Problem 5.1

Problem 5.1#

Integrated Energy Grids

Problem 5.1

Assume we have three buses (Denmark, Netherlands and Germany) with nominal voltage \(V_{nom}\)= 2000 V connected by three transmission lines. In the bus, Denmark there is a wind generator that is producing 100 MW. In the bus Germany, there is a load that is consuming 100 MW of active power and 100 MVAR of reactive power. The transmission lines DK-NL and NL-GE have a unitary resistance \(r\)=0.01 and reactance \(x\)=0.1. The transmission lines DK-GE has a unitary resistance \(r\)=0.02 and reactance \(x\)=0.1. Using Python for Power System Analysis (PyPSA):

a) Calculate the power flows along the transmission lines using AC power flow representation.

b) Calculate the power flows along the transmission lines using a linearized approximation (also known as DC optimal power flow) and discuss the results.

Hint: It is recommended to follow the PyPSA tutorial before trying this problem.

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 three buses corresponding to Denmark, Netherlands and Germany. The added buses can be printed using network.buses

network = pypsa.Network()

network.add("Bus", "bus Denmark", v_nom=2000.0)
network.add("Bus", "bus Netherlands", v_nom=2000.0)
network.add("Bus", "bus Germany", v_nom=2000.0)

network.buses

	v_nom	x	y	carrier	v_mag_pu_set	v_mag_pu_min	v_mag_pu_max	control
Bus
bus Denmark	2000.0	0.0	0.0	AC	1.0	0.0	inf	PQ
bus Netherlands	2000.0	0.0	0.0	AC	1.0	0.0	inf	PQ
bus Germany	2000.0	0.0	0.0	AC	1.0	0.0	inf	PQ

We add the three lines connecting the buses

network.add("Line"," line DK-NL", bus0 = "bus Denmark", bus1= "bus Netherlands", x=0.1, r=0.01)
network.add("Line"," line DK-DE", bus0 = "bus Denmark", bus1= "bus Germany", x=0.1, r=0.02)
network.add("Line"," line NL-DE", bus0 = "bus Netherlands", bus1= "bus Germany", x=0.1, r=0.01)

network.lines

	bus0	bus1	x	r	g	b	s_nom	s_nom_mod	s_nom_extendable	...	v_ang_min	v_ang_max	x_pu	r_pu	g_pu	b_pu	x_pu_eff	r_pu_eff	s_nom_opt
Line
line DK-NL	bus Denmark	bus Netherlands	0.1	0.01	0.0	0.0	0.0	0.0	False	...	-inf	inf	0.0	0.0	0.0	0.0	0.0	0.0	0.0
line DK-DE	bus Denmark	bus Germany	0.1	0.02	0.0	0.0	0.0	0.0	False	...	-inf	inf	0.0	0.0	0.0	0.0	0.0	0.0	0.0
line NL-DE	bus Netherlands	bus Germany	0.1	0.01	0.0	0.0	0.0	0.0	False	...	-inf	inf	0.0	0.0	0.0	0.0	0.0	0.0	0.0

3 rows × 31 columns

Add a generator at the Denmark bus.

network.add("Generator", 
            "generator Denmark", 
            bus="bus Denmark",
            p_set=100)
network.generators

	bus	control	type	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
generator Denmark	bus Denmark	PQ		0.0	0.0	False	0.0	inf	0.0	1.0	...	0	0	1	0	NaN	NaN	1.0	1.0	1.0	0.0

1 rows × 37 columns

network.generators.p_set

Generator
generator Denmark    100.0
Name: p_set, dtype: float64

Add a load at the Germany bus

network.add("Load", 
            "load Germany", 
            bus="bus Germany", 
            p_set=100,
            q_set=100)
network.loads

	bus	carrier	type	p_set	q_set	sign	active
Load
load Germany	bus Germany			100.0	100.0	-1.0	True

We can now solve the non-linear power flow using a Newton-Raphson method.

network.pf()

INFO:pypsa.pf:Performing non-linear load-flow on AC sub-network <pypsa.networks.SubNetwork object at 0x7f463396fb90> for snapshots Index(['now'], dtype='object', name='snapshot')

{'n_iter': SubNetwork  0
 snapshot     
 now         2,
 'error': SubNetwork             0
 snapshot                
 now         8.204964e-09,
 'converged': SubNetwork     0
 snapshot        
 now         True}

ok, the solution converge, we can check now the active power flow on the lines.

network.lines_t.p0

	line DK-NL	line DK-DE	line NL-DE
snapshot
now	31.441059	68.558996	31.441054

We can also check the voltage angles on the buses. We need to apply a conversion factor if we want to show them in degrees.

network.buses_t.v_ang * 180 / np.pi

Bus	bus Denmark	bus Netherlands	bus Germany
snapshot
now	0.0	-0.00004	-0.00008

and their per-unit magnitudes

network.buses_t.v_mag_pu

Bus	bus Denmark	bus Netherlands	bus Germany
snapshot
now	1.0	0.999999	0.999998

b) Calculate the power flows along the transmission lines using a linearized approximation (DC optimal power flow).

We can use linear power flows and get a similar solution

network.lpf()

INFO:pypsa.pf:Performing linear load-flow on AC sub-network <pypsa.networks.SubNetwork object at 0x7f463396fb90> for snapshot(s) Index(['now'], dtype='object', name='snapshot')

network.lines_t.p0

	line DK-NL	line DK-DE	line NL-DE
snapshot
now	33.333333	66.666667	33.333333

In this simple case, the solution that we obtain are very similar.