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Scenario generation of aggregated wind, photovoltaics and small hydro production
for power systems applications
S. Camala, F. Tengb, A. Michiorria, G.Kariniotakisa, L. Badesab
aMINES ParisTech - PSL University, Center for processes, renewable energies and energy systems (PERSEE), Sophia Antipolis, France
bImperial College London, Electrical and Electronic Engineering Department, London, SW7 2AZ, UK
Abstract
This paper proposes a methodology for an efficient generation of correlated scenarios of Wind, Photovoltaics (PV) and
small Hydro production considering the power system application at hand. The merits of scenarios obtained from a
direct probabilistic forecast of the aggregated production are compared with those of scenarios arising from separate
production forecasts for each energy source, the correlations of which are modeled in a later stage with a multivariate
copula. It is found that scenarios generated from separate forecasts reproduce globally better the variability of a
multi-source aggregated production. Aggregating renewable power plants can potentially mitigate their uncertainty
and improve their reliability when they offer regulation services. In this context, the first application of scenarios
consists in devising an optimal day-ahead reserve bid made by a Wind-PV-Hydro Virtual Power Plant (VPP). Scenarios
are fed into a two-stage stochastic optimization model, with chance-constraints to minimize the probability of failing
to deploy reserve in real-time. Results of a case study show that scenarios generated by separately forecasting the
production of each energy source leads to a higher Conditional Value at Risk than scenarios from direct aggregated
forecasting. The alternative forecasting methods can also significantly affect the scheduling of future power systems
with high penetration of weather-dependent renewable plants. The generated scenarios have a second application here
as the inputs of a two-stage stochastic unit commitment model. The case study demonstrates that the direct forecast of
aggregated production can effectively reduce the system operational cost, mainly through better covering the extreme
cases. The comprehensive application-based assessment of scenario generation methodologies in this paper informs the
decision-makers on the optimal way to generate short-term scenarios of aggregated RES production according to their
risk aversion and to the contribution of each source in the aggregation.
Keywords: Ancillary Services, Chance-constrained Optimization, Forecasting, Renewable Energy, Scenarios,
Stochastic Scheduling
1. Nomenclature
1.1. Indices
agg Level of aggregation in the VPP
sIndex of energy source type in the VPP
da Day-ahead variable
gIndex of thermal generators
rt Real-time variable
↑,↓Upward,downward regulation
ωIndex of production scenarios
IThis paper was submitted for review on 03.07.2018. The re-
search was carried as part of the European project REstable (Ref-
erence Number 77872), supported by the ERA-NET Smart Grids
Plus program with financial contribution from the European Com-
mission, ADEME, Juelich Research Center, Fundacao para a Ciencia
e a Tecnologia
Email address: simon.camal@mines-paristech.fr (S. Camal)
1.2. Sets
cCoefficients of the objective function
GSet of thermal generators
OSubset of slow thermal generators
πPrices for energy and reserve
1.3. Variables
aRReserve activation probability
bBinary variable for ramp function
BVector of decision variables for bidding
δgradient indicator function for Brier Score
∆R↑,−Deficit of upward reserve in real-time
∆R↓,−Deficit of downward reserve in real-time
Eda Energy on the day-ahead market
Ert,+Energy surplus on the real-time market
Ert,−Energy deficit on the real-time market
l
l
Preprint submitted to Applied Energy April 12, 2019
FCumulative distribution function
mMarginal distribution function of a copula
oTObjective function for risk-neutral bidding
oβ,T Objective function for risk-averse bidding
ψPositive variable for ramp function
PLS Load-shedding
PgPower produced by generator g
PRC RES curtailment
Rda Reserve on the day-ahead market
Rrt Reserve on the real-time market
Σ Covariance matrix of Normal variables
uVector of uniform i.i.d. variables
UGeneralized cumulative distribution function
XVector of features for production forecasting
Yagg Power production at aggregation level agg
ygBinary variable, commitment decision
for generator g
zgStartup state of generator g
1.4. Parameters
βAversion to risk on revenue
cLS Load-shedding cost
cg
mMarginal cost of generator g
cg
nl No-load cost of generator g
cg
st Startup cost of generator g
Confidence parameter of chance constraints
kHorizon of forecast
Ω Number of scenarios
pωProbability of occurrence for scenario ω
Pg
max Rated power of unit g
Pg
msg Minimum stable generation of unit g
Pg
rd Maximum ramp-down capability of unit g
Pg
ru Maximum ramp-up capability of unit g
PD
ω,t Demand in scenario ωand time-step t
PR
ω,t Power produced by RES in scenario ω
and time-step t
tRuntime of forecast
τtDuration of time-step tin the two-stage SUC
TPeriod of bidding optimization
2. Introduction
Weather-dependent Renewable Energy Sources (RES)
constitute an increasing share of power generation, but the
uncertainty of their production raises challenges for their
interaction with power systems. In particular, the vari-
ability of aggregated RES power production must be ac-
counted for in power system applications which exploit the
complementarity between RES (e.g. bidding on electricity
markets by an aggregator operating RES plants), or con-
sider the systemic impact of aggregated RES plants (e.g.
scheduling of power systems with high RES penetration).
The complementarity between RES is a subject of in-
terest in applications related to the provision of Ancillary
Services (AS). Historically, only conventional dispatchable
power plants provided AS, but recent studies have shown
that RES plants of various sources have the technical ca-
pacity to offer AS, namely Wind [1], run-of-river hydro [2],
and Photovoltaics (PV) [3]. However the uncertainty as-
sociated with the production of a single RES plant is too
large to provide AS with the high level of reliability re-
quired by system operators. A first solution to increase the
reliability of AS delivered by RES is to combine RES plants
with Energy Storage Systems (ESS) which can compensate
forecast error, but with a high investment cost in the case
of static storage [4]. A second solution is to aggregate RES
plants blending energy sources and weather conditions, in
order to reduce the uncertainty of the total production
[5]. The aggregated plants are coordinated via a Virtual
Power Plant (VPP) which ensures sufficient regulation ca-
pacity to comply with the stringent pre-qualification tests
for the provision of AS [6]. Such a VPP can integrate other
agents like flexible consumers or ESS from static storages
and electric mobility [7],[8], [9]. Following the ongoing
development of short-term markets for AS [10], method-
ologies for the optimal offer of energy and reserve have
been proposed for wind farms [11],[12], microgrids [13],
and aggregated flexible loads [14] [15]. The optimization
models of these methodologies rely on probabilistic fore-
casts or trajectories of RES production. Adding chance
constraints to the optimization makes it possible to bal-
ance revenue with the technical risk of not supplying re-
serve [14]. The realization of these constraints depends
on scenarios of RES production, reproducing the expected
uncertainty at short-term. In contrast with the reviewed
works, which considered single energy sources or several
sources independently, the present paper investigates how
to forecast the expected interaction between multiple RES
at short-term and quantify the associated impact on AS
bidding.
