Assessment of Generation Adequacy by Modeling a

Joint Probability Distribution Model

Marie-Louise Kloubert
TransnetBW
Stuttgart, Germany
m.kloubert@transnetbw.de

Abstract—The socio-politically motivated energy transition leads Nowadays, the generation adequacy for a future scenario is
to expansion of renewable energy power plants (REPP) and assessed by e.g. TSOs or departments of governments. In
simultaneous shutdowns of conventional power plants. As the addition, several research paper deal with this topic. They can
electric load always needs to be balanced by sufficient
generation and the energy production by REPP cannot be be distinguished by their approach, the considered time
steered adequate to the demand but depends on weather horizon and the level of view (national vs. continental).
situations, new methods are needed to assess generation Especially in Europe with a growing amount of REPP and
adequacy. These results could be used afterwards to define decreasing conventional generation parks, different
measures to ensure reliable energy supply. examinations by the European Network of Transmission
System Operators for Electricity (ENTSO-E) and national
To evaluate the generation adequacy a joint probability
distribution model is proposed. It models the dependencies of TSOs exist. Since 2011, the German TSOs examine the
weather characteristics, the electrical load and unavailabilities generation adequacy and publish their results [1]. They use a
of conventional power plants using pair-copula constructions. deterministic approach and analyze a synthetic situation with
The model is, then, applied to determine the probability of a national view for a time horizon up to three years in the
generation imbalance. The implementation is shown for future. This situation consists, on the one hand, of the highest
Germany in the scenario year 2030.
measured load demand over the last years and, on the other
hand, of very low energy production by renewable energies.
Index Terms — generation adequacy, system adequacy, Based on historical unavailability indices of conventional
copula, dependencies, probability density function, pair-copula, power plants, they determine a level of available capacity for
solar irradiance, vine copula, wind speed the examined situation. As the situation is synthetic and not
modeled by a consisting approach, the probability of this
I. INTRODUCTION specific situation is unknown. Therefore, the results are
Due to the lack of great storage capacities, the energy difficult to interpret and can hardly be used to deduce
generated by power plants must always balance the electrical measures. In the moment, the German TSOs work on a
load. Conventional power plants can be easily controlled and probabilistic method.
adjusted for different load demands. In contrast, REPP The ENTSO-E analyzes the generation adequacy with
generate energy depending on current weather situations. different approaches and different time horizons on a
With the transition of the energy system, the amount of European level [2], [3] and [4]. Twice a year seasonal outlook
conventional power plants is reduced and replaced by REPP. reports are published to analyze the generation adequacy for a
Therefore, there is a growing need to assess the generation short time horizon of half a year using a deterministic
adequacy and to find measures to ensure reliable energy approach for the whole ENTSO-E area which covers 36
supply in the future even during peak times with very high countries. With known planned and, additional, randomly
load demand. In this paper, generation adequacy is focused as generated unavailabilities of conventional power plants for
part of the system adequacy, which consists of two parts: the next six months and the expected load demands of each
generation adequacy and transmission adequacy. Generation country, an outlook for the generation adequacy of each week
adequacy analyzes to what extent the electrical generation is determined. The probabilities of the analyzed events are
can equal the electrical load, meanwhile transmission unknown.
adequacy proves to what extent generated energy can be In contrast, the annual mid-term forecast (MAF) examine the
transported from the source to the sink. The limiting factors generation adequacy for the next 10 years based on a partly
are the generation park respectively the electrical grid. probabilistic approach. As a database, synthetic climate data

