Problem 7.3#
Integrated Energy Grids
Problem 7.3
[This problem has been slightly modified from the exercise sheet to alleviate solve issues.] Assume the gas transport system that is shown in Figure 2. Gas can be injected in Node 1 at a marginal price of 10 EUR/kg or at Node 5 at 15 EUR/kg. There is a demand of 400 kg/s in Node 4. Finally, Node 2-3 represents a pumping station where gas pressure can be increased by \(k_{23}\)=1.2 at a cost that is proportional to the mass flow \(m_{23}\) by a constant \(o_{23}\) =5 EUR/kg.
To calculate the speed of sound in gas \(c\), assume the universal gas constant \(R\)=8.314 J/molK, the molar mass of methane \(M\)=16 g/mol, compression factor \(Z\)=1.31, and temperature T=25\(^{\circ}\)C. Assume the diameter of the pipelines is \(D\) = 1000 mm, the length \(L_{12}\)=\(L_{34}\)=\(L_{45}\)=1000 m, and calculate the Darcy friction coefficient \(f_{D}\) assuming fully-turbulent flow and roughness \(\epsilon\)=0.001m.
(a) Write the equations of the optimization problem that minimizes the total system cost and use the Weymouth equation to represent pressure drop in the pipelines as a function of the mass flow \(m_{ij}\).
We define the pressure at differet node as \(\pi_i\) and the mass flow at different pipes as \(m_{ij}\) We have 9 variables (\(m_{12}\), \(m_{23}\), \(m_{34}\), \(m_{45}\), \(\pi_1\), \(\pi_2\), \(\pi_3\), \(\pi_4\), \(\pi_5\)).
We impose the energy balance constraint in nodes 1, 4 and 5, and the pipeline mass conservation and Weymouth equation in pipelines 12, 34 and 45.
\[\begin{alignat*}{3}
\min_{x,y} \quad & o_1g_1 + o_5g_5 +o_{23}m_{23} & \\
\text{s.t.} \quad & g_{1}=m_{12} \\
\quad & -d_{4}=-m_{34}+m_{45} \\
\quad & g_{5}=-m_{45} \\
\quad & m_{12}=m_{23} \\
\quad & m_{23}=m_{34} \\
\quad & m_{34}=m_{45} \\
\quad & \pi_{3}=k_{23}\pi_{2} \\
\quad & \pi_{1}^2-\pi_{2}^2=a_{12}m_{12} |m_{12} | \\
\quad & \pi_{3}^2-\pi_{4}^2=a_{34}m_{34} |m_{34} | \\
\quad & \pi_{4}^2-\pi_{5}^2=a_{45}m_{45} |m_{34} | \\
\end{alignat*}\]
where the coefficients \(a_{ij}=\frac{Lf^Dc^2}{DA^2}\) depend on the physical properties of the pipelines and the methane gas.
(b) Use gurobipy to formulate and solve the problem. Solve the problem for two extra cases of : \(o_{23}\) =1 EUR/kg and \(o_{23}\) =10 EUR/kg and compare the results.
Hint: It is usefull to define logical boundaries for the problem to make the optimisation faster. For example, we assume the pressure at all nodes to be between [0, 100 kPa], which is an acceptable range as pressure in gas pipelines is usually in the range of 30-100 kpa.
We start by calculating the cofficients \(a_{ij}\) from the Weymouth equation
import numpy as np
The cross-sectional area \(A\), and the speed of sound in gas \(c\), can be calculated as
D = 1 # m
Z = 1.31
R = 8.314 # J/molk
M = 0.016 # Kg/mol
T = 273+25 # K
A = np.pi*(D/2)**2
c=np.sqrt(Z*R*T/M)
c
450.3900615022494
The Darcy friction coefficient can be calculated as \(\frac{1}{\sqrt f_D}=-2log_{10}(\frac{\epsilon}{3.7D})\)
epsilon = 0.001 #m
f_D=(1/(-2*np.log10(epsilon/(3.7*D))))**2
f_D
0.0196354659355267
The coefficients from the Weymouth equation can be calculated as \(a_{ij}=\frac{Lf_Dc^2}{DA^2}\)
L = 1000 #km
L_12=L
a_12=L_12*f_D*c**2/(D*A**2)
a_34=a_12
a_45=a_12
a_12
6457122.799219107
We import gurobipy and write the problem
import gurobipy as gp
from gurobipy import GRB
k_23 = 1.2
d_4 = 400 #kg/s
o_1 = 10 #EUR/kg
o_5 = 15 #EUR/kg
o_m23 = 5 #EUR/kg
model = gp.Model("My_quadratic_problem")
g_1 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, name="g_1")
g_5 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, name="g_5")
m_12 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, ub=d_4, name="m_12")
m_23 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, ub=d_4, name="m_23")
m_34 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, ub=d_4, name="m_34")
m_45 = model.addVar(vtype=GRB.CONTINUOUS, lb=-d_4, ub=0, name="m_45")