The aggregated production of weather-dependent re-
newable power plants plays also a major role in the schedul-
ing of electrical networks. When scheduling electrical sys-
tems, operators decide the commitment decision of gener-
ators to meet the demand with minimal cost. At present,
AS are scheduled following deterministic rules that involve
imposing pre-defined requirements to tackle the uncertain-
ties associated with forecasting error and equipment out-
ages. As the uncertainty introduced by the total gen-
eration of weather-dependent renewable power plants is
much more significant than that of demand, scheduling
processes performed under deterministic rules become in-
efficient [16] [17] [18] [19]. In contrast, stochastic unit com-
mitment models employ scenarios to model the uncertainty
of weather-dependent renewables [20] [21]. These models
ensure economic efficiency and network security.
In both applications introduced above, short-term de-
cisions are derived from a stochastic optimization which
relies on scenarios of aggregated RES production as in-
puts. It is known that the quality of the scenarios has
a direct impact on the performance of the optimization
model [21]. In the case of short-term horizons, scenar-
ios should combine two properties: reproduce the interde-
2
pendence in the aggregated renewable production process,
and vary as a function of the influence of explanatory vari-
ables such as weather forecasts. The interdependence in
RES production comprises classically the temporal depen-
dencies between successive lead-times, which is generally
not captured by production forecasts [22], and the possi-
ble correlation between power plants. A specific challenge
for scenarios of a multi-source RES aggregation is to model
the dependencies between energy sources, for simultaneous
and successive lead-times, while preserving the conditional
response to explanatory variables.
A popular method to generate scenarios is to build a
multivariate Gaussian copula from probabilistic forecasts
of the marginal distributions (e.g. production of a given
wind farm at different lead times [22], or production of
several PV plants at different lead times [23]). Copulas
are flexible tools to model dependencies in uncertainties,
although in problems focused on extreme regions of the
marginal distributions, analytical models using exponen-
tial functions may be more appropriate [22]. Vine copulas,
which form flexible trees of bivariate copulas, have been
successfully used for the probabilistic forecast of multi-
ple Wind [24] and PV [25] plants. However to the au-
thors’ knowledge, neither Gaussian copulas nor Vine cop-
ulas have been reported in the literature in the context
of scenarios for a multi-source RES aggregation. Hierar-
chical copula models, which base on independent forecasts
of each contributor of an aggregation at several hierarchi-
cal levels, have been proposed in the context of electricity
demand [?] and insurance exposure [26]. While these ap-
proaches are effective on aggregations which follow natu-
rally a tree structure such as electricity networks, their ap-
plication on a multi-source RES aggregation which has no
obvious hierarchical structure would require a large num-
ber of permutations to assess all possible sums between
plants or sources.
Alternative methods to generate scenarios exist. Time
series analysis can derive spatio-temporal models for re-
newable power plants at multiple sites [27]. The machine
learning approach of [28] builds an iterative neural network
that outputs scenarios by random generation of errors and
step-ahead forecasts. For stochastic scheduling of power
systems with high penetration by wind power, [20] develop
a multi-stage scenario tree based on the distribution of
wind forecasting errors. In [21], scenarios of wind forecast-
ing errors are generated by a Levy α-stable distribution.
These approaches are based on deterministic forecasts of
production, which is valid for a single energy source but
would miss important interdependences of uncertainty be-
tween energy sources. The Generative Adversarial Net-
works of [29] produce trajectories with adequate diversity
and similar statistical properties to historical data thanks
to the ability of this unsupervised deep learning to learn
complex non-linearities and classify large inputs. The min-
imization of the Wasserstein distance between the genera-
tor and discriminator provides good climatological proper-
ties of the scenarios, and could be applied to multi-source
RES aggregations. However, the solution of [29] proposes
only a classification of typical conditions (e.g. scenarios
for a sunny winter day). This is suitable for long-term
scenarios but not directly applicable for short-term scenar-
ios where scenarios should reflect the expected conditions
such as weather forecasts.
In the presented literature few efforts have been made
to assess how different scenario generation approaches im-
pact the results of the optimization. More specifically,
the use of aggregated, multi-source RES trajectories in
stochastic optimization has not been frequently investi-
gated. Uncertainty in production is of particular interest
when aggregating power plants of different energy sources,
possibly subject to different climates. The present pa-
per addresses the challenge of generating scenarios of the
output of a mix of power plants harvesting different en-
ergy sources. The dimension of the forecasting problem
is significantly more complex than forecasting single en-
ergy sources. Furthermore, we investigate which scenario
generation method is most suitable considering the frame-
work of the applications, with examples for AS bidding
and power system scheduling.
The framework of the paper, illustrated in Figure 1, is
structured as follows: we start by forecasting the produc-
tion of a portfolio of PV, Wind and small Hydro plants,
separately for each source and directly at the aggregated
level. From these probabilistic forecasts we generate sce-
narios by applying multivariate copulae on the available
marginal distributions over the range of lead-times, and
over the range of sources in the case of separate forecasts.
The Gaussian copula is first applied as a standard spatio-
temporal dependence model, and a Vine copula is pro-
posed to model the dependencies between separate fore-
casts with more flexibility. Then, these scenarios are ap-
plied to two relevant case studies, in which the solution
of the stochastic optimization is highly dependent on the
characterization of the uncertainty. The first case study is
the day-ahead bidding of energy and automatic Frequency
Restoration Reserve (aFRR) by a renewable VPP using
a two-stage stochastic optimization. In the second case
study, we schedule a network with significant penetration
of Wind, PV and small hydro.
The key contributions of this work are identified as
follows:
1. Firstly, this paper addresses the problem of generat-
ing probabilistic forecasts for the aggregated power
of a set of renewable power plants harvesting dif-
ferent energy sources. Despite abundant works in
the literature on models that separately forecast PV,
Wind or Hydro, the problem has not been tackled
jointly.
2. Based on probabilistic forecasts of either the indi-
vidual energy sources or the aggregation, this paper
proposes to generate scenarios of the aggregated pro-
duction, with a multivariate Gaussian copula taking
into account correlations.
3
3. In addition, to evaluate the statistical performance
of the forecasting methods, this paper compares the
methods in two particular applications, reserve bid-
ing and system scheduling. The problem of reserve
bidding is formulated as a chance-constrained opti-
mization to include a maximum frequency of reserve
under-fulfillment. A two-stage stochastic unit com-
mitment model is applied in this paper to evaluate
the impact on the system operational costs.
