Note

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

Optimization with Linopy

In PyPSA v0.22, an additional optimization module was introduced to the package. It is built on Linopy and aims at

• performance as we know it from the native PyPSA optimization (lopf with pyomo=False, now deprecated)

• flexibility as we know from the Pyomo implementation

• usability as we know from pandas/xarray

Linopy is a stand-alone package and works similar to Pyomo, but without the memory overhead and much faster. If you come from an older PyPSA version, you might be familiar with the pyomo-based n.lopf function. The Linopy-based optimization is the successor, is now the default optimization method in PyPSA, and is accessed via the pypsa.Network.optimize accessor. See the migration guide for more information.

For additional information on the Linopy package, have a look at the documentation.

Let’s get started

Now, we demonstrate the behaviour of the optimization with linopy. The core functions for the optimization can be called via the pypsa.Network.optimize accessor. The accessor is used for creating, solving, modifying the optimization problem. Further, it supports to run different optimization formulations and provides helper functions.

At first, we run the ordinary linearized optimal power flow (LOPF). We then extend the formulation by some additional constraints.

import pypsa

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

/tmp/ipykernel_2966/285364342.py:1: DeprecationWarning:

The 'update' and 'from_master' parameters are deprecated and do not have any effect. Example networks are always updated and retrieved for the current version.Deprecated in version 0.35 and will be removed in version 1.0.

INFO:pypsa.io:Retrieving network data from https://github.com/PyPSA/PyPSA/raw/master/examples/networks/ac-dc-meshed/ac-dc-meshed.nc.
INFO:pypsa.io:Imported network ac-dc-meshed.nc has buses, carriers, generators, global_constraints, lines, links, loads

In order to make the network a bit more interesting, we modify its data: We set gas generators to non-extendable,

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

… add ramp limits,

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

…and add an additional store.

n.add("Bus", "storebus", carrier="hydro", x=-5, y=55)
n.add(
    "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.add(
    "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,
);

Run Optimization

The optimization based on linopy mimics the well-known `n.lopf <https://pypsa.readthedocs.io/en/v0.28.0/api_reference.html#pypsa.Network.lopf>`__ optimization. We run it by calling the optimize accessor.

n.optimize()

WARNING:pypsa.consistency:The following stores have carriers which are not defined:
Index(['store'], dtype='object', name='Store')
WARNING:pypsa.consistency:The following links have carriers which are not defined:
Index(['battery_discharge'], dtype='object', name='Link')
WARNING:pypsa.consistency:The following buses have carriers which are not defined:
Index(['storebus'], dtype='object', name='Bus')
WARNING:pypsa.consistency:The following lines have zero x, which could break the linear load flow:
Index(['2', '3', '4'], dtype='object', name='Line')
WARNING:pypsa.consistency:The following lines have zero r, which could break the linear load flow:
Index(['0', '1', '5', '6'], dtype='object', name='Line')
INFO:linopy.model: Solve problem using Highs solver
INFO:linopy.io: Writing time: 0.09s
INFO:linopy.constants: Optimization successful:
Status: ok
Termination condition: optimal
Solution: 300 primals, 748 duals
Objective: 1.41e+07
Solver model: available
Solver message: Optimal

INFO:pypsa.optimization.optimize:The shadow-prices of the constraints Generator-fix-p-lower, Generator-fix-p-upper, Generator-ext-p-lower, Generator-ext-p-upper, Generator-fix-p-ramp_limit_up, Generator-fix-p-ramp_limit_down, Line-ext-s-lower, Line-ext-s-upper, Link-ext-p-lower, Link-ext-p-upper, Store-ext-e-lower, Store-ext-e-upper, StorageUnit-ext-p_dispatch-lower, StorageUnit-ext-p_dispatch-upper, StorageUnit-ext-p_store-lower, StorageUnit-ext-p_store-upper, StorageUnit-ext-state_of_charge-lower, StorageUnit-ext-state_of_charge-upper, StorageUnit-state_of_charge_set, Kirchhoff-Voltage-Law, StorageUnit-energy_balance, Store-energy_balance were not assigned to the network.

Running HiGHS 1.10.0 (git hash: fd86653): Copyright (c) 2025 HiGHS under MIT licence terms
LP   linopy-problem-0udrj21g has 748 rows; 300 cols; 1571 nonzeros
Coefficient ranges:
  Matrix [1e-02, 2e+00]
  Cost   [9e-03, 3e+03]
  Bound  [5e+01, 8e+05]
  RHS    [9e-01, 8e+04]
Presolving model
524 rows, 296 cols, 1324 nonzeros  0s
428 rows, 200 cols, 1410 nonzeros  0s
Dependent equations search running on 53 equations with time limit of 1000.00s
Dependent equations search removed 0 rows and 0 nonzeros in 0.00s (limit = 1000.00s)
400 rows, 199 cols, 1366 nonzeros  0s
Presolve : Reductions: rows 400(-348); columns 199(-101); elements 1366(-205)
Solving the presolved LP
Using EKK dual simplex solver - serial
  Iteration        Objective     Infeasibilities num(sum)
          0    -1.3357027996e+05 Pr: 114(89908.3); Du: 0(2.99875e-11) 0s
        178     1.4121237693e+07 Pr: 0(0); Du: 0(7.10543e-14) 0s
Solving the original LP from the solution after postsolve
Model name          : linopy-problem-0udrj21g
Model status        : Optimal
Simplex   iterations: 178
Objective value     :  1.4121237693e+07
Relative P-D gap    :  3.9571145032e-16
HiGHS run time      :          0.00
Writing the solution to /tmp/linopy-solve-9fvdyh9w.sol

('ok', 'optimal')

We now have a model instance attached to our network. It is a container of all variables, constraints and the objective function. You can modify this as much as you please, by directly adding or deleting variables or constraints etc.

n.model

Linopy LP model
===============

Variables:
----------
 * Generator-p_nom (Generator-ext)
 * Line-s_nom (Line-ext)
 * Link-p_nom (Link-ext)
 * Store-e_nom (Store-ext)
 * StorageUnit-p_nom (StorageUnit-ext)
 * Generator-p (snapshot, Generator)
 * Line-s (snapshot, Line)
 * Link-p (snapshot, Link)
 * Store-e (snapshot, Store)
 * StorageUnit-p_dispatch (snapshot, StorageUnit)
 * StorageUnit-p_store (snapshot, StorageUnit)
 * StorageUnit-state_of_charge (snapshot, StorageUnit)
 * StorageUnit-spill (snapshot, StorageUnit)
 * Store-p (snapshot, Store)
 * objective_constant

Constraints:
------------
 * Generator-ext-p_nom-lower (Generator-ext)
 * Generator-ext-p_nom-upper (Generator-ext)
 * Line-ext-s_nom-lower (Line-ext)
 * Line-ext-s_nom-upper (Line-ext)
 * Link-ext-p_nom-lower (Link-ext)
 * Link-ext-p_nom-upper (Link-ext)
 * Store-ext-e_nom-lower (Store-ext)
 * Store-ext-e_nom-upper (Store-ext)
 * StorageUnit-ext-p_nom-lower (StorageUnit-ext)
 * StorageUnit-ext-p_nom-upper (StorageUnit-ext)
 * Generator-fix-p-lower (snapshot, Generator-fix)
 * Generator-fix-p-upper (snapshot, Generator-fix)
 * Generator-ext-p-lower (snapshot, Generator-ext)
 * Generator-ext-p-upper (snapshot, Generator-ext)
 * Generator-fix-p-ramp_limit_up (snapshot, Generator-fix)
 * Generator-fix-p-ramp_limit_down (snapshot, Generator-fix)
 * Line-ext-s-lower (snapshot, Line-ext)
 * Line-ext-s-upper (snapshot, Line-ext)
 * Link-ext-p-lower (snapshot, Link-ext)
 * Link-ext-p-upper (snapshot, Link-ext)
 * Store-ext-e-lower (snapshot, Store-ext)
 * Store-ext-e-upper (snapshot, Store-ext)
 * StorageUnit-ext-p_dispatch-lower (snapshot, StorageUnit-ext)
 * StorageUnit-ext-p_dispatch-upper (snapshot, StorageUnit-ext)
 * StorageUnit-ext-p_store-lower (snapshot, StorageUnit-ext)
 * StorageUnit-ext-p_store-upper (snapshot, StorageUnit-ext)
 * StorageUnit-ext-state_of_charge-lower (snapshot, StorageUnit-ext)
 * StorageUnit-ext-state_of_charge-upper (snapshot, StorageUnit-ext)
 * StorageUnit-state_of_charge_set (snapshot, StorageUnit-state_of_charge_set_i)
 * Bus-nodal_balance (Bus, snapshot)
 * Kirchhoff-Voltage-Law (snapshot, cycles)
 * StorageUnit-energy_balance (snapshot, StorageUnit)
 * Store-energy_balance (snapshot, Store)
 * GlobalConstraint-co2_limit (snapshot)