21st Power Systems Computation Conference Porto, Portugal — June 29 – July 3, 2020
PSCC 2020
and electrical load data are used. The planned unavailabilities It was introduced by Sklar in 1959 [16]. In contrast to the
are considered by an optimized maintenance schedule. In correlation coefficient by Pearson, the copula can model
addition, unavailabilities of conventional power plants are linear and nonlinear dependencies. It was introduced to
generated randomly. As a result, for each historical data point uncertainty modelling in power systems in recent years to
an unavailability for the generation park is fixed and all these model e.g. the interdependencies between two wind power
historical data points are used to assess the generation generators close to each other. Existing works mainly use
adequacy. An optimization is used to determine an optimal bivariate copulas that take into account the dependence of
dispatch of power plants using a high amount of assumptions. only two locations [17], [18], [19] and [20]. These
The quality of the synthetic climate and electric load data is approaches differ by the selected copula model (Gaussian,
unclear. As the input data are the most important influencing Gumbel etc.) and by the considered uncertainties (wind/wind,
values of the approach, the results of the approaches are wind/load, etc.). In [21] a 15-dimensional Gaussian copula is
difficult to interpret. In addition, correlations between e.g. the built. However, the Gaussian copula is quite inflexible. A
unavailabilities of conventional power plants and the load are more flexible tool are vine copulas [24], which have been
neglected. Besides, the reality is not reflected adequately by applied, e.g. in [22], to model the spatial dependence of wind
an optimized maintenance schedule and an optimal dispatch power forecast errors or in [23] to represent the dependencies
of power plants. The Pentalateral Energy Forum, a of wind speed in a small dimension. Even though copulas
framework for regional cooperation in Central Western have proved to be useful in risk analysis of power systems,
Europe, publishes every year the Pentalateral Generation the existing copulas normally just consider one kind of
Adequacy Assessment (PLEF GAA) with a very similar uncertainties, e.g. wind power, but neglect the necessity to
approach to the MAF approach [5]. The results of PLEF implement a model considering a wide range of uncertainties
GAA and MAF are valued by the index ‘Loss of Load to use it for real problem solutions. This paper proposes a
Expectation’ (LOLE). consistent probabilistic method to evaluate the generation
Various indices exist to measure the reliability of power adequacy using a copula approach. A joint distribution model
systems. They are used to interpret and compare the results of is developed to model the interdependencies of wind speeds,
different approaches. The Loss of Load Probability (LOLP), solar irradiations, electrical load and unavailabilities of
the Loss of Energy Expectation (LOEE), the Expected conventional power plants based on historical data.
Duration per Interruption (EDPI) and LOLE are the most Afterwards, samples are generated based on the model to
commonly used ones [6]. Scientific approaches dealing with assess the generation adequacy for a future scenario.
reliability evaluations can be divided in analytical Exemplarily, the joint probability model to assess generation
approaches, e.g. [7] and [8], and simulation approaches, e.g. adequacy is applied for Germany. The proposed model
Monte Carlo Simulations as used in [9]. They differ mainly includes the interdependencies of wind speed and solar
due to the considered uncertain input variables and their irradiance at 91 stations in Germany, the German electrical
modeling procedure. In [10] a quasi-sequential Monte Carlo load and historical unavailabilities of conventional power
simulation is modelled to analyze the reliability of power plants. The model reflects the characteristics of the marginal
systems with renewable energies. The influence on reliability distributions and the linear and nonlinear dependencies of all
indices by wind power is shown in [11].[12] combines the uncertain variables using copula.
cross-entropy method and the copula approach to evaluate In Section II, the copula concept with a special focus on
generation adequacy for a test case with no consideration of pair-copulas is introduced, in section III the joint probability
historical values. model for Germany is presented. Section IV assesses the
The growing share of REPP in the grid more and more generation adequacy for Germany. Section V concludes.
requires the analysis of the distributions and dependencies of
REPP and electrical load feed-ins not just for generation II. COPULA
adequacy assessment but for all kind of power system
This section gives an overview of copula constructions.
analysis, e.g. probabilistic load flow approaches. In [13] all
Further information can be found in [16] and [24]. Copulas
uncertainties are represented by beta-distributions and
are used to model a joint probability distribution for which
correlations are neglected. Other represent wind speeds by
the marginal probability distribution of each variable is
Weibull distributions and determine the Pearson correlation
uniform. A bivariate copula function
coefficient to capture the linear correlation, e.g. presented in
[14] and [15]. Others use copulas to describe the dependence (1)
of statistical variables. is a distribution on with uniform marginals. The central
Copulas are statistical functions that allow building a joint theorem of copulas is given by Sklar. The equation (2) shows
probability distribution by modelling the marginal the connection of the bivariate distribution functions and their
distribution function and the dependence structure separately. univariate margins.

