555555556*
x40*x37 - 17.3611111111111*sqr(x46) + 8.68055555555556*x46*x49 +
8.68055555555556*x49*x46 + x31 =E= 0;
e33.. 8.68055555555556*x37*x40 + 8.68055555555556*x40*x37 - 17.3611111111111*
sqr(x40) + 8.68055555555556*x46*x49 + 8.68055555555556*x49*x46 -
17.3611111111111*sqr(x49) + x32 =E= 0;
e34.. 5.80204778156997*x40*x45 - 0.68259385665529*x40*x54 + 5.80204778156997*
x45*x40 - 11.5160955631399*sqr(x45) + 0.68259385665529*x45*x49 +
0.68259385665529*x49*x45 + 5.80204778156997*x49*x54 - 0.68259385665529*
x54*x40 + 5.80204778156997*x54*x49 - 11.5160955631399*sqr(x54) + x33
=E= 0;
e35.. 5.80204778156997*x40*x45 - 11.5160955631399*sqr(x40) + 0.68259385665529*
x40*x54 + 5.80204778156997*x45*x40 - 0.68259385665529*x45*x49 -
0.68259385665529*x49*x45 - 11.5160955631399*sqr(x49) + 5.80204778156997*
x49*x54 + 0.68259385665529*x54*x40 + 5.80204778156997*x54*x49 + x34 =E= 0
;
e36.. 2.9875672666543*x44*x45 - 5.82213453330859*sqr(x44) + 0.593802189645574*
x44*x54 + 2.9875672666543*x45*x44 - 0.593802189645574*x45*x53 -
0.593802189645574*x53*x45 - 5.82213453330859*sqr(x53) + 2.9875672666543*
x53*x54 + 0.593802189645574*x54*x44 + 2.9875672666543*x54*x53 + x35 =E= 0
;
e37.. 2.9875672666543*x44*x45 - 0.593802189645574*x44*x54 + 2.9875672666543*x45
*x44 - 5.82213453330859*sqr(x45) + 0.593802189645574*x45*x53 +
0.593802189645574*x53*x45 + 2.9875672666543*x53*x54 - 0.593802189645574*
x54*x44 + 2.9875672666543*x54*x53 - 5.82213453330859*sqr(x54) + x36 =E= 0
;
e38.. sqr(x1) + sqr(x19) =L= 9;
e39.. sqr(x2) + sqr(x20) =L= 9;
e40.. sqr(x3) + sqr(x21) =L= 6.25;
e41.. sqr(x4) + sqr(x22) =L= 6.25;
e42.. sqr(x5) + sqr(x23) =L= 2.25;
e43.. sqr(x6) + sqr(x24) =L= 2.25;
e44.. sqr(x7) + sqr(x25) =L= 2.25;
e45.. sqr(x8) + sqr(x26) =L= 2.25;
e46.. sqr(x9) + sqr(x27) =L= 6.25;
e47.. sqr(x10) + sqr(x28) =L= 6.25;
e48.. sqr(x11) + sqr(x29) =L= 6.25;
e49.. sqr(x12) + sqr(x30) =L= 6.25;
e50.. sqr(x13) + sqr(x31) =L= 6.25;
e51.. sqr(x14) + sqr(x32) =L= 6.25;
e52.. sqr(x15) + sqr(x33) =L= 6.25;
e53.. sqr(x16) + sqr(x34) =L= 6.25;
e54.. sqr(x17) + sqr(x35) =L= 6.25;
e55.. sqr(x18) + sqr(x36) =L= 6.25;
e56.. sqr(x37) + sqr(x46) =L= 1.21;
e57.. sqr(x38) + sqr(x47) =L= 1.21;
e58.. sqr(x39) + sqr(x48) =L= 1.21;
e59.. sqr(x40) + sqr(x49) =L= 1.21;
e60.. sqr(x41) + sqr(x50) =L= 1.21;
e61.. sqr(x42) + sqr(x51) =L= 1.21;
e62.. sqr(x43) + sqr(x52) =L= 1.21;
e63.. sqr(x44) + sqr(x53) =L= 1.21;
e64.. sqr(x45) + sqr(x54) =L= 1.21;
e65.. sqr(x37) + sqr(x46) =G= 0.81;
e66.. sqr(x38) + sqr(x47) =G= 0.81;
e67.. sqr(x39) + sqr(x48) =G= 0.81;
e68.. sqr(x40) + sqr(x49) =G= 0.81;
e69.. sqr(x41) + sqr(x50) =G= 0.81;
e70.. sqr(x42) + sqr(x51) =G= 0.81;
e71.. sqr(x43) + sqr(x52) =G= 0.81;
e72.. sqr(x44) + sqr(x53) =G= 0.81;
e73.. sqr(x45) + sqr(x54) =G= 0.81;
e74.. x55 =L= 2.5;
e75.. x56 =L= 3;
e76.. x57 =L= 2.7;
e77.. x55 =G= 0.1;
e78.. x56 =G= 0.1;
e79.. x57 =G= 0.1;
e80.. x58 =L= 3;
e81.. x59 =L= 3;
e82.. x60 =L= 3;
e83.. x58 =G= -3;
e84.. x59 =G= -3;
e85.. x60 =G= -3;
e86.. x46 =E= 0;
e87.. x13 - x55 =E= 0;
e88.. x10 - x56 =E= 0;
e89.. x1 - x57 =E= 0;
e90.. x31 - x58 =E= 0;
e91.. x28 - x59 =E= 0;
e92.. x19 - x60 =E= 0;
e93.. x11 + x14 + x16 =E= 0;
e94.. x5 + x12 =E= -0.9;
e95.. x2 + x6 + x7 =E= 0;
e96.. x3 + x8 =E= -1;
e97.. x4 + x9 + x17 =E= 0;
e98.. x15 + x18 =E= -1.25;
e99.. x29 + x32 + x34 =E= 0;
e100.. x23 + x30 =E= -0.3;
e101.. x20 + x24 + x25 =E= 0;
e102.. x21 + x26 =E= -0.35;
e103.. x22 + x27 + x35 =E= 0;
e104.. x33 + x36 =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