WP EN2015-10

Selecting representative days forinvestment planning models

K. Poncelet, H. Hoschle, E. Delarue and W. D’haeseleer

TME Working Paper - Energy and Environment

DRAFT VERSION June 1th 2015

An electronic version of the paper may be downloaded

from the TME website:

http://www.mech.kuleuven.be/tme/research/

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Selecting representative days for investment planning

models

Kris Poncelet%,+, Hanspeter Hoschle%,+, Erik Delarue%,+

and William D’haeseleer%,+,*

%University of Leuven (KU Leuven), Energy Institute, Celestijnenlaan 300, B-3001 Leuven, Belgium+EnergyVille, Thor Park, B-3600 Genk, Belgium

*Corresponding author: +32 16 32 25 10, william.dhaeseleer@kuleuven.be

Abstract

Due to computational restrictions, bottom-up energy-system planning models typically repre-sent intra-annual variations in demand and supply by using a limited number of so-called timeslices. Capturing the inherent variable character of intermittent renewable energy sources inthis limited number of time slices is challenging. One approach to do this, is by using a smallset of historical days to represent the entire year (so-called representative days). However, thereare no optimization models available in the literature to optimize the selection of representativedays. This paper presents a novel approach to select a predefined number of representativedays, using mixed integer linear programming. Different temporal aspects of relevance forplanning models are identified, and metrics are introduced to evaluate to what extent thesedifferent aspects are captured by the selected set of representative days. The quality of theresults using the developed approach is contrasted to the results obtained with simple heuristicapproaches. The results indicate that the presented model performs significantly better, forall presented cases. Finally, the model is used to analyze the trade-off between increasing thenumber of representative days and the temporal resolution. The results indicate that using asufficiently high number of representative days (i.e., 8-12) should be prioritized to increasingthe temporal resolution above a 4-hourly resolution.

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Nomenclature

AbbreviationsCE Correlation errorIRES Intermittent renewable energy sourcesIQR Interquartile rangeLT Long-termMILP Mixed integer linear programmingNRMSE Normalized root-mean-square errorPV PhotovoltaicRAE Relative area errorRES Renewable energy sourcesRLDC Residual load duration curve

Setsp ∈ P Set of duration curvesp ∈ OP ⊂ P Set of duration curves of the original time seriesp ∈ DP ⊂ P Set of duration curves of the short-term dynamic fluctuations of the original time seriesp ∈ CP ⊂ P Set of duration curves to improve capturing the correlation between the different time seriesb ∈ B Set of binsd ∈ D Set of potential representative periods (days)m ∈M Set of periods for accounting for medium-term variations

ParametersAp,d,b Parameter indicating the number of data points of day d and time series p that are in bin bBinSizep,b Parameter indicating the number of data points of time series p that are in bin b

Ep,b Mean value of the range of time series p corresponding to bin bEm,p Average value of the period m of time series pNrepr Number of representative periods to be selectedNm Number of periods identified for accounting for medium-term variationsNtotal Number of potential representative periods that can be selectedWp Weight assigned to the errors of profile pα Parameter determining how stringent medium-term variations should be accounted for

Variableserrorp,b ∈ R+

0 error of the approximation of duration curve p at the end of bin bud ∈ {0, 1} variable indicating whether potential representative period d is selected or notwD

d ∈ R+0 weight assigned to day d

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1 Introduction

Bottom-up long-term (LT) energy-system planning models, such as TIMES, PRIMES and EnergyPLAN are fre-quently used to perform scenario analyses, thereby identifying possible transition pathways for energy systems(e.g., [6],[22],[2],[10]). As such, these models can be used to underpin policy advice. In the last decades, a lot ofinterest has gone to the transition towards a sustainable energy system, in which renewable energy sources (RES)are expected to play a key role. The electric power generated by some of these RES, such as solar PV panelsand wind turbines, is location specific and intermittent (i.e., highly variable and only limitedly predictable). Thisintermittent character induces increased cycling of dispatchable plants [5]. As a result, modeling a large-scaleintegration of intermittent RES (IRES) requires a high level of temporal, spatial and technical detail.

Moreover, LT planning models typically span a long time horizon (e.g., 2014-2050), a large geographical region(e.g., Europe) and consider a large set of different technologies. As a result, solving these large-scale planningmodels while maintaining a high level of temporal, spatial and technical detail is computationally challenging,leading to a trade-off between the computational cost and the incorporated level of detail [19].

Regarding the temporal dimension, the level of detail is typically reduced by representing intra-annual variationsin demand and supply by using a limited set of so-called ’time slices’. A typical example of the use of these timeslices can be found in [22], where 4 time slices are introduced to incorporate seasonal variations, and each seasonaltime slice is disaggregated into a day, a night and a peak-demand period to account for diurnal variations indemand and supply (leading to a total of 12 time slices) . The number of time slices used in planning modelstypically ranges between 4-100 for large-scale energy-system planning models, up to more than thousand for powersystem planning models that do not account for the transition pathway. Within each time slice, the supply (e.g.,solar irradiance, wind speed) and demand (e.g., electricity demand) is assigned a single value for each region.

Different approaches exist to assign a value to these time slices [11],[21]. One of the approaches, which issuitable to grasp the variability of IRES, is to select a limited set of historical periods (e.g. days, weeks) that arerepresentative for the entire time series. While this choice of representative periods can have a strong impact onthe model results [21], de Sisternes and Webster [3] point out that there is no consistent criterion to select theserepresentative periods, or to assess the validity of the assumption.