4. The comprehensive application-based assessment of
scenario generation methodologies informs the deci-
sion makers on the optimal way to generate short-
term scenarios of aggregated RES production ac-
cording to their risk aversion and to the contribution
of each source in the aggregation.
The methodology developed for the probabilistic fore-
casting of weather-dependent RES production is presented
in Section 3. Section 4 develops the methodology for gen-
erating scenarios. Then, we develop the evaluation frame-
work for two applications: a reserve bidding optimization
problem in Section 5, and a stochastic unit commitment
problem in Section 6. We present the results in Section 7
and conclude in Section 8.
3. Probabilistic Forecasting of Weather-dependent
Production
The production of weather-dependent renewable power
plants is forecast on a day-ahead horizon with two possi-
ble levels of aggregation: (1) plants of same energy source
and (2) all plants of possibly different energy sources lo-
cated under a common perimeter (e.g. the portfolio of an
aggregator, or a control area). Among the various exist-
ing approaches for the density forecast of renewable pro-
duction, we choose the Quantile Regression Forest (QRF)
model. This is a validated nonparametric model, which
can achieve adequate performance in terms of reliability
[30] and global forecasting skill [31].
The production forecast of an aggregated portfolio agg
at runtime tand horizon kis given in (1) from the explana-
tory variables of all plants under the aggregation level cho-
sen Xagg
t. Features of Wind and PV plants comprise day-
ahead Numerical Weather Predictions (NWP) retrieved at
each power plant site. In the case of run-of-river hydro,
we consider NWP and past production values. Hydro pro-
duction levels in recent weeks inform about features that
are independent from day-ahead weather forecasts, such as
the hydrological conditions of the river or the availability
of the plant. Learning in the QRF model is mainly influ-
enced by three hyper-parameters: the number of variables
randomly selected mtry, the number of trees δand the
minimum number of observations in leaves minobs. The
number of selected variables mtry is kept to 1/3 of the
available variables. The number of trees δmust be sev-
eral hundreds (at least 500) to make sure that important
Figure 1: Methodology framework: aggregated production scenarios
are generated and evaluated on a stochastic optimization problem
(here chance-constrained reserve bidding)
variables are selected a few times. The minimum num-
ber of observations per leaf minobs is kept to 10 to avoid
overfitting. Finally, the aggregated production behaves
differently according to the horizon (e.g. diurnal presence
of PV, intra-day wind patterns...), so the QRF model is
trained on multiple horizon intervals.
ˆ
fYagg
t+k|Xagg
t=QRF (Yagg
t+k, Xagg
t, mtry, δ, minobs) (1)
In the case of direct aggregated forecast, the reliability
is improved by adding variables to the marginal features
4
corresponding to each plant of the aggregation: (1) mini-
mum and maximum values of variables across plants shar-
ing the same source (e.g. minimum and maximum wind
speed) help the QRF model explore more robustly the vari-
ance of the aggregated production process, (2) lagged val-
ues of marginal features add information on the weather
patterns around the time of prediction or on persistent be-
haviour of production for a given energy source. In the case
of separate forecasting by energy source, a preprocessing
treatment of the production variable Yagg
t+khelps reduce the
impact of deterministic trends on learning and obtain con-
ditional variances independent of conditional means [22].
The treatment is specific to each energy source and is car-
ried out before summing up the power production data: for
Wind we apply the logit transform [22], while for PV we
normalize production with a simple analytical model for
the top of atmosphere global horizontal irradiance. The
logit transform could also been applied to Hydro (by anal-
ogy with Wind as a double-bounded production that is
non-linearly dependent on an inflow). However, in this
case the hydro production is not explained by a water
flow forecast but rather by features mixing production and
weather variables, meaning there is no homogeneity in the
non-linear relationships between features and production,
and therefore logit transform is not employed.
At the end of this first step of the methodology, we
obtain at runtime t, two sets of forecasts for all horizons
k∈[1, K] :
1. The total aggregated production for all plants, ˆ
FYtotal
t+k;
2. The production of plants sharing the same energy
source, namely Wind, PV and run-of-river Hydro,
ˆ
FYW ind
t+k,ˆ
FYP V
t+k,ˆ
FYHydro
t+k.
4. Generation of Production Scenarios
The probabilistic forecasts obtained in the previous
section inform us about the predicted levels of produc-
tion and their relative uncertainty. However simple Monte
Carlo sampling on the predictive densities obtained does
not consider correlations resulting thus in non realistic al-
ternative forecast scenarios with close horizons or spatial
correlations between plants or energy sources. In contrast,
adding a model of the correlations between the predictive
densities, such as a multivariate copula [?], produces sce-
narios with more realistic behavior with respect to the real
production patterns.
4.1. Multivariate Copula based on Probabilistic Forecasts
The multivariate copula is a multivariate distribution
function, the marginals of which should be distributed uni-
formly in the rank domain [23]. The marginals are predic-
tive densities of the production for specific dimensions of
the problem, for instance the various horizons or the differ-
ent energy sources of the aggregation. Due to the bound-
edness of intermittent production, the forecasted distribu-
tion function ˆ
FYof an intermittent production Yis not
strictly monotonous, hence a given power observation yobs
can be associated with several quantile values ˆ
FY(yobs),
for instance if yobs is a wind power observation occurring
at wind speeds over the nominal speed value. In order to
obtain marginals uniformly distributed from intermittent
production forecasts, we apply to each forecast the distri-
butional transform developed by Ruschendorf [32], which
generalizes in (2) the property of uniform distribution to
discontinuous Cumulative Distribution Functions (CDFs).
U(y) = ˆ
FY−(y) + V(ˆ
FY(y)−ˆ
FY−(y)) (2)
where Vis a random variable following the uniform dis-
tribution and ˆ
FY−is the left-hand CDF of the production
variable Y. Note that U(y) = ˆ
FY(y) if the CDF is contin-
uous. We can now construct our multivariate copula from
the transformed marginal distributions.
