IEEE TRANSACTIONS ON POWER SYSTEMS 1
Selecting representative days for capturing the
implications of integrating intermittent renewables
in generation expansion planning problems
Kris Poncelet, Student Member, IEEE, Hanspeter H
¨
oschle, Student Member, IEEE, Erik Delarue, Member, IEEE,
Ana Virag, and William D’haeseleer, Member, IEEE,
Abstract—Due to computational restrictions, energy-system
optimization models (ESOMs) and generation expansion planning
models (GEPMs) frequently represent intra-annual variations in
demand and supply by using the data of a limited number of
representative historical days. The vast majority of the current
approaches to select a representative set of days relies on
either simple heuristics or clustering algorithms and compari-
son of different approaches is restricted to different clustering
algorithms. This paper contributes by: (i) proposing criteria
and metrics for evaluating representativeness, (ii) providing a
novel optimization-based approach to select a representative
set of days and (iii) evaluating and comparing the developed
approach to multiple approaches available from the literature.
The developed optimization-based approach is shown to achieve
more accurate results than the approaches available from the
literature. As a consequence, by applying this approach to select
a representative set of days, the accuracy of ESOMs/GEPMs can
be improved without increasing the computational cost. The main
disadvantage is that the approach is computationally costly and
requires an implementation effort.
Index Terms—Energy-system planning, Generation expansion
planning, Power system modeling, Wind energy integration,
Power system economics
NOMENCLATURE
A. Abbreviations
CE Correlation error
DC Duration curve
ESOM Energy-system optimization model
GEP M Generation expansion planning model
IRES Intermittent renewable energy sources
LP Linear programming
MILP Mixed integer linear programming
NRM SE Normalized root-mean-square error
RDC Ramp duration curve
REE Relative energy error
RLDC Residual load duration curve
B. Sets
B (index b) Set of bins
K. Poncelet, E. Delarue and W. D’haeseleer are with the Department of
Mechanical Engineering, KU Leuven and EnergyVille, Belgium. Kris Poncelet
holds a PhD grant of the Flemish Institute of Technological Research (VITO)
H. H
¨
oschle is with the Department of Electrical Engineering, KU Leuven
and EnergyVille, Belgium. H. H
¨
oschle holds a PhD grant of the Research
Foundation - Flanders (FWO) and the Flemish Institute for Technological
Research (VITO).
A. Virag is with the Flemish Institute of Technological Research (VITO)
and EnergyVille, Belgium.
C (index c) Set of duration curves
D (index d) Set of potential representative days
M (index m) Set of medium-term periods
P (index p) Set of original time series
T (index t) Set of time steps
C. Parameters
A
c,b,d
Share of the time of day d during which the
lowest value of the range corresponding to
bin b of duration curve c is exceeded
L
c,b
Share of the time during which the values
of a time series with corresponding duration
curve c exceed the lowest value of the range
corresponding to bin b
N
repr
Number of representative periods to select
N
total
Total number of repitions required to scale up
the duration of a single representative period
to one year
D. Variables
error
c,b
Error in approximating duration curve c at
the bottom of bin b
u
d
Binary selection variable of day d
w
d
Weight assigned to day d, i.e., the number of
times the representative period is assumed to
be repeated within a single year
I. INTRODUCTION
B
OTTOM-UP energy-system optimization models
(ESOMs), such as TIMES [1] and MESSAGE [2],
and generation expansion planning models (GEPMs), such
as LIMES-EU [3] and ReEDS [4] are used frequently to
underpin energy policy by performing scenario analyses
for the transition of the energy/electricity system. In such
models, investment and operational decisions are optimized
simultaneously given certain exogenous parameters, e.g., the
projected evolution of fossil fuel prices.
Due to the fact that ESOMs and GEPMs typically cover
a time horizon of multiple decades, are technology rich
and span a large geographical area, solving these models
is computationally demanding. To maintain tractability, these
models typically use a low level of temporal and geographical
detail. However, due to the highly variable, unpredictable
IEEE TRANSACTIONS ON POWER SYSTEMS 2
and location-specific characteristics of intermittent renewable
energy sources (IRES), such as solar PV panels and wind
turbines, using a high level of temporal and geographical detail
becomes increasingly important. In this regard, Pfenninger et.
al. [5] identify resolving details in time and space as the main
challenge for this group of models.