In this regard, it is essential to define which aspects of the temporal dimension are relevant to consider inplanning models with endogenous investments in both dispatchable technologies and IRES. Ideally, the set ofselected periods should account for:

• Static aspect:

– Average value of each time series

– Probability distribution of each time series

• Dynamic aspect of each time series

• Correlation between different time series

First of all, the set of representative periods should approximate the average value of each time series. A morestringent requirement is that the probability distribution of each time series should be approximated. This isessential to grasp the variable nature of IRES. Moreover, this is important to have a correct representation of theyearly electric energy that can be generated with a specific technology whenever there is a non-linear relationshipbetween the renewable resource (e.g., wind speed), and the electricity generation. Both these static aspects areaccounted for in the duration curve of a time series1.

However, the duration curve merely presents a static picture of the variability of different time series. As theduration curve is found by ordering all values in the time series, chronology of the data is lost. Therefore, theduration curve of the original time series does not contain information about the dynamics of this time series.Ideally, the dynamic aspect of different time series are also reflected in the set of representative days. One canconsider dynamics on different time scales. Short-term fluctuations, on time scales of minutes up to hours, areimportant to account for the limited flexibility of dispatchable power plants (e.g., ramping rates, minimum upand down times), as well as the potential of storage technologies, such as pumped hydro storage plants. Medium-term fluctuations, comprising weekly and seasonal effects, are important to account for longer periods of lowwind speeds and solar irradiance, during which short-term storage technologies might be depleted. As such, thesefluctuations might determine the required amount of back-up capacity. Similarly, grasping longer periods of high

1The duration curve of a time series is found by ordering all values in a time series from high to low.

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IRES generation will influence the the potential of long-term storage technologies, such as power-to-gas, as well asthe amount of IRES curtailment2.

Finally, it is also important to consider the timing of the RES generation with respect to the demand forelectrical energy. First, this will determine the economic value (and the revenue) for RES generation [12]. Second,the timing of the generation will influence the amount of RES electricity generation that will need to be curtailed.Finally, this timing will influence the residual load duration curve (RLDC), which is of particular importancein energy system planning models (see e.g., [23],[24],[21]). To account for this, the correlation between differenttime series should ideally be reflected in the set of representative periods. On the one hand, this comprises thecorrelation between different types of time series (e.g., the correlation between wind generation and electricitydemand in a single region). On the other hand, also the correlation between time series in interconnected regionsis of importance.

In this paper, we present a novel approach, based on mixed-integer linear programming (MILP), to selecta predefined number of historical days and corresponding weights to represent the temporal dimension. Thepresented approach allows to account for the average value and the distribution of values (probability densityfunction) of different time series (e.g., solar irradiance, wind speed, electricity demand). Moreover, the approachcan also accounts for dynamic aspects on different time frames, and the correlation between different time series.Different error metrics are introduced to evaluate the quality of grasping the different temporal aspects. Wefurthermore apply this approach to make the trade-off between the number of representative days used and thetemporal resolution.

The remainder of this paper is structured as follows. Section 2 presents an overview of the available literature.The model formulation, as well as the metrics used to evaluate the quality of the approximation of the differenttemporal aspects, are presented in Section 3. Furthermore, the results are discussed in Section 4. Finally, someconclusions are presented in Section 5.

2 Literature review

The recent literature has shown that the temporal representation has a significant impact on the results obtainedwith LT planning models [20],[4],[13],[21]. Using a temporal representation that insufficiently accounts for theinherent variable character of IRES generation is shown to lead to an overestimation of the potential uptake ofIRES generation, an overestimation of the use of baseload technologies, and an underestimation of total systemcosts. The term temporal representation is used here to refer to both the temporal structure of the planning model(i.e., the structure of the time-slice tree), and the approach used to assign a value to each of these time slices.Although both aspects are important to account for the fluctuations of IRES and the electricity demand, it isimportant to make a distinction between the two. An example of the temporal structure typically applied in LTplanning models is presented in Figure 1. While the majority of the existing literature focuses on the impact ofthe temporal structure and, more specifically, on the impact of the temporal resolution (i.e., the number of diurnaltime slices), the focus in this work is on the data preprocessing step (i.e., how to assign values to a limited set oftime slices, starting from the original time series).

Different approaches can be found in the literature to assign values to these time slices [21]:

1. Taking the average value over a subset of the entire time series, which corresponds to the definition of thetime slice

• Not explicitly taking into account different renewable resource availabilities (e.g., the average wind speedduring the night of all days in the winter period is assigned to the time slice corresponding to nights inthe winter season)(see e.g., [22],[6],[16]).

• Explicitly accounting different resource availabilities (see e.g., [21]).

2. Use the data of some selected historical periods directly in the time slices, thereby assuming that the selectedperiods are representative for the entire time series (i.e., use the data of a well-chosen subset of the originaltime series directly)(see e.g., [10])

3. Generate values based on a statistical analysis of the time series (see e.g., [9])

In [21], it is shown that the first approach is only capable of adequately capturing the variability of IRESwhen the different resource availabilities are explicitly accounted for in the time-slice tree. However, chronology

2It must be noted it is only meaningful to consider these different aspects if the temporal structure of the planning models effectivelyallows to take these aspects into account.

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is lost in this approach, making it ill suited to properly value flexible technologies and storage plants. On theother hand, by using a set of historical days, the variability of IRES can be captured, while maintaining diurnalchronology. Numerous planning models therefore use a set of representative periods (also referred to as typicaldays or type-days) to approximate intra-annual variations in demand and supply. However, while this choice ofrepresentative days can have a strong impact on the model results [21], de Sisternes and Webster [3] point out thatthere is no consistent criterion to select these representative periods, or to assess the validity of the assumption.Moreover, documentation about how these representative periods are selected is often not provided.

