Sampling Strategies for Static Powergrid Models

Stephan Balduin

, Eric MSP Veith

and Sebastian Lehnhoff

OFFIS - Institute for Information Technology, Escherweg 2, Oldenburg, Germany

Keywords:

Machine Learning, Power Grid, Power Flow, Surrogate Models, Sampling, Correlation.

Abstract:

Machine learning and computational intelligence technologies gain more and more popularity as possible

solution for issues related to the power grid. One of these issues, the power ﬂow calculation, is an iterative

method to compute the voltage magnitudes of the power grid’s buses from power values. Machine learning

and, especially, artiﬁcial neural networks were successfully used as surrogates for the power ﬂow calculation.

Artiﬁcial neural networks highly rely on the quality and size of the training data, but this aspect of the process

is apparently often neglected in the works we found. However, since the availability of high quality historical

data for power grids is limited, we propose the Correlation Sampling algorithm. We show that this approach

is able to cover a larger area of the sampling space compared to different random sampling algorithms from

the literature and a copula-based approach, while at the same time inter-dependencies of the inputs are taken

into account, which, from the other algorithms, only the copula-based approach does.

1 INTRODUCTION

The knowledge about the current state of the power

grid is usually limited to information about the power

generation or consumption of the grids’ participants,

either through prognosis or by estimations via default

load proﬁles. However, a stable grid operation re-

quires a certain frequency level (50 Hz in Europe) and

certain voltage levels. Since only the power values are

known, voltage information needs to be calculated,

which is done with Power Flow (PF) analysis (Pow-

ell, 2004). The PF analysis is performed many times

during the operation of power grid,s and the results

can be used, e. g., for market analysis or short-term

operational planning.

Since the PF analysis often requires performing

matrix inversion, a task with a high computational

burden, there are many approaches to reduce this

computation time. Besides improvements for the tra-

ditional methods, the advancement and application

of Machine Learning (ML) models for energy sys-

tems have also increased in the past two decades. Al-

though the number of papers that solely focus on PF

is rather small (Hasan et al., 2020), there are many

works about the closely related optimal PF and proba-

bilistic PF, which are speciﬁc use cases for PF. Artiﬁ-

https://orcid.org/0000-0002-2018-1078

https://orcid.org/0000-0003-2487-7475

https://orcid.org/0000-0003-2340-6807

cial neural networks are used with great performance

for various PF-related problems. However, artiﬁcial

neural networks require a large amount of data, espe-

cially when several hidden layers are used.

In our previous works in (Balduin et al., 2019;

Balduin et al., 2020), we built a deep neural network

to avoid the costly PF for a low voltage power grid

model. One issue we identiﬁed concerns the availabil-

ity of power system data that can be used for training.

Therefore, we decided to take a deeper look at the

available data sets and sampling algorithms in the lit-

erature.

The contribution of this paper is two-fold. First,

we are pointing out the challenges and pitfalls of

retrieving training data for a power grid simulation

model and how different approaches in the literature

handled this. Second, we present the correlation-

based approach that we built to overcome some of

those issues.

The rest of this paper is structured as follows. In

section 2 we present the results of our investigation

and discuss relevant literature. The simulation model

is described in section 3, section 4 provides some of

the basics of sampling strategies, and in section 5 we

discuss the challenges of applying sampling strategies

to power grid models. In section 6, we present our

Correlation Sampling approach, which we compare

and discuss in section 7. We conclude our paper in

section 8.

Balduin, S., Veith, E. and Lehnhoff, S.

Sampling Strategies for Static Powergrid Models.