abs_m_12 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, name="abs_m_12")
abs_m_34 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, name="abs_m_34")
abs_m_45 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, name="abs_m_45")
p_1 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, ub=100*1e5, name="p_1")
p_2 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, ub=100*1e5, name="p_2")
p_3 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, ub=100*1e5, name="p_3")
p_4 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, ub=100*1e5, name="p_4")
p_5 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, ub=100*1e5, name="p_5")
model.setObjective(o_1*g_1 + o_5*g_5 + o_m23*m_23, GRB.MINIMIZE)
constraint_a = model.addLConstr(g_1 == m_12)
constraint_b = model.addLConstr(-g_5 == m_45)
constraint_c = model.addLConstr(-d_4 == -m_34+m_45)
constraint_d = model.addLConstr(m_12 == m_23)
constraint_e = model.addLConstr(m_12 == m_34)
constraint_f = model.addLConstr(p_3 == k_23*p_2)
#constraint_g = model.addLConstr(p_4 == p4)
# We need to define additional constraints to use absolute values
from gurobipy import abs_
model.addConstr(abs_m_12 == abs_(m_12), name='abs_m_12')
model.addConstr(abs_m_34 == abs_(m_34), name='abs_m_34')
model.addConstr(abs_m_45 == abs_(m_45), name='abs_m_45')
constraint_h = model.addQConstr(a_12*m_12*abs_m_12 == p_1**2-p_2**2)
constraint_j = model.addQConstr(a_34*m_34*abs_m_34 == p_3**2-p_4**2)
constraint_k = model.addQConstr(a_45*m_45*abs_m_45 == p_4**2-p_5**2)
model.setParam("Method", 2) # We use the barrier method, Barrier is ideal for quadratic / conic problems and numerically unstable models
model.optimize()
Set parameter TokenServer to value "sophia1.hpc.ait.dtu.dk"
Set parameter Method to value 2
Gurobi Optimizer version 11.0.2 build v11.0.2rc0 (linux64 - "CentOS Linux 7 (Core)")
CPU model: AMD EPYC 7351 16-Core Processor, instruction set [SSE2|AVX|AVX2]
Thread count: 32 physical cores, 32 logical processors, using up to 32 threads
Optimize a model with 6 rows, 14 columns and 12 nonzeros
Model fingerprint: 0xac4bb344
Model has 3 quadratic constraints
Model has 3 general constraints
Variable types: 14 continuous, 0 integer (0 binary)
Coefficient statistics:
Matrix range [1e+00, 1e+00]
QMatrix range [1e+00, 6e+06]
Objective range [5e+00, 2e+01]
Bounds range [4e+02, 1e+07]
RHS range [4e+02, 4e+02]
Presolve removed 5 rows and 8 columns
Presolve time: 0.00s
Presolved: 22 rows, 12 columns, 41 nonzeros
Presolved model has 6 bilinear constraint(s)
Solving non-convex MIQCP
Variable types: 12 continuous, 0 integer (0 binary)
Root barrier log...
Ordering time: 0.00s
Barrier statistics:
AA' NZ : 6.000e+00
Factor NZ : 1.000e+01
Factor Ops : 3.000e+01 (less than 1 second per iteration)
Threads : 1
Objective Residual
Iter Primal Dual Primal Dual Compl Time
0 6.00000000e+03 -6.31598434e+03 6.79e+04 0.00e+00 7.31e+03 0s
1 6.00000000e+03 -3.40669849e+03 0.00e+00 6.94e-18 9.41e+02 0s
2 6.00000000e+03 5.99059330e+03 0.00e+00 6.94e-18 9.41e-01 0s
3 6.00000000e+03 5.99999059e+03 0.00e+00 0.00e+00 9.41e-04 0s
4 6.00000000e+03 5.99999999e+03 0.00e+00 4.37e-24 9.41e-07 0s
5 6.00000000e+03 6.00000000e+03 0.00e+00 3.04e-24 9.41e-10 0s
Barrier solved model in 5 iterations and 0.09 seconds (0.00 work units)
Optimal objective 6.00000000e+03
Root relaxation: objective 6.000000e+03, 4 iterations, 0.01 seconds (0.00 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 6000.00000 0 6 - 6000.00000 - - 0s
0 0 6000.00000 0 6 - 6000.00000 - - 0s
0 2 6000.00000 0 6 - 6000.00000 - - 0s
H 192 100 6000.0000000 6000.00000 0.00% 1.3 0s
Explored 245 nodes (312 simplex iterations) in 0.15 seconds (0.00 work units)
Thread count was 32 (of 32 available processors)
Solution count 1: 6000
Optimal solution found (tolerance 1.00e-04)
Best objective 6.000000000000e+03, best bound 6.000000000000e+03, gap 0.0000%