Status:
-------
ok

Modify model, optimize and feed back to network

When you have a fresh network and you just want to create the model instance, run

n.optimize.create_model()

WARNING:pypsa.consistency:The following stores have carriers which are not defined:
Index(['store'], dtype='object', name='Store')
WARNING:pypsa.consistency:The following links have carriers which are not defined:
Index(['battery_discharge'], dtype='object', name='Link')
WARNING:pypsa.consistency:The following buses have carriers which are not defined:
Index(['storebus'], dtype='object', name='Bus')
WARNING:pypsa.consistency:The following lines have zero x, which could break the linear load flow:
Index(['2', '3', '4'], dtype='object', name='Line')
WARNING:pypsa.consistency:The following lines have zero r, which could break the linear load flow:
Index(['0', '1', '5', '6'], dtype='object', name='Line')

Linopy LP model
===============

Variables:
----------
 * Generator-p_nom (Generator-ext)
 * Line-s_nom (Line-ext)
 * Link-p_nom (Link-ext)
 * Store-e_nom (Store-ext)
 * StorageUnit-p_nom (StorageUnit-ext)
 * Generator-p (snapshot, Generator)
 * Line-s (snapshot, Line)
 * Link-p (snapshot, Link)
 * Store-e (snapshot, Store)
 * StorageUnit-p_dispatch (snapshot, StorageUnit)
 * StorageUnit-p_store (snapshot, StorageUnit)
 * StorageUnit-state_of_charge (snapshot, StorageUnit)
 * StorageUnit-spill (snapshot, StorageUnit)
 * Store-p (snapshot, Store)
 * objective_constant

Constraints:
------------
 * Generator-ext-p_nom-lower (Generator-ext)
 * Generator-ext-p_nom-upper (Generator-ext)
 * Line-ext-s_nom-lower (Line-ext)
 * Line-ext-s_nom-upper (Line-ext)
 * Link-ext-p_nom-lower (Link-ext)
 * Link-ext-p_nom-upper (Link-ext)
 * Store-ext-e_nom-lower (Store-ext)
 * Store-ext-e_nom-upper (Store-ext)
 * StorageUnit-ext-p_nom-lower (StorageUnit-ext)
 * StorageUnit-ext-p_nom-upper (StorageUnit-ext)
 * Generator-fix-p-lower (snapshot, Generator-fix)
 * Generator-fix-p-upper (snapshot, Generator-fix)
 * Generator-ext-p-lower (snapshot, Generator-ext)
 * Generator-ext-p-upper (snapshot, Generator-ext)
 * Generator-fix-p-ramp_limit_up (snapshot, Generator-fix)
 * Generator-fix-p-ramp_limit_down (snapshot, Generator-fix)
 * Line-ext-s-lower (snapshot, Line-ext)
 * Line-ext-s-upper (snapshot, Line-ext)
 * Link-ext-p-lower (snapshot, Link-ext)
 * Link-ext-p-upper (snapshot, Link-ext)
 * Store-ext-e-lower (snapshot, Store-ext)
 * Store-ext-e-upper (snapshot, Store-ext)
 * StorageUnit-ext-p_dispatch-lower (snapshot, StorageUnit-ext)
 * StorageUnit-ext-p_dispatch-upper (snapshot, StorageUnit-ext)
 * StorageUnit-ext-p_store-lower (snapshot, StorageUnit-ext)
 * StorageUnit-ext-p_store-upper (snapshot, StorageUnit-ext)
 * StorageUnit-ext-state_of_charge-lower (snapshot, StorageUnit-ext)
 * StorageUnit-ext-state_of_charge-upper (snapshot, StorageUnit-ext)
 * StorageUnit-state_of_charge_set (snapshot, StorageUnit-state_of_charge_set_i)
 * Bus-nodal_balance (Bus, snapshot)
 * Kirchhoff-Voltage-Law (snapshot, cycles)
 * StorageUnit-energy_balance (snapshot, StorageUnit)
 * Store-energy_balance (snapshot, Store)
 * GlobalConstraint-co2_limit (snapshot)

Status:
-------
initialized

Through the model instance we gain a lot of flexibility. Let’s say for example we want to remove the Kirchhoff Voltage Law constraint, thus convert the model to a transport model. This can be done via

n.model.constraints.remove("Kirchhoff-Voltage-Law")

Now, we want to optimize the altered model and feed to solution back to the network. Here again, we use the optimize accessor.

n.optimize.solve_model()

INFO:linopy.model: Solve problem using Highs solver
INFO:linopy.io: Writing time: 0.09s
INFO:linopy.constants: Optimization successful:
Status: ok
Termination condition: optimal
Solution: 300 primals, 728 duals
Objective: 1.41e+07
Solver model: available
Solver message: Optimal

INFO:pypsa.optimization.optimize:The shadow-prices of the constraints Generator-fix-p-lower, Generator-fix-p-upper, Generator-ext-p-lower, Generator-ext-p-upper, Generator-fix-p-ramp_limit_up, Generator-fix-p-ramp_limit_down, Line-ext-s-lower, Line-ext-s-upper, Link-ext-p-lower, Link-ext-p-upper, Store-ext-e-lower, Store-ext-e-upper, StorageUnit-ext-p_dispatch-lower, StorageUnit-ext-p_dispatch-upper, StorageUnit-ext-p_store-lower, StorageUnit-ext-p_store-upper, StorageUnit-ext-state_of_charge-lower, StorageUnit-ext-state_of_charge-upper, StorageUnit-state_of_charge_set, StorageUnit-energy_balance, Store-energy_balance were not assigned to the network.

Running HiGHS 1.10.0 (git hash: fd86653): Copyright (c) 2025 HiGHS under MIT licence terms
LP   linopy-problem-vv3zz9dh has 728 rows; 300 cols; 1511 nonzeros
Coefficient ranges:
  Matrix [1e-02, 2e+00]
  Cost   [9e-03, 3e+03]
  Bound  [5e+01, 8e+05]
  RHS    [9e-01, 8e+04]
Presolving model
504 rows, 296 cols, 1264 nonzeros  0s
426 rows, 218 cols, 1302 nonzeros  0s
Dependent equations search running on 52 equations with time limit of 1000.00s
Dependent equations search removed 0 rows and 0 nonzeros in 0.00s (limit = 1000.00s)
399 rows, 218 cols, 1248 nonzeros  0s
Presolve : Reductions: rows 399(-329); columns 218(-82); elements 1248(-263)
Solving the presolved LP
Using EKK dual simplex solver - serial
  Iteration        Objective     Infeasibilities num(sum)
          0    -2.4998861248e+00 Ph1: 190(204422); Du: 2(2.49989) 0s
        236     1.4121074568e+07 Pr: 0(0) 0s
Solving the original LP from the solution after postsolve
Model name          : linopy-problem-vv3zz9dh
Model status        : Optimal
Simplex   iterations: 236
Objective value     :  1.4121074568e+07
Relative P-D gap    :  7.9143204302e-16
HiGHS run time      :          0.00
Writing the solution to /tmp/linopy-solve-hmfn8y44.sol

('ok', 'optimal')

Here, we followed the recommended way to run altered models:

1. Create the model instance - n.optimize.create_model()

2. Modify the model to your needs

3. Solve and feed back - n.optimize.solve_model()

For compatibility reasons the optimize function, also allows passing a extra_funcionality argument, as we know it from the lopf function. The above behaviour with use of the extra functionality is obtained through

def remove_kvl(n, sns):
    print("KVL removed!")
    n.model.constraints.remove("Kirchhoff-Voltage-Law")


n.optimize(extra_functionality=remove_kvl)

WARNING:pypsa.consistency:The following stores have carriers which are not defined:
Index(['store'], dtype='object', name='Store')
WARNING:pypsa.consistency:The following links have carriers which are not defined:
Index(['battery_discharge'], dtype='object', name='Link')
WARNING:pypsa.consistency:The following buses have carriers which are not defined:
Index(['storebus'], dtype='object', name='Bus')
WARNING:pypsa.consistency:The following lines have zero x, which could break the linear load flow:
Index(['2', '3', '4'], dtype='object', name='Line')
WARNING:pypsa.consistency:The following lines have zero r, which could break the linear load flow:
Index(['0', '1', '5', '6'], dtype='object', name='Line')
INFO:linopy.model: Solve problem using Highs solver

KVL removed!