DOI: 10.5220/0011306400003274

In Proceedings of the 12th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2022), pages 319-326

ISBN: 978-989-758-578-4; ISSN: 2184-2841

319

2 RELATED WORK

The Open Power System Data platform (Wiese et al.,

2019) provides a hub for different data sets that can be

used for electricity system modeling. In their work,

the authors criticize that the quality and accessibil-

ity of publicly available data sets is often inadequate,

require different ﬁles to download, have poor docu-

mentation, or are erroneous. This is different for the

data sets provided or linked on the Open Power Sys-

tem Data platform; however most concern power gen-

eration. The available load data sets are highly ag-

gregated hourly or monthly time series or small-scale

household data sets. Another good overview of data

sets, especially for distribution grids, can be found in

the wiki of the openmod initiative

. The Simbench

project (Spalthoff et al., 2019) provides a large data

set, ranging over all German voltage-levels, contain-

ing time series for loads and generation. Finally, there

are the IEEE test cases, which mainly focus on North-

American-style systems.

Besides using publicly available datasets some

works propose methodologies to create synthetic

datasets. (H

ulk et al., 2017) used annual consumption

data to generate a synthetic data set of the German

energy system. This was extended by (Amme et al.,

2018) with a focus on the medium-voltage grid level.

Likewise, in the research project SmartNord (Blank

et al., 2015), a methodology was proposed to generate

synthetic household loads that, once aggregated, fol-

low the German default load proﬁle H0, which grid

operators use.

When neither of the above mentioned data sets

ﬁt or the data set is not large enough, sampling may

be the solution. While this originates in the ﬁeld of

probabilistic PF, where inputs of the PF calculation

are modeled as random variables (Chen et al., 2008),

sampling is used in other PF-related ﬁelds as well.

(Cai et al., 2013) use polynomial normal transforma-

tion together with Latin Hypercube sampling to build

probability distribution models for probabilistic PF.

Their models were able to handle correlated inputs

and achieved better results on the IEEE 14-bus and

118-bus systems compared to a Simple Random Sam-

pling (SRS) approach. Also in the ﬁeld of probabilis-

tic PF, (Huang et al., 2020) sampled with Latin Hy-

percube sampling as well but used D-vine copulas to

model the inter-dependencies of wind speed between

four wind farms. They evaluated the approach on a

modiﬁed IEEE 33-bus system against SRS.

(Lei et al., 2020) used a Monte-Carlo simulation

https://wiki.openmod-initiative.org/wiki/

Distribution network datasets, retrieved on 07 Apr.

2022

approach combined with an interior point algorithm

to obtain feasible samples for optimal PF. They also

did a sample pre-classiﬁcation to group samples that

share the same active constraints. The test cases were

carried out on the IEEE 39, 57, and 118-bus systems

as well as on a Polish 2383-bus system. Some works,

especially in the ﬁeld of optimal PF simply use the

base load values provided with most power grid mod-

els, e. g., the works in (Guha et al., 2019) and (Pan

et al., 2019) use 10% and (Zamzam and Baker, 2020)

even 70% deviation of the base load, although they, at

least, did not sample from a uniform distribution.

In (Thayer and Overbye, 2020), the authors sam-

pled a variation of the overall consumption and indi-

vidual scaling factors for each load on the IEEE 14-

bus system. Afterwards, loads are summed up and lin-

early scaled to match the overall consumption. Their

use case was voltage control based on deep reinforce-

ment learning. Quite similar is the work of (Diao

et al., 2019). However, the authors used the base load

of the IEEE 14-bus system and created a load ﬂuctua-

tion between 80 % and 120 % of the base load values.

From this literature research we conclude that

there are a couple of data sets available as well as

several ways to generate synthetic data sets. Unfor-

tunately, those data sets comprise not more than one

year of data. Furthermore, we found different ap-

proaches to directly sample the power grid model,

predominantly one of the IEEE test cases, from dif-

ferent research ﬁelds. Some of the works we have

discussed consider actual time series of, e. g., wind

farms for generation, others simply used the base load

for sampling. The resulting sampling data has a high

chance to have completely different distributions than

realistic (or synthetic) data, which can affect the qual-

ity of a prediction model. To this end, we propose

a methodology that takes into account realistic time

series and their inter-dependencies while at the same

time preserve the ﬂexibility of the sampling proce-

dure.