print(str(g_1.VarName) + ' ' + str(g_1.x))
print(str(g_5.VarName) + ' ' + str(g_5.x))
print(str(m_12.VarName) + ' ' + str(m_12.x))
print(str(m_23.VarName) + ' ' + str(m_23.x))
print(str(m_34.VarName) + ' ' + str(m_34.x))
print(str(m_45.VarName) + ' ' + str(m_45.x))
print(str(p_1.VarName) + ' ' + str(round(p_1.x)) + ' Pa')
print(str(p_2.VarName) + ' ' + str(round(p_2.x)) + ' Pa')
print(str(p_3.VarName) + ' ' + str(round(p_3.x)) + ' Pa')
print(str(p_4.VarName) + ' ' + str(round(p_4.x)) + ' Pa')
print(str(p_5.VarName) + ' ' + str(round(p_5.x)) + ' Pa')
g_1 200.0
g_5 200.0
m_12 200.0
m_23 200.0
m_34 200.0
m_45 -200.0
p_1 2170327 Pa
p_2 2109985 Pa
p_3 2531981 Pa
p_4 2480453 Pa
p_5 2531981 Pa
We can also solve for different compressor costs:
flows_g1=[]
flows_g5=[]
for o_m23 in [0,1,2,3,4, 5, 6 ,10, 15] : #EUR/kg
#model.setParam("MIPFocus", 1)
#model.setParam("Threads", 1)
#model.setParam("Seed", 0)
model = gp.Model("My_quadratic_problem")
g_1 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, name="g_1")
g_5 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, name="g_5")
m_12 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, ub=d_4, name="m_12")
m_23 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, ub=d_4, name="m_23")
m_34 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, ub=d_4, name="m_34")
m_45 = model.addVar(vtype=GRB.CONTINUOUS, lb=-d_4, ub=0, name="m_45")
abs_m_12 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, name="abs_m_12")
abs_m_34 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, name="abs_m_34")
abs_m_45 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, name="abs_m_45")
p_1 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, ub=100*1e5, name="p_1")
p_2 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, ub=100*1e5, name="p_2")
p_3 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, ub=100*1e5, name="p_3")
p_4 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, ub=100*1e5, name="p_4")
p_5 = model.addVar(vtype=GRB.CONTINUOUS, lb=0, ub=100*1e5, name="p_5")
model.setObjective(o_1*g_1 + o_5*g_5 + o_m23*m_23, GRB.MINIMIZE)
constraint_a = model.addLConstr(g_1 == m_12)
constraint_b = model.addLConstr(-g_5 == m_45)
constraint_c = model.addLConstr(-d_4 == -m_34+m_45)
constraint_d = model.addLConstr(m_12 == m_23)
constraint_e = model.addLConstr(m_12 == m_34)
constraint_f = model.addLConstr(p_3 == k_23*p_2)
#constraint_g = model.addLConstr(p_4 == p4)
# We need to define additional constraints to use absolute values
from gurobipy import abs_
model.addConstr(abs_m_12 == abs_(m_12), name='abs_m_12')
model.addConstr(abs_m_34 == abs_(m_34), name='abs_m_34')
model.addConstr(abs_m_45 == abs_(m_45), name='abs_m_45')
constraint_h = model.addQConstr(a_12*m_12*abs_m_12 == p_1**2-p_2**2)
constraint_j = model.addQConstr(a_34*m_34*abs_m_34 == p_3**2-p_4**2)
constraint_k = model.addQConstr(a_45*m_45*abs_m_45 == p_4**2-p_5**2)
model.setParam("Method", 2) # barrier
model.optimize()
flows_g1.append(g_1.x)
flows_g5.append(g_5.x)
Set parameter Method to value 2
Gurobi Optimizer version 11.0.2 build v11.0.2rc0 (linux64 - "CentOS Linux 7 (Core)")
CPU model: AMD EPYC 7351 16-Core Processor, instruction set [SSE2|AVX|AVX2]
Thread count: 32 physical cores, 32 logical processors, using up to 32 threads
Optimize a model with 6 rows, 14 columns and 12 nonzeros
Model fingerprint: 0xcf560096
Model has 3 quadratic constraints
Model has 3 general constraints
Variable types: 14 continuous, 0 integer (0 binary)
Coefficient statistics:
Matrix range [1e+00, 1e+00]
QMatrix range [1e+00, 6e+06]
Objective range [1e+01, 2e+01]
Bounds range [4e+02, 1e+07]
RHS range [4e+02, 4e+02]
Presolve removed 5 rows and 8 columns
Presolve time: 0.00s
Presolved: 22 rows, 12 columns, 41 nonzeros
Presolved model has 6 bilinear constraint(s)
Solving non-convex MIQCP
Variable types: 12 continuous, 0 integer (0 binary)
Root barrier log...