INFO:linopy.io: Writing time: 0.09s
INFO:linopy.constants: Optimization successful:
Status: ok
Termination condition: optimal
Solution: 300 primals, 728 duals
Objective: 1.41e+07
Solver model: available
Solver message: Optimal

INFO:pypsa.optimization.optimize:The shadow-prices of the constraints Generator-fix-p-lower, Generator-fix-p-upper, Generator-ext-p-lower, Generator-ext-p-upper, Generator-fix-p-ramp_limit_up, Generator-fix-p-ramp_limit_down, Line-ext-s-lower, Line-ext-s-upper, Link-ext-p-lower, Link-ext-p-upper, Store-ext-e-lower, Store-ext-e-upper, StorageUnit-ext-p_dispatch-lower, StorageUnit-ext-p_dispatch-upper, StorageUnit-ext-p_store-lower, StorageUnit-ext-p_store-upper, StorageUnit-ext-state_of_charge-lower, StorageUnit-ext-state_of_charge-upper, StorageUnit-state_of_charge_set, StorageUnit-energy_balance, Store-energy_balance were not assigned to the network.

Running HiGHS 1.10.0 (git hash: fd86653): Copyright (c) 2025 HiGHS under MIT licence terms
LP   linopy-problem-0jne_wdu has 728 rows; 300 cols; 1511 nonzeros
Coefficient ranges:
  Matrix [1e-02, 2e+00]
  Cost   [9e-03, 3e+03]
  Bound  [5e+01, 8e+05]
  RHS    [9e-01, 8e+04]
Presolving model
504 rows, 296 cols, 1264 nonzeros  0s
426 rows, 218 cols, 1302 nonzeros  0s
Dependent equations search running on 52 equations with time limit of 1000.00s
Dependent equations search removed 0 rows and 0 nonzeros in 0.00s (limit = 1000.00s)
399 rows, 218 cols, 1248 nonzeros  0s
Presolve : Reductions: rows 399(-329); columns 218(-82); elements 1248(-263)
Solving the presolved LP
Using EKK dual simplex solver - serial
  Iteration        Objective     Infeasibilities num(sum)
          0    -2.4998861248e+00 Ph1: 190(204422); Du: 2(2.49989) 0s
        236     1.4121074568e+07 Pr: 0(0) 0s
Solving the original LP from the solution after postsolve
Model name          : linopy-problem-0jne_wdu
Model status        : Optimal
Simplex   iterations: 236
Objective value     :  1.4121074568e+07
Relative P-D gap    :  7.9143204302e-16
HiGHS run time      :          0.00
Writing the solution to /tmp/linopy-solve-ofmi12g9.sol

('ok', 'optimal')

Additional constraints

In the following, we exemplarily present a set of additional constraints. Note, the dual values of the additional constraints won’t be stored in default data fields in the PyPSA network. But in any case, they are stored in the linopy.Model.

Again, we first build the optimization model, add our constraints and finally solve the network. For the first step, we use again our accessor optimize to access the function create_model. This returns the linopy model that we can modify.

m = n.optimize.create_model()  # the return value is the model, let's use it directly!

WARNING:pypsa.consistency:The following stores have carriers which are not defined:
Index(['store'], dtype='object', name='Store')
WARNING:pypsa.consistency:The following links have carriers which are not defined:
Index(['battery_discharge'], dtype='object', name='Link')
WARNING:pypsa.consistency:The following buses have carriers which are not defined:
Index(['storebus'], dtype='object', name='Bus')
WARNING:pypsa.consistency:The following lines have zero x, which could break the linear load flow:
Index(['2', '3', '4'], dtype='object', name='Line')
WARNING:pypsa.consistency:The following lines have zero r, which could break the linear load flow:
Index(['0', '1', '5', '6'], dtype='object', name='Line')

1. Minimum for state of charge

Assume we want to set a minimum state of charge of 50 MWh in our storage unit. This is done by:

sus = m.variables["StorageUnit-state_of_charge"]
m.add_constraints(sus >= 50, name="StorageUnit-minimum_soc")

Constraint `StorageUnit-minimum_soc` [snapshot: 10, StorageUnit: 2]:
--------------------------------------------------------------------
[2015-01-01 00:00:00, su]: +1 StorageUnit-state_of_charge[2015-01-01 00:00:00, su]   ≥ 50.0
[2015-01-01 00:00:00, su2]: +1 StorageUnit-state_of_charge[2015-01-01 00:00:00, su2] ≥ 50.0
[2015-01-01 01:00:00, su]: +1 StorageUnit-state_of_charge[2015-01-01 01:00:00, su]   ≥ 50.0
[2015-01-01 01:00:00, su2]: +1 StorageUnit-state_of_charge[2015-01-01 01:00:00, su2] ≥ 50.0
[2015-01-01 02:00:00, su]: +1 StorageUnit-state_of_charge[2015-01-01 02:00:00, su]   ≥ 50.0
[2015-01-01 02:00:00, su2]: +1 StorageUnit-state_of_charge[2015-01-01 02:00:00, su2] ≥ 50.0
[2015-01-01 03:00:00, su]: +1 StorageUnit-state_of_charge[2015-01-01 03:00:00, su]   ≥ 50.0
                ...
[2015-01-01 06:00:00, su2]: +1 StorageUnit-state_of_charge[2015-01-01 06:00:00, su2] ≥ 50.0
[2015-01-01 07:00:00, su]: +1 StorageUnit-state_of_charge[2015-01-01 07:00:00, su]   ≥ 50.0
[2015-01-01 07:00:00, su2]: +1 StorageUnit-state_of_charge[2015-01-01 07:00:00, su2] ≥ 50.0
[2015-01-01 08:00:00, su]: +1 StorageUnit-state_of_charge[2015-01-01 08:00:00, su]   ≥ 50.0
[2015-01-01 08:00:00, su2]: +1 StorageUnit-state_of_charge[2015-01-01 08:00:00, su2] ≥ 50.0
[2015-01-01 09:00:00, su]: +1 StorageUnit-state_of_charge[2015-01-01 09:00:00, su]   ≥ 50.0
[2015-01-01 09:00:00, su2]: +1 StorageUnit-state_of_charge[2015-01-01 09:00:00, su2] ≥ 50.0

The return value of the add_constraints function is a array with the labels of the constraints. You can access the constraint now through:

m.constraints["StorageUnit-minimum_soc"]

Constraint `StorageUnit-minimum_soc` [snapshot: 10, StorageUnit: 2]:
--------------------------------------------------------------------
[2015-01-01 00:00:00, su]: +1 StorageUnit-state_of_charge[2015-01-01 00:00:00, su]   ≥ 50.0
[2015-01-01 00:00:00, su2]: +1 StorageUnit-state_of_charge[2015-01-01 00:00:00, su2] ≥ 50.0
[2015-01-01 01:00:00, su]: +1 StorageUnit-state_of_charge[2015-01-01 01:00:00, su]   ≥ 50.0
[2015-01-01 01:00:00, su2]: +1 StorageUnit-state_of_charge[2015-01-01 01:00:00, su2] ≥ 50.0
[2015-01-01 02:00:00, su]: +1 StorageUnit-state_of_charge[2015-01-01 02:00:00, su]   ≥ 50.0
[2015-01-01 02:00:00, su2]: +1 StorageUnit-state_of_charge[2015-01-01 02:00:00, su2] ≥ 50.0
[2015-01-01 03:00:00, su]: +1 StorageUnit-state_of_charge[2015-01-01 03:00:00, su]   ≥ 50.0
                ...
[2015-01-01 06:00:00, su2]: +1 StorageUnit-state_of_charge[2015-01-01 06:00:00, su2] ≥ 50.0
[2015-01-01 07:00:00, su]: +1 StorageUnit-state_of_charge[2015-01-01 07:00:00, su]   ≥ 50.0
[2015-01-01 07:00:00, su2]: +1 StorageUnit-state_of_charge[2015-01-01 07:00:00, su2] ≥ 50.0
[2015-01-01 08:00:00, su]: +1 StorageUnit-state_of_charge[2015-01-01 08:00:00, su]   ≥ 50.0
[2015-01-01 08:00:00, su2]: +1 StorageUnit-state_of_charge[2015-01-01 08:00:00, su2] ≥ 50.0
[2015-01-01 09:00:00, su]: +1 StorageUnit-state_of_charge[2015-01-01 09:00:00, su]   ≥ 50.0
[2015-01-01 09:00:00, su2]: +1 StorageUnit-state_of_charge[2015-01-01 09:00:00, su2] ≥ 50.0

and inspects its attributes like lhs, sign and rhs, e.g.

m.constraints["StorageUnit-minimum_soc"].rhs

<xarray.DataArray 'rhs' (snapshot: 10, StorageUnit: 2)> Size: 160B
array([[50., 50.],
       [50., 50.],
       [50., 50.],
       [50., 50.],
       [50., 50.],
       [50., 50.],
       [50., 50.],
       [50., 50.],
       [50., 50.],
       [50., 50.]])
Coordinates:
  * snapshot     (snapshot) datetime64[ns] 80B 2015-01-01 ... 2015-01-01T09:0...
  * StorageUnit  (StorageUnit) object 16B 'su' 'su2'

xarray.DataArray

'rhs'

• snapshot: 10
• StorageUnit: 2

• 50.0 50.0 50.0 50.0 50.0 50.0 50.0 ... 50.0 50.0 50.0 50.0 50.0 50.0

  array([[50., 50.],
         [50., 50.],
         [50., 50.],
         [50., 50.],
         [50., 50.],
         [50., 50.],
         [50., 50.],
         [50., 50.],
         [50., 50.],
         [50., 50.]])