At this point, we propose two distinct methods. The
first method, entitled ”Direct Gaussian” (DG), constructs
a multivariate temporal dependence model between the
variables Ytotal
k, which represent the values taken by the
total aggregated production at the successive horizons k∈
[1, K]. We start by collecting a series of observed aggre-
gated productions ytotal
t+k, not included in the training set
of the forecasting model, for all horizons k. The position
of these observed productions in the forecast distribution
ˆ
FYtotal
t+kat each horizon kis evaluated according to (2), and
constitutes a realization utotal
t+kof the uniformly distributed
marginal Utotal
k.
utotal
t+k=Utotal
k(ytotal
t+k),∀t, ∀k(3)
All marginals are then converted in (4) into normally dis-
tributed variables Ztotal
kusing the probit function Φ [22],
forming a multivariate variable Ztotal normally distributed
with a zero mean vector and covariance Σtotal of dimension
K×K.
ztotal
t+k=Ztotal
k(ytotal
t+k) = Φ(utotal
t+k),∀t, ∀k(4)
Ztotal = (Ztotal
1, Ztotal
2, ..., Ztotal
k, ..., Ztotal
K) (5)
Ztotal ∼ MV N (0,Σtotal) (6)
The density of Ztotal forms a Gaussian copula parametrized
by the mean vector and the covariance matrix, which is
computed here as the empirical covariance matrix on the
observed normally transformed marginals. A final step
consists in drawing samples from the copula to generate
trajectories which reproduce the temporal correlation be-
tween horizons. To generate Ω distinct scenarios of to-
tal aggregated production for a given period of interest
[t+ 1, t +K] we:
1. Draw Ω i.i.d.random vectors sωfollowing the uni-
form distribution U(0,1)K, where ω∈[1,Ω];
2. Convert them into realizations ztotal
ωof Ztotal;
3. Generate trajectories ˆytotal,DG
ω,t+kfor the period of in-
terest by applying in (7) the quantile values given by
each zωto the marginal forecasts ˆ
FYtotal
t+k:
5
ˆytotal,DG
ω,t+k=ˆ
F−1
Ytotal
t+k
(ztotal
ω,t+k)∀ω, ∀t, ∀k(7)
A second method is based on the separate forecasting
by energy source instead of the direct forecasting of the
total aggregated production. This approach, entitled ”In-
direct Gaussian” (IG), models the dependencies between
productions of the different energy sources, over all hori-
zons. The observed productions are now collected and
aggregated for each energy source separately, resulting in
an observation vector ysour ces
t+kof dimension S, S being the
number of sources (in the present case S=3 with Wind, PV
and small Hydro). The multivariate copula of the variable
Zsources is constructed in the same way as for the DG
method, giving a covariance matrix Σsources of dimension
SK ×SK. Scenarios are generated by sampling the covari-
ance matrix and affecting the resulting quantiles zsources
ω
of dimension SK to the probabilistic forecasts of the re-
spective energy sources for the period of interest. Lastly
the obtained equiprobable trajectories are summed across
all sources in (8) to form Ω trajectories of total aggregated
production.
ˆytotal,IG
ω,t+k=X
s=[1,S]
ˆ
F−1
Ys
t+k(zsources,s
ω,t+k)∀ω∈[1,Ω],∀t, ∀k(8)
A variant of this method, entitled ”Indirect Vine” (IV),
consists in replacing the Gaussian copula by a regular Vine
copula to model non-Gaussian dependencies between hori-
zons and energy sources. A regular Vine copula is formed
sequentially by joining bivariate copulas into trees. The
selected tree among all possible combinations is the tree
that maximizes the sum of empirical rank correlations over
the possible pairs (Maximum Spanning Tree algorithm, see
[24]). To generate a number of scenarios Ω from the Vine
copula at horizon t+k(omitted in notations below for the
sake of simplicity) we:
1. Draw Ω i.i.d random vectors sωfollowing the uniform
distribution U(0,1)SK
2. Retrieve the uniform marginal CDF value mω,d, d ∈
SK of the production variable Yd, conditioned by
the other variables, by inverting the h-function of
the Vine copula [24]
mω,d =ˆ
F−1
d|d−1,...,1(sω,d|mω ,d−1, ..., mω,1) (9)
3. Invert the CDF of the marginal production variable
ˆ
F(d)
Yto obtain the production trajectory.
yω,d =ˆ
F−1
Yd(zω,d) (10)
In the next section, we evaluate the quality of the trajec-
tories of total aggregated production obtained by direct
aggregated forecasting and separate forecasting by energy
source.
4.2. Evaluation of Trajectories
The generated trajectories must reproduce correlations
between horizons, locations and energy sources. We assess
the quality of trajectories by a proper score, the Variogram-
based score (VS) [33], to determine whether trajectories
reproduce correctly the main moments of the original pro-
duction [23]. The VS of order γcan be expressed in (11) as
the quadratic difference between the Variogram of the orig-
inal production data yand the Variogram of the forecast
trajectories ˆyω t. The latter is approximated by the mean
of the score over the scenarios. Here, pairs of points are
equally weighted, wij = 1. Points with a low correlation
and thus a low signal-to-noise ratio are therefore not pe-
nalized [33]. The discrimination ability of the score could
be lower than with a correlation model fitted on data, but
we choose to use equal weights to investigate the whole
range of correlations including long intervals (production
gradients over several hours are an important input for
reserve bidding).
V S(γ)
t=X
i,j∈M
wij (|yt,i −yt,j |γ−
1
ΩX
ω∈[1,Ω]
(|ˆyωt,i −ˆyωt,j |γ))2(11)
Beyond similarity in the trajectory distributions, tra-
jectories should also exhibit characteristic events of the
original time series, such as gradients. Gradients up to a
few hours are of particular interest when offering reserve
capacities: the validity period (contracted duration of ca-
pacity) of secondary reserve (aFRR) goes from 15 min in
the Netherlands to 1h in Portugal and 4h in Germany [34].
We evaluate the similarity of gradients observed in the
original time series with gradients in scenarios by means of
a Brier Score (BS) defined in (13). The events considered
in the score are production gradients δtover an interval
∆t, which are higher than a threshold value r, taken as
the average observed gradient over the interval.
δt(y; ∆t) = 1(|yt+∆t−yt| ≥ r) (12)
BS =1
T
T
X
t=1
(1
ΩX
ω∈[1,Ω]
δt(ˆyω; ∆t)−δt(y; ∆t))2(13)
The generated scenarios are now applied to two stochas-
tic optimization problems, reserve bidding and network
scheduling.
5. Case Study 1: Day-ahead Bidding of Energy
and Automatic Frequency Restoration Reserve
A VPP aggregating Wind, PV and small Hydro power
plants jointly offers energy and symmetrical automatic
Frequency Restoration Reserve (aFRR) on a day-ahead
market. It earns revenues for reserve capacities and energy
activated for upward reserve (which could not be sold in
the energy market), while it pays the grid operator for the
activated energy during downward reserve (to compensate
for the energy not delivered). We consider no uncertainty
6
of market conditions, in order to individuate the impact
of production scenarios on the result.
A specific market condition of this problem is the acti-
vation of the VPP: considering that the TSO activates the
aFRR under a merit-order scheme, what is the probabil-
ity of the VPP being activated and at what intensity? In
a simplified approach we assume that the VPP has sim-
ilar marginal costs to its competitors. The activation is
then modeled by an activation probability, denoted below
aR: the probability of activation equals the effective aFRR
demand divided by the tendered demand [5].
