Problem 5.3

Problem 5.3#

Integrated Energy Grids

Problem 5.3

Note: This is equivalent to Problem 2.2 which we solved using linopy, but here we will use PyPSA.

Consider the following economic dispatch problem:

• we have three generators: solar, wind and gas

• solar and wind have no marginal costs, and gas has fuel costs of 60 EUR/MWh.

• we need to cover a electricity demand of 13.2 MWh

• the installed capacities are 15 MW, 20 MW and 20 MW for wind, solar, and gas, respectively

• assume the capacity factor for solar is 0.17 and for wind 0.33.

Use PYPSA to find the optimal solution as well and obtain the electricity price (Lagrange multiplier for the energy balance constraint)

Note

If you have not yet set up Python on your computer, you can execute this tutorial in your browser via Google Colab. Click on the rocket in the top right corner and launch “Colab”. If that doesn’t work download the .ipynb file and import it in Google Colab.

Then install the following packages by executing the following command in a Jupyter cell at the top of the notebook.

!pip install numpy pypsa

import numpy as np
import pypsa

We start by creating the network object and adding the bus.

network = pypsa.Network()

network.add("Bus", "my bus") 
network.buses

	v_nom	type	x	y	carrier	unit	location	v_mag_pu_set	v_mag_pu_min	v_mag_pu_max	control	generator	sub_network
Bus
my bus	1.0		0.0	0.0	AC			1.0	0.0	inf	PQ

We add the generators

network.add("Generator", 
            "solar", 
            bus="my bus", 
            p_nom=15, 
            p_max_pu=0.17, #capacity factor
            marginal_cost=0) 
    
network.add("Generator", 
            "wind", 
            bus="my bus", 
            p_nom=20, 
            p_max_pu=0.33, #capacity factor
            marginal_cost=0) 

network.add("Generator", 
            "gas", 
            bus="my bus", 
            p_nom=20, 
            marginal_cost=60) #EUR/MWh_elec
network.generators

	bus	control	p_nom	p_nom_mod	p_nom_extendable	p_nom_min	p_nom_max	p_min_pu	p_max_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
Generator
solar	my bus	PQ	15.0	0.0	False	0.0	inf	0.0	0.17	...	0	0	1	0	NaN	NaN	1.0	1.0	1.0	0.0
wind	my bus	PQ	20.0	0.0	False	0.0	inf	0.0	0.33	...	0	0	1	0	NaN	NaN	1.0	1.0	1.0	0.0
gas	my bus	PQ	20.0	0.0	False	0.0	inf	0.0	1.00	...	0	0	1	0	NaN	NaN	1.0	1.0	1.0	0.0

3 rows × 37 columns

We add the load

network.add("Load", 
            "electricity demand", 
            bus="my bus", 
            p_set=13.2)
network.loads

	bus	carrier	type	p_set	q_set	sign	active
Load
electricity demand	my bus			13.2	0.0	-1.0	True

Calculate the optimal economic dispatch

network.optimize()

WARNING:pypsa.consistency:The following buses have carriers which are not defined:
Index(['my bus'], dtype='object', name='Bus')

INFO:linopy.model: Solve problem using Highs solver

INFO:linopy.io: Writing time: 0.01s

INFO:linopy.constants: Optimization successful: 
Status: ok
Termination condition: optimal
Solution: 3 primals, 7 duals
Objective: 2.43e+02
Solver model: available
Solver message: Optimal

INFO:pypsa.optimization.optimize:The shadow-prices of the constraints Generator-fix-p-lower, Generator-fix-p-upper 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-flthrja8 has 7 rows; 3 cols; 9 nonzeros
Coefficient ranges:
  Matrix [1e+00, 1e+00]
  Cost   [6e+01, 6e+01]
  Bound  [0e+00, 0e+00]
  RHS    [3e+00, 2e+01]
Presolving model
0 rows, 0 cols, 0 nonzeros  0s
0 rows, 0 cols, 0 nonzeros  0s
Presolve : Reductions: rows 0(-7); columns 0(-3); elements 0(-9) - Reduced to empty
Solving the original LP from the solution after postsolve
Model name          : linopy-problem-flthrja8
Model status        : Optimal
Objective value     :  2.4300000000e+02
Relative P-D gap    :  0.0000000000e+00
HiGHS run time      :          0.00
Writing the solution to /tmp/linopy-solve-985oe5uf.sol

('ok', 'optimal')

Now we can show the optimal dispatch and the electricity price (marginal cost at the bus)

network.generators_t.p

Generator	solar	wind	gas
snapshot
now	2.55	6.6	4.05

network.buses_t.marginal_price

Bus	my bus
snapshot
now	60.0