21st Power Systems Computation Conference Porto, Portugal — June 29 – July 3, 2020
PSCC 2020
 (2) Distributions can be unimodal or multimodal. The main
C is called the copula; it describes the dependence between distribution models and the non-parametric estimation are
shortly introduced:
two variables and with distribution functions F and G.
H is called joint distribution. The equation can be extended 1) Normal distribution
from the bivariate case to a multivariate one. A copula is, The Normal distribution is the most frequently used
therefore, a flexible and multifunctional tool to model a probability distribution function. A variable X is normally
multivariate distribution. The main advantage of copulas are distributed with the mean value ( and variance , i.e.
the possibility to model marginal distributions and the joint , when the probability density function (PDF) can
dependence structure separately. In addition, it is not be calculated with
restricted to specific parametric models such as the (3)
multivariate normal distribution. Copulas may be determined .
in a parametric approach. It can be distinguished between
different copula classes, which in turn consist of different 2) Weibull distribution
families. The most common classes are the Archimedean A variable X is Weibull distributed, when the PDF can be
copulas and the elliptical copulas. The Archimedean copulas calculated with formula (4). Herewith, is called the scale
include e.g. Gumbel, Joe, Frank and Clayton family. The parameter and the shape parameter, with and
elliptical copulas consist of Gaussian and Student´s-t family. The PDF is defined as
Examples of bivariate Gaussian copulas are shown in Figure
1. (4)
1 1 .
0.8 0.8

0.6 0.6 3) Multimodal distribution

A multimodal distribution is a PDF with more than one
u2

u2

0.4 0.4

0.2 0.2 local maximum, which results in various modes. Often

0 0
multimodal distributions are mixtures of two or more
different unimodal distributions. They often arise due to the
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
u1 u1

contemporaneous consideration of different groups.

Figure 1: Examples of bivariate Gaussian copulas with the rank correlation
4) Non-parametric estimation by Kernel estimator
(left) and (right)
A basic and well known probability density estimator is
Different classes and families can express different the histogram. Its main disadvantages are the resulting
dependence structures. Elliptical copulas have radial noncontinuous density functions and the lack of rule for
symmetries, resulting in the same upper and lower tail estimating, e.g., the bin width. A more flexible method for
dependence. So their application is restricted to cases without non-parametric estimation is the Kernel density estimator.
different tail behaviour. In contrast, the Archimedean copulas The Kernel density estimator for estimating the density of a
allow a greater range of dependence structures. E.g., the random sample ( ) is defined by
Gumbel family represents upper tail dependence, while . (5)
Clayton family represents lower tail dependence [16]. The The value describes the bandwidth and is also called the
Archimedean are mainly used for bivariate copulas, as they smoothing parameter. The Kernel function K must satisfy (6).
are not practical for higher dimensions and entail one-
parametric Archimedean copulas, two-parametric (6)
Archimedean copulas and rotated versions of the Normally, known density functions, e.g. the Normal
Archimedean used for negative dependencies [25]. The joint distribution, are selected for the Kernel function [26].
probability model is determined in two steps: First, marginal B. Dependence Modelling Using Pair-Copula Constructions
probability distribution functions are determined and, second,
Pair-copula constructions are flexible and suitable for
the dependence structure is captured by the copula.
higher dimensions. They can handle high dimensions with
A. Marginal Distribution Functions heterogeneous dependencies. The main idea is the
In order to determine marginal distributions, two main decomposition of a multivariate copula into various bivariate
approaches can be used: parametric and non-parametric copulas. Consequently, they allow different dependence
estimation. The parametric estimation assumes a particular structures between pairs of variables as the copula is built up
form for the underlying distribution function first, while the by products of marginal densities and bivariate copula
non-parametric avoids any classification to parametric densities. The pair-copula construction was introduced in
families, e.g. Normal or Weibull distribution [26]. [24]. For high-dimensional distributions, a great variety of