Nevertheless, different approaches to select representative periods can be identified from the literature. Theseapproaches can be divided into two, fundamentally different, groups. The reasoning behind the first group ofapproaches starts from the fact that there are a lot of days with similar conditions. By selecting days whichare significantly different from each other, different situations and types of events are captured. Implicitly, theassumption made in these approaches is that better capturing this variety of events leads to a better approximationof the entire time series. To select different days, a heuristic approach is typically employed. Often, these heuristicsare relatively simple. An example can be found in [8], where 12 representative days are selected. For every even-numbered month, the day with the highest peak-load, and a second random day is selected. Another example canbe found in [10], where based on the season, and the the average wind speed, a set of 24 candidates of representativedays is chosen (in a later stage, 12 out of these 24 candidates are selected). Other examples can a.o. be foundin [18],[1],[14]. However, some heuristic algorithms are more advanced. For example, the approach described byNahmmacher et al. [17] uses a clustering algorithm to cluster days with similar load and resource availabilitypatterns, until a predefined number of remaining clusters remains. From each resulting clusters, a single day isselected, and the weight assigned to each day corresponds to the size of the cluster.

A second approach aims to directly optimize the choice of a set of periods, based on a predefined criterion. Incontrast to the heuristic approaches presented above, a set of periods is evaluated. As an example, de Sisternesand Webster [3] select the combination of weeks which gives the best approximation of the RLDC by testing allpossible combinations. However, as the number of combinations strongly increases with the number of candidateperiods and the amount of periods to select, testing all possible combinations can only be done for selecting up to5 weeks out of 52. Moreover, the approach is not suited for models with endogenous investments in IRES.

The main advantage of the optimization-based approach is that it evaluates directly the entire set of selectedperiods. If a good criterion can be formulated to express whether a set of periods is representative, an optimalchoice of periods can be selected. In contrast, in the heuristic approaches, specific periods are selected based on it’scharacteristics (or the similarity with the characteristics of other periods). As a result, the set of selected periodis not evaluated within the algorithm, and a good result cannot be guaranteed. Moreover, approaches based onoptimization typically have a higher flexibility than heuristic approaches, making it easier to incorporate additionalconstraints. However, heuristic approaches have a lower computational cost. Up to our knowledge, there existsno optimization-based approach to select a high number of representative periods (for investment models withendogenous investments in IRES).

A different classification of the different approaches can be made according to which aspects are considered inthe selection process. Table 1 presents for the most advanced approaches found in the literature, which aspectsare explicitly considered in selecting the representative periods (indicated by a X). It is important to note thatthis table merely reflects which aspects are explicitly accounted for in the approach, and not the quality ofthe approximation. E.g., the approach used by Nahmmacher does not explicitly account for the static aspects.Nonetheless, the results indicate that the heuristic used implicitly approximates the static aspects with a highaccuracy.

Explicitly accounted Static Aspects Dynamic Aspects Correlationin selection of days Mean Prob.

distr.ST MT Type Regions

de Sisternes and Webster [3] X XGolling [9] X X X XNahmmacher et al. [17]Poncelet et al. X X X X X

Table 1: Overview of the temporal aspects explicitly considered when selecting representative days in different approachesavailable in the literature, as well as the approach presented in this paper.

In this paper, we present a novel approach to select representative days based on mixed integer linear pro-gramming. With respect to the approaches available in the literature, the approach presented here has several

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advantages. First, while the approaches available in the literature are either based on heuristics, or can only beused for selecting a small set of representative periods, the model presented here allows to use an optimizationroutine to select a desired amount of representative periods. As a result, the approach presented here makesoptimal use of the available resources (i.e., given a limited number of representative days, the selected days basedon our approach captures the temporal dimension as good as possible). Second, it allows to consider not onlythe probability density function of the load and different renewable resources (e.g., wind speed, solar irradiance),but also dynamic aspects on different time-scales (e.g., short-term fluctuations and seasonal effects), as well as thecorrelation between the availability of different resources (e.g., wind speed, solar irradiance) and different regions.

Figure 1: Example of the temporal structure used in LT planning models [15]. In this figure, a year is split up intofour seasons. Each season is in turn disaggregated into a time slice corresponding to weekend days, and a time slicecorresponding to week days. Finally, each day is disaggregated into a day and a night time slice. The indicators on theright refer to the terminology used for the different time slice levels in the TIMES model.

3 Methods and assumptions

In this section, we first present the model used to selected representative periods. Second, we define the metrics usedto analyze the quality of the approximation. Different error metrics are introduced to reflect the approximation ofthe different temporal aspects.

3.1 Model formulation

Static aspects

As discussed in Section 1, the duration curve3 of a time series contains information about both the averagevalue, and the probability distribution of a time series. A duration curve can be constructed for the entire timeseries, as well as based on the selected representative periods (with associated weights). The model aims to selecta combination of representative periods and associated weights, such that the deviations between the original andthe approximated duration curve are minimized, for all considered time series4. An optimization model shouldtherefore be capable of selecting a set of representative periods (and associated weights), construct the durationcurve based on the selected set of periods, and evaluate the quality of the approximation. However, obtaining theapproximated duration curve requires ordering the values of the selected periods. However, this ordering operationcannot be integrated into a MILP formulation (or similar non-linear formulations).

To overcome this issue, the data points of the original time series p ∈ P are divided into a number of binsb ∈ B, where b is a specific bin corresponding to the set of all bins B. This is visualized in Figure 2. Each binthus corresponds to values within a specific range (the highest values belong to the first bin, the lowest valuescorrespond to the last bin). For the original time series, the number of data points corresponding to each bin isexactly known. This is reflected in the parameter BinSizep,b. From this, the number of data points having a valuegreater than or equal to the lowest value corresponding to a bin b can be derived as

∑b′≤bBinSizep,b′ .

Similarly, for every potential representative period d ∈ D (e.g., a day or a week), the number of data pointsbelonging to each bin is known. This is reflected in the parameter Ap,b,d. Assume that a set of representativeperiods D′ ⊂ D is selected, and a weight wd is assigned to each selected representative period d ∈ D′. In this case,

3The duration curve of a time series is found by ordering all values in the time series from high to low.4Relevant time series are for example wind speed and solar irradiation time series. Moreover, planning models with multiple regions

can have a separate time series for wind speeds in each region.