Ordering time: 0.00s
Barrier statistics:
AA' NZ : 6.000e+00
Factor NZ : 1.000e+01
Factor Ops : 3.000e+01 (less than 1 second per iteration)
Threads : 1
Objective Residual
Iter Primal Dual Primal Dual Compl Time
0 5.35609179e+03 -8.31598434e+03 8.21e+04 0.00e+00 8.00e+03 0s
1 4.89547077e+03 -5.52790984e+03 0.00e+00 6.94e-18 1.04e+03 0s
2 4.71641154e+03 3.85457112e+03 0.00e+00 0.00e+00 8.62e+01 0s
3 4.01781083e+03 3.97814377e+03 0.00e+00 1.83e-19 3.97e+00 0s
4 4.00001744e+03 3.99997777e+03 0.00e+00 2.39e-18 3.97e-03 0s
5 4.00000002e+03 3.99999998e+03 0.00e+00 6.61e-19 3.97e-06 0s
6 4.00000000e+03 4.00000000e+03 0.00e+00 1.73e-18 3.97e-09 0s
7 4.00000000e+03 4.00000000e+03 0.00e+00 0.00e+00 3.97e-12 0s
Barrier solved model in 7 iterations and 0.02 seconds (0.00 work units)
Optimal objective 4.00000000e+03
Extra simplex iterations after uncrush: 1
Root relaxation: objective 4.000000e+03, 5 iterations, 0.01 seconds (0.00 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 4000.00000 0 4 - 4000.00000 - - 0s
0 0 4000.00000 0 4 - 4000.00000 - - 0s
0 2 4000.00000 0 4 - 4000.00000 - - 0s
* 5709 4512 86 4001.2022489 4000.00000 0.03% 1.6 0s
* 8022 4512 75 4001.1937597 4000.00000 0.03% 1.6 0s
H 9520 3579 4000.0000000 4000.00000 0.00% 1.6 0s
Explored 9520 nodes (15651 simplex iterations) in 0.29 seconds (0.01 work units)
Thread count was 32 (of 32 available processors)
Solution count 3: 4000 4001.19 4001.2
Optimal solution found (tolerance 1.00e-04)
Warning: max constraint violation (7.8125e-03) exceeds tolerance
Best objective 4.000000000002e+03, best bound 4.000000000000e+03, gap 0.0000%
Set parameter Method to value 2
Gurobi Optimizer version 11.0.2 build v11.0.2rc0 (linux64 - "CentOS Linux 7 (Core)")
CPU model: AMD EPYC 7351 16-Core Processor, instruction set [SSE2|AVX|AVX2]
Thread count: 32 physical cores, 32 logical processors, using up to 32 threads
Optimize a model with 6 rows, 14 columns and 12 nonzeros
Model fingerprint: 0x66dccb5e
Model has 3 quadratic constraints
Model has 3 general constraints
Variable types: 14 continuous, 0 integer (0 binary)
Coefficient statistics:
Matrix range [1e+00, 1e+00]
QMatrix range [1e+00, 6e+06]
Objective range [1e+00, 2e+01]
Bounds range [4e+02, 1e+07]
RHS range [4e+02, 4e+02]
Presolve removed 5 rows and 8 columns
Presolve time: 0.00s
Presolved: 22 rows, 12 columns, 41 nonzeros
Presolved model has 6 bilinear constraint(s)
Solving non-convex MIQCP
Variable types: 12 continuous, 0 integer (0 binary)
Root barrier log...
Ordering time: 0.00s
Barrier statistics:
AA' NZ : 6.000e+00
Factor NZ : 1.000e+01
Factor Ops : 3.000e+01 (less than 1 second per iteration)
Threads : 1
Objective Residual
Iter Primal Dual Primal Dual Compl Time
0 5.48366463e+03 -7.91598434e+03 8.18e+04 0.00e+00 7.98e+03 0s
1 5.12990779e+03 -5.26355510e+03 0.00e+00 1.04e-17 1.04e+03 0s
2 5.01409384e+03 4.27185040e+03 0.00e+00 3.47e-18 7.42e+01 0s
3 4.41859263e+03 4.37739470e+03 0.00e+00 1.73e-18 4.12e+00 0s
4 4.40001817e+03 4.39997693e+03 0.00e+00 2.17e-19 4.12e-03 0s
5 4.40000002e+03 4.39999998e+03 0.00e+00 4.70e-13 4.12e-06 0s
6 4.40000000e+03 4.40000000e+03 0.00e+00 4.98e-16 4.12e-09 0s
7 4.40000000e+03 4.40000000e+03 0.00e+00 1.08e-18 4.12e-12 0s
Barrier solved model in 7 iterations and 0.02 seconds (0.00 work units)
Optimal objective 4.40000000e+03
Extra simplex iterations after uncrush: 1
Root relaxation: objective 4.400000e+03, 5 iterations, 0.01 seconds (0.00 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 4400.00000 0 4 - 4400.00000 - - 0s
0 0 4400.00000 0 4 - 4400.00000 - - 0s
H 0 0 4400.0067500 4400.00000 0.00% - 0s
Explored 1 nodes (5 simplex iterations) in 0.03 seconds (0.00 work units)
Thread count was 32 (of 32 available processors)
Solution count 1: 4400.01
Optimal solution found (tolerance 1.00e-04)
Best objective 4.400006749972e+03, best bound 4.400000000000e+03, gap 0.0002%
Set parameter Method to value 2
Gurobi Optimizer version 11.0.2 build v11.0.2rc0 (linux64 - "CentOS Linux 7 (Core)")
CPU model: AMD EPYC 7351 16-Core Processor, instruction set [SSE2|AVX|AVX2]
Thread count: 32 physical cores, 32 logical processors, using up to 32 threads
Optimize a model with 6 rows, 14 columns and 12 nonzeros
Model fingerprint: 0xa6aa52a3
Model has 3 quadratic constraints
Model has 3 general constraints
Variable types: 14 continuous, 0 integer (0 binary)
Coefficient statistics:
Matrix range [1e+00, 1e+00]
QMatrix range [1e+00, 6e+06]
Objective range [2e+00, 2e+01]
Bounds range [4e+02, 1e+07]
RHS range [4e+02, 4e+02]
Presolve removed 5 rows and 8 columns
Presolve time: 0.00s
Presolved: 22 rows, 12 columns, 41 nonzeros
Presolved model has 6 bilinear constraint(s)
Solving non-convex MIQCP
Variable types: 12 continuous, 0 integer (0 binary)
Root barrier log...