21st Power Systems Computation Conference Porto, Portugal — June 29 – July 3, 2020
PSCC 2020
possible pair-constructions can be determined, i.e. the unavailabilities of conventional power plants. The
decomposition is not unique. The graphical models – interdependencies are nonlinear, the distributions are non-
Regular-vines (R-vine), Canonical vines (C-vines) and normal and, due to the great amount of considered
Drawable vines (D-vines) – can be used to organize them uncertainties, the structure becomes high-dimensional. The
[25]. With d variables, pair-copulas arise. C-vine trees problem is modelled with a C-vine copula, which can cope
with the mentioned characteristics. To develop the joint
have a star structure. Figure 2 shows the structure of a C-vine
probability distribution, a great amount of data is needed. As
copula for four variables, which results in six pair-copulas.
the copula models the marginal distributions and the
Three trees arise with a unique node connected to all other
dependence structure separately, the data are used to estimate,
nodes of the tree. In general, a root node is chosen in each
on the one hand, the marginal distributions and, on the other
tree and the pair-wise dependencies to all other nodes are
hand, the dependence structure. The latter requires
determined conditioned on all previous root nodes [27].
contemporaneous available data. In this section the marginal
Formula 7 shows the connection of pair-copula densities and
distribution functions are presented, followed by the
marginal densities for the C-vine approach
description of the copula. For the analysis, the software R and
the package CDVine is used.
(7)
A. Marginal Distribution Functions
where denotes the marginal densities and , 1) Weather Characteristics
In a first step, the weather characteristics are investigated.
z is defined as
Wind speed data are taken from the database of the German
Weather Forecast Service (DWD). They comprise measured
and presents the bivariate copula densities with hourly mean wind speed data. Datasets can entail missing
their parameter(s) . data points due to e.g. defective measurement equipment or
The joint density for the example with four variables results false data classification. Considering 20 years (1995 to 2014),
in 91 out of 410 German measurement stations are selected.
. They are chosen considering simultaneous availability of data
and measurement consistency, e.g. no change of station
Copulas can be fitted e.g. by a method of moments
height etc. during the measurement time. Their locations are
inversion of Kendall´s or by the maximum likelihood
shown in Figure 3.
estimation. With the Akaike Information Criteria (AIC) or the
Bayesian Information Criteria (BIC) the copula family is
chosen.

2
12 3
13

1 14 tree 1
4

13
23|1

12 24|1 14 tree 2

34|12
23|1 24|1 tree 3

Figure 2: Structure of a four dimensional C-vine structure

III. JOINT PROBABILITY MODEL TO ASSESS GENERATION

ADEQUACY
A joint probability model is established to assess the
generation adequacy for Germany based on the model Figure 3: Considered measurement stations in Germany
described in [28]. This model includes the uncertainties due
to weather characteristics, the electrical load and For each station a Weibull distribution is approximated
where parameters are estimated by maximum likelihood. The

21st Power Systems Computation Conference Porto, Portugal — June 29 – July 3, 2020
PSCC 2020
fitting is based on around 70,000 observations for each Figure 3. They cannot be categorized using any parametric
station. To confirm the fitting, the χ2-test is used, which families. Instead, the non-parametric kernel estimator is used
requires statistically independent observations. This is to estimate the PDF of the data.
achieved by selecting observations with time distance 2) Electrical Load
considering their autocorrelation function [29]. For the German electrical load, only measurements for
As an example, an approximated Weibull distributions for whole Germany are published, i.e. no regional information of
station 49, in northern Germany, is shown in Figure 4. The electrical load is available. Hence, the hourly total load data
station is marked in a light blue in Figure 3. published by ENTSO-E are used. They consist of
measurements for the years 2006 to 2014. The total amount
of available data are nearly 80,000 observations. The
histograms show two significant modes. The load data are
estimated by a bimodal distribution, consisting of two Normal
distributions as seen in Figure 6. The whole electrical load is
strongly influenced by industrial electrical load. It peaks
during weekdays and daytime, but gets small at weekends
and during the night. This results in two groups with mean
values of ca. 48 GW and 65 GW.