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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

minp

maxp

BinSizep,b∑b′≤b BinSizep,b′

∑b′≤b Ap,b′,d ∗ wD

d

errorp,b

Duration [%]

Duration

curvevalues

[-]

Original duration curve Approximated duration curve

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

minp

maxp

BinSizep,b∑b′≤b BinSizep,b′

∑b′≤b Ap,b′,d ∗ wD

d

errorp,b

Duration [%]

Duration

curvevalues

[-]

Original duration curve Approximated duration curve

Figure 2: Methodology using different bins to approximate the duration curve. The blue curve corresponds to theoriginal duration curve, while the green curve is formed by selecting a set of representative days. The bins are indicatedby the orange rectangles.

the number of data points of the approximated profile having a value greater than or equal to the lowest valuecorresponding to bin b is also known, i.e.,

∑b′≤b

∑d∈D′ wd ·Ap,b′,d.

At the end of each bin, the distance between the original and the approximated duration curve is taken as theapproximation error5 (Equation 2). The objective function of the model is to minimize the weighted sum of allthese distances (i.e., errors), for all considered time series p ∈ P (Equation 1). A weight Wp can be assigned toeach time series.

Minud,wd(∑

p∈PWp

∑

b∈Berrorp,b), (1)

subject to

errorp,b = |∑

b′≤bBinSizep,b′ −

∑

b′≤b

∑

d∈Dwd ·Ap,b′,d|, ∀p ∈ P, b ∈ B, (2)

∑

d∈Dud = Nrepr, (3)

wd ≤ ud ·Ntotal, ∀d ∈ D, (4)

∑

d

wd = Ntotal, (5)

ud ∈ {0, 1}, ∀d ∈ D; wd ∈ R+0 , ∀d ∈ D; errorp,b ∈ R+

0 , ∀p ∈ P, b ∈ B. (6)

Equation 3 imposes that the number of selected periods corresponds to the predefined number of representativeperiods Nrepr. Equation 4 restricts nonzero weights to selected periods, by using a binary variable ud whichindicates whether day d is selected or not. Moreover, the maximum weight that can be assigned to a singleselected period is fixed to the total number of periods Ntotal. The weight from all selected periods can therefore bechosen freely, which is important to account for rare events. Finally, Equation 5 guarantees that the total lengthof the approximated curve corresponds to the original curve.

3.2 Dynamic aspects

It must be noted that the possibility to account for short-term, as well as medium-term variability is stronglydetermined by the temporal structure of the planning model. Accounting for short-term fluctuations requiresa planning model with a sufficiently high temporal resolution (i.e., a high number of time slices at the diurnallevel). Regarding the integration of medium-term variability, the structure of the time-slice tree at the higher

5Note that all deviations between the original, and the approximated curve within a bin are not taken into account within theoptimization (i.e., the vertical distance between both curves in Figure 2). It is reasoned that by using a sufficiently high number ofbins, these errors will become very small, such that minimizing the horizontal distances between the curves at the end of each bin is agood metric for the quality of the approximation.

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time-slice levels is important. Indeed, the temporal structure at these higher time-slice levels will determine howthe representative periods are linked to each other. The temporal structure of the planning model is clearly ofimportance, and should be considered when selecting representative periods. However, the aim of this work ison finding a set of representative periods which reflects the different temporal aspects. Therefore, in the furthermodel description, it is assumed that the planning model is set up in such a way that these dynamic effects canbe accounted for.

Regarding the short-term fluctuations, the representative periods would ideally reflect the distribution of short-term fluctuations of different time series. Within the optimization, this can be accounted for by introducing, foreach original time series, an additional duration curve representing the change in value of the original time series(i.e., the so-called ramp). An example of the Belgian wind generation ramp curve is displayed in Figure 3. As canbe seen, this duration curve comprises both positive values, corresponding to an increase in wind generation, andnegative values, corresponding to a decrease in wind generation.

To account for variations in the medium term (multiple weeks to seasonal effects), the original time series canbe split up into a number of medium-term periods m ∈ M (e.g., seasons), of which each consist of a number Nm

of potentially representative periods d ∈ Dm (e.g., days). Additional constraints can be added to the model suchthat the selected periods reflect these medium-term variations6:

∑

d∈Dm

wd = Nm, ∀m ∈M, (7)

|∑

d∈Dm

(wd ·Ap,b,d · Ep,b)−Nm · Em,p| ≤ α ·Nm · Em,p, ∀m ∈M. (8)

Equation 7 implies that at least one period must be selected from each medium-term period, and the weightsshould be selected to reflect the duration of medium-term period m. Dm is the set of potential representativeperiods belonging to medium-term period m. Moreover, Equation 8 imposes that the average value of the selectedperiods belonging to medium-term period m should be similar to the average value of the original medium-termperiod. Here, Ep,b reflects the mean value of the range corresponding to bin b, while Em,p is the average valueof the original medium-term period. Finally, α is a parameter that can be set between zero and one. Very lowvalues enforce that medium-term variations are accounted for, but might lead to infeasibilies and the eliminationof solutions which perform well for the static indicators. On the other hand, when using high α values, themedium-term variations might not be sufficiently accounted for.

0 20 40 60 80 100−0.2

−0.1

0

0.1

0.2

Duration [%]

∆P

e

Pinst·15m

in[1/min]

Duration curve of short-term fluctuationsof onshore wind generation in Belgium

Figure 3: Duration curve of the observed changes in electrical power generated by onshore wind power turbines over 15minute intervals in Belgium (data from 2014 [7]). The observed changes in electrical power ∆Pe are normalized to theinstalled capacity Pinst. Positive values correspond to an increase in wind generation, while negative values correspondto a decrease in wind generation.