5.1. Mathematical Formulation
This bidding problem can be formulated as a two-stage
stochastic linear optimization. In a first stage, the VPP
offers volumes of energy and reserve in their respective
day-ahead (da) markets for each market time unit tof the
optimization period T. Then in a recourse stage occuring
in real time (rt), the VPP compensates its imbalances in
the energy market, delivers the requested reserve and any
penalties if it fails to do so. At this stage the decisions
of the VPP are computed for each scenario of index ω
Assuming that the VPP is price-taker and risk-neutral,
the objective function writes:
min oT=E(f(B, ω)) = X
ω∈[1,Ω]
pωf(B, ω) (14)
where the bidding net penalty for scenario ωis
f(B, ω) = X
t∈[1,T ]
[cT
da,t.Bda
t+cT
rt,t.Br t
ωt] (15)
with the following decision variables:
Bda
t= (Eda
t, R↑,da
t, R↓,da
t) (16)
Brt
ωt = (Ert,−
ωt , Ert,+
ωt , R↑,rt
ωt , R↓,rt
ωt ,∆R↓,−
ωt ,∆R↑,−
ωt ) (17)
and their corresponding costs:
cT
da,t = (−πda
E,t ,−πda
R↑,t,−πda
R↓,t)
cT
rt,t = (πrt,−
E,t ,−πrt,+
E,t , art
R↑,t.πrt
R↑,t,
−art
R↓,t.πrt
R↓,t,−πrt,−
R↑,t ,−πrt,−
R↓,t ) (18)
This problem is subject to the following constraints:
1. The simulated production of the VPP must match
the sum of energy and reserve, considering day-ahead
and real-time deviations.
Eda
t+Ert,+
ωt −Ert,−
ωt +art
R↑,t.R↑,rt
ωt
−art
R↓,t.R↓,rt
ωt =Yagg
ωt (19)
2. We add the possibility for the operator to offer less
reserve capacity than contracted: this reserve deficit
equals the deviation between day-ahead reserve offer
and real-time reserve offer
∆R↑,−
ωt =R↑,da
t−R↑,rt
ωt ∀ω,t (20)
∆R↓,−
ωt =R↓,da
t−R↓,rt
ωt ∀ω,t (21)
3. The offer is symmetrical: the upward day-ahead re-
serve equals the downward day-ahead reserve.
4. The total power offered on energy and reserve mar-
kets can not exceed the installed capacity of the
VPP.
5. The downward reserve can not exceed the energy
offered.
This problem is easily generalized into a risk-averse
formulation inserting an economic Conditional Value-at-
Risk (CVaR), where the Value-at-Risk θda is the upper
bound of revenues rhaving the probability 1 −αto be
exceeded for all scenarios. The CVaR is linear with respect
to the variables, so the problem remains linear. In this
formulation we add in (25) a non-anticipativity constraint
to ensure that the day-ahead decisions are independent
from the outcomes of the production scenarios.
max oβ,T = (1 −β).E(r(B, ω)) + (22)
β.(θda −1
1−αX
ω∈[1,Ω]
pωρω)
s.t.
θda −r(B, ω)≤ρω,∀ω∈Ω (23)
ρω≥0∀ω∈Ω (24)
Bda
ω,t =Bda
ω0,t ∀ω, ω0∈Ω (25)
(13)-(15) (26)
5.2. Chance-constrained Stochastic Optimization
A reserve offer that is unfulfilled too frequently might
be discarded by network operators. Although reserve penal-
ties may be in force, these are mainly considered to be
a dissuasive signal rather than an arbitrage opportunity.
The purely economic approach described above tends to
minimize the volume of reserve offered to hedge against
high penalties. A more balanced behavior between revenue
and risk of underfulfillment can be obtained by adding
chance constraints to the optimization problem. A solu-
tion is deemed feasible if the constraints representing the
underfulfillment have a very low probability of occurrence
over the scenario set. We add to the previous model a
chance constraint on upward reserve (27) and downward
reserve (28) to ensure that the reserve in the real-time is
at least equal to the day-ahead reserve volume (i.e. no
reserve deficit) with a probability of 1 −.
P r(∆R↑,−
ωt ≤0) ≥1−∀ω,t (27)
P r(∆R↓,−
ωt ≤0) ≥1−∀ω,t (28)
A chance constraint is difficult to solve in its general
form because it is not convex. The uncertain parameters
are here the productions of each plant. These parameters
are not normally distributed, so we can not easily con-
vert it into a second-order cone constraint by inverting
the Gaussian distribution function [13]. One option is to
7
derive the φ-divergence between the distribution of pro-
duction and the normal distribution, then apply a distri-
butionally robust chance-constrained programming model
using this φdivergence [14]. As the distributions of re-
newable production show significant divergences with the
Gaussian distribution (right skews and fat tails), we opt
for an alternative approach which is distribution-free and
scenario-oriented, i.e. the constraint is approximated by
a non-decreasing convex function [35]. The constraint is
conservatively approximated by a technical CVaR function
on the distribution of reserve deficit CVaR∆R↑,−
ωt such that:
CVaR∆R↑,−
ωt (1 −) =
inf
α(E([∆R↑,−
ωt +α]+)
)−α(29)
Then the chance constraint (27) can be conservatively
linearized as in (30). The expected value of the ramp func-
tion is obtained in (31) by averaging its value over the
number of scenarios.
E([∆R↑,−
ωt +α↑
t]+)≤α↑
t. ∀t (30)
E([∆R↑,−
ωt +α↑
t]+) = X
ω∈[1,Ω]
pω[∆R↑,−
ωt +α↑
t]+(31)
The positive ramp function appearing in (30) is ap-
proached using the big M constraint technique: we insert
binary variables bωt and positive variables ψω t for each
scenario ωsuch that :
−M.(1 −b↑
ωt)≤∆R↑,−
ωt +α↑
t≤M.b↑
ωt (32)
ψ↑
ωt =b↑
ωt.(∆R↑,−
ωt +α↑
t) (33)
−M.(1 −b↑
ωt)≤ψ↑
ωt −(∆R↑,−
ωt +α↑
t)≤M.(1 −b↑
ωt)(34)
−M.b↑
ωt ≤ψ↑
ωt ≤M.b↑
ωt (35)
6. Case Study 2: Stochastic Unit Commitment
The impact that the two forecasting methods have on
the power system operation is studied in this section. The
system operation is simulated by a two-stage Stochastic
Unit Commitment (SUC) framework. This SUC obtains
the optimal scheduling and dispatch for the system genera-
tors in a 24h-horizon, using the forecasted RES generation
scenarios as an input. For each time-step in the two-stage
SUC, the first stage corresponds to the Day-Ahead (DA)
decisions, while the second stage is the Real-Time (RT)
decision. In the DA, the commitment decision for slow
generators (which cannot suddenly start generating in RT)
is fixed for all scenarios. The RT decision corresponds to
the dispatch of online slow units, the commitment state
and dispatch of fast generators, the RES curtailment and
the load shedding. The computational efficiency of the
SUC model is a key element for its practical applications.