Traditionally, large-scale ESOMs represent seasonal and
diurnal variations in demand and supply by disaggregating
a year into a limited number of so-called time slices (e.g.,
[4], [6], [7]). For each variable time series (e.g., the load or
wind speed), the value assigned to a specific time slice thus
corresponds to the average value of that part of the time series
corresponding to the specific time slice. While such a stylized
representation of the temporal dimension achieves a reasonable
accuracy for systems with a low penetration of IRES, several
authors have recently shown that for systems with a high
penetration of IRES, this approach leads to an underestimation
of the variability of IRES, and hence to an overestimation of
the potential uptake of IRES, an overestimation of the use of
baseload technologies and an underestimation of the value of
flexible technologies [8]–[10].
Multiple ways to improve the modeling of the temporal
dimension have recently been developed. The current literature
mainly focuses on increasing the temporal resolution, i.e.,
increasing the number of diurnal time slices (e.g., [8], [9]).
Nevertheless, it has been shown that increasing the temporal
resolution is not sufficient to grasp the inherent variability of
IRES [8], [10]. Different approaches to model the temporal
dimension to account for the variability of IRES have been
analyzed in [10], where it is shown that using the data of a
well-chosen set of historical days to represent an entire year
can be a suitable approach. However, a justified selection of a
representative set of historical periods is not straightforward.
Nevertheless, several planning models make use of some sort
of representative periods to reduce the computational cost.
Well-known examples are a.o. the US-REGEN model devel-
oped by the Electric Power Research Institute (EPRI) [11],
the POTEnCIA model recently developed by The Institute
for Prospective Technological Studies (IPTS) of the European
Commission’s Joint Research Centre [12] and the LIMES-EU
model developed by the Potsdam Institute of Climate Impact
Research [3]. Other examples can be found in [13], [14].
The literature contains various approaches to select a rep-
resentative set of historical periods. Nevertheless, frequently a
set of representative days (also referred to as typical days or
type-days) is used in planning models without documenting
how these days are selected, e.g., [15], [16]. In other work,
the set of representative days is obtained by using simple
heuristics, e.g., [17]–[20], sometimes supplemented by ran-
domly selecting some additional days, e.g., [14], [21]. As
pointed out by de Sisternes [22], a consistent criterion to select
these representative periods or to assess the validity of the
approximation is lacking. In general, the idea behind most
of these simple heuristic approaches is to select a number of
periods with different load and/or meteorological conditions in
order to capture a variety of different events. As an example,
to select three representative days, Belderbos et al. [18] select
the day that contains the minimum demand level of the year,
the day that contains the maximum demand level and the day
that contains the largest demand spread in 24 hours.
More advanced approaches to select a representative set of
historical periods can be divided into two groups. The first
and by far the largest group employs clustering algorithms
to cluster periods with similar load, wind speed and/or solar
irradiance patterns into clusters. For every resulting cluster,
either the cluster’s centroid or a single historical period from
that cluster is taken as the representative period for that cluster.
The weight assigned to each representative period, i.e., the
number of periods that are represented by this selected period,
corresponds to the number of periods that are grouped into its
parent cluster. Clustering approaches thus implicitly determine
the weight assigned to every selected representative period,
which allows to appropriately account for both common and
rare events. This is a major advantage compared to the heuris-
tic approaches discussed earlier. To perform the clustering,
different algorithms are employed which can be classified
into hierarchical and partitional clustering algorithms. A more
detailed overview of clustering algorithms is presented in [23].
The goal of all these algorithms is to minimize the sum of the
distances between every object (i.e., a period) and the cluster’s
centroid or median. For the GEPM LIMES-EU, Nahmmacher
et al. [24] use Ward’s hierarchical clustering algorithm. A
similar clustering technique is used in the US-REGEN model
to select additional representative periods, after having first
used heuristics to select a number of periods containing
extreme events [11]. Partitional clustering algorithms, such
as k-medoids [23] and k-means [25]–[27] are also frequently
used. The performance of the k-means, fuzzy C-means and
hierarchical Wards clustering algorithm are evaluated in [23],
but the differences between these algorithms were found to be
minor for the presented case. Besides clustering algorithms,
scenario reduction techniques following a similar philoso-
phy as the clustering approaches, such as the fast-backward
method, are also employed to select representative periods,
e.g., [28].