3 SIMULATION MODEL

We used the Python library pandapower (Thurner

et al., 2018), which allows to model arbitrary power

grid topologies and is able to perform a power ﬂow

calculation for that topology given a set of input data

for all relevant nodes. To setup the simulation, a

grid model is instantiated and a data set is loaded.

We used a power grid from the Simbench project

(1-LV-rural3--0-sw), because they have data sets

included that are explicitly tailored for the grid topol-

ogy. The simulation loop consists of assigning input

SIMULTECH 2022 - 12th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

320

values (active and reactive power) from the data set to

the corresponding nodes of the power grid, perform-

ing the power ﬂow calculation, and then saving the

results from the buses: voltage magnitude per unit

(although not used for this paper), active, and reac-

tive power. This process is repeated until all entries

of the data set are simulated.

4 SAMPLING STRATEGIES

Since the model described above is a computer-based

simulation model, the design of experiments literature

would recommend space-ﬁlling designs (Dean and

Voss, 1999). Such designs aim to spread the sample

points for each input evenly in the sample space. This

can be achieved with Monte Carlo Sampling (some-

times also called Simple Random Sampling (SRS)),

i. e., using the uniform distribution independently for

each input. Given that enough samples were drawn,

this approach creates nearly orthogonal sampling de-

signs i. e., the inputs are uncorrelated.

In general, orthogonality and uniformly dis-

tributed inputs are desired properties of a sampling

design since they can improve the validity of the pre-

diction model created from that design. However,

there are cases where some of the inputs in the origi-

nal system-under-investigation are correlated and the

power grid is a prime example for this. To build a

model that captures this behavior, the sample distri-

butions for those inputs need to be correlated as well.

One solution is to use Copulas, which were ﬁrst pro-

posed by (Sklar, 1959). Copulas can handle marginal

distributions of random variables and dependencies

separately. That is why an increasing number of pub-

lications that have to deal with dependencies in power

system modeling use Copulas.

5 CHALLENGES OF POWER

GRID SAMPLING

The power grid is a complex system, i. e., the more

basic approaches from the design of experiments lit-

erature for sampling and analysis cannot be applied

to the power grid model without modiﬁcations. The

complex inter-dependencies between the parts of the

power grid make it hard to guarantee properties like

orthogonality or uniformly distributed marginal dis-

tributions of the inputs without risking a decrease of

the quality of the prediction model. Not considering

speciﬁc correlations could even lead to ill-conditioned

states of the power grid, where the PF calculation

fails.

(Gerster et al., 2021) investigated sampling strate-

gies for the determination of ﬂexibility potentials at

vertical system interconnections. One of their ma-

jor conclusions concerns the application of uniform

sampling for each of the inputs. With an increasing

number of inputs, the samples suffer more and more

from the convolution problem (Bremer and Lehn-

hoff, 2018), i. e., at the vertical system interconnec-

tion point the actually covered space on the P-Q plane

gets smaller the more inputs are involved.

Another challenge concerns the deﬁnition of the

sample space of the inputs. While for most inputs,

zero can be considered as minimum value, the maxi-

mum is not clearly deﬁned. The base value attached

to the publicly available power grid models and test

cases may serve as reference value. However, it is not

a maximum value since calculating the PF using base

loads usually results in a healthy or, depending on the

test case, slightly violated system state. It is also not

an average value, which can be seen at the distribu-

tions of realistic load or generation proﬁles.

The advantage of using the base load as reference

and creating samples around those values with a spe-

ciﬁc deviation is that just the grid model itself without

any time series data is required. This makes it conve-

nient if only the general capabilities of an ML model

should be explored. However, unless the ML model

is at least evaluated on realistic data, the model may

only be a showcase for a certain ML algorithm on an

environment that happens to be a power grid model.