6In the model formulation presented here, medium-term variations are reflected only in terms of the average value of each medium-term period. Different constraints could easily be formulated to consider also the distribution of a time series within each medium-termperiod. However, more binding constraints for each medium-term interval will lead to a less accurate approximation of the other aspects.For this reason, the distribution within each medium-term period is not considered here.

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3.3 Correlation

Up to now, the model does not account for the correlation between different time series. The sample correlationcorrp1,p2 between two time series p1, p2 ∈ P is defined as follows:

corrp1,p2 =

∑Tt

((p1t − p1) · (p2t − p2)

)√∑T

t (p1t − p1)2 ·∑Tt (p2t − p2)2

. (9)

Here, the index t is used to refer to a specific sample of the time series of size T. Moreover, p1 and p2 indicate themean value of time series p1 and p2 respectively. It is important to note that both factors in the denominator areapproximated implicitly by approximating the duration curves of time series p1 and p2. However, the numeratoris not. Therefore, an additional time series (p1t − p1) · (p2t − p2) is introduced. Again, a duration curve of thistime series can be constructed, and approximated by dividing this curve into a number of bins. An example ofsuch a duration curve of this additional time series for the correlation between wind generation and PV generationin Belgium, is shown in Figure 4. An additional duration curve is introduced for every combination of time seriesfor which the correlation is relevant to consider.

0 20 40 60 80 100

0

0.2

Duration [%]

(PP

Vt

−P

PV)·(P

WO

nt

−P

WO

n)

PP

V·P

WO

n[-]

Duration curve for capturing the correlation betweenPV generation and onshore wind generation in Belgium (2014)

Figure 4: Additional duration curve introduced in the optimization to capture the correlation between solar PV andwind generation. Positive values correspond to data points contributing to a positive correlation, i.e., data points forwhich either both time series have a value above their mean value, or both time series have a value below their meanvalue. On the contrary, negative values correspond to data points contributing to a negative correlation. The integralof this curve (i.e., the sum of all values) provides information about the correlation between both time series. As can beseen from this figure, there is a small negative correlation between solar PV and wind generation.

3.4 Error Metrics

As discussed in Section 1, there are different temporal aspects which can be of importance for energy-systemplanning models. To evaluate how well these aspects are captured by the selected set of representative periods,different error metrics are defined.

Static aspects

The duration curve of a time series contains both information about the average value and the distribution ofvalues of a time series. Therefore, the normalized root-mean-square error (NRMSE) between the original durationcurve, and the approximated duration curve is used as a metric for the capturing the static aspects of a time series.The NRMSE between an original duration curve DC of a time series p, and the approximation of this durationcurve (represented by a tilde) is defined as follows:

NRMSEp =

√1T ·∑T

t=1(DCp,t − DCp,t)2

(max(DCp)−min(DCp)). (10)

A good approximation of the duration curve generally corresponds to a good approximation of the average value ofthe time series. However, as the deviations are squared, different approximations could obtain the same NRMSE,by being either systematically above, below or fluctuating around the original curve. Therefore, we introduce asecond error metric for the total area under the duration curve (which directly corresponds to the average valueof the time series), the relative area error (RAE):

RAEp = |∑T

t=1(DCp,t)−∑T

t=1(DCp,t)∑Tt=1(DCp,t)

|. (11)

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Dynamic aspects

Following a similar reasoning, the NRMSE of the duration curve of the short-term fluctuations of a time seriesis the metric used to evaluate to what extent the short-term dynamics are captured. Regarding the medium-termaspects, the error metric used is:

RAEMTm,p =

|∑d∈Dm(wd ·Ap,b,d · Ep,b)−Nm · Em,p|

Nm · Em,p

, (12)

i.e. the value of α for which the equality sign in Equation 8 would hold.

Correlation

To analyze how well the correlation between different time series are grasped, we define the correlation errorCE as the difference between the sample correlation based on the entire time series, and the sample correlationbased on the selected representative periods:

CEp1,p2 = |corrp1,p2 − corrp1,p2|. (13)

Note that the correlation error lies in the range [0, 2], where a value of zero indicates that the correlation of theapproximated profiles corresponds to the correlation of the entire time series, while higher values indicate a highercorrelation error.

3.5 Assumptions

In Section 4, the results of the model to select representative days, will be analyzed. The original time series usedin the model include a time series for the electricity demand, a time series for the onshore wind generation and atime series for solar PV generation. All data corresponds to the Belgian electricity system in the year 2014, and isprovided by the Belgian transmission system operator at a 15-minute resolution [7]. As one cannot a priori knowwhether or not the year 2014 is a representative year for the different time series, it is advised to use multiple yearsof data to construct the different duration curves (and medium-term average values). However, as the goal in thiswork is to analyze to what extent the presented model is capable of selecting representative periods to approximatea given original time series, it is reasoned that the size of the original time series will not significantly influencethe presented results.

Throughout the different model runs presented in Section 4, a number of bins |B| , equal to 40, is used for allprofiles. Every bin is constructed such that the range of values (i.e., the height of the bin in Figure 2) in each bin isidentical. For a time series p with maximum value maxp and minimum value minp, each bin spans a range of valuesof (maxp − minp)/|B|. Moreover, every considered profile is given an identical weight. All runs are performedwith an optimality gap of 1%, and a time out of 6 hours. All runs are performed on a Intel R©CoreTMQuad CPUQ9550 @ 2.83GHz×4, with a memory of 13.5GiB, and a 64-bit system.

While the duration of the potential representative periods can be freely determined by the user, the onlypotential representative periods considered here are days. Finally, it is reasoned that capturing medium-termvariations only becomes important as long-term storage technologies become mature. At this point, this is not thecase. For this reason, the extent to which medium-term variations can be captures is not further analyzed.

