Line parameters for linear power flow

Power flow calculations rely on many parameters, related to the existing infrastructure of lines (branches), nodes (buses) and transformers. Detailed information is scarce and especially in the composition of open energy models, it is required to abstract and interpolate the existing data to guess the electric properties as accurate as needed. While for applications in electrical engineering, almost all parameters are required and there is almost no way around detailed data collection, for linearized power flow calculations, like in pomato, only a subset of electric properties are relevant for the calculations. This allows to extrapolate the needed data with a higher degree of freedom, which does not mean the derivation process shouldn’t be rooted on a solid electrical engineering foundation but allows to work with substantial larger, automatically generated, datasets that are not available otherwise.

In the following sections we document the process for the data set used in POMATO for the purpose of transparency and to iron out mistakes or inaccuracies along the way.

Original Data

The data originates from the ENTSO-E Gridmap which was scraped using the GridKit utility [2] and is currently forked and maintained within the PyPSA Project.

The data comes with very limited information, as it is solely based on geographic information. The following data is included:

nodes	node_id, station_id, voltage, dc, terminal, symbol, position (let/lon)
lines	line_id, node_i, node_j, voltage, circuits, length, underground
dclines	dcline_id, node_i, node_i, length, underground
transformers	tranformer_id, node_i, node_i

Stations can consist of multiple nodes (buses), and will ultimately be omitted. Transformers are not actually part of the dataset, but extrapolated based on the substations and their nodes.

Some statistics regarding the data:

GridKit Data Summary

		Total	CWE	DE
nodes		8011	1612	604
lines	voltage	9856	1875	663
	132		10	9
	220		1076	211
	380		789	443
transformers		1061	213	96
dclines		51	19	13

Required Data

Pomato remains in the realm of linear programming, therefore the only parameter that is required is the imaginary part of the complex impedance \(Z = R + jX\) of a network element, the reactance $X$ which relates approximately to the susceptance \(B \approx \frac{1}{X}\) which is the only remaining parameter of the linear power equations.

This parameter is dependant on multiple factors, e.g. materials of the line, configuration of circuits, voltage level and many more. While it is possible to research these properties for a individual line, the effort is unreasonable for an entire system and given the purpose of a techno economic analysis. The next sections document the process of using default parameters to find reasonable assumptions for impedances of network elements.

Lines

For transmission lines [1] provides a detailed overview of how parameters manifest depending on material- and construction properties. The level of detail is however far too great to be useful in this application as we hardly know anything about the transmission lines.

Table 9.7 however provides parameters for voltage levels 110KV to 380kV in single and double line configurations for the commonly used overhead line tower (Donaumast). This table combines many assumptions that are difficult to make: type of ropes used (German Beseilung), mounting and bundle configuration. The following table contains the parameters. Note: for consistency always the single circuit values are used. The ropes are assumed to be Al/St-Ropes of type 243-A1/39-St1A with a nominal current \(I_{nom}\) of 645 as per Table 9.2 in [1].

Exemplary overhead line parameters, from Table 9.2 in [1].

Voltage	Configuration	Positive Sequence Impedance		\(I_{nom}\)
kV		r_per_km \(\Omega\)/km	x_per_km \(\Omega\)/km	A
110	Single Rope	0.12	0.387	645
220	2-Bundle	0.06	0.301	1290
380	4-Bundle	0.03	0.246	2580

Based on these values and a line length $l$ in km the apparent power for a transmission the thermal capacity, resistance and reactance are calculated as:

\[\begin{split}S &= \sqrt{3} \cdot U_d \cdot I_{nom} \\ x &= x\_per\_km \cdot l \\ r &= r\_per\_km \cdot l \\\end{split}\]

In linear power flow only active power is accounted for. While also representing a strong assumption the full apparent power is used as thermal capacity and therefore as limit to active power in the calculations.

The values for reactance and resistance in per unit (p.u.), in reference to \(S_B = 1\) MVA and the nominal line voltage \(U_d\) become:

\[\begin{split}z_{base} &= U_d^2 / S_B \\ x\_pu &= x / z_{base} \\ r\_pu &= r / z_{base}\end{split}\]

Cables

The dataset includes an underground identifier, which allows to categorize transmission lines as cables. Given that cables have different parameters, especially regarding their impedance it makes sense to include that information. However, the line parameters are highly dependant on operating temperature, which is impossible to know or too specific to assume. Generally cables would provide lower impedance than overhead but lines, but their operation differs significantly due to high capacity to earth and difficult heat dispersion through the isolation. At this point we prefer using the types for overhead lines. This needs more input.

Transformers

Transformers are used to connect nodes of different voltage levels which share the same substation. All transformers in the dataset connect two voltage levels. The relevant types are 110kV/220kV, 110kV/380kV and 220kV/380kV. Transformers are modeled as lines in pomato, and linear power flow in general, where the impedance can be calculated using short circuit voltages and the ohmic voltage drop (\(u_{kr}\), \(u_{Rr}`$ in [1]_, :math:`v_{sc}\), \(v_{scr}\) in the PyPSA documentation (at least to my understanding). These parameters are from from the pyPSA project [3] as standard types mostly equivalent to the ones used in pandapower [4] and SimBench [5] [6], the latter included higher rated transformers and a 220/380 type. The following table contains the relevant types, with reference.