It does not necessarily imply that this model still per-

forms well if realistic data is used.

Figure 1: Active power time series of the load connected to

bus 42 (randomly selected) over one year of simulated time.

Taken from the Simbench grid 1-LV-rural3--0-sw.

Figure 1 illustrates the active power of a randomly

selected household of the power grid described in sec-

tion 3. The maximum peak power is 3 kW, which is

the nominal power of the corresponding load in the

grid model. In Figure 2 we plotted the histogram of

Sampling Strategies for Static Powergrid Models

321

this time series. Now it becomes obvious that most

of the data is between 0.0 and 0.5 kW. Actually, the

mean value is ≈ 0.2657, the standard deviation ≈

0.2781, and the median is ≈ 0.1835. Sampling around

the base load of 3 kW would result in samples that,

although valid, do not represent the original data and,

consequently, which do not contain the necessary in-

formation for a ML model that should make predic-

tions based on realistic input data.

Figure 2: Histogram of the same active power time series as

in Figure 1.

Next, we simulated the power grid for one year

of simulated time. We followed (Gerster et al., 2021)

and plotted active against reactive power at the slack

bus to get an estimate of the distribution of all of the

simulation data and to be able to detect possible con-

volution problems. This can be seen in Figure 3.

Now, we wanted to evaluate how well different

sampling algorithms from the literature perform for

this data set. We started with two variants of the SRS

method; the ﬁrst samples between 0 and the base load

(Equation 1) and the second samples around the base

load with a certain δ (Equation 2).

∗

∼ Uniform[0,p

] (1)

∗

∼ Uniform[(1 − δ) · p

,(1 + δ) · p

] (2)

Here, p

is the vector of base loads in the grid, δ is

the deviation from the base load, and p

∗

is the vector

of sampled power values. We used this formulas to

generate 5000 samples for active and reactive power

consumption as well as active power generation (the

generators of the grid in-use had reactive power set to

zero) with a δ of 0.5 in the second case. Afterwards,

we calculated the PF for all samples to obtain the

active and reactive power for the slack bus, just like

above. The results can be seen in Figure 3. While all

of the samples were feasible (i. e., the PF converged),

we see that those distributions did not match at all.

Figure 3: Plot of the P-Q plane at the slack bus. The lower

dot cloud represents the results from original data sets. The

middle dot cloud represents the results from the ﬁrst SRS

variant (sampling between zero and the base load) and the

upper dot cloud the results from the second SRS variant

(sampling around the base load).

A more advanced sampling strategy was used by

(Thayer and Overbye, 2020). First, the authors used

Equation 1 to sample active power on the interval [0.0,

1.0). Next, they varied the total active power loading

uniformly between 60 % and 140% of the total ac-

tive power loading P calculated from the base load.

Each of the loads is scaled linearly with the factor

∗

where P

∗

is the total active power calculated

from the samples p

∗

. For reactive power Q, a power

factor p f for each load is drawn uniformly on the in-

terval [0.8, 1.0) and Q is calculated with

Q = P · tan(arccos(p f )) · L,L ∈ {−1, 1}. (3)

The factor L is a random variable with a chance of

10% to be -1 and, therefore, to ﬂip the sign of Q. Like

before, we created 5000 samples and calculated the

PF results. The P-Q plot is shown in Figure 4. While

a much larger part of the sampling space is covered,

the areas of the original data and the sample data were

completely different.

Finally, we created a Gaussian copula to perform

the same task. We used the python package copulas

which provides appropriate functions. The result is

shown in Figure 4. The copula samples cover most of

the space that is covered by the original data as well.