• Coordinates: (2)
  • snapshot
    (snapshot)
    datetime64[ns]
    2015-01-01 ... 2015-01-01T09:00:00

    array(['2015-01-01T00:00:00.000000000', '2015-01-01T01:00:00.000000000',
           '2015-01-01T02:00:00.000000000', '2015-01-01T03:00:00.000000000',
           '2015-01-01T04:00:00.000000000', '2015-01-01T05:00:00.000000000',
           '2015-01-01T06:00:00.000000000', '2015-01-01T07:00:00.000000000',
           '2015-01-01T08:00:00.000000000', '2015-01-01T09:00:00.000000000'],
          dtype='datetime64[ns]')

  • StorageUnit
    (StorageUnit)
    object
    'su' 'su2'

    array(['su', 'su2'], dtype=object)

• Indexes: (2)
  • snapshot
    PandasIndex

    PandasIndex(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))

  • StorageUnit
    PandasIndex

    PandasIndex(Index(['su', 'su2'], dtype='object', name='StorageUnit'))

• Attributes: (0)

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

The battery in our system is modelled with two links and a store. We should make sure that its charging and discharging capacities, meaning their links, are somehow coupled.

capacity = m.variables["Link-p_nom"]
eff = n.links.at["battery_power", "efficiency"]
lhs = capacity.loc["battery_power"] - eff * capacity.loc["battery_discharge"]
m.add_constraints(lhs == 0, name="Link-battery_fix_ratio")

Constraint `Link-battery_fix_ratio`
-----------------------------------
+1 Link-p_nom[battery_power] - 0.9 Link-p_nom[battery_discharge] = -0.0

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

For this, we use the linopy function groupby_sum which follows the pattern from pandas/xarray’s groupby function.

total_demand = n.loads_t.p_set.sum().sum()
buses = n.generators.bus.to_xarray()
prod_per_bus = m.variables["Generator-p"].groupby(buses).sum().sum("snapshot")
m.add_constraints(prod_per_bus >= total_demand / 5, name="Bus-minimum_production_share")

Constraint `Bus-minimum_production_share` [bus: 3]:
---------------------------------------------------
[Frankfurt]: +1 Generator-p[2015-01-01 00:00:00, Frankfurt Wind] + 1 Generator-p[2015-01-01 01:00:00, Frankfurt Wind] + 1 Generator-p[2015-01-01 02:00:00, Frankfurt Wind] ... +1 Generator-p[2015-01-01 07:00:00, Frankfurt Gas] + 1 Generator-p[2015-01-01 08:00:00, Frankfurt Gas] + 1 Generator-p[2015-01-01 09:00:00, Frankfurt Gas]        ≥ 6509.525616820881
[Manchester]: +1 Generator-p[2015-01-01 00:00:00, Manchester Wind] + 1 Generator-p[2015-01-01 01:00:00, Manchester Wind] + 1 Generator-p[2015-01-01 02:00:00, Manchester Wind] ... +1 Generator-p[2015-01-01 07:00:00, Manchester Gas] + 1 Generator-p[2015-01-01 08:00:00, Manchester Gas] + 1 Generator-p[2015-01-01 09:00:00, Manchester Gas] ≥ 6509.525616820881
[Norway]: +1 Generator-p[2015-01-01 00:00:00, Norway Wind] + 1 Generator-p[2015-01-01 01:00:00, Norway Wind] + 1 Generator-p[2015-01-01 02:00:00, Norway Wind] ... +1 Generator-p[2015-01-01 07:00:00, Norway Gas] + 1 Generator-p[2015-01-01 08:00:00, Norway Gas] + 1 Generator-p[2015-01-01 09:00:00, Norway Gas]                             ≥ 6509.525616820881

con = prod_per_bus >= total_demand / 5

con

Constraint (unassigned) [bus: 3]:
---------------------------------
[Frankfurt]: +1 Generator-p[2015-01-01 00:00:00, Frankfurt Wind] + 1 Generator-p[2015-01-01 01:00:00, Frankfurt Wind] + 1 Generator-p[2015-01-01 02:00:00, Frankfurt Wind] ... +1 Generator-p[2015-01-01 07:00:00, Frankfurt Gas] + 1 Generator-p[2015-01-01 08:00:00, Frankfurt Gas] + 1 Generator-p[2015-01-01 09:00:00, Frankfurt Gas]        ≥ 6509.525616820881
[Manchester]: +1 Generator-p[2015-01-01 00:00:00, Manchester Wind] + 1 Generator-p[2015-01-01 01:00:00, Manchester Wind] + 1 Generator-p[2015-01-01 02:00:00, Manchester Wind] ... +1 Generator-p[2015-01-01 07:00:00, Manchester Gas] + 1 Generator-p[2015-01-01 08:00:00, Manchester Gas] + 1 Generator-p[2015-01-01 09:00:00, Manchester Gas] ≥ 6509.525616820881
[Norway]: +1 Generator-p[2015-01-01 00:00:00, Norway Wind] + 1 Generator-p[2015-01-01 01:00:00, Norway Wind] + 1 Generator-p[2015-01-01 02:00:00, Norway Wind] ... +1 Generator-p[2015-01-01 07:00:00, Norway Gas] + 1 Generator-p[2015-01-01 08:00:00, Norway Gas] + 1 Generator-p[2015-01-01 09:00:00, Norway Gas]                             ≥ 6509.525616820881

… and now let’s solve the network again.

n.optimize.solve_model()

INFO:linopy.model: Solve problem using Highs solver
INFO:linopy.io: Writing time: 0.1s
INFO:linopy.constants: Optimization successful:
Status: ok
Termination condition: optimal
Solution: 300 primals, 772 duals
Objective: 1.43e+07
Solver model: available
Solver message: Optimal

INFO:pypsa.optimization.optimize:The shadow-prices of the constraints Generator-fix-p-lower, Generator-fix-p-upper, Generator-ext-p-lower, Generator-ext-p-upper, Generator-fix-p-ramp_limit_up, Generator-fix-p-ramp_limit_down, Line-ext-s-lower, Line-ext-s-upper, Link-ext-p-lower, Link-ext-p-upper, Store-ext-e-lower, Store-ext-e-upper, StorageUnit-ext-p_dispatch-lower, StorageUnit-ext-p_dispatch-upper, StorageUnit-ext-p_store-lower, StorageUnit-ext-p_store-upper, StorageUnit-ext-state_of_charge-lower, StorageUnit-ext-state_of_charge-upper, StorageUnit-state_of_charge_set, Kirchhoff-Voltage-Law, StorageUnit-energy_balance, Store-energy_balance, StorageUnit-minimum_soc were not assigned to the network.