Transformer parameters from various sources.

	\(S_n\)	\(U_H\)	\(U_L\)	\(u_{kr}\)	\(u_{Rr}\)	Source
	MVA	kV	kV	%	%
160 MVA 380/110 kV	160	110	380	12.2	0.25	pandapower/pyPSA
100 MVA 220/110 kV	100	110	220	12	0.26	pandapower/pyPSA
300MVA220/110	300	110	220	12	0.128	SimBench
350MVA380/110	350	110	380	22	0.257	SimBench
Typx380/220	600	220	380	18.5	0.25	SimBench

Given these parameters, we can calculate the transformer impedance following equations (8.3 - 8.5) from [1]:

\[\begin{split}z &= \dfrac{u_{kr} \cdot U_H^2}{100 \cdot S_n} \\ r &= \dfrac{u_{Rr} \cdot U_H^2}{100 \cdot S_n} \\ x &= \sqrt{z^2 - r^2}\end{split}\]

The respective p.u. values are obtained with rated power $S_n$ in reference to the base $S_B = 1$MVA:

\[\begin{split}x\_pu &= x \cdot S_B / S_n \\ r\_pu &= r \cdot S_B / S_n\end{split}\]

DC Lines

DC lines represent active network elements are do not interact with the parameterization of linear power flow. Therefore they require no parametrization for power flow calculations. The rated capacity is not included in the data, but given the limited amount of elements and ease of research, these values can be manually added.

Validation

To validate the parameters we can look into the static grid models that are published on TSO websites. For example the German TSO 50Hertz. publishes the data for their system including nominal current, nominal voltage and impedance for each element. The following table shows two lines (220 and 380kV) and two transformers.

Public data from 50Hertz’s static gridmodel.

		\(U_n\)	L	\(I_r\)	R1	X1
Line	Redwitz - Remptendorf	380	56.0	3600	1.6526	14.9690
Line	Neuenhagen - Marzahn	380	16.9	2400	0.7606	4.3135
Line	Neuenhagen - Hennigsdorf	220	45.9	1070	3.1905	13.1680
		Ur1	Ur2	Sr	R1	X1
Transformer	Wolmirstedt	400	231	400	0.9171	63.1933
Transformer	Röhrsdorf	380	231	800	0.3328	22.5167

Given the parameters from the previous sections and the dataset which does include length l and voltage level \(U_d\), the nominal current \(I_d\) we would estimate the following resistance and reactance.

Calculated (estimated) parameters for a sample of network elements.

		\(U_d\)	\(l\)	\(I_{nom}\)	r	x
Line	Redwitz - Remptendorf	380	57.7	2580	1.73027	14.1882
Line	Neuenhagen - Marzahn	380	17.5	2580	0.525349	4.30786
Line	Neuenhagen - Hennigsdorf	220	70.7	1290	4.24497	21.2956
		Ur1	Ur2	Sr	r	x
Transformer	Wolmirstedt	380	220	600	0.601667	44.5193
Transformer	Röhrsdorf	380	220	600	0.601667	44.5193

The comparison shows that line parameters are fairly accurate in terms of impedance, given that the length is accurate (which it isn’t in the Henningsdorf line) but rather imprecise in term of nominal voltage and therefore capacity. This is no surprise as the nominal current that depends on how lines are mounted and there are huge differences. For example the Redwitz line is a know congestion, therefore it contains larger bundles/stronger ropes than other 380 lines.

Similarly the transformer parameters are in the ballpark but not super accurate. Again, the differences between transformers of the same type, namely differences in rated current yield large differences in capacity and impedances.

However, given that all parameters are derived from a handful of standard types, the results are satisfactory. More precise calibration, based on the static grid models or specific information is always possible.

[1] (1,2,3,4)

Oeding, D. and B.R. Oswald (2016). Elektrische Kraftwerke und Netze. Springer Berlin Heidelberg. doi: 10.1007/978-3-642-19246-3

[2]

Wiegmans, Bart (2016). GridKit extract of ENTSO-E interactive map doi:10.5281/zenodo.55853

[3]

Brown, T., J. Horsch, and D. Schlachtberger (2018). PyPSA: Python for PowerSystem Analysis. Journal of Open Research Software. doi: 10.5334/jors.188. eprint:1707.09913

[4]

Thurner, L. et al. (2018). pandapower - an Open Source Python Tool for Convenient Modeling, Analysis and Optimization of Electric Power Systems IEEE Transactions on Power Systems. doi: 10.1109/TPWRS.2018.2829021

[5]

Meinecke, Steffen et al. (2020). SimBench—A Benchmark Dataset of Electric Power Systems to Compare Innovative Solutions based on Power Flow Analysis. Energies. doi: 10.3390/en13123290

[6]

Müller, Ulf Philipp et al. (2018) The eGo grid model: An open source approach towards a model of German high and extra-high voltage power grid. Journal of Physics. doi: 10.1088/1742-6596/977/1/012003