Ordering time: 0.00s
Barrier statistics:
AA' NZ : 6.000e+00
Factor NZ : 1.000e+01
Factor Ops : 3.000e+01 (less than 1 second per iteration)
Threads : 1
Objective Residual
Iter Primal Dual Primal Dual Compl Time
0 5.61130636e+03 -7.51598434e+03 8.14e+04 0.00e+00 7.95e+03 0s
1 5.35762853e+03 -4.99243168e+03 0.00e+00 6.94e-18 1.04e+03 0s
2 5.29184568e+03 4.69491452e+03 0.00e+00 0.00e+00 5.97e+01 0s
3 4.81814867e+03 4.77608199e+03 0.00e+00 3.25e-19 4.21e+00 0s
4 4.80001789e+03 4.79997550e+03 0.00e+00 1.63e-18 4.24e-03 0s
5 4.80000002e+03 4.79999998e+03 0.00e+00 6.88e-19 4.24e-06 0s
6 4.80000000e+03 4.80000000e+03 0.00e+00 0.00e+00 4.24e-09 0s
7 4.80000000e+03 4.80000000e+03 0.00e+00 3.87e-19 4.24e-12 0s
Barrier solved model in 7 iterations and 0.02 seconds (0.00 work units)
Optimal objective 4.80000000e+03
Extra simplex iterations after uncrush: 1
Root relaxation: objective 4.800000e+03, 5 iterations, 0.01 seconds (0.00 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 4800.00000 0 4 - 4800.00000 - - 0s
0 0 4800.00000 0 4 - 4800.00000 - - 0s
0 2 4800.00000 0 4 - 4800.00000 - - 0s
* 8908 4732 76 4800.6758245 4800.00000 0.01% 1.6 0s
Explored 10351 nodes (16856 simplex iterations) in 0.25 seconds (0.01 work units)
Thread count was 32 (of 32 available processors)
Solution count 1: 4800.68
Optimal solution found (tolerance 1.00e-04)
Warning: max constraint violation (2.4414e-04) exceeds tolerance
Best objective 4.800675824507e+03, best bound 4.800675824507e+03, gap 0.0000%
Set parameter Method to value 2
Gurobi Optimizer version 11.0.2 build v11.0.2rc0 (linux64 - "CentOS Linux 7 (Core)")
CPU model: AMD EPYC 7351 16-Core Processor, instruction set [SSE2|AVX|AVX2]
Thread count: 32 physical cores, 32 logical processors, using up to 32 threads
Optimize a model with 6 rows, 14 columns and 12 nonzeros
Model fingerprint: 0xf89c6754
Model has 3 quadratic constraints
Model has 3 general constraints
Variable types: 14 continuous, 0 integer (0 binary)
Coefficient statistics:
Matrix range [1e+00, 1e+00]
QMatrix range [1e+00, 6e+06]
Objective range [3e+00, 2e+01]
Bounds range [4e+02, 1e+07]
RHS range [4e+02, 4e+02]
Presolve removed 5 rows and 8 columns
Presolve time: 0.00s
Presolved: 22 rows, 12 columns, 41 nonzeros
Presolved model has 6 bilinear constraint(s)
Solving non-convex MIQCP
Variable types: 12 continuous, 0 integer (0 binary)
Root barrier log...
Ordering time: 0.00s
Barrier statistics:
AA' NZ : 6.000e+00
Factor NZ : 1.000e+01
Factor Ops : 3.000e+01 (less than 1 second per iteration)
Threads : 1
Objective Residual
Iter Primal Dual Primal Dual Compl Time
0 5.73910355e+03 -7.11598434e+03 8.07e+04 0.00e+00 7.90e+03 0s
1 5.57850009e+03 -4.70322109e+03 0.00e+00 1.39e-17 1.03e+03 0s
2 5.54896076e+03 5.12499969e+03 0.00e+00 6.94e-18 4.24e+01 0s
3 5.21555996e+03 5.18053470e+03 0.00e+00 2.17e-19 3.50e+00 0s
4 5.20001580e+03 5.19998001e+03 0.00e+00 2.53e-19 3.58e-03 0s
5 5.20000002e+03 5.19999998e+03 0.00e+00 3.68e-13 3.58e-06 0s
6 5.20000000e+03 5.20000000e+03 0.00e+00 4.58e-16 3.58e-09 0s
7 5.20000000e+03 5.20000000e+03 0.00e+00 4.34e-19 3.58e-12 0s
Barrier solved model in 7 iterations and 0.02 seconds (0.00 work units)
Optimal objective 5.20000000e+03
Extra simplex iterations after uncrush: 1
Root relaxation: objective 5.200000e+03, 5 iterations, 0.01 seconds (0.00 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 5200.00000 0 4 - 5200.00000 - - 0s
0 0 5200.00000 0 4 - 5200.00000 - - 0s
0 2 5200.00000 0 4 - 5200.00000 - - 0s
* 4948 5005 118 5200.0000000 5200.00000 0.00% 1.6 0s
Explored 11567 nodes (18646 simplex iterations) in 0.25 seconds (0.01 work units)
Thread count was 32 (of 32 available processors)
Solution count 1: 5200
Optimal solution found (tolerance 1.00e-04)
Warning: max constraint violation (3.8147e-04) exceeds tolerance
Best objective 5.200000000021e+03, best bound 5.200000000021e+03, gap 0.0000%
Set parameter Method to value 2
Gurobi Optimizer version 11.0.2 build v11.0.2rc0 (linux64 - "CentOS Linux 7 (Core)")
CPU model: AMD EPYC 7351 16-Core Processor, instruction set [SSE2|AVX|AVX2]
Thread count: 32 physical cores, 32 logical processors, using up to 32 threads
Optimize a model with 6 rows, 14 columns and 12 nonzeros
Model fingerprint: 0xcee932d4
Model has 3 quadratic constraints
Model has 3 general constraints
Variable types: 14 continuous, 0 integer (0 binary)
Coefficient statistics:
Matrix range [1e+00, 1e+00]
QMatrix range [1e+00, 6e+06]
Objective range [4e+00, 2e+01]
Bounds range [4e+02, 1e+07]
RHS range [4e+02, 4e+02]
Presolve removed 5 rows and 8 columns
Presolve time: 0.00s
Presolved: 22 rows, 12 columns, 41 nonzeros
Presolved model has 6 bilinear constraint(s)
Solving non-convex MIQCP
Variable types: 12 continuous, 0 integer (0 binary)
Root barrier log...