4 Results and discussion

In this section, the error metrics will be used to assess the quality of the approximation of the different temporalaspects. Different optimization runs will be tested, differing in the number of days that can be selected and thedifferent temporal aspects considered in the optimization. In this regard, we can distinguish between the durationcurves of the original time series (p ∈ OP ⊂ P ), the duration curves of the short-term fluctuations (p ∈ DP ⊂ P ),and the additional duration curves based on the correlation between different time series (p ∈ CP ⊂ P ). Here,we consider 3 original time series (load, wind, PV), which additionally give lead to 3 dynamic time series and 3correlation time series.

4.1 Static aspects

First, the focus is on the static aspects (i.e., the optimization only aims to approximate the duration curves of timeseries p ∈ OP ), as these are considered to be the most important. The performance of the presented optimizationmodel will be compared to a selection of representative days using simple heuristics. The heuristics applied are

11

presented in Table A1 in the Appendix. Moreover, the results are contrasted to a number of random selections ofrepresentative days.

Figures 5 and 6 present for a varying number of representative days the average NRMSE, and the averageRAE respectively. The errors obtained using the simple heuristics, the random selections, and the optimizationprocedure are highlighted.

2 4 8 12 240

5

10

15

20

25

days [#]

NRM

SE

OP

avg[%

]

Optimization (40 bins, 0.25h) Heuristics Random (10 000 samples)

Figure 5: Average normalized root-mean-square error (NRMSE) of the original time series, for a varying number ofrepresentative days. The quality of the approximation using the presented optimization model (blue dots) is comparedto the quality of the approximation using simple heuristics (red squares) and the quality of the approximation obtainedby using a number of random selection of days (orange boxplots). The details of the heuristics applied are presented inTable A1 in the Appendix. The boxplots presented highlight the median value, the 25th and 75th percentile (box). Thewhiskers correspond to the lowest value within 1.5 interquartile range (IQR) of the 25th percentile, and the highest valuewithin 1.5 IQR of the 75th percentile. Outliers above the upper whisker are not shown, while no outliers below the lowerwhisker could be observed.

As can be seen in Figure 5, the optimization model obtains an approximation of the duration curves witha significantly higher accuracy than the approach based on heuristics for a given number of days. Seen from adifferent perspective, this implies that by using the optimization model to select representative days, the numberof days considered can be lowered while maintaining a reasonable accuracy. A clear example of this can be seenin Figure 5, where the average NRMSE obtained with the heuristics using 24 days, is higher than the averageNRMSE for a set of 2 representative days obtained using the optimization model. Remarkably, the best set ofrepresentative days, obtained by a number of random selection of a days, performs significantly better than theheuristic approach. Moreover, the applied heuristics are shown to perform worse than the 75th percentile of therandom selections in all our cases. Nevertheless, the optimization approach performs better than the randomselections in all cases.

Figure 5 furthermore indicates that the difference between the heuristic approach and the optimization approachbecomes smaller as the number of representative days is increased. However, note that this effect will become lesspronounced when additional temporal aspects or multiple regions are considered. Similarly, one can expect thatthe performance of the approach using a number of random selections of days will decrease when additional aspectsor regions are considered. Indeed, these additional requirements will reduce the number of combinations of daysyielding a good approximation, thereby reducing the probability of obtaining a good solution using a number ofrandom selections.

Regarding the average value of each time series, it can be seen from Figure 6 that a good approximation ofthe duration curves of the different time series yields a good approximation of the average value of each timeseries. Already by using a minimum of 2 representative days using the optimization model, the average RAE iswell below 5%. The heuristic approach again falls short of obtaining a good approximation of the average value inthe majority of the cases.

12

2 4 8 12 240

20

40

60

80

days [#]

RAE

OP

avg[%

]Optimization (40 bins, 0.25h) Heuristics Random (10 000 samples)

Figure 6: Average relative area error (RAE) of the original time series, for a varying number of representative days.The quality of the approximation using the presented optimization model (blue dots) is compared to the quality of theapproximation using simple heuristics (red squares). The details of the heuristics applied are presented in Table A1 inthe Appendix. The boxplots presented indicate the median value, the 25th and 75th percentile (box). The whiskerscorrespond to the lowest value within 1.5 interquartile range (IQR) of the 25th percentile, and the highest value within1.5 IQR of the 75th percentile. Outliers above the upper whisker are not shown, outliers below the lower whisker do notoccur.

4.2 Dynamic aspects

Regarding the dynamic aspects, the performance of the runs with and without the dynamic profiles p ∈ DPincluded in the optimization are presented in Figure 7. From this figure, it can be observed that, in general, theerrors on the dynamic duration curves are rather small, even if a low number of representative days is selectedwithout including the dynamic profiles in the optimization. Moreover, including the dynamic profiles is shown tohave a limited effect, both on the approximation of the original duration curves, and on the approximation of thedynamic duration curves. Therefore, we can conclude that the added value of including these dynamic profiles intothe optimization is limited.

Remarkably, adding the dynamic profiles to the optimization not necessarily improves the approximation ofthe dynamic aspect (this can be observed when 8 or 12 days are selected). This is possible due to the factthat the optimization cannot directly optimize the NRMSE of the profiles. While minimizing the errors used inthe optimization (see Equation 2) is a good proxy for minimizing the NRMSE, both errors are not completelyequivalent. It is therefore possible that a higher objective function value in the optimization yields a slightly lowerNRMSE. This is what has occurred here.

4.3 Correlation

The influence of including the correlation time series p ∈ CP on the different error metrics is presented in Figure 8.From this figure, it can be clearly observed that not explicitly accounting for the correlation between the differenttime series (dashed line), can give rise to significant approximation errors for the correlation. While generally, thiserror will decrease as the number of days included in the model is increased, the correlation error is shown to bedrastically reduced by including the correlation time series in the optimization.