Figure 4: Wind speed measurements at station 49, in northern Germany,

fitted by a Weibull distribution

Figure 6: Total electrical load of Germany fitted by a bimodal distribution

consisting of two Normal distributions

3) Unavailabilities of conventional power plants

Due to maintenance or failures, conventional power plants
can be unavailable. Unavailabilities can be distinguished by
planned and unscheduled ones. Every power plant needs to be
maintained in a regular period. These unavailabilities
Figure 5: Solar irradiance data at station 43, in central Germany, fitted by the normally last a few weeks and are planned beforehand. The
non-parametric kernel estimator owner of power plants need to inform their local TSO about
this measure. These unavailabilities are not independent. Next
The measured solar irradiance data published by the DWD
to this, unscheduled unavailabilities occur. Again, they can be
are not sufficient for developing a joint probability
subdivided in disposable and not disposable. If they are
distribution model. Therefore, modelled data from
disposable, they can be delayed for at least a few hours or
EUMETSAT’s Satellite Application Facility on Climate even weeks. Unplanned unavailabilities are expected to be
Monitoring (CM SAF) are used. The data are hourly independent.
modelled values based on satellite information of Meteosat Since 2011 unavailabilities of conventional power plants in
satellites. They are available for a 0.05 x 0.05 degree grid Germany need to be published on the EEX transparency
[30]. To synchronize these data with the wind speed data, the platform. The notifications include both, planned and
closest available values for the locations (Figure 1) are used. unscheduled unavailabilities. Each notification consists of,
In a detailed analysis, the data of the same period as the wind among other things, the company’s name, the duration, the
speed data (1995 to 2014) are investigated. For this time energy source, the type, the reason and the amount of non-
period, around 150,000 contemporaneous observations for available power. A mapping to an individual power plants is
each station are available. As an example, the fitting for the impossible as no unique identifier is given. The data cannot
station 43 is shown (Figure 5) and marked in a light blue in be fitted adequately by a parametric distribution; therefore,

21st Power Systems Computation Conference Porto, Portugal — June 29 – July 3, 2020
PSCC 2020
they are approximated by a non-parametric distribution as For this paper, 5,000,000 samples were generated and
seen in Figure 7. probabilities for specific events were calculated. The samples
B. C-Vine-Copula were generated using the copula with the algorithm presented
in [31]. Then, the probabilities based on the real observations
The joint distribution model is established with the C-vine were contrasted.
copula approach. The bivariate copulas are selected with the
AIC from 33 different copula families listed in [27]. They
include elliptical and Archimedean copulas and, as
subclasses, one-parametric, two-parametric and rotated
Archimedean copulas. All copulas are fitted by the maximum
likelihood estimation. Based on the AIC, the best fitting
copula family is selected. With an amount of 184 variables,
16,836 copula families were selected. The variables consist of
91 wind speed measurements, 91 solar irradiance model
based data, the measured total electrical load and the
notifications of unavailabilities for conventional power
plants. So the dependencies of the different regional
influences are fully captured. The dependence structure is
determined with 10,500 contemporaneous observations. In
Figure 8 an overview of the chosen families is given.

Figure 8: Overview of the selected families for the vine copula

As the generation of the renewable energies is the
interesting value instead of weather phenomena. The specific
events are selected for different wind speeds, solar irradiance
and electric load to allow an interpretation of the events. Two
cases for wind speed are regarded: less than and more
than . As the measurements are not recorded on the same
height, they are all transformed to the same height of a typical
wind energy generator of 120 m using the Hellman
exponential law. The Hellmann exponential law is defined
Figure 7: The unavailabilities of conventional power plants fitted by a by (12) [32].
nonparametric distribution
(12)
From 33 available families [27], only 24 different were
selected. It can be seen, that one-third of all selected families The parameter is called friction coefficient or Hellmann
are t-copula and Gaussian copula, which have radial exponent. It describes the surface roughness that depends on
symmetries. All other belong to the Archimedean class, the topography at a specific site. The measured wind speed on
which express asymmetric and/or nonlinear dependence. E.g., height is described by , the wind speed on hub´s height
Frank copulas show less dependence for more extreme values
is .
than for the middle range. Clayton copula expresses more
dependence for smaller values, while survival Clayton In the examination three cases are considered: the defined
expresses stronger dependence for greater values. Nearly one- criteria should be true for
third of all copulas are rotated copulas expressing negative i) at least 90% of all stations
dependence. The results of the classes depend on the ii) at least 50% of all stations
decomposition. iii) the stations in the North of Germany (NG), marked
in black in Figure 3
C. Verification of the developed model Exemplarily, the results of six events are shown in Table 1.
The model is verified by the comparison of the determined It can be seen that the modelled probabilities and empirical
probability for specific events using the copula and using the probabilities are close to each other. Event 1 and event 5
real observations. For the copula, the probability of these show very low modelled probabilities, meanwhile the
events can be estimated by simulating a high number of empirical probabilities are zero. Therefore, these events may
samples and extracting the samples that meet predetermined occur but have not been observed yet in the historical
criteria for these events using the mathematical definition of database. In a next step, the model is used to analyze the
the copula. generation adequacy for a future scenario.