Ordering time: 0.00s
Barrier statistics:
AA' NZ : 6.000e+00
Factor NZ : 1.000e+01
Factor Ops : 3.000e+01 (less than 1 second per iteration)
Threads : 1
Objective Residual
Iter Primal Dual Primal Dual Compl Time
0 5.86739250e+03 -6.71598434e+03 7.89e+04 0.00e+00 7.81e+03 0s
1 5.79243327e+03 -4.35146449e+03 0.00e+00 4.51e-17 1.01e+03 0s
2 5.78491928e+03 5.56129062e+03 0.00e+00 6.29e-18 2.24e+01 0s
3 5.60951100e+03 5.58885407e+03 0.00e+00 0.00e+00 2.07e+00 0s
4 5.60001029e+03 5.59998854e+03 0.00e+00 3.25e-19 2.18e-03 0s
5 5.60000001e+03 5.59999999e+03 0.00e+00 6.27e-13 2.18e-06 0s
6 5.60000000e+03 5.60000000e+03 0.00e+00 7.72e-16 2.18e-09 0s
7 5.60000000e+03 5.60000000e+03 0.00e+00 8.67e-19 2.18e-12 0s
Barrier solved model in 7 iterations and 0.02 seconds (0.00 work units)
Optimal objective 5.60000000e+03
Extra simplex iterations after uncrush: 1
Root relaxation: objective 5.600000e+03, 5 iterations, 0.01 seconds (0.00 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 5600.00000 0 4 - 5600.00000 - - 0s
0 0 5600.00000 0 4 - 5600.00000 - - 0s
0 2 5600.00000 0 4 - 5600.00000 - - 0s
H 4467 1927 5600.0006928 5600.00000 0.00% 1.6 0s
Explored 4467 nodes (7189 simplex iterations) in 0.19 seconds (0.01 work units)
Thread count was 32 (of 32 available processors)
Solution count 1: 5600
Optimal solution found (tolerance 1.00e-04)
Warning: max constraint violation (3.9062e-03) exceeds tolerance
Best objective 5.600000692807e+03, best bound 5.600000000000e+03, gap 0.0000%
Set parameter Method to value 2
Gurobi Optimizer version 11.0.2 build v11.0.2rc0 (linux64 - "CentOS Linux 7 (Core)")
CPU model: AMD EPYC 7351 16-Core Processor, instruction set [SSE2|AVX|AVX2]
Thread count: 32 physical cores, 32 logical processors, using up to 32 threads
Optimize a model with 6 rows, 14 columns and 12 nonzeros
Model fingerprint: 0xac4bb344
Model has 3 quadratic constraints
Model has 3 general constraints
Variable types: 14 continuous, 0 integer (0 binary)
Coefficient statistics:
Matrix range [1e+00, 1e+00]
QMatrix range [1e+00, 6e+06]
Objective range [5e+00, 2e+01]
Bounds range [4e+02, 1e+07]
RHS range [4e+02, 4e+02]
Presolve removed 5 rows and 8 columns
Presolve time: 0.00s
Presolved: 22 rows, 12 columns, 41 nonzeros
Presolved model has 6 bilinear constraint(s)
Solving non-convex MIQCP
Variable types: 12 continuous, 0 integer (0 binary)
Root barrier log...
