Note

You can download this example as a Jupyter notebook or start it in interactive mode.

Screening curve analysis

Compute the long-term equilibrium power plant investment for a given load duration curve (1000-1000z for z \(\in\) [0,1]) and a given set of generator investment options.

[1]:
import pypsa
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

%matplotlib inline

Generator marginal (m) and capital (c) costs in EUR/MWh - numbers chosen for simple answer.

[2]:
generators = {
    "coal": {"m": 2, "c": 15},
    "gas": {"m": 12, "c": 10},
    "load-shedding": {"m": 1012, "c": 0},
}

The screening curve intersections are at 0.01 and 0.5.

[3]:
x = np.linspace(0, 1, 101)
df = pd.DataFrame(
    {key: pd.Series(item["c"] + x * item["m"], x) for key, item in generators.items()}
)
df.plot(ylim=[0, 50], title="Screening Curve", figsize=(9, 5))
plt.tight_layout()
../_images/examples_generation-investment-screening-curve_5_0.png
[4]:
n = pypsa.Network()

num_snapshots = 1001
n.snapshots = np.linspace(0, 1, num_snapshots)
n.snapshot_weightings = n.snapshot_weightings / num_snapshots

n.add("Bus", name="bus")

n.add("Load", name="load", bus="bus", p_set=1000 - 1000 * n.snapshots.values)

for gen in generators:
    n.add(
        "Generator",
        name=gen,
        bus="bus",
        p_nom_extendable=True,
        marginal_cost=float(generators[gen]["m"]),
        capital_cost=float(generators[gen]["c"]),
    )
[5]:
n.loads_t.p_set.plot.area(title="Load Duration Curve", figsize=(9, 5), ylabel="MW")
plt.tight_layout()
../_images/examples_generation-investment-screening-curve_7_0.png
[6]:
n.lopf(solver_name="cbc")
n.objective
WARNING:pypsa.components:Solving optimisation problem with pyomo.In PyPSA version 0.21 the default will change to ``n.lopf(pyomo=False)``.Explicitly set ``n.lopf(pyomo=True)`` to retain current behaviour.
INFO:pypsa.opf:Performed preliminary steps
INFO:pypsa.opf:Building pyomo model using `kirchhoff` formulation
INFO:pypsa.opf:Solving model using cbc
INFO:pypsa.opf:Optimization successful
# ==========================================================
# = Solver Results                                         =
# ==========================================================
# ----------------------------------------------------------
#   Problem Information
# ----------------------------------------------------------
Problem:
- Name: unknown
  Lower bound: 14706.19381
  Upper bound: 14706.19381
  Number of objectives: 1
  Number of constraints: 7008
  Number of variables: 3007
  Number of nonzeros: 3002
  Sense: minimize
# ----------------------------------------------------------
#   Solver Information
# ----------------------------------------------------------
Solver:
- Status: ok
  User time: -1.0
  System time: 0.18
  Wallclock time: 0.11
  Termination condition: optimal
  Termination message: Model was solved to optimality (subject to tolerances), and an optimal solution is available.
  Statistics:
    Branch and bound:
      Number of bounded subproblems: None
      Number of created subproblems: None
    Black box:
      Number of iterations: 1512
  Error rc: 0
  Time: 0.12796783447265625
# ----------------------------------------------------------
#   Solution Information
# ----------------------------------------------------------
Solution:
- number of solutions: 0
  number of solutions displayed: 0
[6]:
14706.19381

The capacity is set by total electricity required.

NB: No load shedding since all prices are below 10 000.

[7]:
n.generators.p_nom_opt.round(2)
[7]:
Generator
coal             500.0
gas              490.0
load-shedding     10.0
Name: p_nom_opt, dtype: float64
[8]:
n.buses_t.marginal_price.plot(title="Price Duration Curve", figsize=(9, 4))
plt.tight_layout()
../_images/examples_generation-investment-screening-curve_11_0.png

The prices correspond either to VOLL (1012) for first 0.01 or the marginal costs (12 for 0.49 and 2 for 0.5)

Except for (infinitesimally small) points at the screening curve intersections, which correspond to changing the load duration near the intersection, so that capacity changes. This explains 7 = (12+10 - 15) (replacing coal with gas) and 22 = (12+10) (replacing load-shedding with gas).