We performed this experiment with different grid

models and different time series and got similar re-

sults. Our conclusion is that SRS-based approaches

https://sdv.dev/, retrieved on 14 Apr. 2022

SIMULTECH 2022 - 12th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

322

Figure 4: The results of the advanced and the copula-based

sampling were added to the plot. The darker dot cloud on

top of the dot cloud of original data shows that copulas were

able to reproduce the behavior of the original data.

are ﬁne when no realistic data sets are available or the

model prediction model will not not be used with re-

alistic data sets. In any other case, copulas allow to

create samples that represent the realistic data set.

However, there is one additional concern related

to our speciﬁc use case. The copula samples might

match the realistic data too well. One of the de-

sired properties for sampling designs is that the sam-

ple points are evenly spread over the whole sample

space. Although we don’t know the real boundaries of

the sampling space, the SRS-based approaches cover

valid areas of the sampling space that are not covered

by the copula samples. We address this shortcoming

with our sampling algorithm.

6 CORRELATION SAMPLING

The correlation sampling approach consists of two

parts. In the ﬁrst part, the correlations between the

inputs are calculated and, int the second part, those

correlations are used to create a sampling design.

6.1 Correlations

Naturally, the different entities that are connected to

the power grid have inter-dependencies. Households

follow similar patterns although there are different

types of proﬁles. Photovoltaic modules are heavily

dependent on the time of the day and weather condi-

tions like cloudiness and solar radiation, which results

in high correlations at spatially close positioned mod-

ules. Correlation can also be found between commer-

cial facilities like different super markets or between

several heating devices, which are dependent on tem-

perature conditions.

Utilizing those inter-dependencies is also done in

(Huang et al., 2020) to sample wind power plants for

probabilistic PF and in (Blank, 2015) to assess the re-

liability of coalitions for the provision of ancillary ser-

vices. Those inter-dependencies can also be found in

the time series data sets for power grids, at least when

the data set aims to be realistic. Therefore, we de-

cided to use correlations, or, more speciﬁc partial cor-

relations, to generate samples. The widely used cor-

relation coefﬁcient by Pearson (Benesty et al., 2009)

is deﬁned as

cov(X,Y )

(4)

with X,Y being random variables, σ

,σ

the stan-

dard deviation of X and Y and cov is the covariance.

When more than two random variables are involved,

other variables Z = (Z

,...,Z

) may have correlation

to X and Y as well. Especially, Z

might be related to

both X and Y . To get the unbiased correlation between

X and Y , the partial correlation can be calculated with

XY |Z

− r

· r

Y Z

1 − r

Y Z

(5)

This can be described as two linear regression

problems, the ﬁrst between Z

and X and the second

between Z

and Y (Whittaker, 2009). Since the resid-

uals of those linear regressions are uncorrelated to Z

the sample correlation can be calculated to obtain the

partial correlation between X and Y .

We will illustrate this using the data set from the

Simbench grid that was already used in the previous

chapter. The Partial Correlation Matrix (PCM) C be-

tween all of the inputs for the power grid over the en-

tire data set, displayed as heat map, can be seen in

Figure 5.

Although this PCM is sufﬁcient for our sampling

algorithm, we still used the whole data set to calcu-

late the correlations. To overcome this, we selected a

subset of the data set containing 2500 samples

and

calculated the PCM as C

again. This heat map can be

seen in Figure 6.

If you take a close look at both heat maps, you

probably recognize similar ”patterns”. In fact, those

PCMs are quite similar with a correlation factor of

= 0.98, with duplicates (the lower left triangle

of the matrix) included. The accumulated point-wise

This number is arbitrarily chosen and may only ﬁt the

current use case.

Sampling Strategies for Static Powergrid Models

323

Figure 5: Heat map to illustrate the partial correlations

between the inputs of data set provided by the Simbench

power grid.

