Meshed AC-DC Networks¶

This example demonstrates how to optimise meshed AC-DC networks in PyPSA. The example has a 3-node AC network coupled via AC-DC converters to a 3-node DC network. There is also a single point-to-point DC connection using the Link component.

In [2]:

import pypsa

n = pypsa.examples.ac_dc_meshed()
n.links.loc["Norwich Converter", "p_nom_extendable"] = False

import pypsa n = pypsa.examples.ac_dc_meshed() n.links.loc["Norwich Converter", "p_nom_extendable"] = False

INFO:pypsa.network.io:Imported network 'AC-DC-Meshed' has buses, carriers, generators, global_constraints, lines, links, loads

In [3]:

line_color = n.lines.bus0.map(n.buses.carrier).map(
    lambda ct: "r" if ct == "DC" else "b"
)

n.plot(
    line_color=line_color,
    link_color="c",
    title="AC (blue) - DC (red) - P2P-DC (cyan)",
    jitter=0.4,
)

line_color = n.lines.bus0.map(n.buses.carrier).map( lambda ct: "r" if ct == "DC" else "b" ) n.plot( line_color=line_color, link_color="c", title="AC (blue) - DC (red) - P2P-DC (cyan)", jitter=0.4, )

Out[3]:

{'nodes': {'Bus': <matplotlib.collections.PatchCollection at 0x725fa31a9a90>},
 'branches': {'Link': <matplotlib.collections.LineCollection at 0x725fa31a9be0>,
  'Line': <matplotlib.collections.LineCollection at 0x725fc490e990>},
 'flows': {}}

We inspect the topology of the network. Therefore, use n.determine_network_topology() and inspect the subnetworks in n.sub_networks.

In [4]:

n.determine_network_topology()
n.sub_networks["n_branches"] = [len(sn.branches()) for sn in n.sub_networks.obj]
n.sub_networks["n_buses"] = [
    len(sn.components.buses.static) for sn in n.sub_networks.obj
]

n.sub_networks

n.determine_network_topology() n.sub_networks["n_branches"] = [len(sn.branches()) for sn in n.sub_networks.obj] n.sub_networks["n_buses"] = [ len(sn.components.buses.static) for sn in n.sub_networks.obj ] n.sub_networks

Out[4]:

	carrier	slack_bus	obj	n_branches	n_buses
name
0	AC	Manchester	<pypsa.SubNetwork object at 0x725fa31...	3	3
1	DC	Norwich DC	<pypsa.SubNetwork object at 0x725fa31...	3	3
2	AC	Frankfurt	<pypsa.SubNetwork object at 0x725fa31...	1	2
3	AC	Norway	<pypsa.SubNetwork object at 0x725fa31...	0	1

The network covers 10 time steps. These are given by the snapshots attribute.

In [5]:

n.snapshots

n.snapshots

Out[5]:

DatetimeIndex(['2015-01-01 00:00:00', '2015-01-01 01:00:00',
               '2015-01-01 02:00:00', '2015-01-01 03:00:00',
               '2015-01-01 04:00:00', '2015-01-01 05:00:00',
               '2015-01-01 06:00:00', '2015-01-01 07:00:00',
               '2015-01-01 08:00:00', '2015-01-01 09:00:00'],
              dtype='datetime64[ns]', name='snapshot', freq=None)

There are 6 generators in the network, 3 wind and 3 gas. All are attached to AC buses:

In [6]:

n.generators

n.generators

Out[6]:

	bus	control	type	p_nom	p_nom_mod	p_nom_extendable	p_nom_min	p_nom_max	p_nom_set	p_min_pu	...	min_up_time	min_down_time	up_time_before	down_time_before	ramp_limit_up	ramp_limit_down	ramp_limit_start_up	ramp_limit_shut_down	weight	p_nom_opt
name
Manchester Wind	Manchester	Slack		80.0	0.0	True	100.0	inf	NaN	0.0	...	0	0	1	0	NaN	NaN	NaN	NaN	1.0	0.0
Manchester Gas	Manchester	PQ		50000.0	0.0	True	0.0	inf	NaN	0.0	...	0	0	1	0	NaN	NaN	NaN	NaN	1.0	0.0
Norway Wind	Norway	Slack		100.0	0.0	True	100.0	inf	NaN	0.0	...	0	0	1	0	NaN	NaN	NaN	NaN	1.0	0.0
Norway Gas	Norway	PQ		20000.0	0.0	True	0.0	inf	NaN	0.0	...	0	0	1	0	NaN	NaN	NaN	NaN	1.0	0.0
Frankfurt Wind	Frankfurt	Slack		110.0	0.0	True	100.0	inf	NaN	0.0	...	0	0	1	0	NaN	NaN	NaN	NaN	1.0	0.0
Frankfurt Gas	Frankfurt	PQ		80000.0	0.0	True	0.0	inf	NaN	0.0	...	0	0	1	0	NaN	NaN	NaN	NaN	1.0	0.0

6 rows × 42 columns

We see that the generators have different capital and marginal costs. All of them have a p_nom_extendable set to True, meaning that capacities can be extended in the optimisation. The wind generators have a per unit limit for each time step, given by the weather potentials at the site.

In [7]:

n.generators_t.p_max_pu.plot()

n.generators_t.p_max_pu.plot()

Out[7]:

<Axes: xlabel='snapshot'>

Alright now we know how the network looks like, where the generators and lines are. Now, let's perform a optimization of the operation and capacities.

In [8]:

n.optimize();

n.optimize();

/tmp/ipykernel_3368/1450685117.py:1: FutureWarning: The default value of `include_objective_constant` will change from True to False in version 2.0. Set `include_objective_constant` explicitly to suppress this warning. Using False improves LP numerical conditioning by not including the objective constant as a variable.
  n.optimize();
WARNING:pypsa.consistency:The following lines have zero x, which could break the linear load flow:
Index(['2', '3', '4'], dtype='object', name='name')

WARNING:pypsa.consistency:The following lines have zero r, which could break the linear load flow:
Index(['0', '1', '5', '6'], dtype='object', name='name')

INFO:linopy.model: Solve problem using Highs solver

INFO:linopy.model:Solver options:
 - log_to_console: False

INFO:linopy.io: Writing time: 0.07s

INFO:linopy.constants: Optimization successful: 
Status: ok
Termination condition: optimal
Solution: 187 primals, 467 duals
Objective: -3.47e+06
Solver model: available
Solver message: Optimal

INFO:pypsa.optimization.optimize:The shadow-prices of the constraints Generator-ext-p-lower, Generator-ext-p-upper, Line-ext-s-lower, Line-ext-s-upper, Link-fix-p-lower, Link-fix-p-upper, Link-ext-p-lower, Link-ext-p-upper, Kirchhoff-Voltage-Law were not assigned to the network.

The objective is given by:

In [9]:

n.objective

n.objective

Out[9]:

-3474094.130816234

Why is this number negative? It considers the starting point of the optimisation, thus the existent capacities given by n.generators.p_nom are taken into account.

The real system cost are given by

In [10]:

n.objective + n.objective_constant

n.objective + n.objective_constant

Out[10]:

18440973.38746291

The optimal capacities are given by p_nom_opt for generators, links and storages and s_nom_opt for lines.

Let's look how the optimal capacities for the generators look like.

In [11]:

n.generators.p_nom_opt.div(1e3).round(2).sort_values()

n.generators.p_nom_opt.div(1e3).round(2).sort_values()

Out[11]:

name
Manchester Gas    -0.00
Norway Gas        -0.00
Frankfurt Gas      0.98
Norway Wind        1.53
Frankfurt Wind     1.67
Manchester Wind    4.09
Name: p_nom_opt, dtype: float64

Their production is again given as a time-series in n.generators_t.

In [12]:

n.generators_t.p.div(1e3).plot.area(stacked=True, lw=0, ylabel="GW")

n.generators_t.p.div(1e3).plot.area(stacked=True, lw=0, ylabel="GW")

Out[12]:

<Axes: xlabel='snapshot', ylabel='GW'>

What are the locational marginal prices in the network? From the optimisation these are given for each bus and snapshot.