A second group of approaches aims to optimize the selection
of representative periods with respect to a predetermined, user-
defined criterion (external validity indices). In this approach,
the selection procedure is directly based on evaluating the full
set of representative periods using external validity indices,
whereas in the heuristic and clustering approaches, the se-
lection is based on the characteristics of individual histori-
cal periods or the ”similarity” between individual historical
periods; this is a clear fundamental distinction. To the best
of our knowledge, the only optimization-based approach in
the field of energy research is presented by de Sisternes
and Webster [22]. In their approach, the set of weeks which
best approximates the residual load duration curve (RLDC)
is selected by enumerating all possible combinations of a
predetermined number of representative weeks. While this
approach is shown to achieve good results, it has a number of
limitations. First, the number of combinations for selecting
k representative periods out of n candidate periods equals
n!
(k!(n−k)!
, and thus strongly increases with both the number
of candidate periods and the number of periods to select. As
a consequence, enumeration is only computationally feasible
IEEE TRANSACTIONS ON POWER SYSTEMS 3
for selecting up to 5 weeks out of 52. Therefore, using this
approach to optimally select a number of representative days
instead of weeks is computationally infeasible. Second, the
approach does not determine the optimal weights for each
selected period. Finally, the approximation of the RLDC is
used as a decision criterion, but the RLDC is dependent on
the investments in IRES. Therefore, the approach cannot be
used for models with endogenous investments in IRES.
Although multiple approaches to selecting representative
periods are available from the literature, there is no consistent
comparison of the quality of these different approaches. In
this regard, the current literature is restricted to comparing
different clustering algorithms. More complete information
on the quality of different approaches is vital for ESOMs
and GEPMs as a better selection of a representative set of
historical periods allows to improve the accuracy of these
models without increasing computational complexity.
Moreover, despite the multitude of different approaches to
select representative periods, there is not a single optimization-
based approach in the field of energy research that can be used
to select a sufficiently high number of representative periods.
The aim of this paper is to identify a sound approach
for selecting representative historical periods. To this end, (i)
criteria and metrics for representativeness are proposed, (ii) a
novel optimization-based approach is presented and (iii) this
approach is compared to different approaches available in the
literature in terms of both accuracy and ease of use.
1
The remainder of this paper is structured as follows. Sec-
tion II discusses the different temporal aspects which are
important to capture in ESOMs and GEPMs, and derives
corresponding metrics to evaluate the representativeness of
the selected periods. Section III provides an overview of the
different approaches considered in this work and presents
our novel optimization-based approach. Next, the data and
assumptions are presented in Section IV, while the results of
the different approaches are discussed in Section V. In Section
VI, these different approaches are applied to a test case to
illustrate the value of a good selection of representative days.
Finally, the main conclusions are presented in Section VII.
II. TEMPORAL ASPECTS
Fig. 1 illustrates the concept of using a representative set of
historical periods (e.g., days or weeks) in ESOMs/GEPMs. As
is illustrated in this figure, the tool to select a representative
set of periods takes different time series as input, for instance
quarter-hourly load and wind generation data of multiple
years. The output is a representative set of periods and the
weights given to each of these representative periods, i.e.,
the number of times the representative period is assumed
to be repeated within a single year. In the ESOM/GEPM,
balance of generation (gen) and demand (DEM) is imposed
in every time step t (e.g., quarter-hour) of every selected
period d. Power generation gen
g
by every technology/plant
g is restricted by the installed capacity (cap
g
). The fixed
1
Ease of use comprises the required effort for implementing the approach,
the computational cost of executing the approach as well as the flexibility to
incorporate user-specific constraints.