However, a trade-off between approximating the duration curves of the original time series and approximatingthe correlation between these time series can be expected. This can be clearly observed in Figure 8. Moreover, thistrade-off can be expected to be more pronounced when a low number of days is selected. Indeed, as the number ofselected days is increased, there are more combinations of days which yield a good approximation of the durationcurves of the original time series. Therefore, there is more opportunity to find a set of days that also approximatesthe correlation between the different time series without having to sacrifice the quality of the approximation ofthe duration curves of the original time series. Figure 8 shows that when 12 or more representative days areselected, the impact on the approximation of the original duration curves is negligible, while the correlation errorcan be significantly reduced. Moreover, even when a low number of days is selected, the gains in terms of a betterapproximation of the correlation seem to outweigh the negative impact on the approximation of the duration curvesoriginal time series.

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2 4 8 12 240

1

2

3

4

5

Days

NRM

SE

OP

avg[%

]with Dynamic Profiles without Dynamic Profileswith Dynamic Profiles without Dynamic Profiles

0

1

2

3

4

5

NRM

SE

DP

avg[%

]

Figure 7: Impact of including the dynamic profiles in the optimization. The approximation error of the duration curvesof the original time series is presented on the left (blue) axis. The metric used here to evaluate the approximationerror is the average normalized root-mean-square error (NRMSE) of the duration curves of the original time series. Theapproximation of the short-term dynamic fluctuations of the time series is presented on the right (red) axis. The metricused here to evaluate the approximation error is the average NRMSE of the duration curves of dynamic fluctuations ofthe original time series. The details of the metrics used are presented in section 3.4.

In this respect, it can again be expected that when multiple regions are considered, including the additionalcorrelation profiles in the optimization will again cause a higher increase of the approximation error of the durationcurves of the original time series. To limit this effect, while still providing an incentive for approximating thecorrelation, the weight W assigned to the correlation time series can be lowered.

2 4 8 12 240

1

2

3

4

5

Days

NRM

SE

OP

avg[%

]

with Correlation Profiles without Correlation Profileswith Correlation Profiles without Correlation Profiles

0.00

0.04

0.08

0.12

0.16

0.20

CE

avg[-]

Figure 8: Impact of including the correlation profiles in the optimization. The approximation error of the durationcurves of the original time series is presented on the left (blue) axis. The metric used here to evaluate the approximationerror is the average normalized root-mean-square error (NRMSE) of the duration curves of the original time series. Theimpact on the approximation of the correlation between the different time series is presented on the right (red) axis. Themetric used to evaluate the approximation of the correlation between the different time series is the average correlationerror (CE). The details of the metrics used are presented in section 3.4.

As discussed in Section 1, accounting for the correlation between different profiles is a.o. important to have agood approximation of the RLDC. This is visualized in Figure 9, which displays the RLDC, and the approximationsof this RLDC when 8 representative days are selected, both with and without including the additional correlationduration curves. As can be observed from this figure, the approximation of the RLDC is better when the correlationprofiles are included in the optimization.

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0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

−5

0

5

10

Duration [%]

ResidualLoad[G

W]

Residual Load Duration Curve (Orig)

Residual Load Duration Curve (8 days, w/o Correlation Profiles)

Residual Load Duration Curve (8 days, with Correlation Profiles)

Figure 9: Approximation of the residual load duration curve (RLDC) for 8 selected representative days. Eight represen-tative days are selected by the model, once with, and once without including the additional duration curves to accountfor the correlation between different profiles. Based on the Belgian electricity demand of 2014, an installed capacity ofonshore wind turbines of 12 000 MW, and an installed capacity of PV panels of 6 000 MW, the RLDC is constructed.The approximation of the RLDC for both sets of 8 representative days is also shown. Including the correlation profilesinto the optimization is shown to lead to a better approximation of the RLDC.

4.4 Impact of the temporal resolution

Up to now, all results are based on selecting a varying number of representative days, all having the original,15-minute, resolution. Every representative day is therefore represented by 96 time slices. This implies that, givena restriction for the total number of time slices, more representative days could be selected if the resolution wouldbe lowered. As the goal is to make optimal use of the available number of time slices, the trade-off between thenumber of representative days and the temporal resolution is analyzed7.

Figure 10 presents the average NRMSE of the original profiles as a function of the total number of timeslices considered. In general, one can clearly see that the marginal gain of using an additional day decreaseswith the number of days considered (see Figure 5). Similarly, the marginal gain of increasing the resolution alsodecreases as the resolution increases (this can be seen by connecting the points corresponding to the same numberof representative days). Therefore, there will be a trade-off between the number of representative days and theresolution.

From these curves, the efficient frontier can be deduced, i.e., a collection of points for which the highest accuracycorresponding to a specific number of time slices is obtained. From Figure 10, it can be seen that a reasonablenumber of representative days should be prioritized to using a high resolution. Only once a reasonable numberof days is obtained (i.e., the marginal value of increasing the number of days is sufficiently reduced), increasingthe resolution becomes relevant. Based on these results for a single region, when the number of time slices of theplanning model is restricted to a value below 72, it is advisable to use a low resolution, such that a sufficientlyhigh number of days can be taken into account. Within the range of 72 to 288 time slices, using a number ofrepresentative days with a 2-hourly resolution is shown to be optimal. For a higher number of time slices, theresolution can be further increased to hourly values.

While the resolution is shown to have a limited effect on the approximation of the duration curves of the originaltime series, this is not necessarily the case for the other error metrics. Table 2 shows the impact of the resolutionon the different error metrics if all days of the original time series are selected. This gives an indication about theimpact of lowering the resolution on the different error metrics. As can be observed, the resolution predominantlyinfluences the quality of the approximation of the short-term dynamic aspects.

