Note
You can download this example as a Jupyter notebook or start it in interactive mode.
Power to Gas with Heat Coupling#
This is an example for power to gas with optional coupling to heat sector (via boiler OR Combined-Heat-and-Power (CHP))
A location has an electric, gas and heat bus. The primary source is wind power, which can be converted to gas. The gas can be stored to convert into electricity or heat (with either a boiler or a CHP).
[1]:
import pypsa
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from pyomo.environ import Constraint
%matplotlib inline
[2]:
import logging
logging.basicConfig(level="INFO")
Combined-Heat-and-Power (CHP) parameterisation#
This setup follows http://www.ea-energianalyse.dk/reports/student-reports/integration_of_50_percent_wind%20power.pdf pages 35-6 which follows http://www.sciencedirect.com/science/article/pii/030142159390282K
[3]:
# ratio between max heat output and max electric output
nom_r = 1.0
# backpressure limit
c_m = 0.75
# marginal loss for each additional generation of heat
c_v = 0.15
Graph for the case that max heat output equals max electric output
[4]:
fig, ax = plt.subplots(figsize=(9, 5))
t = 0.01
ph = np.arange(0, 1.0001, t)
ax.plot(ph, c_m * ph)
ax.set_xlabel("P_heat_out")
ax.set_ylabel("P_elec_out")
ax.grid(True)
ax.set_xlim([0, 1.1])
ax.set_ylim([0, 1.1])
ax.text(0.1, 0.7, "Allowed output", color="r")
ax.plot(ph, 1 - c_v * ph)
for i in range(1, 10):
k = 0.1 * i
x = np.arange(0, k / (c_m + c_v), t)
ax.plot(x, k - c_v * x, color="g", alpha=0.5)
ax.text(0.05, 0.41, "iso-fuel-lines", color="g", rotation=-7)
ax.fill_between(ph, c_m * ph, 1 - c_v * ph, facecolor="r", alpha=0.5)
fig.tight_layout()
Optimisation#
[5]:
network = pypsa.Network()
network.set_snapshots(pd.date_range("2016-01-01 00:00", "2016-01-01 03:00", freq="H"))
network.add("Bus", "0", carrier="AC")
network.add("Bus", "0 gas", carrier="gas")
network.add("Carrier", "wind")
network.add("Carrier", "gas", co2_emissions=0.2)
network.add("GlobalConstraint", "co2_limit", sense="<=", constant=0.0)
network.add(
"Generator",
"wind turbine",
bus="0",
carrier="wind",
p_nom_extendable=True,
p_max_pu=[0.0, 0.2, 0.7, 0.4],
capital_cost=1000,
)
network.add("Load", "load", bus="0", p_set=5.0)
network.add(
"Link",
"P2G",
bus0="0",
bus1="0 gas",
efficiency=0.6,
capital_cost=1000,
p_nom_extendable=True,
)
network.add(
"Link",
"generator",
bus0="0 gas",
bus1="0",
efficiency=0.468,
capital_cost=400,
p_nom_extendable=True,
)
network.add("Store", "gas depot", bus="0 gas", e_cyclic=True, e_nom_extendable=True)
Add heat sector
[6]:
network.add("Bus", "0 heat", carrier="heat")
network.add("Carrier", "heat")
network.add("Load", "heat load", bus="0 heat", p_set=10.0)
network.add(
"Link",
"boiler",
bus0="0 gas",
bus1="0 heat",
efficiency=0.9,
capital_cost=300,
p_nom_extendable=True,
)
network.add("Store", "water tank", bus="0 heat", e_cyclic=True, e_nom_extendable=True)
Add CHP constraints
[7]:
# Guarantees ISO fuel lines, i.e. fuel consumption p_b0 + p_g0 = constant along p_g1 + c_v p_b1 = constant
network.links.at["boiler", "efficiency"] = (
network.links.at["generator", "efficiency"] / c_v
)
model = network.optimize.create_model()
link_p = model.variables["Link-p"]
link_p_nom = model.variables["Link-p_nom"]
# Guarantees heat output and electric output nominal powers are proportional
model.add_constraints(
network.links.at["generator", "efficiency"] * nom_r * link_p_nom["generator"]
- network.links.at["boiler", "efficiency"] * link_p_nom["boiler"]
== 0,
name="heat-power output proportionality",
)
# Guarantees c_m p_b1 \leq p_g1
model.add_constraints(
c_m * network.links.at["boiler", "efficiency"] * link_p.sel(Link="boiler")
- network.links.at["generator", "efficiency"] * link_p.sel(Link="generator")
<= 0,
name="backpressure",
)
# Guarantees p_g1 +c_v p_b1 \leq p_g1_nom
model.add_constraints(
link_p.sel(Link="boiler")
+ link_p.sel(Link="generator")
- link_p_nom.sel({"Link-ext": "generator"})
<= 0,
name="top_iso_fuel_line",
)
network.optimize.solve_model()
INFO:linopy.model: Solve problem using Glpk solver
INFO:linopy.io: Writing time: 0.11s
INFO:linopy.constants: Optimization successful:
Status: ok
Termination condition: optimal
Solution: 38 primals, 83 duals
Objective: 1.62e+05
Solver model: not available
Solver message: optimal
INFO:pypsa.optimization.optimize:The shadow-prices of the constraints Generator-ext-p-lower, Generator-ext-p-upper, Link-ext-p-lower, Link-ext-p-upper, Store-ext-e-lower, Store-ext-e-upper, Store-energy_balance, backpressure, top_iso_fuel_line were not assigned to the network.
GLPSOL--GLPK LP/MIP Solver 5.0
Parameter(s) specified in the command line:
--lp /tmp/linopy-problem-g3vqvmyh.lp --output /tmp/linopy-solve-02iflpo6.sol
Reading problem data from '/tmp/linopy-problem-g3vqvmyh.lp'...
83 rows, 38 columns, 159 non-zeros
449 lines were read
GLPK Simplex Optimizer 5.0
83 rows, 38 columns, 159 non-zeros
Preprocessing...
52 rows, 37 columns, 127 non-zeros
Scaling...
A: min|aij| = 2.000e-01 max|aij| = 3.120e+00 ratio = 1.560e+01
Problem data seem to be well scaled
Constructing initial basis...
Size of triangular part is 50
0: obj = 0.000000000e+00 inf = 1.377e+02 (11)
22: obj = 1.703538022e+05 inf = 0.000e+00 (0)
* 25: obj = 1.622539996e+05 inf = 0.000e+00 (0)
OPTIMAL LP SOLUTION FOUND
Time used: 0.0 secs
Memory used: 0.1 Mb (104191 bytes)
Writing basic solution to '/tmp/linopy-solve-02iflpo6.sol'...
[7]:
('ok', 'optimal')
[8]:
network.objective
[8]:
162253.9996
Inspection#
[9]:
network.loads_t.p
[9]:
| Load | load | heat load |
|---|---|---|
| snapshot | ||
| 2016-01-01 00:00:00 | 5.0 | 10.0 |
| 2016-01-01 01:00:00 | 5.0 | 10.0 |
| 2016-01-01 02:00:00 | 5.0 | 10.0 |
| 2016-01-01 03:00:00 | 5.0 | 10.0 |
[10]:
network.links.p_nom_opt
[10]:
Link
P2G 58.6489
generator 28.4900
boiler 4.2735
Name: p_nom_opt, dtype: float64
[11]:
# CHP is dimensioned by the heat demand met in three hours when no wind
4 * 10.0 / 3 / network.links.at["boiler", "efficiency"]
[11]:
4.273504273504273
[12]:
# elec is set by the heat demand
28.490028 * 0.15
[12]:
4.2735042
[13]:
network.links_t.p0
[13]:
| Link | P2G | generator | boiler |
|---|---|---|---|
| snapshot | |||
| 2016-01-01 00:00:00 | 5.0000 | 21.3675 | 4.2735 |
| 2016-01-01 01:00:00 | 23.1854 | 21.3675 | 4.2735 |
| 2016-01-01 02:00:00 | 58.6489 | 0.0000 | 0.0000 |
| 2016-01-01 03:00:00 | 41.3708 | 21.3675 | 4.2735 |
[14]:
network.links_t.p1
[14]:
| Link | P2G | generator | boiler |
|---|---|---|---|
| snapshot | |||
| 2016-01-01 00:00:00 | -3.00000 | -9.99999 | -13.33332 |
| 2016-01-01 01:00:00 | -13.91124 | -9.99999 | -13.33332 |
| 2016-01-01 02:00:00 | -35.18934 | -0.00000 | -0.00000 |
| 2016-01-01 03:00:00 | -24.82248 | -9.99999 | -13.33332 |
[15]:
pd.DataFrame({attr: network.stores_t[attr]["gas depot"] for attr in ["p", "e"]})
[15]:
| p | e | |
|---|---|---|
| snapshot | ||
| 2016-01-01 00:00:00 | 22.64100 | 11.7298 |
| 2016-01-01 01:00:00 | 11.72980 | 0.0000 |
| 2016-01-01 02:00:00 | -35.18930 | 35.1893 |
| 2016-01-01 03:00:00 | 0.81854 | 34.3708 |
[16]:
pd.DataFrame({attr: network.stores_t[attr]["water tank"] for attr in ["p", "e"]})
[16]:
| p | e | |
|---|---|---|
| snapshot | ||
| 2016-01-01 00:00:00 | -3.33333 | 6.66667 |
| 2016-01-01 01:00:00 | -3.33333 | 10.00000 |
| 2016-01-01 02:00:00 | 10.00000 | 0.00000 |
| 2016-01-01 03:00:00 | -3.33333 | 3.33333 |
[17]:
pd.DataFrame({attr: network.links_t[attr]["boiler"] for attr in ["p0", "p1"]})
[17]:
| p0 | p1 | |
|---|---|---|
| snapshot | ||
| 2016-01-01 00:00:00 | 4.2735 | -13.33332 |
| 2016-01-01 01:00:00 | 4.2735 | -13.33332 |
| 2016-01-01 02:00:00 | 0.0000 | -0.00000 |
| 2016-01-01 03:00:00 | 4.2735 | -13.33332 |
[18]:
network.stores.loc["gas depot"]
[18]:
attribute
bus 0 gas
type
carrier gas
e_nom 0.0
e_nom_extendable True
e_nom_min 0.0
e_nom_max inf
e_min_pu 0.0
e_max_pu 1.0
e_initial 0.0
e_initial_per_period False
e_cyclic True
e_cyclic_per_period True
p_set 0.0
q_set 0.0
sign 1.0
marginal_cost 0.0
marginal_cost_quadratic 0.0
capital_cost 0.0
standing_loss 0.0
build_year 0
lifetime inf
e_nom_opt 35.1893
Name: gas depot, dtype: object
[19]:
network.generators.loc["wind turbine"]
[19]:
attribute
bus 0
control PQ
type
p_nom 0.0
p_nom_extendable True
p_nom_min 0.0
p_nom_max inf
p_min_pu 0.0
p_max_pu 1.0
p_set 0.0
q_set 0.0
sign 1.0
carrier wind
marginal_cost 0.0
marginal_cost_quadratic 0.0
build_year 0
lifetime inf
capital_cost 1000.0
efficiency 1.0
committable False
start_up_cost 0.0
shut_down_cost 0.0
stand_by_cost 0.0
min_up_time 0
min_down_time 0
up_time_before 1
down_time_before 0
ramp_limit_up NaN
ramp_limit_down NaN
ramp_limit_start_up 1.0
ramp_limit_shut_down 1.0
weight 1.0
p_nom_opt 90.927
Name: wind turbine, dtype: object
[20]:
network.links.p_nom_opt
[20]:
Link
P2G 58.6489
generator 28.4900
boiler 4.2735
Name: p_nom_opt, dtype: float64
Calculate the overall efficiency of the CHP
[21]:
eta_elec = network.links.at["generator", "efficiency"]
r = 1 / c_m
# P_h = r*P_e
(1 + r) / ((1 / eta_elec) * (1 + c_v * r))
[21]:
0.9099999999999999