Ordering time: 0.00s
Barrier statistics:
AA' NZ : 6.000e+00
Factor NZ : 1.000e+01
Factor Ops : 3.000e+01 (less than 1 second per iteration)
Threads : 1
Objective Residual
Iter Primal Dual Primal Dual Compl Time
0 6.00000000e+03 -6.31598434e+03 6.79e+04 0.00e+00 7.31e+03 0s
1 6.00000000e+03 -3.40669849e+03 0.00e+00 6.94e-18 9.41e+02 0s
2 6.00000000e+03 5.99059330e+03 0.00e+00 6.94e-18 9.41e-01 0s
3 6.00000000e+03 5.99999059e+03 0.00e+00 0.00e+00 9.41e-04 0s
4 6.00000000e+03 5.99999999e+03 0.00e+00 4.37e-24 9.41e-07 0s
5 6.00000000e+03 6.00000000e+03 0.00e+00 3.04e-24 9.41e-10 0s
Barrier solved model in 5 iterations and 0.01 seconds (0.00 work units)
Optimal objective 6.00000000e+03
Root relaxation: objective 6.000000e+03, 4 iterations, 0.01 seconds (0.00 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 6000.00000 0 6 - 6000.00000 - - 0s
0 0 6000.00000 0 6 - 6000.00000 - - 0s
0 2 6000.00000 0 6 - 6000.00000 - - 0s
H 192 100 6000.0000000 6000.00000 0.00% 1.3 0s
Explored 245 nodes (312 simplex iterations) in 0.05 seconds (0.00 work units)
Thread count was 32 (of 32 available processors)
Solution count 1: 6000
Optimal solution found (tolerance 1.00e-04)
Best objective 6.000000000000e+03, best bound 6.000000000000e+03, gap 0.0000%
Set parameter Method to value 2
Gurobi Optimizer version 11.0.2 build v11.0.2rc0 (linux64 - "CentOS Linux 7 (Core)")
CPU model: AMD EPYC 7351 16-Core Processor, instruction set [SSE2|AVX|AVX2]
Thread count: 32 physical cores, 32 logical processors, using up to 32 threads
Optimize a model with 6 rows, 14 columns and 12 nonzeros
Model fingerprint: 0x1c94ba1b
Model has 3 quadratic constraints
Model has 3 general constraints
Variable types: 14 continuous, 0 integer (0 binary)
Coefficient statistics:
Matrix range [1e+00, 1e+00]
QMatrix range [1e+00, 6e+06]
Objective range [6e+00, 2e+01]
Bounds range [4e+02, 1e+07]
RHS range [4e+02, 4e+02]
Presolve removed 5 rows and 8 columns
Presolve time: 0.00s
Presolved: 22 rows, 12 columns, 41 nonzeros
Presolved model has 6 bilinear constraint(s)
Solving non-convex MIQCP
Variable types: 12 continuous, 0 integer (0 binary)
Root barrier log...
Ordering time: 0.00s
Barrier statistics:
AA' NZ : 6.000e+00
Factor NZ : 1.000e+01
Factor Ops : 3.000e+01 (less than 1 second per iteration)
Threads : 1
Objective Residual
Iter Primal Dual Primal Dual Compl Time
0 6.14782407e+03 -5.91598434e+03 6.65e+04 1.95e-03 7.23e+03 0s
1 6.20341015e+03 -3.06879754e+03 0.00e+00 1.04e-17 9.27e+02 0s
2 6.19527275e+03 5.95560926e+03 0.00e+00 0.00e+00 2.40e+01 0s
3 6.02018308e+03 5.97508089e+03 0.00e+00 0.00e+00 4.51e+00 0s
4 6.00021295e+03 5.99996873e+03 0.00e+00 2.68e-19 2.44e-02 0s
5 6.00000021e+03 5.99999999e+03 0.00e+00 8.91e-11 2.44e-05 0s
6 6.00000000e+03 6.00000000e+03 0.00e+00 9.59e-14 2.44e-08 0s
7 6.00000000e+03 6.00000000e+03 0.00e+00 9.58e-17 2.44e-11 0s
Barrier solved model in 7 iterations and 0.02 seconds (0.00 work units)
Optimal objective 6.00000000e+03
Extra simplex iterations after uncrush: 1
Root relaxation: objective 6.000000e+03, 5 iterations, 0.01 seconds (0.00 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 6000.00000 0 4 - 6000.00000 - - 0s
0 0 6000.00000 0 4 - 6000.00000 - - 0s
H 0 0 6000.0067499 6000.00000 0.00% - 0s
Explored 1 nodes (5 simplex iterations) in 0.03 seconds (0.00 work units)
Thread count was 32 (of 32 available processors)
Solution count 1: 6000.01
Optimal solution found (tolerance 1.00e-04)
Warning: max constraint violation (3.9062e-03) exceeds tolerance
Best objective 6.000006749886e+03, best bound 6.000000000000e+03, gap 0.0001%
Set parameter Method to value 2
Gurobi Optimizer version 11.0.2 build v11.0.2rc0 (linux64 - "CentOS Linux 7 (Core)")
CPU model: AMD EPYC 7351 16-Core Processor, instruction set [SSE2|AVX|AVX2]
Thread count: 32 physical cores, 32 logical processors, using up to 32 threads
Optimize a model with 6 rows, 14 columns and 12 nonzeros
Model fingerprint: 0xb4494f8e
Model has 3 quadratic constraints
Model has 3 general constraints
Variable types: 14 continuous, 0 integer (0 binary)
Coefficient statistics:
Matrix range [1e+00, 1e+00]
QMatrix range [1e+00, 6e+06]
Objective range [1e+01, 2e+01]
Bounds range [4e+02, 1e+07]
RHS range [4e+02, 4e+02]
Presolve removed 5 rows and 8 columns
Presolve time: 0.00s
Presolved: 22 rows, 12 columns, 41 nonzeros
Presolved model has 6 bilinear constraint(s)
Solving non-convex MIQCP
Variable types: 12 continuous, 0 integer (0 binary)
Root barrier log...