Running HiGHS 1.10.0 (git hash: fd86653): Copyright (c) 2025 HiGHS under MIT licence terms
LP   linopy-problem-fpdnd_pg has 772 rows; 300 cols; 1653 nonzeros
Coefficient ranges:
  Matrix [1e-02, 2e+00]
  Cost   [9e-03, 3e+03]
  Bound  [5e+01, 8e+05]
  RHS    [9e-01, 8e+04]
Presolving model
527 rows, 296 cols, 1384 nonzeros  0s
431 rows, 200 cols, 1470 nonzeros  0s
Dependent equations search running on 53 equations with time limit of 1000.00s
Dependent equations search removed 0 rows and 0 nonzeros in 0.00s (limit = 1000.00s)
403 rows, 199 cols, 1426 nonzeros  0s
Presolve : Reductions: rows 403(-369); columns 199(-101); elements 1426(-227)
Solving the presolved LP
Using EKK dual simplex solver - serial
  Iteration        Objective     Infeasibilities num(sum)
          0    -1.3357133799e+05 Pr: 126(125956); Du: 0(3.07779e-11) 0s
        172     1.4337112555e+07 Pr: 0(0) 0s
Solving the original LP from the solution after postsolve
Model name          : linopy-problem-fpdnd_pg
Model status        : Optimal
Simplex   iterations: 172
Objective value     :  1.4337112555e+07
Relative P-D gap    :  5.1967092874e-16
HiGHS run time      :          0.00
Writing the solution to /tmp/linopy-solve-z_9kgkla.sol

('ok', 'optimal')

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.model.constraints

linopy.model.Constraints
------------------------
 * Generator-ext-p_nom-lower (Generator-ext)
 * Generator-ext-p_nom-upper (Generator-ext)
 * Line-ext-s_nom-lower (Line-ext)
 * Line-ext-s_nom-upper (Line-ext)
 * Link-ext-p_nom-lower (Link-ext)
 * Link-ext-p_nom-upper (Link-ext)
 * Store-ext-e_nom-lower (Store-ext)
 * Store-ext-e_nom-upper (Store-ext)
 * StorageUnit-ext-p_nom-lower (StorageUnit-ext)
 * StorageUnit-ext-p_nom-upper (StorageUnit-ext)
 * Generator-fix-p-lower (snapshot, Generator-fix)
 * Generator-fix-p-upper (snapshot, Generator-fix)
 * Generator-ext-p-lower (snapshot, Generator-ext)
 * Generator-ext-p-upper (snapshot, Generator-ext)
 * Generator-fix-p-ramp_limit_up (snapshot, Generator-fix)
 * Generator-fix-p-ramp_limit_down (snapshot, Generator-fix)
 * Line-ext-s-lower (snapshot, Line-ext)
 * Line-ext-s-upper (snapshot, Line-ext)
 * Link-ext-p-lower (snapshot, Link-ext)
 * Link-ext-p-upper (snapshot, Link-ext)
 * Store-ext-e-lower (snapshot, Store-ext)
 * Store-ext-e-upper (snapshot, Store-ext)
 * StorageUnit-ext-p_dispatch-lower (snapshot, StorageUnit-ext)
 * StorageUnit-ext-p_dispatch-upper (snapshot, StorageUnit-ext)
 * StorageUnit-ext-p_store-lower (snapshot, StorageUnit-ext)
 * StorageUnit-ext-p_store-upper (snapshot, StorageUnit-ext)
 * StorageUnit-ext-state_of_charge-lower (snapshot, StorageUnit-ext)
 * StorageUnit-ext-state_of_charge-upper (snapshot, StorageUnit-ext)
 * StorageUnit-state_of_charge_set (snapshot, StorageUnit-state_of_charge_set_i)
 * Bus-nodal_balance (Bus, snapshot)
 * Kirchhoff-Voltage-Law (snapshot, cycles)
 * StorageUnit-energy_balance (snapshot, StorageUnit)
 * Store-energy_balance (snapshot, Store)
 * GlobalConstraint-co2_limit (snapshot)
 * StorageUnit-minimum_soc (snapshot, StorageUnit)
 * Link-battery_fix_ratio (Link-ext)
 * Bus-minimum_production_share (bus)

The last three entries show our constraints. 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	1835.745612	1000.000000
2015-01-01 01:00:00	1836.063354	490.091300
2015-01-01 02:00:00	1886.063354	490.091300
2015-01-01 03:00:00	1936.063354	490.091300
2015-01-01 04:00:00	1986.063354	1000.000000
2015-01-01 05:00:00	50.000000	50.000000
2015-01-01 06:00:00	50.000000	50.000000
2015-01-01 07:00:00	100.000000	156.725142
2015-01-01 08:00:00	150.000000	50.000000
2015-01-01 09:00:00	50.000000	50.000000

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

/tmp/ipykernel_2966/2985613193.py:1: FutureWarning:

DataFrame.groupby with axis=1 is deprecated. Do `frame.T.groupby(...)` without axis instead.

bus
Frankfurt     0.200000
Manchester    0.200000
Norway        0.637048
dtype: float64

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

n.model.dual

/home/docs/checkouts/readthedocs.org/user_builds/pypsa/envs/latest/lib/python3.13/site-packages/linopy/common.py:437: UserWarning:

Coordinates across variables not equal. Perform outer join.

<xarray.Dataset> Size: 7kB
Dimensions:                                (Generator-ext: 3, Line-ext: 7,
                                            Link-ext: 6, Store-ext: 1,
                                            StorageUnit-ext: 2, snapshot: 10,
                                            Generator-fix: 3,
                                            StorageUnit-state_of_charge_set_i: 1,
                                            Bus: 10, cycles: 2, StorageUnit: 2,
                                            Store: 1, bus: 3)
Coordinates: (12/13)
  * Generator-ext                          (Generator-ext) object 24B 'Manche...
  * Line-ext                               (Line-ext) object 56B '0' '1' ... '6'
  * Link-ext                               (Link-ext) object 48B 'Norwich Con...
  * Store-ext                              (Store-ext) object 8B 'store'
  * StorageUnit-ext                        (StorageUnit-ext) object 16B 'su' ...
  * snapshot                               (snapshot) datetime64[ns] 80B 2015...
    ...                                     ...
  * StorageUnit-state_of_charge_set_i      (StorageUnit-state_of_charge_set_i) object 8B ...
  * Bus                                    (Bus) object 80B 'London' ... 'sto...
  * cycles                                 (cycles) int64 16B 0 1
  * StorageUnit                            (StorageUnit) object 16B 'su' 'su2'
  * Store                                  (Store) object 8B 'store'
  * bus                                    (bus) object 24B 'Frankfurt' ... '...
Data variables: (12/37)
    Generator-ext-p_nom-lower              (Generator-ext) float64 24B -0.0 ....
    Generator-ext-p_nom-upper              (Generator-ext) float64 24B nan .....
    Line-ext-s_nom-lower                   (Line-ext) float64 56B -0.0 ... -0.0
    Line-ext-s_nom-upper                   (Line-ext) float64 56B nan ... nan
    Link-ext-p_nom-lower                   (Link-ext) float64 48B -0.0 ... -0.0
    Link-ext-p_nom-upper                   (Link-ext) float64 48B nan ... -527.1
    ...                                     ...
    StorageUnit-energy_balance             (snapshot, StorageUnit) float64 160B ...
    Store-energy_balance                   (snapshot, Store) float64 80B 521....
    GlobalConstraint-co2_limit             float64 8B -950.5
    StorageUnit-minimum_soc                (snapshot, StorageUnit) float64 160B ...
    Link-battery_fix_ratio                 float64 8B -567.9
    Bus-minimum_production_share           (bus) float64 24B 35.2 45.42 -0.0

xarray.Dataset

• Dimensions:
  • Generator-ext: 3
  • Line-ext: 7
  • Link-ext: 6
  • Store-ext: 1
  • StorageUnit-ext: 2
  • snapshot: 10
  • Generator-fix: 3
  • StorageUnit-state_of_charge_set_i: 1
  • Bus: 10
  • cycles: 2
  • StorageUnit: 2
  • Store: 1
  • bus: 3
• Coordinates: (13)
  • Generator-ext
    (Generator-ext)
    object
    'Manchester Wind' ... 'Frankfurt...

    array(['Manchester Wind', 'Norway Wind', 'Frankfurt Wind'], dtype=object)

  • Line-ext
    (Line-ext)
    object
    '0' '1' '2' '3' '4' '5' '6'

    array(['0', '1', '2', '3', '4', '5', '6'], dtype=object)

  • Link-ext
    (Link-ext)
    object
    'Norwich Converter' ... 'battery...

    array(['Norwich Converter', 'Norway Converter', 'Bremen Converter', 'DC link',
           'battery_power', 'battery_discharge'], dtype=object)

  • Store-ext
    (Store-ext)
    object
    'store'

    array(['store'], dtype=object)

  • StorageUnit-ext
    (StorageUnit-ext)
    object
    'su' 'su2'

    array(['su', 'su2'], dtype=object)

  • snapshot
    (snapshot)
    datetime64[ns]
    2015-01-01 ... 2015-01-01T09:00:00

    array(['2015-01-01T00:00:00.000000000', '2015-01-01T01:00:00.000000000',
           '2015-01-01T02:00:00.000000000', '2015-01-01T03:00:00.000000000',
           '2015-01-01T04:00:00.000000000', '2015-01-01T05:00:00.000000000',
           '2015-01-01T06:00:00.000000000', '2015-01-01T07:00:00.000000000',
           '2015-01-01T08:00:00.000000000', '2015-01-01T09:00:00.000000000'],
          dtype='datetime64[ns]')