Note: What remains unclear is what is causing :nbsphinx-math:`l `= 0… it should be 2.

[9]:
n.buses_t.marginal_price.round(2).sum(axis=1).value_counts()
[9]:
2.0       499
12.0      489
1012.0     10
22.0        1
7.0         1
0.0         1
dtype: int64
[10]:
n.generators_t.p.plot(ylim=[0, 600], title="Generation Dispatch", figsize=(9, 5))
plt.tight_layout()
../_images/examples_generation-investment-screening-curve_14_0.png

Demonstrate zero-profit condition.

  1. The total cost is given by

[11]:
(
    n.generators.p_nom_opt * n.generators.capital_cost
    + n.generators_t.p.multiply(n.snapshot_weightings.generators, axis=0).sum()
    * n.generators.marginal_cost
)
[11]:
Generator
coal             8249.750250
gas              6400.839161
load-shedding      55.604396
dtype: float64
  1. The total revenue by

[12]:
(
    n.generators_t.p.multiply(n.snapshot_weightings.generators, axis=0)
    .multiply(n.buses_t.marginal_price["bus"], axis=0)
    .sum(0)
)
[12]:
Generator
coal             8249.750198
gas              6400.839108
load-shedding      55.604395
dtype: float64

Now, take the capacities from the above long-term equilibrium, then disallow expansion.

Show that the resulting market prices are identical.

This holds in this example, but does NOT necessarily hold and breaks down in some circumstances (for example, when there is a lot of storage and inter-temporal shifting).

[13]:
n.generators.p_nom_extendable = False
n.generators.p_nom = n.generators.p_nom_opt
[14]:
n.lopf();
WARNING:pypsa.components:Solving optimisation problem with pyomo.In PyPSA version 0.21 the default will change to ``n.lopf(pyomo=False)``.Explicitly set ``n.lopf(pyomo=True)`` to retain current behaviour.
INFO:pypsa.opf:Performed preliminary steps
INFO:pypsa.opf:Building pyomo model using `kirchhoff` formulation
INFO:pypsa.opf:Solving model using glpk
INFO:pypsa.opf:Optimization successful
# ==========================================================
# = Solver Results                                         =
# ==========================================================
# ----------------------------------------------------------
#   Problem Information
# ----------------------------------------------------------
Problem:
- Name: unknown
  Lower bound: 2306.19380619383
  Upper bound: 2306.19380619383
  Number of objectives: 1
  Number of constraints: 1002
  Number of variables: 3004
  Number of nonzeros: 3004
  Sense: minimize
# ----------------------------------------------------------
#   Solver Information
# ----------------------------------------------------------
Solver:
- Status: ok
  Termination condition: optimal
  Statistics:
    Branch and bound:
      Number of bounded subproblems: 0
      Number of created subproblems: 0
  Error rc: 0
  Time: 0.06878948211669922
# ----------------------------------------------------------
#   Solution Information
# ----------------------------------------------------------
Solution:
- number of solutions: 0
  number of solutions displayed: 0
[15]:
n.buses_t.marginal_price.plot(title="Price Duration Curve", figsize=(9, 5))
plt.tight_layout()
../_images/examples_generation-investment-screening-curve_22_0.png
[16]:
n.buses_t.marginal_price.sum(axis=1).value_counts()
[16]:
2.0       500
12.0      490
1012.0     10
0.0         1
dtype: int64

Demonstrate zero-profit condition. Differences are due to singular times, see above, not a problem

  1. Total costs

[17]:
(
    n.generators.p_nom * n.generators.capital_cost
    + n.generators_t.p.multiply(n.snapshot_weightings.generators, axis=0).sum()
    * n.generators.marginal_cost
)
[17]:
Generator
coal             8249.750250
gas              6400.839161
load-shedding      55.604396
dtype: float64
  1. Total revenue

[18]:
(
    n.generators_t.p.multiply(n.snapshot_weightings.generators, axis=0)
    .multiply(n.buses_t.marginal_price["bus"], axis=0)
    .sum()
)
[18]:
Generator
coal             8242.257742
gas              6395.944056
load-shedding      55.604396
dtype: float64