High-quality scenario generation may reduce the number
of scenarios that are needed to describe the uncertainty,
leading to reduced computational time for SUC.
6.1. Mathematical Formulation
The SUC minimizes the expected operational cost over
all scenarios and time-steps:
min X
t∈[1,T] X
ω∈[1,Ω]
pω
X
g∈G
Cg
ω,t +CLS
ω,t
(36)
Where the operating cost for each generator Cg
ω,t and the
load-shedding cost CLS
ω,t are defined as:
Cg
ω,t = cg
st ·zg
ω,t +τtcg
nl ·yg
ω,t + cg
m·Pg
ω,t(37)
CLS
ω,t =τt·cLS ·PLS
ω,t (38)
The problem is subject to the following constraints:
X
g∈G
Pg
ω,t + PR
ω,t −PRC
ω,t = PD
ω,t −PLS
ω,t ∀ω, t (39)
yg
ω,t ·Pg
msg ≤Pg
ω,t ≤yg
ω,t ·Pg
max ∀g, ω, t (40)
−τt−1·Pg
rd ≤Pg
ω,t −Pg
ω,t−1
≤τt−1·Pg
ru ∀g, ω, t (41)
zg
ω,t ≥yg
ω,t −yg
ω,t−1∀g, ω , t (42a)
zg
ω,t ≥0∀g, ω , t (42b)
yg
ω,t =yg
1,t ∀g∈ O,∀ω, t (43)
Constraint (39) enforces the power balance, (40) enforces
generation limits, (41) enforces ramp limits, (42a)-(42b)
define the startup state of generators and (43) is the non-
anticipativity condition, which fixes the commitment de-
cision for slow generators in the DA. Frequency response
modeling is not considered in this model, as the key objec-
tive is to analyze the quality of the generated scenario by
alternative methods. Nevertheless, it is straightforward
to include the various forms of frequency response con-
straints.
7. Evaluation of the methodology
In this section, we evaluate the proposed methodology,
starting with the forecasting of the aggregated production
and the generation of scenarios. The obtained scenarios
are then applied to the two case studies, namely reserve
bidding and stochastic scheduling.
7.1. Forecasting of Aggregated Production
We forecast the production of a following VPP for the
day ahead comprising 3 Wind farms, 3 small Hydro plants
and 9 PV farms, all located in France within the same
control area but with different climates. To assess the sen-
sitivity of our generation method to the relative propor-
tion of each energy source (Wind, PV, Hydro), we scale
the installed capacities of the farms to obtain two different
VPP configurations in Table 1. The first VPP (VPP1) is
dominated by Wind, whereas the second VPP (VPP2) is
8
Configuration Wind PV Hydro
VPP1 32 9 12
VPP2 12.4 36 4.6
Table 1: Installed capacities of VPP configurations in MW
dominated by PV. VPP2 shows the same capacity ratio
between Wind and Hydro as for VPP1.
The following NWP are retrieved from the ECMWF
forecasting center: for the geographical location of each
PV plant, surface solar radiation oriented downwards, to-
tal cloud coverage, hourly rainfall; for each Wind plant,
zonal and meridional wind speeds at 10 m; and for each
Hydro plant, daily cumulated rainfall, surface solar radi-
ation downwards, air temperature at 2 m. The produc-
tion forecast is assumed to be done before noon when the
day-ahead energy market closes, so we use NWP issued
at 00h00 of the previous day. The resulting forecasting
horizon is thus comprised between 24h and 48h. We fore-
cast the production of all plants of the same source (PV,
Wind and Hydro) with the QRF model. Then, we forecast
the aggregated production (53 MW capacity) directly, this
time without preprocessing of the production variables to
preserve coherence between energy sources.
The QRF model is trained on 6 horizon intervals of 4
hours each, and evaluated using a 7-fold cross-validation:
one day of the week is chosen alternatively as the test set
on which the forecasting is evaluated and the model is
trained on the remaining days of the week. The available
dataset consists in 200 days of power production between
June 2015 and March 2016, with a 30-min resolution for
each plant (9600 points). The reliability diagram in Figure
2 indicates that direct forecasting of the aggregated pro-
duction of VPP1 shows adequate reliability. The separate
forecasting is also reliable: the observed frequencies of the
forecast distribution transform for Hydro (see Figure 3)
are distributed almost uniformly (a similar performance is
obtained for PV and Wind, not shown).
The Root Mean Square Error (RMSE) and the Contin-
uous Ranked Probability Score (CRPS) in Table 2 indicate
that the QRF model has comparable performance with re-
spect to the state of the art in forecasting of Wind and PV
production [36]. Note that the direct forecast of the ag-
gregated production shows slightly lower errors than the
forecasts for each energy source, because the aggregated
production has a smoother profile and because the QRF
model is able to learn from explanatory variables of large
dimension, even if they reflect different energy sources.
The VPP dominated by PV (VPP2) shows lower forecast-
ing error at night because the total available capacity is
lower (no PV production).
7.2. Generation of Scenarios
The covariance matrix obtained from separate forecasts
on each energy source show in Figure 4 that the correla-
tions between energy sources are low but existing: posi-
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Nominal quantiles
Observed quantiles
Figure 2: Reliability diagram for the direct forecast of the total
aggregated production in VPP1. Consistency bars are 5%-quantile
and 95%-quantile of resampled forecasts
Nominal proportions
Observed frequencies [%]
0
2
4
6
8
10
0.0 0.2 0.4 0.6 0.8 1.0
Figure 3: Observed frequencies of distribution transform for Hy-
dro plants in VPP1. The dashed line represents an ideal uniform
distribution
VPP Level NRMSE NCRPS
- - (0-24h) (0-6h) (6-12h)(12-18h)(18-24h)
VPP1 PV 0.045 −0.046 0.048 −
VPP1 Wind 0.095 0.044 0.045 0.048 0.051
VPP1 Hydro 0.081 0.032 0.047 0.044 0.044
VPP1 Aggreg. 0.065 0.030 0.036 0.036 0.035
VPP2 Aggreg. 0.046 0.012 0.036 0.037 0.014
Table 2: Forecasting results, normalized by installed capacity.