Input time series
Wind
PV
Load
Selection of representative periods
Energy system optimization model (ESOM) /
Generation expansion planning model (GEPM)
min
cap,gen
fixed cost + variable cost,
s.t.:
fixed cost = F C
g
· cap
g
variable cost =
X
d∈D
0
w
d
·
X
g,t
(V C
g
· gen
g,d,t
· ∆
t
)
gen
g,d,t
≤ cap
g
∀g, d, t
X
g
gen
g,d,t
= DEM
d,t
∀d, t
. . .
Set of representative periods
d ∈ D
0
with weights w
d
Fig. 1. Schematic of the use of a set of representative historical periods in
ESOMs/GEPMs.
costs relate to the construction and fixed operations and
maintenance of this capacity. Variable costs, comprising fuel
costs, variable operations and maintenance costs and taxes
are related to the generation levels of every technology/plant
in the selected periods. The weights of each representative
period are used to scale the variable costs incurred in the
selected periods to an equivalent annual cost. Similarly, fuel
consumption and emissions during the selected periods can be
scaled to equivalent annual amounts. Thus, the representative
set of periods is used to endogenously determine a good
approximation of the amount of electricity that is generated by
different technologies/units and the associated costs, emissions
and fuel use without requiring to optimize the operations over
an entire year.
To effectively quantify the accuracy of approximating dif-
ferent time series (e.g., load, wind generation) by a set of
representative periods, appropriate metrics must be defined.
To this end, the different temporal aspects that impact the
results of ESOMs/GEPMs are identified. From the literature
[11], [24], [29], we synthesize the following list of temporal
aspects:
1) the annual load and average IRES capacity factors;
2) the distribution of values for each time series
3) the correlation between the different time series;
4) the variability of each time series.
First, the selected set of periods should preserve the annual
electricity demand and the average IRES capacity factors for
each model region. To evaluate the quality of the approxi-
mation in this respect, the average value (over all considered
time series p ∈ P) of the relative errors in approximating
IEEE TRANSACTIONS ON POWER SYSTEMS 4
the average value of each time series is used as a metric, see
Eq. (1). Since for the case presented here, the relative error in
the average value of a time series is identical to the relative
error of the energy content of a time series, we refer to this
metric as the relative energy error (REE
av
) in the remainder
of this text. Note that we use |.| to refer to the absolute value,
while k.k is used to refer to the cardinality of a set.
Second, a more stringent requirement is that the distribution
of load and IRES generation levels, and their respective
frequency of occurrence correspond to the one observed in the
entire time series. Regarding the time series for IRES genera-
tion, it is crucial to account for both periods of very high IRES
generation, during which partial curtailment might be required,
and periods of near-zero IRES generation, which determine the
need for back-up capacity. Moreover, capturing the distribution
of IRES generation is required to account for the reduction in
operating hours of different types of dispatchable power plants.
Thus, by capturing the distribution of each time series, major
challenges related to the integration of IRES are accounted
for. Therefore, this criterion, which has also been used in
[24], [26], is considered to be the most important criterion
for evaluating a set of representative periods. The information
regarding the distribution of values and their respective fre-
quency of occurrence can be represented by the duration curve
(DC) of the time series.
2
Therefore, the average normalized
root-mean-square error (NRMSE) of the approximation of the
DC of each time series is used as a second metric, to which
we refer as NRM SE
DC
av
(Eq. (2)). The approximation of the
duration curve,
g
DC
p
, can be constructed by sorting the data of
the selected periods from high to low while correcting for the
fraction of a year that each selected period represents. Below,
the index t ∈ T is used to refer to a specific time step of the
original time series (e.g., quarter-hourly or hourly interval).
