Note

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

Optimization without pyomo

In this example we demonstrate the behaviour of the Linear Optimal Power Flow (LOPF) calculation without using pyomo. This requires to set pyomo to False in the lopf function. Then, the communication with the solvers happens via in-house functions which leads to a much faster solving process.

import pypsa
import pandas as pd
import os

n = pypsa.examples.ac_dc_meshed(from_master=True)

WARNING:pypsa.io:Importing network from PyPSA version v0.17.1 while current version is v0.22.1. Read the release notes at https://pypsa.readthedocs.io/en/latest/release_notes.html to prepare your network for import.
INFO:pypsa.io:Imported network ac-dc-meshed.nc has buses, carriers, generators, global_constraints, lines, links, loads

Modify the network a bit: We set gas generators to non-extendable

n.generators.loc[n.generators.carrier == "gas", "p_nom_extendable"] = False

Add ramp limit

n.generators.loc[n.generators.carrier == "gas", "ramp_limit_down"] = 0.2
n.generators.loc[n.generators.carrier == "gas", "ramp_limit_up"] = 0.2

Add additional storage units (cyclic and non-cyclic) and fix one state_of_charge

n.add(
    "StorageUnit",
    "su",
    bus="Manchester",
    marginal_cost=10,
    inflow=50,
    p_nom_extendable=True,
    capital_cost=10,
    p_nom=2000,
    efficiency_dispatch=0.5,
    cyclic_state_of_charge=True,
    state_of_charge_initial=1000,
)

n.add(
    "StorageUnit",
    "su2",
    bus="Manchester",
    marginal_cost=10,
    p_nom_extendable=True,
    capital_cost=50,
    p_nom=2000,
    efficiency_dispatch=0.5,
    carrier="gas",
    cyclic_state_of_charge=False,
    state_of_charge_initial=1000,
)

n.storage_units_t.state_of_charge_set.loc[n.snapshots[7], "su"] = 100

Add an additional store

n.add("Bus", "storebus", carrier="hydro", x=-5, y=55)
n.madd(
    "Link",
    ["battery_power", "battery_discharge"],
    "",
    bus0=["Manchester", "storebus"],
    bus1=["storebus", "Manchester"],
    p_nom=100,
    efficiency=0.9,
    p_nom_extendable=True,
    p_nom_max=1000,
)
n.madd(
    "Store",
    ["store"],
    bus="storebus",
    e_nom=2000,
    e_nom_extendable=True,
    marginal_cost=10,
    capital_cost=10,
    e_nom_max=5000,
    e_initial=100,
    e_cyclic=True,
);

Extra functionalities:

from pypsa.linopt import get_var, linexpr, join_exprs, define_constraints

One of the most important functions is linexpr which take one or more tuples of coefficient and variable pairs which should go into the left hand side (lhs) of the constraint.

1. Add mimimum for state_of_charge

def minimal_state_of_charge(n, snapshots):
    vars_soc = get_var(n, "StorageUnit", "state_of_charge")
    lhs = linexpr((1, vars_soc))
    define_constraints(n, lhs, ">", 50, "StorageUnit", "soc_lower_bound")

2. Fix the ratio between ingoing and outgoing capacity of the Store

def fix_link_cap_ratio(n, snapshots):
    vars_link = get_var(n, "Link", "p_nom")
    eff = n.links.at["battery_power", "efficiency"]
    lhs = linexpr(
        (1, vars_link["battery_power"]), (-eff, vars_link["battery_discharge"])
    )
    define_constraints(n, lhs, "=", 0, "battery_discharge", attr="fixratio")

3. Every bus must in total produce the 20% of the total demand

This requires the function pypsa.linopt.join_exprs which sums up arrays of linear expressions

def fix_bus_production(n, snapshots):
    total_demand = n.loads_t.p_set.sum().sum()
    prod_per_bus = (
        linexpr((1, get_var(n, "Generator", "p")))
        .groupby(n.generators.bus, axis=1)
        .apply(join_exprs)
    )
    define_constraints(
        n, prod_per_bus, ">=", total_demand / 5, "Bus", "production_share"
    )

Combine them …

def extra_functionalities(n, snapshots):
    minimal_state_of_charge(n, snapshots)
    fix_link_cap_ratio(n, snapshots)
    fix_bus_production(n, snapshots)

…and run the lopf with pyomo=False

n.lopf(
    pyomo=False,
    extra_functionality=extra_functionalities,
    keep_shadowprices=["Bus", "battery_discharge", "StorageUnit"],
)

INFO:pypsa.linopf:Prepare linear problem
INFO:pypsa.linopf:Total preparation time: 0.3s
INFO:pypsa.linopf:Solve linear problem using Glpk solver
INFO:pypsa.linopf:Optimization successful. Objective value: 1.43e+07

('ok', 'optimal')

The keep_shadowprices argument in the lopf now decides which shadow prices (SP) should be retrieved. It can either be set to True, then all SP are kept. It also can be a list of names of the constraints. Therefore the name argument in define_constraints is necessary, in our case ‘battery_discharge’, ‘StorageUnit’ and ‘Bus’.