Figure 6: In contrast to Figure 5, the correlations are calcu-

lated from a small subset of the data.

difference C - C

sums up to -391.6 with a mean of -

0.006, which indicates that the reduced PCM slightly

over-estimates positive correlations. With a standard

deviation of 0.055, we concluded that the reduced

PCM is similar enough

6.2 Sampling

The next step concerned how to integrate the PCM C

into the sampling procedure. Most of the partial cor-

relations are lower than 0.5 but there is another clus-

ter between ≈ 0.85 and 1.0. To not suppress the ran-

domness of the sampling, we deﬁned a threshold t of

0.85 and ignored all correlations that were lower with

their absolute value. Each sample s is initially gen-

erated with the Dirichlet distribution, which was used

by (Gerster et al., 2021) with good results. For each

sample s

in s, all subsequent entries s

with i < j, are

adapted depending on their partial correlation C

i j

This may depend on the data sets in-use. For our use

case, the similarity was sufﬁcient











, |C

i j

| < t

+ s

· (1 −C

i j

), C

i j

> 0

1 − s

+ s

· (1 +C

i j

), C

i j

< 0

(6)

The general idea is to pull the value of sample s

towards the value of sample s

if they’re highly corre-

lated. In Equation 6, cases two and three account for

positive and negative correlation respectively.

We also applied some additional optimizations to

better suit the current use case. First of all, we multi-

plied s

with a normal distributed noise factor of 10%

in cases where the correlation exceeds the threshold

to relax the linear dependency towards s

. Second, to

overcome some of the issues we’ve seen at the other

sampling approaches, we calculated the sum of all

values of this sample s and compared it to an interval

min

, s

max

]. When s is not in [s

min

, s

max

], s is discarded

and sampled again.

The values s

min

and s

max

are derived from the data

set again, by normalizing each time series individu-

ally, then building the sum for each time step, and,

ﬁnally, assigning the minimum value to s

min

and the

maximum value to s

max

. However, since we used a re-

duced data set and, therefore, the interval [s

min

, s

max

]

may be to small, we extended it by 20 % in each di-

rection.

7 EVALUATION

7.1 Results

We used the described methodology to create samples

like we did for the other sampling strategies. The P-

Q plot at the slack bus is shown in Figure 7. It can

be seen that the correlation samples not only cover

the space of the real data but are also located in the

regions largely around the real data. This even in-

cludes most of the space covered by the other sam-

pling strategies.

The correlation between the copula-sampled PCM

and the original full-data PCM is ≈ 0.98, which

matches the correlation of the PCM of the reduced

data set. For the correlation-sampled PCM, the is ≈

0.864, which is less than the reduced data set but still

very high. However, this difference may be one of the

reasons why a larger area is covered.

We repeated this comparison with another power

grid model, the CIGRE low voltage benchmark grid,

in combination with synthetic time series data from

Smart Nord, since we used this model in our previous

works already. The results are shown in Figure 8 and

this resembles all the issues and conclusions we iden-

SIMULTECH 2022 - 12th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

324

Figure 7: The results of the correlation samples are added

to the P-Q plot. A large area around the original data is

covered, which includes even the areas of most of the other

sampling algorithms.

Figure 8: The P-Q plot with different sampling algorithms

for the CIGRE low voltage grid.

tiﬁed for the Simbench grid, although the data sets are

completely independent of each other.

7.2 Discussion

The correlation sampling approach is, in its current

state, a subject of experimentation. There are sev-

eral parameters, like the correlation threshold or the

number of samples that are used to calculate the par-

tial correlations, which were determined experimen-

tally and not for a special mathematical reason. There

might be parameters that even lead to better results.

Furthermore, only linear correlations are considered

but in the data sets might be nonlinear correlations

as well, which could be utilized to get more accurate

samples.

On the other side, correlation sampling solved the

issue we had with other sampling strategies. It covers

the areas of the original data in the output space and,

at the same, the regions beyond as well. In theory, this

improves a prediction model’s capabilities to general-

ize when some parameters in the grid conﬁguration

have changed. However, this is beyond the scope of

this paper.