Ordering time: 0.00s
Barrier statistics:
AA' NZ : 6.000e+00
Factor NZ : 1.000e+01
Factor Ops : 3.000e+01 (less than 1 second per iteration)
Threads : 1
Objective Residual
Iter Primal Dual Primal Dual Compl Time
0 6.73912034e+03 -4.31598434e+03 6.65e+04 9.77e-03 7.19e+03 0s
1 6.94437023e+03 -2.13574332e+03 0.00e+00 6.94e-18 9.08e+02 0s
2 6.74363195e+03 5.93954897e+03 0.00e+00 9.54e-18 8.04e+01 0s
3 6.00795756e+03 5.96047329e+03 0.00e+00 8.67e-19 4.75e+00 0s
4 6.00000812e+03 5.99996016e+03 0.00e+00 1.67e-18 4.80e-03 0s
5 6.00000001e+03 5.99999996e+03 0.00e+00 4.34e-19 4.80e-06 0s
6 6.00000000e+03 6.00000000e+03 0.00e+00 0.00e+00 4.80e-09 0s
Barrier solved model in 6 iterations and 0.02 seconds (0.00 work units)
Optimal objective 6.00000000e+03
Extra simplex iterations after uncrush: 1
Root relaxation: objective 6.000000e+03, 5 iterations, 0.01 seconds (0.00 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 6000.00000 0 4 - 6000.00000 - - 0s
0 0 6000.00000 0 4 - 6000.00000 - - 0s
H 0 0 6000.0000000 6000.00000 0.00% - 0s
0 0 6000.00000 0 4 6000.00000 6000.00000 0.00% - 0s
Explored 1 nodes (5 simplex iterations) in 0.03 seconds (0.00 work units)
Thread count was 32 (of 32 available processors)
Solution count 1: 6000
Optimal solution found (tolerance 1.00e-04)
Best objective 6.000000000002e+03, best bound 6.000000000002e+03, gap 0.0000%
Set parameter Method to value 2
Gurobi Optimizer version 11.0.2 build v11.0.2rc0 (linux64 - "CentOS Linux 7 (Core)")
CPU model: AMD EPYC 7351 16-Core Processor, instruction set [SSE2|AVX|AVX2]
Thread count: 32 physical cores, 32 logical processors, using up to 32 threads
Optimize a model with 6 rows, 14 columns and 12 nonzeros
Model fingerprint: 0xacefb630
Model has 3 quadratic constraints
Model has 3 general constraints
Variable types: 14 continuous, 0 integer (0 binary)
Coefficient statistics:
Matrix range [1e+00, 1e+00]
QMatrix range [1e+00, 6e+06]
Objective range [1e+01, 2e+01]
Bounds range [4e+02, 1e+07]
RHS range [4e+02, 4e+02]
Presolve removed 5 rows and 8 columns
Presolve time: 0.00s
Presolved: 22 rows, 12 columns, 41 nonzeros
Presolved model has 6 bilinear constraint(s)
Solving non-convex MIQCP
Variable types: 12 continuous, 0 integer (0 binary)
Root barrier log...
Ordering time: 0.00s
Barrier statistics:
AA' NZ : 6.000e+00
Factor NZ : 1.000e+01
Factor Ops : 3.000e+01 (less than 1 second per iteration)
Threads : 1
Objective Residual
Iter Primal Dual Primal Dual Compl Time
0 7.47824067e+03 -2.31598434e+03 6.65e+04 1.95e-02 7.14e+03 0s
1 7.69605683e+03 -1.01943136e+03 0.00e+00 1.73e-17 8.72e+02 0s
2 6.96692162e+03 5.95616533e+03 0.00e+00 6.94e-18 1.01e+02 0s
3 6.00230082e+03 5.99002797e+03 0.00e+00 8.67e-19 1.23e+00 0s
4 6.00000228e+03 5.99999001e+03 0.00e+00 6.98e-19 1.23e-03 0s
5 6.00000000e+03 5.99999999e+03 0.00e+00 0.00e+00 1.23e-06 0s
6 6.00000000e+03 6.00000000e+03 0.00e+00 0.00e+00 1.23e-09 0s
Barrier solved model in 6 iterations and 0.02 seconds (0.00 work units)
Optimal objective 6.00000000e+03
Extra simplex iterations after uncrush: 1
Root relaxation: objective 6.000000e+03, 5 iterations, 0.01 seconds (0.00 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 6000.00000 0 4 - 6000.00000 - - 0s
0 0 6000.00000 0 4 - 6000.00000 - - 0s
0 2 6000.00000 0 4 - 6000.00000 - - 0s
* 3830 3154 88 6000.3514970 6000.00000 0.01% 1.5 0s
Explored 6989 nodes (10653 simplex iterations) in 0.15 seconds (0.00 work units)
Thread count was 32 (of 32 available processors)
Solution count 1: 6000.35
Optimal solution found (tolerance 1.00e-04)
Warning: max constraint violation (3.0518e-05) exceeds tolerance
Best objective 6.000351497047e+03, best bound 6.000000000000e+03, gap 0.0059%
import matplotlib.pyplot as plt
plt.plot([0,1,2,3,4,5,6, 10, 15],flows_g1, color='red', label='g1 (w compressor)')
plt.plot([0,1,2,3,4,5,6, 10, 15],flows_g5, color='green', label='g5')
plt.xticks([0,1,5, 10, 15, 20])
plt.xlabel('compressor price')
plt.legend()
<matplotlib.legend.Legend at 0x7ffea7d285c0>