REE
av
=
P
p∈P
P
t∈T
DC
p,t
−
P
t∈T
g
DC
p,t
P
t∈T
DC
p,t
!
kPk
(1)
NRM SE
av
=
P
p∈P
r
1
kT k
·
P
t∈T
(DC
p,t
−
g
DC
p,t
)
2
max(DC
p
)−min(DC
p
)
!
kPk
(2)
Third, the correlation between different time series can
impact results. Within a single region, this correlation (e.g.,
between the load and solar PV generation) influences the
RLDC, and therefore the expected number of operating hours
of different thermal generation technologies. In addition, it
impacts the need for curtailment of IRES, as well as their
market value [30]. Moreover, the correlation between different
regions is important to account for geographical smoothing
effects of the load, solar PV generation, and particularly wind
generation, and the corresponding value of transmission grids
[29]. As a metric to quantify whether the actual correlation
is captured by the selected representative periods, the average
absolute difference between the correlation based on the data
2
The DC is found by sorting the entire time series from high to low values.
of the entire time series, and the correlation based on the data
in the selected representative periods is used. This is referred
to as the average correlation error (CE
av
) in the remainder
of this text (Eq. (3)). The Pearson correlation coefficient is
used to quantify the correlation corr
p
1
,p
2
between two time
series p
1
, p
2
∈ P (Eq. (4)). Here, V
p1,t
represents the value of
time series p
1
in time step t. Moreover, V
p
1
and V
p
2
indicate
the mean value of time series p
1
and p
2
respectively. As the
Pearson correlation coefficient has a value of 1 in case of total
positive correlation, a value of 0 in case of no correlation and
a value of -1 in case of total negative correlation, the values
for CE
av
lie in the range [0,2].
CE
av
=
2
kPk · (kPk − 1)
·
X
p
i
∈P
X
p
j
∈P,j>i
corr
p
i
,p
j
− gcorr
p
i
,p
j
(3)
corr
p
1
,p
2
=
P
t∈T
(V
p
1
,t
− V
p
1
) · (V
p
2
,t
− V
p
2
)
r
P
t∈T
(V
p
1
,t
− V
p
1
)
2
·
P
t∈T
(V
p
2
,t
− V
p
2
)
2
. (4)
Fourth, the dynamics of fluctuating load and IRES genera-
tion time series can impact results. Short-term fluctuations, on
time scales of minutes up to hours, are important to account
for the limited flexibility of dispatchable power plants (e.g.,
maximum ramp rates, minimum up and down times), as well
as the potential of storage technologies. To quantify to what
extent the distribution of short-term fluctuations is captured,
we introduce the concept of a ramp duration curve (RDC).
The RDC for each time series is found by differentiating and
subsequently sorting the original time series. Accordingly, the
metric used is the average NRMSE of the approximation of
the RDC (N RM SE
RD C
av
):
NRM SE
RD C
av
=
P
p∈P
r
1
kT k
·
P
t∈T
(RD C
p,t
−
^
RD C
p,t
)
2
max(RD C
p
)−min(RD C
p
)
!
kPk
(5)
Medium-term fluctuations, comprising weekly and seasonal
fluctuations, are important to account for the limited energy
storage capacities of different storage technologies. For exam-
ple, longer periods of low wind speeds and solar irradiance,
during which stored energy might be exhausted, can determine
the need for firm back-up capacity. To what extent medium-
term fluctuations are captured depends mainly on the input
parameters used for selecting representative periods, rather
than the used approach in itself. These input parameters are
closely related to the temporal structure of the ESOM/GEPM.
Examples of such input parameters include the time interval
to which the approach for selecting representative periods
is applied (e.g., representative periods can be selected for
each month, season or year) and the choice of the duration
of each individual selected period (e.g., representative hours,
days or weeks). As this paper focuses on approaches to
select representative periods rather than the temporal structure
IEEE TRANSACTIONS ON POWER SYSTEMS 5
of ESOMs/GEPMs, no metric is introduced for capturing
medium-term dynamics.
III. METHODOLOGY
A. General overview
Different approaches to select representative days are evalu-
ated by comparing all four metrics presented in Section II. The
results of this evaluation will be shown for an increasing num-
ber of representative days (N
repr
). The following approaches
to select representative days are evaluated:
1) Heuristics (H);
2) Ward’s hierarchical clustering algorithm (CA);
3) Random selection (RS);
4) MILP optimization model (OPT);
5) Hybrid approach: random selection followed by optimal
weighting (HYB).
The simple heuristics (H) employed in this work are presented
in Tab. I. The total number of days selected is presented in
the utmost left column. These days are obtained by selecting
for every period (indicated in the second column), the days
corresponding to the criteria presented in the third to fifth
column.