21st Power Systems Computation Conference Porto, Portugal — June 29 – July 3, 2020
PSCC 2020
 renewables and pump storages are always available.
Table 1: Results of probability for specific events and
group of stations using the developed joint probability 0.016
model
0.014

vwind osolar Pload Group pmod [%] pemp [%]

[m/s] [W/m²] [GW] 0.012

1 > 10 - <55 i 0.003 0 0.01

2 > 10 - - ii 1.74 1.86

Density
3 > 10 - - iii 0.39 0.32 0.008

4 <3 < 100 - i 1.02 0.16 0.006

5 <3 - - iii 0.004 0
6 - < 100 - i 58.03 57.54 0.004

0.002

IV. GENERATION ADEQUACY ASSESSMENT FOR GERMANY 0

-140 -120 -100 -80 -60 -40 -20 0 20 40 60

Assuming a generation park for a future scenario, the Residual Load [GW]

generation adequacy can be assessed with the developed

model. Figure 9: Distribution of the residual load for the scenario 2030
Here, the installed capacity of the scenario B for the year
For all 5,000,000 samples the renewable generation is
2030, designed by the German TSOs, is used [34]. The
determined and contrasted to the load. Here, Germany is
installed capacity is shown in Table II
monitored separately, i.e. without possible import or export to
Table II: Installed capacity for each energy source for its neighboring countries. The resulting residual load is
scenario B in 2030 [34] shown in Figure 9. Exemplarily, the probability of specific
Energy source Installed capacity [GW] events are presented in Table III. The probability of a residual
Thermal conv. power plants load greater than 0 GW is 62.1 %, meanwhile the probability
61.6 of a residual load exceeding 57.5 GW is less than 0.05 %.
(including capacity reserve)
Pump storage 11.6 Table III: Residual Load
Wind onshore 81.5
Wind offshore 17 Residual Load [GW] Probability [%]
>0 62.1
PV 91.3
> 40 5
Biomass 6
> 45 2.3
Hydro power 5.6 > 50 0.8
Other renewables 1.3 > 55 0.15
> 57.5 0.05
The given scenario defines the installed capacity, only, the
location is not fixed. To analyze the generation adequacy and
the influence of different locations for wind speeds, the place
of location for the REPP is needed. The information of the 0.016

location of existing wind power generators and solar power 0.014

generators in Germany can be found from the database of the
regulatory authority in [35]. The additional generators are 0.012

regionalized based on the energy potential using the method 0.01

from [33].
Density

To determine the wind power for each wind generator, a 0.008

typical power curve is assumed. The transformed data on 0.006

hub´s height are considered appropriate to the description

before. For solar power generator, a linear behavior to the
0.004

measured solar irradiance is presumed. As there are no or not 0.002

sufficient data available for biomass, hydro power, other

0
renewables and pump power, they were neglected in the 0 20 40 60 80 100 120 140 160 180 200

copula model. Assumptions are made for these uncertainties: Remaining Capacity [GW]

80 % of the installed capacity of biomass, hydro power, other Figure 10: Distribution of the remaining capacity for the scenario 2030

