Two formulations of DCOPF#

As suggested in [1], the PTDF formulation is mainly used for ISO/RTO market clearing. Thus, DCOPF2 is developed using PTDF formulation for market clearing analysis.

In this demo, we will compare it with the DCOPF using B-theta formulation.

References:

1. Chen et al., “Security-Constrained Unit Commitment for Electricity Market: Modeling, Solution Methods, and Future Challenges,” in IEEE Transactions on Power Systems, vol. 38, no. 5, pp. 4668-4681, Sept. 2023, doi: 10.1109/TPWRS.2022.3213001.

[1]:

import pandas as pd

import ams

[2]:

ams.config_logger(stream_level=20)

A small-size PJM 5-bus case with ESD1 is used in this example.

[3]:

sp = ams.load(ams.get_case('5bus/pjm5bus_demo.xlsx'),
              setup=True,
              no_output=True)

Parsing input file "/Users/jinningwang/work/ams/ams/cases/5bus/pjm5bus_demo.xlsx"...
Input file parsed in 0.0720 seconds.
Zero Line parameters detected, adjusted to default values: rate_b, rate_c.
All bus type are PQ, adjusted given load and generator connection status.
System set up in 0.0026 seconds.

Whether the dual values are available depends on the solver/method. If you find the dual values are not fully computed, please try other solvers/methods.

Next, we can solve the two routines.

For DCOPF2, it is a good practice to build the PTDF matrix first.

You can also load a calculated PTDF matrix from a file, but AMS do not check the consistency between the PTDF matrix and the network data.

[4]:

sp.mats.build_ptdf()

Building system matrices

[4]:

<7x5 sparse matrix of type '<class 'numpy.float64'>'
        with 28 stored elements in List of Lists format>

[5]:

sp.DCOPF2.run(solver='CLARABEL')

Parsing OModel for <DCOPF2>
Evaluating OModel for <DCOPF2>
Finalizing OModel for <DCOPF2>
<DCOPF2> initialized in 0.0392 seconds.
<DCOPF2> solved as optimal in 0.0192 seconds, converged in 7 iterations with CLARABEL.

[5]:

True

The system energy price is calculated after solving the optimization, using pie, an ExpressionCalc object defined in the routine.

[36]:

sp.DCOPF2.pie.v

[36]:

0.15705242184924276

The congestion price is also calculated, using pic, another ExpressionCalc object defined in the routine.

[37]:

sp.DCOPF2.pic.v

[37]:

array([-0.07942907, -0.14705242,  0.14294758,  0.        , -0.06534746])

Then, we can also run the DCOPF.

[6]:

sp.DCOPF.run(solver='CLARABEL')

Parsing OModel for <DCOPF>
Evaluating OModel for <DCOPF>
Finalizing OModel for <DCOPF>
<DCOPF> initialized in 0.0123 seconds.
<DCOPF> solved as optimal in 0.0206 seconds, converged in 8 iterations with CLARABEL.

[6]:

True

We can compare the Bus results and Line results as follows.

[57]:

resBus = pd.DataFrame()
resBus['idx'] = sp.Bus.idx.v
resBus['pi-2 [$/MW]'] = sp.DCOPF2.get(src='pi', idx=resBus['idx']) / sp.config.mva
resBus['pi [$/MW]'] = sp.DCOPF.get(src='pi', idx=resBus['idx']) / sp.config.mva
resBus['aBus-2 [rad]'] = sp.DCOPF2.get(src='aBus', idx=resBus['idx'])
resBus['aBus [rad]'] = sp.DCOPF.get(src='aBus', idx=resBus['idx'])
resBus.round(8)

[57]:

	idx	pi-2 [$/MW]	pi [$/MW]	aBus-2 [rad]	aBus [rad]
0	0	0.000776	0.000776	0.023989	0.023989
1	1	0.000100	0.000100	0.034668	0.034668
2	2	0.003000	0.003000	0.013068	0.013068
3	3	0.001571	0.001571	0.000000	-0.000000
4	4	0.000917	0.000917	0.022896	0.022896

[61]:

resLine = pd.DataFrame()
resLine['idx'] = sp.Line.idx.v
resLine['ul'] = sp.DCOPF2.get(src='ul', idx=resLine['idx'])
resLine['rate_a [MW]'] = sp.DCOPF2.get(src='rate_a', idx=resLine['idx']) * sp.config.mva
resLine['plf-2 [MW]'] = sp.DCOPF2.get(src='plf', idx=resLine['idx']) * sp.config.mva
resLine['plf [MW]'] = sp.DCOPF.get(src='plf', idx=resLine['idx']) * sp.config.mva
resLine['mu1-2 [$/MW]'] = sp.DCOPF2.get(src='mu1', idx=resLine['idx']) / sp.config.mva
resLine['mu1 [$/MW]'] = sp.DCOPF.get(src='mu1', idx=resLine['idx']) / sp.config.mva
resLine['mu2-2 [$/MW]'] = sp.DCOPF2.get(src='mu2', idx=resLine['idx']) / sp.config.mva
resLine['mu2 [$/MW]'] = sp.DCOPF.get(src='mu2', idx=resLine['idx']) / sp.config.mva
resLine.round(8)

[61]:

	idx	ul	rate_a [MW]	plf-2 [MW]	plf [MW]	mu2-2 [$/MW]	mu2 [$/MW]
0	Line_1	1.0	400.0	-38.000806	-38.000806	0.00000	0.00000
1	Line_2	1.0	200.0	78.912154	78.912153	0.00000	0.00000
2	Line_3	1.0	200.0	17.089459	17.089459	0.00000	0.00000
3	Line_4	1.0	200.0	200.000000	200.000000	0.00342	0.00342
4	Line_5	1.0	200.0	43.998388	43.998388	0.00000	0.00000
5	Line_6	1.0	240.0	-77.089459	-77.089459	0.00000	0.00000
6	Line_7	1.0	400.0	-38.000806	-38.000806	0.00000	0.00000