The clustering algorithm (CA) used is Ward’s hierarchical
clustering algorithm. Some information was provided in Sec-
tion I, for a full description of the algorithm, we refer to [24].
The third approach is to repeatedly select a random subset
of days (RS), and retain from all these subsets the subset
which obtained the lowest errors. This approach is closely
related to the enumerative approach used to select a set of
representative weeks proposed in [22]. However, calculating
the error metrics for all possible subsets of days from a single
year is computationally infeasible if the cardinality of the
subset exceeds 3. Therefore, the number of randomly selected
subsets of days is restricted to 50 000.
The fourth approach (OPT) is a newly developed approach
that employs a MILP optimization model to identify which
days are selected (binary variables) as well as the weight
assigned to each day (linear variables). The model formulation
is presented in Section III-B.
Finally, another new and novel, hybrid, approach (HYB)
that combines features of the RS and the OPT approach is
developed. In this approach, a number of random subsets of
days are taken and for each subset, the weight given to each
day is optimized. The set of weighted days that achieves the
lowest errors is retained. Again, 50 000 randomly selected
subsets are taken.
B. Optimization model formulation
1) Basic model: As discussed in Section II, primarily, the
set of representative days should accurately represent the DC
of each time series. An optimization model should therefore be
capable of selecting a set of representative days (and associated
weights), construct the approximation of the DC based on
the selected days and corresponding weights, and calculate
a metric for the approximation error that can be minimized.
Note that the number of steps of the approximated DC depend
TABLE I
OVERVIEW OF THE SIMPLE HEURISTIC USED TO SELECT A NUMBER OF
REPRESENTATIVE DAYS.
N
repr
Period Load Wind PV
2 Year Highest peak,
lowest valley
- -
4 Year Highest peak,
lowest valley
Highest and
lowest avg.
generation
-
8 Summer,
Winter
Highest peak,
lowest valley
Highest and
lowest avg.
generation
-
12 Summer,
Winter,
Intermediate
Highest peak,
lowest valley
Highest and
lowest avg.
generation
-
24 Spring,
Summer,
Fall, Winter
Highest peak,
lowest valley
Highest and
lowest avg.
generation
Highest and
lowest avg.
generation
0 10 20 30 40 50 60 70 80 90 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
L
c,b
P
d∈D
w
d
N
total
· A
c,b,d
error
c,b
error
c,b
Duration [%]
Duration curve values [-]
Original duration curve c Approximated duration curve c
0 10 20 30 40 50 60 70 80 90 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
L
c,b
P
d∈D
w
d
N
total
· A
c,b,d
error
c,b
error
c,b
Duration [%]
Duration curve values [-]
Original duration curve c Approximated duration curve c
Fig. 2. Visualization of the error term error
c,b
. The duration curve is divided
into 10 bins. The error at the bottom of the bin is displayed for bin b = 8.
on the number of days selected and the resolution of the data
of each day. For example, the approximated DC displayed in
Fig. 2 is constructed by selecting 2 representative days with a
2-hourly resolution, resulting in a total of 24 steps. However,
obtaining the approximation of the DC requires sorting the
values of the selected days which is difficult to integrate in a
single optimization framework.
Nevertheless, it is possible to get a clear view on what the
approximated DC looks like which does not require sorting the
data of the selected days. To this end, each DC c ∈ C is divided
into a number of bins b ∈ B, as visualized by the dashed lines
in Fig. 2. Each bin thus corresponds to values within a specific
range (the highest values belong to the first bin, the lowest
values correspond to the last bin). As the original time series
is known, the share of time during which this time series has
a value greater than or equal to the lowest value in the range
corresponding to bin b is known (marked by a in Fig. 2). For
a DC c ∈ C, this value is represented by the parameter L
c,b
.
Similarly, for every potential representative day d ∈ D, the
share of time in day d during which the time series exceeds the
lowest value of the range corresponding to a bin b is known.
This information is represented by the parameter A
c,b,d
. A
graphical representation of this parameter for Belgian load data
of 2014 and a number of bins equal to 10 is shown in Fig. 3.
This figure shows that, as can be expected, in every day, the