8 CONCLUSION

In this paper, we presented a small literature research

about available data sets for power system modeling,

where to ﬁnd them, and discussed some of the issues

some of those data sets have. We also reviewed al-

gorithms from the literature that were used to sam-

ple data sets for power grid simulation models and

pointed out advantages and disadvantages.

The main issue of those strategies that neglect

inter-dependencies is that created samples cover en-

tirely different areas of the output space considering

the P-Q plane at the grid interconnection point. A pre-

diction model trained with those samples will most

probably fail when more realistic type of data will

be used as prediction input. On the other side, cop-

ulas resemble the original data very well and we rec-

ommend them as ﬁrst choice whenever a prediction

model should be used in a context with realistic data.

Furthermore, we presented our Correlation Sam-

pling approach that aims to not only cover the ”re-

alistic” areas of the output space by taking inter-

dependencies between the inputs into account. But

also to cover the regions beyond to improve the gen-

eralization capabilities of the model.

Although those ﬁrst results looks promising, we

see a lot of potential for improvements of the algo-

rithm. Additionally, the data set created with correla-

tion sampling still needs to be used to build a surro-

gate model, which is the primary purpose we devel-

oped that algorithm. We will present the results from

those experiments in future work.

ACKNOWLEDGEMENTS

This work was funded by the German Federal Min-

istry of Education and Research through the project

PYRATE (01IS19021A).

Sampling Strategies for Static Powergrid Models

325

REFERENCES

Amme, J., Pleßmann, G., B

uhler, J., H

ulk, L., K

otter, E.,

and Schwaegerl, P. (2018). The ego grid model: An

open-source and open-data based synthetic medium-

voltage grid model for distribution power supply sys-

tems. In Journal of Physics: Conference Series, vol-

ume 977, page 012007. IOP Publishing.

Balduin, S., Tr

oschel, M., and Lehnhoff, S. (2019). Towards

domain-speciﬁc surrogate models for smart grid co-

simulation. Energy Informatics, 2(1):1–19.

Balduin, S., Westermann, T., and Puiutta, E. (2020). Eval-

uating different machine learning techniques as sur-

rogate for low voltage grids. Energy Informatics,

3(1):1–12.

Benesty, J., Chen, J., Huang, Y., and Cohen, I. (2009).

Pearson correlation coefﬁcient. In Noise reduction in

speech processing, pages 37–40. Springer.

Blank, M. (2015). Reliability Assessment of Coalitions for

the Provision of Ancillary Services. PhD thesis, Uni-

versit

at Oldenburg.

Blank, M., Breithaupt, T., Bremer, J., Dammasch, A.,

Garske, S., Klingenberg, T., Koch, S., L

unsdorf, O.,

Niesse, A., Scherfke, S., Hofmann, L., and Sonnen-

schein, M. (2015). Smart nord ﬁnal report. Technical

report, Uni Hannover.

Bremer, J. and Lehnhoff, S. (2018). Unfolding ensemble

training sets for improved support vector decoders in

energy management. In ICAART (2), pages 322–329.

Cai, D., Shi, D., and Chen, J. (2013). Probabilistic load ﬂow

computation with polynomial normal transformation

and latin hypercube sampling. IET Generation, Trans-

mission & Distribution, 7(5):474–482.

Chen, P., Chen, Z., and Bak-Jensen, B. (2008). Probabilistic

load ﬂow: A review. In 2008 Third International Con-

ference on Electric Utility Deregulation and Restruc-

turing and Power Technologies, pages 1586–1591.

IEEE.

Dean, A. and Voss, D. (1999). Design and analysis of ex-

periments. Springer.

Diao, R., Wang, Z., Shi, D., Chang, Q., Duan, J., and

Zhang, X. (2019). Autonomous voltage control for

grid operation using deep reinforcement learning. In

2019 IEEE Power & Energy Society General Meeting

(PESGM), pages 1–5. IEEE.