  • Generator-fix
    (Generator-fix)
    object
    'Manchester Gas' ... 'Frankfurt ...

    array(['Manchester Gas', 'Norway Gas', 'Frankfurt Gas'], dtype=object)

  • StorageUnit-state_of_charge_set_i
    (StorageUnit-state_of_charge_set_i)
    object
    'su'

    array(['su'], dtype=object)

  • Bus
    (Bus)
    object
    'London' 'Norwich' ... 'storebus'

    array(['London', 'Norwich', 'Norwich DC', 'Manchester', 'Bremen', 'Bremen DC',
           'Frankfurt', 'Norway', 'Norway DC', 'storebus'], dtype=object)

  • cycles
    (cycles)
    int64
    0 1

    array([0, 1])

  • StorageUnit
    (StorageUnit)
    object
    'su' 'su2'

    array(['su', 'su2'], dtype=object)

  • Store
    (Store)
    object
    'store'

    array(['store'], dtype=object)

  • bus
    (bus)
    object
    'Frankfurt' 'Manchester' 'Norway'

    array(['Frankfurt', 'Manchester', 'Norway'], dtype=object)

• Data variables: (37)
  • Generator-ext-p_nom-lower
    (Generator-ext)
    float64
    -0.0 -0.0 -0.0

    array([-0., -0., -0.])

  • Generator-ext-p_nom-upper
    (Generator-ext)
    float64
    nan nan nan

    array([nan, nan, nan])

  • Line-ext-s_nom-lower
    (Line-ext)
    float64
    -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0

    array([-0., -0., -0., -0., -0., -0., -0.])

  • Line-ext-s_nom-upper
    (Line-ext)
    float64
    nan nan nan nan nan nan nan

    array([nan, nan, nan, nan, nan, nan, nan])

  • Link-ext-p_nom-lower
    (Link-ext)
    float64
    -0.0 -0.0 -0.0 0.442 -0.0 -0.0

    array([-0.        , -0.        , -0.        ,  0.44201919, -0.        ,
           -0.        ])

  • Link-ext-p_nom-upper
    (Link-ext)
    float64
    nan nan nan nan -0.0 -527.1

    array([          nan,           nan,           nan,           nan,
             -0.        , -527.10106054])

  • Store-ext-e_nom-lower
    (Store-ext)
    float64
    -0.0

    array([-0.])

  • Store-ext-e_nom-upper
    (Store-ext)
    float64
    -0.0

    array([-0.])

  • StorageUnit-ext-p_nom-lower
    (StorageUnit-ext)
    float64
    -0.0 -0.0

    array([-0., -0.])

  • StorageUnit-ext-p_nom-upper
    (StorageUnit-ext)
    float64
    nan nan

    array([nan, nan])

  • Generator-fix-p-lower
    (snapshot, Generator-fix)
    float64
    320.4 355.3 327.4 ... 43.23 15.62

    array([[320.40087249, 355.27570163, 327.39947441],
           [ 20.        ,  54.34208543,  26.73388221],
           [132.25233154, 166.59441698, 138.98621375],
           [132.25233154, 166.59441698, 138.98621375],
           [320.40087249, 354.74295792, 327.33475469],
           [ -0.        ,  34.20770924,   6.59950601],
           [ -0.        ,  34.34208543,   6.73388221],
           [314.84531693, 349.22031664, 321.57342232],
           [  8.88888889,  43.23097432,  15.41165998],
           [  8.88888889,  43.23097432,  15.6227711 ]])

  • Generator-fix-p-upper
    (snapshot, Generator-fix)
    float64
    -0.0 -0.0 -0.0 ... -0.0 -0.0 -0.0

    array([[-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.]])

  • Generator-ext-p-lower
    (snapshot, Generator-ext)
    float64
    -0.0 -0.0 -0.0 ... -0.0 -0.0 -0.0

    array([[-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.]])

  • Generator-ext-p-upper
    (snapshot, Generator-ext)
    float64
    -335.7 -289.8 ... -601.8 -637.0

    array([[-335.70944742, -289.77812878, -325.23671755],
           [-636.1103199 , -590.71174497, -625.90230974],
           [-523.85798836, -478.45941343, -513.6499782 ],
           [-523.85798836, -478.45941343, -513.6499782 ],
           [-335.70944742, -290.31087249, -325.30143726],
           [-656.1103199 , -610.84612117, -646.03668594],
           [-656.1103199 , -610.71174497, -645.90230974],
           [-341.26500297, -295.83351377, -331.06276963],
           [-647.22143101, -601.82285608, -637.22453197],
           [-647.22143101, -601.82285608, -637.01342086]])

  • Generator-fix-p-ramp_limit_up
    (snapshot, Generator-fix)
    float64
    nan nan nan -0.0 ... -0.0 -0.0 -0.0

    array([[nan, nan, nan],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.]])

  • Generator-fix-p-ramp_limit_down
    (snapshot, Generator-fix)
    float64
    nan nan nan -0.0 ... -0.0 -0.0 -0.0

    array([[nan, nan, nan],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.],
           [-0., -0., -0.]])

  • Line-ext-s-lower
    (snapshot, Line-ext)
    float64
    -0.0 -0.0 -0.0 ... -0.0 -0.0 -0.0

    array([[-0.        , -0.        , -0.        ,  0.12912605, -0.        ,
            -0.        , -0.        ],
           [-0.        , -0.        , -0.        , -0.        , -0.        ,
            -0.        , -0.        ],
           [-0.        , -0.        , -0.        , -0.        , -0.        ,
            -0.        , -0.        ],
           [-0.        , -0.        , -0.        , -0.        , -0.        ,
            -0.        , -0.        ],
           [-0.        , -0.        , -0.        , -0.        , -0.        ,
            -0.        ,  0.2       ],
           [ 0.13671576, -0.        , -0.        , -0.        , -0.        ,
            -0.        , -0.        ],
           [-0.        , -0.        , -0.        , -0.        , -0.        ,
            -0.        , -0.        ],
           [-0.        , -0.        , -0.        , -0.        , -0.        ,
            -0.        , -0.        ],
           [-0.        , -0.        , -0.        , -0.        , -0.        ,
            -0.        , -0.        ],
           [-0.        , -0.        , -0.        , -0.        , -0.        ,
            -0.        , -0.        ]])

  • Line-ext-s-upper
    (snapshot, Line-ext)
    float64
    -0.0 -0.0 -0.008673 ... -0.0 -0.0

    array([[-0.        , -0.        , -0.00867342, -0.        , -0.        ,
            -0.        , -0.        ],
           [-0.        , -0.        , -0.        , -0.        , -0.        ,
            -0.        , -0.        ],
           [-0.        , -0.        , -0.        , -0.        , -0.        ,
            -0.        , -0.        ],
           [-0.        , -0.        , -0.        , -0.        , -0.        ,
            -0.        , -0.        ],
           [-0.        , -0.        , -0.        , -0.        , -0.        ,
            -0.        , -0.        ],
           [-0.        , -0.13349168, -0.        , -0.        , -0.        ,
            -0.        , -0.        ],
           [-0.        , -0.        , -0.        , -0.        , -0.        ,
            -0.        , -0.        ],
           [-0.        , -0.        , -0.        , -0.        , -0.06242987,
            -0.02185245, -0.        ],
           [-0.        , -0.        , -0.        , -0.        , -0.        ,
            -0.        , -0.        ],
           [-0.        , -0.        , -0.        , -0.        , -0.        ,
            -0.        , -0.        ]])

  • Link-ext-p-lower
    (snapshot, Link-ext)
    float64
    0.2333 -0.0 -0.0 ... 114.4 -0.0

    array([[ 2.33333333e-01, -0.00000000e+00, -0.00000000e+00,
             2.64719713e-01, -0.00000000e+00,  2.70360785e+02],
           [-0.00000000e+00, -0.00000000e+00, -0.00000000e+00,
            -0.00000000e+00,  1.12252332e+02, -0.00000000e+00],
           [-0.00000000e+00, -0.00000000e+00, -0.00000000e+00,
            -0.00000000e+00, -0.00000000e+00,  1.01027098e+02],
           [-0.00000000e+00, -0.00000000e+00, -0.00000000e+00,
            -0.00000000e+00, -0.00000000e+00,  1.01027098e+02],
           [-0.00000000e+00, -0.00000000e+00, -0.00000000e+00,
            -0.00000000e+00, -0.00000000e+00,  2.70360785e+02],
           [-0.00000000e+00, -0.00000000e+00, -0.00000000e+00,
             5.39440867e-04,  1.23252332e+02, -0.00000000e+00],
           [-0.00000000e+00, -0.00000000e+00, -0.00000000e+00,
            -0.00000000e+00,  1.23252332e+02, -0.00000000e+00],
           [-0.00000000e+00, -0.00000000e+00, -0.00000000e+00,
             6.42419146e-03, -0.00000000e+00,  2.75360785e+02],
           [-0.00000000e+00, -0.00000000e+00,  2.11111111e-01,
            -0.00000000e+00,  1.14363443e+02, -0.00000000e+00],
           [-0.00000000e+00, -0.00000000e+00, -0.00000000e+00,
            -0.00000000e+00,  1.14363443e+02, -0.00000000e+00]])