Analysing the constraints

Let’s see if the system got our own constraints. We look at n.constraints which combines summarises constraints going into the linear problem

n.constraints

		pnl	specification
component	name
Generator	mu_lower	True	non_ext, p
	mu_upper	True	non_ext, p
Line	mu_upper	True	s
	mu_lower	True	s
Link	mu_upper	True	p
	mu_lower	True	p
Store	mu_upper	True	e
	mu_lower	True	e
StorageUnit	mu_upper	True	p_dispatch, p_store, state_of_charge
	mu_lower	True	p_dispatch, p_store, state_of_charge
	mu_state_of_charge_set	True
Generator	mu_ramp_limit_up	True	nonext.
	mu_ramp_limit_down	True	nonext.
StorageUnit	mu_state_of_charge	True
Store	mu_state_of_charge	True
SubNetwork	mu_kirchhoff_voltage_law	True
Bus	marginal_price	True
GlobalConstraint	mu	False	co2_limit
StorageUnit	soc_lower_bound	True
battery_discharge	fixratio	False
Bus	production_share	False

The last three entries show our constraints. As ‘soc_lower_bound’ is time-dependent, the pnl value is set to True.

Let’s check whether out two custom constraint are fulfilled:

n.links.loc[["battery_power", "battery_discharge"], ["p_nom_opt"]]

	p_nom_opt
Link
battery_power	900.0
battery_discharge	1000.0

n.storage_units_t.state_of_charge

StorageUnit	su	su2
snapshot
2015-01-01 00:00:00	1835.74	1000.000
2015-01-01 01:00:00	1326.16	1000.000
2015-01-01 02:00:00	1376.16	1000.000
2015-01-01 03:00:00	1426.16	1000.000
2015-01-01 04:00:00	1986.06	1000.000
2015-01-01 05:00:00	50.00	50.000
2015-01-01 06:00:00	50.00	50.000
2015-01-01 07:00:00	100.00	156.727
2015-01-01 08:00:00	50.00	140.120
2015-01-01 09:00:00	50.00	50.000

n.generators_t.p.groupby(n.generators.bus, axis=1).sum().sum() / n.loads_t.p.sum().sum()

bus
Frankfurt     0.200000
Manchester    0.200000
Norway        0.637047
dtype: float64

Looks good! Now, let’s see which dual values were parsed. Therefore we have a look into n.dualvalues

n.dualvalues

		in_comp	pnl
component	name
StorageUnit	mu_upper	True	True
	mu_lower	True	True
	mu_state_of_charge_set	True	True
	mu_state_of_charge	True	True
Bus	marginal_price	True	True
StorageUnit	soc_lower_bound	True	True
battery_discharge	fixratio	False	False
Bus	production_share	True	False

Again we see the last two entries reflect our constraints (the values in the columns play only a minor role). Having a look what the values are:

from pypsa.linopt import get_dual

get_dual(n, "StorageUnit", "soc_lower_bound")

StorageUnit	su	su2
snapshot
2015-01-01 00:00:00	-0.00000	-0.00000
2015-01-01 01:00:00	-0.00000	-0.00000
2015-01-01 02:00:00	-0.00000	-0.00000
2015-01-01 03:00:00	-0.00000	-0.00000
2015-01-01 04:00:00	-0.00000	-0.00000
2015-01-01 05:00:00	-0.00000	-0.00000
2015-01-01 06:00:00	-157.42300	-4.44444
2015-01-01 07:00:00	-0.00000	-0.00000
2015-01-01 08:00:00	-0.00000	-0.00000
2015-01-01 09:00:00	-5.55556	-67.84860

get_dual(n, "battery_discharge", "fixratio")

0    567.89
Name: fixratio, dtype: float64

get_dual(n, "Bus", "production_share")

Bus
London            NaN
Norwich           NaN
Norwich DC        NaN
Manchester   -45.4185
Bremen            NaN
Bremen DC         NaN
Frankfurt    -35.2007
Norway        -0.0000
Norway DC         NaN
storebus          NaN
Name: production_share, dtype: float64

Side note

Some of the predefined constraints are stored in components itself like n.lines_t.mu_upper or n.buses_t.marginal_price, this is the case if their are designated columns are spots for those. All other dual are under the hook stored in n.duals