Quickstart 1 - Markets¶

Problem Description¶

Consider the following simple electricity market with two zones, each with a generator and a load. The demand in zone 1 is 500 MW and in zone 2 is 1500 MW. The generators have a nominal capacity of 2000 MW each, with cost functions defined as follows:

• Zone 1: C1(g1) = 10 g1 + 0.005 g1²
• Zone 2: C2(g2) = 13 g2 + 0.01 g2²

Find the least-cost dispatch of the generators to meet the load while respecting the transmission capacity. Identify the marginal prices at each bus and the flow on the transmission line. Calculate the congestion rent.

PyPSA Solution¶

For installation instructions, consult the Installation section first.

The first step is always to import the pypsa module:

In [2]:

import pypsa

import pypsa

A new PyPSA network instance can be created with the pypsa.Network constructor.

In [3]:

n = pypsa.Network()

n = pypsa.Network()

Components like buses can be added with n.add() and registered under an arbitrary name, e.g. "zone_1":

In [4]:

n.add("Bus", "zone_1")
n.add("Bus", "zone_2")
n.buses

n.add("Bus", "zone_1") n.add("Bus", "zone_2") n.buses

Out[4]:

	v_nom	type	x	y	carrier	unit	location	v_mag_pu_set	v_mag_pu_min	v_mag_pu_max	control	generator	sub_network
name
zone_1	1.0		0.0	0.0	AC			1.0	0.0	inf	PQ
zone_2	1.0		0.0	0.0	AC			1.0	0.0	inf	PQ

Next, add the loads to the network, where p_set specifies the power demand in MW at the corresponding bus:

In [5]:

n.add("Load", "load_1", bus="zone_1", p_set=500)
n.add("Load", "load_2", bus="zone_2", p_set=1500)
n.loads

n.add("Load", "load_1", bus="zone_1", p_set=500) n.add("Load", "load_2", bus="zone_2", p_set=1500) n.loads

Out[5]:

	bus	carrier	type	p_set	q_set	sign	active
name
load_1	zone_1			500.0	0.0	-1.0	True
load_2	zone_2			1500.0	0.0	-1.0	True

Generators are added in a similar way, where p_nom specifies the nominal capacity, and marginal_cost and marginal_cost_quadratic specify the linear and quadratic coefficients of the cost function:

In [6]:

n.add(
    "Generator",
    "gen_1",
    bus="zone_1",
    p_nom=2000,
    marginal_cost=10,
    marginal_cost_quadratic=0.005,
)
n.add(
    "Generator",
    "gen_2",
    bus="zone_2",
    p_nom=2000,
    marginal_cost=13,
    marginal_cost_quadratic=0.01,
)
n.generators

n.add( "Generator", "gen_1", bus="zone_1", p_nom=2000, marginal_cost=10, marginal_cost_quadratic=0.005, ) n.add( "Generator", "gen_2", bus="zone_2", p_nom=2000, marginal_cost=13, marginal_cost_quadratic=0.01, ) n.generators

Out[6]:

	bus	control	type	p_nom	p_nom_mod	p_nom_extendable	p_nom_min	p_nom_max	p_nom_set	p_min_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
name
gen_1	zone_1	PQ		2000.0	0.0	False	0.0	inf	NaN	0.0	...	0	0	1	0	NaN	NaN	1.0	1.0	1.0	0.0
gen_2	zone_2	PQ		2000.0	0.0	False	0.0	inf	NaN	0.0	...	0	0	1	0	NaN	NaN	1.0	1.0	1.0	0.0

2 rows × 38 columns

Lines connecting two buses bus0 and bus1 are added with a nominal capacity s_nom in MW and a reactance x in Ohm (which is required for modelling power flow):

In [7]:

n.add("Line", "line_1", bus0="zone_1", bus1="zone_2", x=0.01, s_nom=400)
n.lines

n.add("Line", "line_1", bus0="zone_1", bus1="zone_2", x=0.01, s_nom=400) n.lines

Out[7]:

	bus0	bus1	type	x	r	g	b	s_nom	s_nom_mod	s_nom_extendable	...	v_ang_min	v_ang_max	sub_network	x_pu	r_pu	g_pu	b_pu	x_pu_eff	r_pu_eff	s_nom_opt
name
line_1	zone_1	zone_2		0.01	0.0	0.0	0.0	400.0	0.0	False	...	-inf	inf		0.0	0.0	0.0	0.0	0.0	0.0	0.0

1 rows × 32 columns

With all components added, the network can be optimised with the n.optimize() method. This function constructs the optimisation problem with the linopy library, solves it with a selected solver ("highs" as default), and stores the results in the network instance n.

In [8]:

n.optimize(solver_name="highs", log_to_console=False)

n.optimize(solver_name="highs", log_to_console=False)

WARNING:pypsa.consistency:The following buses have carriers which are not defined. Run n.sanitize() to add them. Components with undefined carriers:
Index(['zone_1', 'zone_2'], dtype='object', name='name')

WARNING:pypsa.consistency:The following lines have carriers which are not defined. Run n.sanitize() to add them. Components with undefined carriers:
Index(['line_1'], dtype='object', name='name')

WARNING:pypsa.consistency:The following lines have zero r, which could break the linear load flow:
Index(['line_1'], dtype='object', name='name')

INFO:linopy.model: Solve problem using Highs solver

INFO:linopy.model:Solver options:
 - log_to_console: False

INFO:linopy.io: Writing time: 0.02s

INFO:linopy.constants: Optimization successful: 
Status: ok
Termination condition: optimal
Solution: 3 primals, 8 duals
Objective: 3.94e+04
Solver model: available
Solver message: Optimal

INFO:pypsa.optimization.optimize:The shadow-prices of the constraints Generator-fix-p-lower, Generator-fix-p-upper, Line-fix-s-lower, Line-fix-s-upper were not assigned to the network.

Out[8]:

('ok', 'optimal')

The optimised generators dispatch can be accessed with:

In [9]:

n.generators_t.p

n.generators_t.p

Out[9]:

name	gen_1	gen_2
snapshot
now	900.0	1100.0

The market clearing prices per bus can be accessed with:

In [10]:

n.buses_t.marginal_price

n.buses_t.marginal_price

Out[10]:

name	zone_1	zone_2
snapshot
now	19.00009	35.00011

The optimised flows on transmission lines can be accessed with:

In [11]:

n.lines_t.p1

n.lines_t.p1

Out[11]:

name	line_1
snapshot
now	-400.0

Here, n.lines_t.p0 denotes flow from bus0 to bus1 if values are positive. The related attribute n.lines_t.p1 denotes flow from bus1 to bus0 if values are positive.

The congestion rent can be calculated as the product of the flow on the line and the price difference between the two buses:

In [12]:

n.buses_t.marginal_price.eval("zone_2 - zone_1") * n.lines_t.p0["line_1"]

n.buses_t.marginal_price.eval("zone_2 - zone_1") * n.lines_t.p0["line_1"]

Out[12]:

snapshot
now    6400.008
dtype: float64

This example is based on Tom Brown's Energy Systems course, taken from the lecture on Complex Markets, slides 37ff.