In [13]:

n.buses_t.marginal_price.mean(axis=1).plot(figsize=(8, 3), ylabel="€/MWh")

n.buses_t.marginal_price.mean(axis=1).plot(figsize=(8, 3), ylabel="€/MWh")

Out[13]:

<Axes: xlabel='snapshot', ylabel='€/MWh'>

We can inspect further quantities as the active power of AC-DC converters and HVDC link.

In [14]:

n.links_t.p0.round(2)

n.links_t.p0.round(2)

Out[14]:

name	Norwich Converter	Norway Converter	Bremen Converter	DC link
snapshot
2015-01-01 00:00:00	-250.84	674.58	-423.74	-318.00
2015-01-01 01:00:00	315.07	-116.73	-198.34	-318.00
2015-01-01 02:00:00	350.76	581.97	-932.73	-318.00
2015-01-01 03:00:00	-85.77	272.56	-186.79	-318.00
2015-01-01 04:00:00	317.37	-79.75	-237.62	-318.00
2015-01-01 05:00:00	386.75	-494.20	107.45	-318.00
2015-01-01 06:00:00	900.00	-257.52	-642.48	318.00
2015-01-01 07:00:00	123.68	971.92	-1095.60	-86.86
2015-01-01 08:00:00	244.72	850.88	-1095.60	318.00
2015-01-01 09:00:00	820.02	-86.85	-733.17	-83.68

In [15]:

n.lines_t.p0.round(2)

n.lines_t.p0.round(2)

Out[15]:

name	0	1	2	3	4	5	6
snapshot
2015-01-01 00:00:00	79.47	-38.11	-52.97	-303.81	370.78	-202.73	-534.34
2015-01-01 01:00:00	-449.46	787.04	-211.27	103.80	-12.93	209.36	-823.21
2015-01-01 02:00:00	-181.27	520.56	-505.91	-155.15	426.82	-248.68	173.90
2015-01-01 03:00:00	-45.47	234.47	-34.04	-119.81	152.75	-232.72	-743.79
2015-01-01 04:00:00	-73.21	295.42	-227.20	90.17	10.42	-240.11	-883.82
2015-01-01 05:00:00	-594.50	1198.08	-125.90	260.84	-233.35	19.36	-1030.85
2015-01-01 06:00:00	-661.29	1378.42	-632.37	267.63	10.11	-53.45	319.39
2015-01-01 07:00:00	-383.77	540.91	-469.93	-346.25	625.67	393.72	600.73
2015-01-01 08:00:00	-778.28	1444.07	-522.12	-277.40	573.48	229.29	501.35
2015-01-01 09:00:00	-465.58	899.57	-632.37	187.65	100.80	78.62	-248.57

...or the active power injection per bus.

In [16]:

n.buses_t.p.round(2)

n.buses_t.p.round(2)

Out[16]:

name	London	Norwich	Norwich DC	Manchester	Bremen	Bremen DC	Frankfurt	Norway	Norway DC
snapshot
2015-01-01 00:00:00	282.20	-164.62	-250.84	-117.58	-534.34	-423.74	534.34	0.0	674.58
2015-01-01 01:00:00	-658.83	-577.67	315.07	1236.50	-823.21	-198.34	823.21	0.0	-116.73
2015-01-01 02:00:00	67.41	-769.24	350.76	701.83	173.90	-932.73	-173.90	0.0	581.97
2015-01-01 03:00:00	187.24	-467.19	-85.77	279.94	-743.79	-186.79	743.79	0.0	272.56
2015-01-01 04:00:00	166.90	-535.53	317.37	368.63	-883.82	-237.62	883.82	0.0	-79.75
2015-01-01 05:00:00	-613.86	-1178.72	386.75	1792.58	-1030.85	107.45	1030.85	0.0	-494.20
2015-01-01 06:00:00	-607.85	-1431.87	900.00	2039.72	319.39	-642.48	-319.39	0.0	-257.52
2015-01-01 07:00:00	-777.48	-147.19	123.68	924.68	600.73	-1095.60	-600.73	0.0	971.92
2015-01-01 08:00:00	-1007.58	-1214.77	244.72	2222.35	501.35	-1095.60	-501.35	0.0	850.88
2015-01-01 09:00:00	-544.19	-820.95	820.02	1365.14	-248.57	-733.17	248.57	0.0	-86.85