NRMSE average on horizon, NCRPS by intervals of day
tive correlations between 0.2 and 0.5 can be observed be-
tween Wind and Hydro around noon, while PV and Wind
alternate between low positive and negative correlations.
Negative correlations are interesting for the present appli-
cation: a source ramping up (eg: Wind speed rising at a
given Wind farm) can potentially substitute a drop in pro-
duction of another source (eg: cloud passing upon a PV
plant).
9
Properties and scores of scenarios for VPP1 and VPP2
obtained by the scenarios from DG and IG methods, with
Gaussian copula, are displayed as a function of the hourly
horizon in Figure 5 (results of DG scenarios are repre-
sented with solid lines and denoted as ”aggregated”, re-
sults of IG scenarios are represented with dashed lines and
denoted as ”separate”). The mean values of scenarios are
close for both methods on VPP1 and VPP2. This proves
that the methods DG and IG are coherent, as expected:
the summed expectation of energy sources is equal to the
expectation of the sum. The amplitude of the scenario
set (difference between minimum and maximum values of
scenarios at each time step) is lower for the IG method,
for both VPPs. This is due to the fact that extreme ag-
gregated production levels observed during training have
been considered directly in the dependence model of the
DG method, whereas they are only reconstructed a poste-
riori by the IG method. In terms of bias, the IG method
is more biased around noon when PV production is max-
imum, which is probably related to a higher bias in the
separate PV forecast.
For VPP1 (wind-dominated), the average VS of sce-
narios is of similar level for both DG and IG methods.
For VPP2 (PV-dominated), the VS of the IG approach is
significantly lower, which indicates a more realistic vari-
ability of scenarios from separate forecasts by source. The
aggregated production profile depends here more on the
horizon than for the wind-dominated case. Despite the
lower performance of separate probabilistic forecasts (cf.
2), the IG method compensates with a high number of
possible combinations between sources. In contrast, the
covariance of the DG method summarizes the variability
with less versatility. In addition, the DG method ignores
the saturations occuring for each source, which creates an
underestimation of the smoothing effect.
Concerning gradients of aggregated production, the cor-
responding BS are reported in Figure 5 for intervals of
1h to 4h. The events evaluated by the BS are gradients
that exceed the average gradient values for each interval.
For VPP1, we observe that IG scores are slightly better
than DG except for the 1-hour gradient at night. The 1-h
auto-correlation is higher for Hydro than for Wind during
this period, and the IG method seems to overestimate the
weight of Hydro in this case. The auto-correlations are
lower at further lags, so this effect disappears for BS at
intervals superior to 1 h. For VPP2, where dependence
on the horizon is more pronounced, the DG method has
a better BS when production is high because it considers
more extreme production, and worse BS at night when
production is more stable.
Finally, the distribution of errors for VPP2 in Figure
6 shows that the variance of the error is slightly reduced
for the IG method compared to the DG method. In sum-
mary, scenarios from separate forecasts show better aver-
age properties than scenarios from direct aggregated fore-
cast for stochastic optimization, especially when the ag-
gregated production profile depends on the horizon (e.g.
high PV share in the aggregation).
PV.0h
PV.12h
W.0h
W.12h
H.0h
H.12h
PV.0hPV.12h W.0h W.12h H.0h H.12h
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Figure 4: Covariance matrix of the multivariate Gaussian Copula,
in the rank domain of Wind (W), Hydro (H) and PV (PV) for each
hour of the day
7.3. Case Study 1: Reserve Bidding
The optimization is computed for 100 days using price
data from Portugal, with 100 trajectories of aggregated
production of VPP1, generated from the Direct Aggre-
gated production forecast and Gaussian Copula (DG), the
Indirect aggregated production forecast and Gaussian Cop-
ula (IG) and Vine Copula (IV). The Mcoefficient is set
to a number close to the aggregated installed capacity (60
MW vs 53 MW) and is set to 1%. We consider a mod-
erately risk-averse VPP, β= 0.5. The penalties for not
supplying reserve are taken equal to two times the price
of restoration reserve (instead of 1.5 times in current Por-
tuguese rules). These high penalties lead the optimization
model to face higher penalties on the imbalance energy
market in order to be able to supply reserve.
The frequency of reserve deficit simulated by the model
is null for all scenarios, which was not the case for a model
without chance-constraints. Economic and technical re-
sults are reported in Table 3. The use of the direct ag-
gregated forecast, which generates scenarios with higher
amplitudes, reduces the amount of reserve offered to avoid
penalties for reserve underfulfillment. Considering that in
the case study the day-ahead price for energy is higher
than the day-ahead price for reserve, the average revenue
obtained in the objective function increases up to 6% when
comparing with the scenarios from separated forecasts. In
contrast, scenarios from separate forecasting give a higher
5%-Conditional Value at Risk (CVaR), up to 18%. This
is associated with a more conservative bidding in the day-
ahead market: more reserve capacity and less energy is
offered. The CVaR increases as less penalties are to be
paid for energy deficit in real-time. This is due to the
lower Mean Absolute Error (MAE) and Root Mean Square
Error (RMSE) of scenarios based on separate forecasting.
10
NRMSE
sdValue_MW
VS_mean
VS_sd
BS_3h
BS_4h
meanValue_MW
NMAE
averageAmplitude_MW
Bias
BS_1h
BS_2h
0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25
0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25
0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25
0.05
0.10
0.15
0.20
0.03
0.06
0.09
0.18
0.21
0.24
0.27
0.1
0.2
0.3
5
10
15
20
0.3
0.4
0.5
−0.01
0.00
0.01
0.02
0.05
0.10
0.15
0.20
0.25
4
6
8
5
10
15
20
0.05
0.10
0.15
0.20
0.25
0.05
0.10
Hour of Day
value
vppType
VPP1
VPP2
forecastType
aggregated
separated
Figure 5: Properties and scores of scenarios generated with Gaussian Copula, directly aggregated from the DG method (”aggregated”, solid
line) and indirectly aggregated by separate production forecast from the IG method (”separated”, dashed-line). Starting at top-left: average
Amplitude of the scenario set, in MW; average bias of scenarios, scaled by installed capacity; Brier Score for ramps on several intervals, from
1 hour to 4 hours; mean value of scenarios, in MW; Normalized Mean Average Error of scenarios (NMAE), scaled by installed capacity;
Normalized Root Mean Square Error of scenarios (NMRSE), scaled by installed capacity; standard deviation of the scenarios, in MW; mean
value of the Variogram Score with moment = 0.5; standard deviation of the Variogram Score with moment = 0.5
Lastly, the flexible dependence model of the Vine Copula
generates more extreme values of aggregated production
from the separate forecast than the Gaussian Copula. In
return, the results for the method IV is intermediary be-
tween the results of DG and IG: higher average revenue
than IG but lower CVar.