  • Link-ext-p-upper
    (snapshot, Link-ext)
    float64
    -0.0 -0.2222 -0.0 ... -0.0 -0.0

    array([[  -0.        ,   -0.22222222,   -0.        ,   -0.        ,
            -188.14854094,   -0.        ],
           [  -0.        ,   -0.        ,   -0.        ,   -0.        ,
              -0.        ,   -0.        ],
           [  -0.        ,   -0.        ,   -0.        ,   -0.        ,
              -0.        ,   -0.        ],
           [  -0.        ,   -0.        ,   -0.        ,   -0.        ,
              -0.        ,   -0.        ],
           [  -0.        ,   -0.        ,   -0.        ,   -0.        ,
            -188.14854094,   -0.        ],
           [  -0.        ,   -0.        ,   -0.        ,   -0.        ,
              -0.        ,   -8.        ],
           [  -0.        ,   -0.        ,   -0.        ,   -0.        ,
              -0.        ,   -8.        ],
           [  -0.        ,   -0.        ,   -0.        ,   -0.        ,
            -191.59298539,   -0.        ],
           [  -0.        ,   -0.        ,   -0.        ,   -0.21111111,
              -0.        ,   -0.        ],
           [  -0.        ,   -0.        ,   -0.        ,   -0.        ,
              -0.        ,   -0.        ]])

  • Store-ext-e-lower
    (snapshot, Store-ext)
    float64
    -0.0 -0.0 -0.0 ... -0.0 -0.0 10.0

    array([[-0.],
           [-0.],
           [-0.],
           [-0.],
           [-0.],
           [-0.],
           [-0.],
           [-0.],
           [-0.],
           [10.]])

  • Store-ext-e-upper
    (snapshot, Store-ext)
    float64
    -0.0 -0.0 -0.0 ... -0.0 -0.0 -0.0

    array([[ -0.],
           [ -0.],
           [ -0.],
           [ -0.],
           [-10.],
           [ -0.],
           [ -0.],
           [ -0.],
           [ -0.],
           [ -0.]])

  • StorageUnit-ext-p_dispatch-lower
    (snapshot, StorageUnit-ext)
    float64
    300.4 220.4 -0.0 ... -0.0 -0.0 -0.0

    array([[300.40087249, 220.40087249],
           [ -0.        ,  -0.        ],
           [112.25233154, 112.25233154],
           [112.25233154, 112.25233154],
           [300.40087249, 300.40087249],
           [ -0.        ,  -0.        ],
           [ -0.        ,  -0.        ],
           [ -0.        , 305.95642804],
           [ -0.        ,  -0.        ],
           [ -0.        ,  -0.        ]])

  • StorageUnit-ext-p_dispatch-upper
    (snapshot, StorageUnit-ext)
    float64
    -0.0 -0.0 -0.0 ... -0.0 -0.0 -0.0

    array([[-0., -0.],
           [-0., -0.],
           [-0., -0.],
           [-0., -0.],
           [-0., -0.],
           [-0., -0.],
           [-0., -0.],
           [-0., -0.],
           [-0., -0.],
           [-0., -0.]])

  • StorageUnit-ext-p_store-lower
    (snapshot, StorageUnit-ext)
    float64
    -0.0 40.0 300.4 ... 306.0 306.0

    array([[-0.00000000e+00,  4.00000000e+01],
           [ 3.00400872e+02,  3.00400872e+02],
           [ 1.88148541e+02,  1.88148541e+02],
           [ 1.88148541e+02,  1.88148541e+02],
           [-5.68434189e-14, -0.00000000e+00],
           [ 3.10400872e+02,  3.10400872e+02],
           [ 3.10400872e+02,  3.10400872e+02],
           [ 1.52978214e+02, -0.00000000e+00],
           [ 3.05956428e+02,  3.05956428e+02],
           [ 3.05956428e+02,  3.05956428e+02]])

  • StorageUnit-ext-p_store-upper
    (snapshot, StorageUnit-ext)
    float64
    -0.0 -0.0 -0.0 ... -0.0 -0.0 -0.0

    array([[-0., -0.],
           [-0., -0.],
           [-0., -0.],
           [-0., -0.],
           [-0., -0.],
           [-0., -0.],
           [-0., -0.],
           [-0., -0.],
           [-0., -0.],
           [-0., -0.]])

  • StorageUnit-ext-state_of_charge-lower
    (snapshot, StorageUnit-ext)
    float64
    -0.0 -0.0 -0.0 ... -0.0 -0.0 -0.0

    array([[-0., -0.],
           [-0., -0.],
           [-0., -0.],
           [-0., -0.],
           [-0., -0.],
           [-0., -0.],
           [-0., -0.],
           [-0., -0.],
           [-0., -0.],
           [-0., -0.]])

  • StorageUnit-ext-state_of_charge-upper
    (snapshot, StorageUnit-ext)
    float64
    -0.0 -40.0 -0.0 ... -0.0 -0.0 -0.0

    array([[ -0., -40.],
           [ -0.,  -0.],
           [ -0.,  -0.],
           [ -0.,  -0.],
           [-10., -10.],
           [ -0.,  -0.],
           [ -0.,  -0.],
           [ -0.,  -0.],
           [ -0.,  -0.],
           [ -0.,  -0.]])

  • StorageUnit-state_of_charge_set
    (snapshot, StorageUnit-state_of_charge_set_i)
    float64
    nan nan nan nan ... -153.0 nan nan

    array([[          nan],
           [          nan],
           [          nan],
           [          nan],
           [          nan],
           [          nan],
           [          nan],
           [-152.97821402],
           [          nan],
           [          nan]])

  • Bus-nodal_balance
    (Bus, snapshot)
    float64
    290.4 590.8 478.5 ... 541.7 541.7

    array([[290.40087249, 590.80174497, 478.54941343, 478.54941343,
            290.40087249, 610.93666061, 610.80174497, 295.96862905,
            601.91285608, 601.91285608],
           [290.40087249, 590.80174497, 478.54941343, 478.54941343,
            290.40087249, 610.93612117, 610.80174497, 295.95043287,
            601.91285608, 601.91285608],
           [290.16753915, 590.80174497, 478.54941343, 478.54941343,
            290.40087249, 610.93612117, 610.80174497, 295.95043287,
            601.91285608, 601.91285608],
           [290.40087249, 590.80174497, 478.54941343, 478.54941343,
            290.40087249, 610.80174497, 610.80174497, 295.95642804,
            601.91285608, 601.91285608],
           [290.13615277, 590.80174497, 478.54941343, 478.54941343,
            290.40087249, 610.93612117, 610.80174497, 295.96220486,
            602.1239672 , 601.91285608],
           [290.13615277, 590.80174497, 478.54941343, 478.54941343,
            290.40087249, 610.93612117, 610.80174497, 295.96220486,
            601.91285608, 601.91285608],
           [290.13615277, 590.80174497, 478.54941343, 478.54941343,
            290.20087249, 610.93612117, 610.80174497, 295.96220486,
            602.1239672 , 601.91285608],
           [289.86812878, 590.80174497, 478.54941343, 478.54941343,
            290.40087249, 610.93612117, 610.80174497, 295.92351377,
            601.91285608, 601.91285608],
           [290.090351  , 590.80174497, 478.54941343, 478.54941343,
            290.40087249, 610.93612117, 610.80174497, 295.92351377,
            601.91285608, 601.91285608],
           [531.72157048, 531.72157048, 531.72157048, 531.72157048,
            531.72157048, 541.72157048, 541.72157048, 541.72157048,
            541.72157048, 541.72157048]])

  • Kirchhoff-Voltage-Law
    (snapshot, cycles)
    float64
    5.684e-14 0.04273 ... -0.0 -0.0

    array([[ 5.68434189e-14,  4.27328442e-02],
           [-0.00000000e+00, -0.00000000e+00],
           [-0.00000000e+00, -0.00000000e+00],
           [-0.00000000e+00, -0.00000000e+00],
           [-0.00000000e+00, -0.00000000e+00],
           [ 3.26194088e-03, -5.68434189e-14],
           [-0.00000000e+00, -0.00000000e+00],
           [-2.21090865e-02, -2.21481764e-02],
           [-0.00000000e+00, -0.00000000e+00],
           [-0.00000000e+00, -0.00000000e+00]])