7In terms of the optimization procedure, using a different temporal resolution for the representative days only impacts the inputparameter Ap,b,d

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12 24 48 72 96 144 192 288 384 576 768 1152 23040

1

2

3

4

5

time slices [#]

NRM

SE

OP

avg[%

]0.25h 1.00h 2.00h 4.00h

Figure 10: Average normalized root-mean-square error (NRMSE) of the approximation of the duration curves of theoriginal time series for an increasing number of time slices. For each resolution, a varying number of representativedays are selected, yielding a total number of time slices. A trade-off between the number of representative days and theresolution of the selected days can be observed.

Error Resolution [h]Metric 0.25 1 2 4

NRMSEOPavg 0 0.123 0.346 1.076

NRMSEDPavg 0 1.120 1.664 2.428

CEavg 0 0.001 0.003 0.009

Table 2: Impact of the temporal resolution on the different error metrics in case all days of the original time series areselected.

4.5 Optimality and computational time

As mentioned in Section 3.5, the optimality gap (or MIP gap) below which the optimization is terminated is setto 1%. For a minimization problem, the optimality gap at a specific time is defined as the relative gap betweenthe highest valid lower bound, and the lowest upper bound of the problem found at that time. This upper boundcorresponds to the best feasible solution found at that time (the so-called incumbent), while the lower boundcorresponds to the lowest objective function value obtained in all the leaf nodes of the branching tree (so-calledbest bound).

0 2 000 4 000 6 000 8 000 10 000 12 000 14 000 16 000 18 000 20 000 21 6000

2

4

6

8

·104

Computation Time [s]

ObjectiveValue[-]

Incumbent Best Bound

20

40

60

80

100

OptCr

OptimalityGap

[%]

Figure 11: Overview of the solving process for a model run aiming to select 8 representative days. The figure shows theevolution of the incumbent solution, the best bound and the optimality gap as a function of the computation time.

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Figure 11 displays the evolution of the incumbent, the best bound and the optimality gap as a function of thecomputing time for a run selecting 8 representative days.

The incumbent objective value is shown to decline rapidly in the beginning but relatively quickly starts tostagnate. For this case, only marginal improvements of the objective values can be observed after 1000 seconds.On the other hand, the best bound increases only very gradually. Indeed, only if a high number of integer variablesis forced to take integer values in the relaxed problem (i.e., if these variables are branched upon), the best boundwill start to increase. However, the size of the branching tree grows exponentially with the number of variablesthat are branched upon. As a result, it is difficult to find good values for the best bound, and therefore, toguarantee that a feasible solution can be considered optimal. However, it must be realized that obtaining theoptimal solution is not a necessary requirement for this model. The goal is to provide a set of representative days,and corresponding weights, that give a good approximation of the different temporal aspects. In this regard, theabove presented results have shown that the developed model is capable of obtaining a good set of representativedays. Nevertheless, further research will be devoted to investigate alternative model formulations in order todecrease the computational cost of the presented model.

5 Summary and conclusions

Bottom-up long-term energy-system planning models are often used to analyze potential transition pathways forthe evolution of the energy system. These models typically span a long time horizon, a large geographical area,and consider a large set of energy technologies. To maintain computational tractability, the level of temporal,geographical and technical detail used in these models is limited. In terms of the temporal dimension, intra-annualvariations in demand and supply are typically represented by using a number of so-called time slices.

Capturing the inherent variable character of intermittent renewable energy sources (IRES) in this limitednumber of time slices is challenging. One approach to do this, is by using a small set of days to represent the entireyear (so-called representative days).

In this paper, we define different temporal aspects that would ideally be captured by the set of representativedays. First, the set of representative days should provide a good approximation of the average value of each timeseries, and the distribution of values over the year. Second, these representative days should capture the dynamicfluctuations of each time series on different time scales. Finally, also the correlation between different time seriesshould be reflected in the set of representative days. For each of these aspects, error metrics are introduced toevaluate how well a specific set of representative days approximates these aspects.

A novel model is presented that optimizes the set of representative days, and corresponding weights, in orderto capture these different temporal aspects. The number of representative days can be specified freely by the user.The problem is formulated using mixed integer linear programming.

The performance of the presented model is evaluated, and compared to a simple heuristic approach to selectrepresentative days, and to an approach that randomly selects sets of representative days. Compared to theseapproaches, the presented model is shown to obtain significantly better results.

The impact of explicitly accounting for the short-term dynamic variations of the different time series is ana-lyzed, and shown to have a limited impact. On the other hand, explicitly accounting for the correlation betweendifferent time series when selecting representative days, is shown to have a significant contribution to capturingthe correlation between different time series.

Finally, the model has been used to analyze the trade-off between the number of representative days, and thetemporal resolution. The results indicate that having a sufficiently high number of representative days (that reflectdifferent situations) should be prioritized to having a high resolution. However, once a sufficiently high number ofdays (8-24) is obtained, it becomes worthwhile to start increasing the resolution.

Further research involves analyzing the amount of representative days that are required in investment planningmodels to obtain good results.

Acknowledgements

The research of Kris Poncelet is supported by a PhD grant provided by VITO. Hanspeter Hoschle holds a pre-doctoral research fellowship by the Research Foundation - Flanders (FWO) and VITO. Erik Delarue is a post-doctoral research fellow of the Research Foundation - Flanders (FWO).

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Appendix A

Nrepr Period Load Wind PV

2 Year Highest peak,lowest valley

- -

4 Year Highest peak,lowest valley

Highest and lowestav. generation

-

8 Su, Wi Highest peak,lowest valley

Highest and lowestav. generation

-

12 Su, Wi, Int. Highest peak,lowest valley

Highest and lowestav. generation

-

24 Su, Fa, Wi, Sp Highest peak,lowest valley

Highest and lowestav. generation

Highest and lowestav. generation

Table A1: Overview of the simple heuristic used to select a number of representative days. The total number of daysselected using the heuristic approach is presented in the utmost left column. These days are obtained by selecting forevery period of the year, the days corresponding to the criteria presented in the 3rd to 5th column.

20