Gerster, J., Lehnhoff, S., Sarstedt, M., Hofmann, L., and

Veith, E. M. (2021). Comparison of random sampling

and heuristic optimization-based methods for deter-

mining the ﬂexibility potential at vertical system in-

terconnections. In 2021 IEEE PES Innovative Smart

Grid Technologies Europe (ISGT Europe), pages 1–6.

IEEE.

Guha, N., Wang, Z., Wytock, M., and Majumdar, A. (2019).

Machine learning for ac optimal power ﬂow. arXiv

preprint arXiv:1910.08842.

Hasan, F., Kargarian, A., and Mohammadi, A. (2020). A

survey on applications of machine learning for opti-

mal power ﬂow. In 2020 IEEE Texas Power and En-

ergy Conference (TPEC), pages 1–6.

Huang, Y., Chen, S., Chen, Z., Hu, W., and Huang, Q.

(2020). Improved probabilistic load ﬂow method

based on d-vine copulas and latin hypercube sampling

in distribution network with multiple wind genera-

tors. IET Generation, Transmission & Distribution,

14(5):893–899.

ulk, L., Wienholt, L., Cußmann, I., M

uller, U. P., Matke,

C., and K

otter, E. (2017). Allocation of annual elec-

tricity consumption and power generation capacities

across multiple voltage levels in a high spatial reso-

lution. International Journal of Sustainable Energy

Planning and Management, 13:79–92.

Lei, X., Yang, Z., Yu, J., Zhao, J., Gao, Q., and Yu, H.

(2020). Data-driven optimal power ﬂow: A physics-

informed machine learning approach. IEEE Transac-

tions on Power Systems, 36(1):346–354.

Pan, X., Zhao, T., and Chen, M. (2019). Deepopf: Deep

neural network for dc optimal power ﬂow. In 2019

IEEE International Conference on Communications,

Control, and Computing Technologies for Smart Grids

(SmartGridComm), pages 1–6. IEEE.

Powell, L. (2004). Power system load ﬂow analysis. Mc-

Graw Hill professional.

Sklar, A. (1959). Fonctions d

e repartition

a n dimension

et leurs marges. Paris: Publications de l’Institut de

statistique de l’Universit

e de Paris, 8:229–231.

Spalthoff, C., Sarajlic, D., Kittl, C., Drauz, S., Kneiske, T.,

Rehtanz, C., and Braun, M. (2019). Simbench: Open

source time series of power load, storage and genera-

tion for the simulation of electrical distribution grids.

In International ETG-Congress 2019; ETG Sympo-

sium, pages 1–6. VDE.

Thayer, B. L. and Overbye, T. J. (2020). Deep reinforce-

ment learning for electric transmission voltage con-

trol. In 2020 IEEE Electric Power and Energy Con-

ference (EPEC), pages 1–8. IEEE.

Thurner, L., Scheidler, A., Sch

afer, F., Menke, J., Dol-

lichon, J., Meier, F., Meinecke, S., and Braun, M.

(2018). pandapower — an open-source python tool for

convenient modeling, analysis, and optimization of

electric power systems. IEEE Transactions on Power

Systems, 33(6):6510–6521.

Whittaker, J. (2009). Graphical models in applied multi-

variate statistics. Wiley Publishing.

Wiese, F., Schlecht, I., Bunke, W.-D., Gerbaulet, C., Hirth,

L., Jahn, M., Kunz, F., Lorenz, C., M

uhlenpfordt, J.,

Reimann, J., and Schill, W.-P. (2019). Open power

system data – frictionless data for electricity system

modelling. Applied Energy, 236:401–409.

Zamzam, A. S. and Baker, K. (2020). Learning optimal so-

lutions for extremely fast ac optimal power ﬂow. In

2020 IEEE International Conference on Communica-

tions, Control, and Computing Technologies for Smart

Grids (SmartGridComm), pages 1–6. IEEE.

SIMULTECH 2022 - 12th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

326