  • StorageUnit-energy_balance
    (snapshot, StorageUnit)
    float64
    290.4 250.4 290.4 ... 296.0 296.0

    array([[290.40087249, 250.40087249],
           [290.40087249, 290.40087249],
           [290.40087249, 290.40087249],
           [290.40087249, 290.40087249],
           [290.40087249, 290.40087249],
           [300.40087249, 300.40087249],
           [300.40087249, 300.40087249],
           [142.97821402, 295.95642804],
           [295.95642804, 295.95642804],
           [295.95642804, 295.95642804]])

  • Store-energy_balance
    (snapshot, Store)
    float64
    521.7 521.7 521.7 ... 531.7 531.7

    array([[521.72157048],
           [521.72157048],
           [521.72157048],
           [521.72157048],
           [521.72157048],
           [531.72157048],
           [531.72157048],
           [531.72157048],
           [531.72157048],
           [531.72157048]])

  • GlobalConstraint-co2_limit
    ()
    float64
    -950.5

    array(-950.45004064)

  • StorageUnit-minimum_soc
    (snapshot, StorageUnit)
    float64
    -0.0 -0.0 -0.0 ... -0.0 5.556 67.85

    array([[ -0.        ,  -0.        ],
           [ -0.        ,  -0.        ],
           [ -0.        ,  -0.        ],
           [ -0.        ,  -0.        ],
           [ -0.        ,  -0.        ],
           [ -0.        ,  -0.        ],
           [157.42265847,   4.44444444],
           [ -0.        ,  -0.        ],
           [ -0.        ,  -0.        ],
           [  5.55555556,  67.84841829]])

  • Link-battery_fix_ratio
    ()
    float64
    -567.9

    array(-567.89006727)

  • Bus-minimum_production_share
    (bus)
    float64
    35.2 45.42 -0.0

    array([35.20056477, 45.41857493, -0.        ])

• Indexes: (13)
  • Generator-ext
    PandasIndex

    PandasIndex(Index(['Manchester Wind', 'Norway Wind', 'Frankfurt Wind'], dtype='object', name='Generator-ext'))

  • Line-ext
    PandasIndex

    PandasIndex(Index(['0', '1', '2', '3', '4', '5', '6'], dtype='object', name='Line-ext'))

  • Link-ext
    PandasIndex

    PandasIndex(Index(['Norwich Converter', 'Norway Converter', 'Bremen Converter', 'DC link',
           'battery_power', 'battery_discharge'],
          dtype='object', name='Link-ext'))

  • Store-ext
    PandasIndex

    PandasIndex(Index(['store'], dtype='object', name='Store-ext'))

  • StorageUnit-ext
    PandasIndex

    PandasIndex(Index(['su', 'su2'], dtype='object', name='StorageUnit-ext'))

  • snapshot
    PandasIndex

    PandasIndex(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='h'))

  • Generator-fix
    PandasIndex

    PandasIndex(Index(['Manchester Gas', 'Norway Gas', 'Frankfurt Gas'], dtype='object', name='Generator-fix'))

  • StorageUnit-state_of_charge_set_i
    PandasIndex

    PandasIndex(Index(['su'], dtype='object', name='StorageUnit-state_of_charge_set_i'))

  • Bus
    PandasIndex

    PandasIndex(Index(['London', 'Norwich', 'Norwich DC', 'Manchester', 'Bremen', 'Bremen DC',
           'Frankfurt', 'Norway', 'Norway DC', 'storebus'],
          dtype='object', name='Bus'))

  • cycles
    PandasIndex

    PandasIndex(Index([0, 1], dtype='int64', name='cycles'))

  • StorageUnit
    PandasIndex

    PandasIndex(Index(['su', 'su2'], dtype='object', name='StorageUnit'))

  • Store
    PandasIndex

    PandasIndex(Index(['store'], dtype='object', name='Store'))

  • bus
    PandasIndex

    PandasIndex(Index(['Frankfurt', 'Manchester', 'Norway'], dtype='object', name='bus'))

• Attributes: (0)

n.model.dual["StorageUnit-minimum_soc"]

<xarray.DataArray 'StorageUnit-minimum_soc' (snapshot: 10, StorageUnit: 2)> Size: 160B
array([[ -0.        ,  -0.        ],
       [ -0.        ,  -0.        ],
       [ -0.        ,  -0.        ],
       [ -0.        ,  -0.        ],
       [ -0.        ,  -0.        ],
       [ -0.        ,  -0.        ],
       [157.42265847,   4.44444444],
       [ -0.        ,  -0.        ],
       [ -0.        ,  -0.        ],
       [  5.55555556,  67.84841829]])
Coordinates:
  * snapshot     (snapshot) datetime64[ns] 80B 2015-01-01 ... 2015-01-01T09:0...
  * StorageUnit  (StorageUnit) object 16B 'su' 'su2'

xarray.DataArray

'StorageUnit-minimum_soc'

• snapshot: 10
• StorageUnit: 2

• -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 -0.0 ... -0.0 -0.0 -0.0 -0.0 5.556 67.85

  array([[ -0.        ,  -0.        ],
         [ -0.        ,  -0.        ],
         [ -0.        ,  -0.        ],
         [ -0.        ,  -0.        ],
         [ -0.        ,  -0.        ],
         [ -0.        ,  -0.        ],
         [157.42265847,   4.44444444],
         [ -0.        ,  -0.        ],
         [ -0.        ,  -0.        ],
         [  5.55555556,  67.84841829]])

• Coordinates: (2)
  • snapshot
    (snapshot)
    datetime64[ns]
    2015-01-01 ... 2015-01-01T09:00:00

    array(['2015-01-01T00:00:00.000000000', '2015-01-01T01:00:00.000000000',
           '2015-01-01T02:00:00.000000000', '2015-01-01T03:00:00.000000000',
           '2015-01-01T04:00:00.000000000', '2015-01-01T05:00:00.000000000',
           '2015-01-01T06:00:00.000000000', '2015-01-01T07:00:00.000000000',
           '2015-01-01T08:00:00.000000000', '2015-01-01T09:00:00.000000000'],
          dtype='datetime64[ns]')

  • StorageUnit
    (StorageUnit)
    object
    'su' 'su2'

    array(['su', 'su2'], dtype=object)

• Indexes: (2)
  • snapshot
    PandasIndex

    PandasIndex(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='h'))

  • StorageUnit
    PandasIndex

    PandasIndex(Index(['su', 'su2'], dtype='object', name='StorageUnit'))

• Attributes: (0)

n.model.dual["Link-battery_fix_ratio"]

<xarray.DataArray 'Link-battery_fix_ratio' ()> Size: 8B
array(-567.89006727)

xarray.DataArray

'Link-battery_fix_ratio'

• -567.9

  array(-567.89006727)

• Coordinates: (0)
• Indexes: (0)
• Attributes: (0)

n.model.dual["Bus-minimum_production_share"]

<xarray.DataArray 'Bus-minimum_production_share' (bus: 3)> Size: 24B
array([35.20056477, 45.41857493, -0.        ])
Coordinates:
  * bus      (bus) object 24B 'Frankfurt' 'Manchester' 'Norway'

xarray.DataArray

'Bus-minimum_production_share'

• bus: 3

• 35.2 45.42 -0.0

  array([35.20056477, 45.41857493, -0.        ])

• Coordinates: (1)
  • bus
    (bus)
    object
    'Frankfurt' 'Manchester' 'Norway'

    array(['Frankfurt', 'Manchester', 'Norway'], dtype=object)

• Indexes: (1)
  • bus
    PandasIndex

    PandasIndex(Index(['Frankfurt', 'Manchester', 'Norway'], dtype='object', name='bus'))

• Attributes: (0)

These are the basic functionalities of the optimize accessor. There are many more functions like abstract optimziation formulations (security constraint optimization, iterative transmission expansion optimization, etc.) or helper functions (fixing optimized capacities, adding load shedding). Try them out if you want!

print("\n".join([func for func in n.optimize.__dir__() if not func.startswith("_")]))

n
expressions
create_model
solve_model
assign_solution
assign_duals
post_processing
optimize_transmission_expansion_iteratively
optimize_security_constrained
optimize_with_rolling_horizon
optimize_mga
optimize_and_run_non_linear_powerflow
fix_optimal_capacities
fix_optimal_dispatch
add_load_shedding