21st Power Systems Computation Conference Porto, Portugal — June 29 – July 3, 2020
PSCC 2020
The residual load needs to be balanced by conventional V. CONCLUSION
power plants and pump storages. Considering the modelled The energy system faces new challenges. Well controllable
unavailabilities and the installed capacity for the future conventional power plants are substituted by renewable
scenario, the distribution of the remaining capacity is shown energies. Therefore, the question arises: how reliable is the
in Figure 10. The probability of negative remaining capacity energy system? This paper proposes a joint probability model
is 0.005 %. The maximum negative remaining capacity is to represent the influencing variables on generation adequacy.
-10.5 GW. In these hours, Germany depends on imports. The The main uncertainies are wind speed, solar irradiance,
distribution of the negative remaining capacity is shown in electrical load and unavailabilities of conventional power
Figure 11. plants. The uncertain variables are captured by a C-vine
copula model consisting of 184 dimensions. Weibull
distributions, multimodal distributions and non-parametric
0.5
kernel density estimator are used for fitting their marginal
density functions. The plurality of these different modelling
0.4 approaches allows a precise fitting of all participants. Then,
the model is used to generate samples and to assess the
0.3
generation adequacy for the year 2030. First, the distribution
of the residual load is analyzed. Then, the remaining capacity
Density

considering conventional power plants is determined.

0.2
Therefore, a probability for the imbalance of generation and
load can be determined. The model can be easily extended to
0.1 analyse the generation adequacy for whole Europe through
the integration of the data for the neighboring countries.
Besides, biomass, hydro power, other renewables and pump
storages could be integrated in the copula model, when
0
-10 -8 -6 -4 -2 0

Negative remaining Capacity [GW] reliable historical data are available. The availability of pump
Figure 11: Distribution of the negative remaining capacity for the scenario storages depend on the level of reservoirs, which itself
2030 depend on weather conditions and the market. These
interdependencies could be added in a further step in the
As Germany plans phasing out coal energy, the model. In addition, changes in the load behavior due to
scenario B 2030 is adjusted for a sensitivity analysis. Instead electrical vehicles, could be modelled adequately. The results
of 61.6 GW of thermal conventional power plants, the can be used in a further step to define measures to cope with
thermal capacity is reduced by 6.5 GW to 55.1 GW, possible imbalance. This research focuses on generation
meanwhile the installed capacity of REPP and storages is adequacy. To evaluate system adequacy, the transmission
kept constant. The probability of negative remaining capacity adequacy needs to be considered. This is needed, to prove if
is in this sensitivity 0.06 %, which is equal to the generation can be transported from the source to the sink.
. The distribution of the negative remaining For this purpose, the approach could be combined with a
capacity can be seen in Figure 12. probabilistic load flow approach.

ACKNOWLEDGMENT
0.35

The author would like to thank Dr. Alexander Dürre,

0.3
research associate at the Department of Statistics, TU
Dortmund University, for helpful comments and discussions
0.25
regarding this work.
0.2
Density

REFERENCES
0.15
[1] 50Hertz, Amprion, Tennet, TransnetBW, „Bericht der deutschen
Ü bertragungsnetzbetreiber zur Leistungsbilanz 2016-2020“,
0.1
available
from https://www.netztransparenz.de/portals/1/Content/Ver%C3
0.05 %B6ffentlichungen/Bericht_zur_Leistungsbilanz_2017.pdf, 2018.
[2] Entso-e, “Winter Outlook Report 2016/2017 and Summer Review
0 2016”, available from:
-16 -14 -12 -10 -8 -6 -4 -2 0
https://docstore.entsoe.eu/Documents/Publications/SDC/2016-
Negative remaining Capacity [GW]
wor_report.pdf, 2016.
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GW installed thermal conventional power plants available from https://docstore.entsoe.eu/Documents/SDC%20doc
uments/SOAF/summer-Outlook-2018-with-cover.pdf, 2017.

21st Power Systems Computation Conference Porto, Portugal — June 29 – July 3, 2020
PSCC 2020
 [4] Entso-e, “Mid-term Adequacy Forecast”, available from: [27] E. Brechmann, U. Schepsmeier, “Modeling Dependence with C-
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21st Power Systems Computation Conference Porto, Portugal — June 29 – July 3, 2020
PSCC 2020