Scenarios oβ,T C V ar5% Eda Ert,−Ert,+Rda
-[€/MWh] [€/MWh] [MWh] [MWh] [MWh] [% Pn]
DG 73 48 9.21.95.26.6
IG 71 58 7.91.46.17.5
IV 69 54 7.51.46.2 6.8
Table 3: Average profits and volumes of energy and reserve for the
optimized bidding, depending on the scenario generation method
7.4. Case Study 2: Stochastic scheduling
To evaluate the impact of different forecasting methods
on power system operation, a two-step approach is applied:
1. First, the two-stage SUC is used to obtain the opti-
mal scheduling of the system for the forecasted RES-
generation scenarios. After solving the optimization,
the DA decisions are recorded.
2. Then, a real time dispatch programme is carried out
over the real RES production data, with the DA de-
cisions obtained in the first part fixed.
This two-step methodology is applied to evaluate the two
scenario generation methods, namely the scenarios issued
from aggregated wind, PV and hydro forecast proposed
in this paper and the scenarios from separate production
forecasts for each energy source. We use here the Gaus-
sian copula in both cases to generate scenarios from the
probabilistic forecasts.
A test case is designed to evaluate the two methods:
Directly aggregated with Gaussian Copula (DG) and In-
directly aggregated with Gaussian Copula (IG). The char-
acteristics of thermal generators are included in Table 4,
where Combined Cycle Gas Turbines (CCGT) are treated
as slow generators and Open Cycle Gas Turbines (OCGT)
11
12
16
20
0
4
8
−30 0 30 −30 0 30 −30 0 30
0
2500
5000
7500
0
5000
10000
15000
20000
0
5000
10000
15000
20000
0
5000
10000
15000
20000
25000
0
10000
20000
0
2500
5000
7500
Scenario Error [% Pn]
Observed Counts for VPP2
forecastType
DG
IG
Figure 6: Distribution of scenario error in MW for VPP2, for selected horizons between 0h and 20h
CCGT OCGT
Number of Units 10 5
Rated Power (MW) 50 10
Min Stable Generation (MW) 30 5
No-Load Cost cnl
g(£/h) 540 300
Marginal Cost cm
g(£/MWh) 47 200
Startup Cost cst
g(£) 1000 0
Table 4: Characteristics of thermal generators
as fast generators. The two VPPs in Table 1 are consid-
ered, while the same 100 RES-generation trajectories as in
section 7.3 are used but scaled up by a factor of 10, corre-
sponding then to installed capacities of 530MW. A scaled
demand profile corresponding to Great Britain’s equiva-
lent consumption is used, with minimum and maximum
demand of 200MW and 600MW, respectively. An MIP
gap of 0.1% is considered for the optimizations and cLS is
set to 30k£/MWh.
A simulation spanning 60 consecutive days was car-
ried out, for which the average daily net-demand (demand
minus RES generation) was of roughly 290MWh for both
VPPs. The average daily savings due to using DG when
compared to IG for the stochastic scheduling are of 1.6%
and 5.1%, for VPP1 and VPP2 respectively. Interestingly,
IG achieves slightly lower operating costs for most of the
days, but the overall advantage of DG is driven by a few
incidents that high cost load-shedding happens when us-
ing IG. This is due to the fact that the lower forecasting
variance in IG tends to lead less CCGTs to be started in
the day-ahead. Although the bias is slight higher in IG,
it is already covered, for most of the cases, by the online
CCGTs. On the other hand, DG schedules more CCGTs
due to the higher variance, and therefore typically induces
higher part-loading and higher operation cost. However,
there are few incidents with particularly low variance for
IG, in which very few CCGTs are started up in the day
ahead scheduling, and when it is combined with large pos-
itive error (much lower RES generation realised than fore-
casted), even to start up all OCGTs in the real-time op-
eration is not sufficient to maintain the system operation
without costly load-shedding. In summary, for the point
of system scheduling, the main advantage of DG is from
its capability to reduce costs while at the same time cover
the extreme errors in the generated scenarios.
8. Conclusion
A QRF model directly derives a short-term probabilis-
tic forecast of the aggregated production of dispersed Wind,
PV and small hydro plants. The same model applied to a
separate forecasting of the energy sources composing the
aggregation results in higher forecast errors than the di-
rect forecast of the aggregated production. This is due to
the efficient learning capacity of the QRF in high dimen-
sion (many explanatory variables from multiple produc-
tion sites), but also to the smoothing effect of aggregation
which reduces the variability and intrinsically reduces the
12
uncertainty in production. An extension of this work could
consist in forecasting the multi-source aggregated produc-
tion with a deep quantile regression approach, with the
objective to outperform the performance of the machine
learning model used here and scale to potentially larger
aggregations.
Scenarios of aggregated production have been gener-
ated from these probabilistic forecasts. The impact of
choosing either a direct forecast of the aggregated pro-
duction or a separate forecast for each energy source is
assessed on two power system applications requiring such
scenarios. A multivariate copula, of type either Gaussian
or Vine, is applied to obtain scenarios from the day-ahead
production forecasts. The generation methods are evalu-
ated on two different aggregations with different capacity
shares for the different energy sources.
It is found on one hand that scenarios from the ag-
gregated forecast have a higher amplitude, because the
scenario generation method incorporates directly past ex-
treme levels of aggregated production, meanwhile scenar-
ios from separate forecasts reconstruct extreme aggregated
levels from the marginal contributions of each source in the
aggregation. On the other hand, scenarios issued from the
separate forecasting of each source reproduce more accu-
rately the variability of the aggregated production when
the production is highly dependent on the horizon, for in-
stance when PV is dominant in the total production. This
is quantified by a sensible gain on the Variogram Score for
the PV-dominated aggregation, whereas the gain is close
to zero for a Wind-dominated aggregation. This is due to
a better reproduction of the various conditions leading to
the observed smoothing effect in aggregated production.
In this case the available information about the variability
for each source is preserved, whereas a given level of to-
tal production can arise from multiple distinct production
patterns at the level of each source, hence loosing infor-
mation on the diversity of production profiles.
When used in both stochastic applications of reserve
bidding and unit commitment, scenarios from direct ag-
gregated production forecast generate more average value
(increased profit or reduced costs) than those from sepa-
rate forecasts, because these applications are sensible to
the amplitude of the scenario set. Extreme levels of aggre-
gated production are more present in the scenarios from
direct aggregated production forecast, which can secure
highly risk-averse decisions (e.g. unit commitment under
extreme RES aggregated production) but also hinders de-
cisions that could be valuable for the agent (e.g. scenar-
ios of high amplitude will limit the offer of reserve of an
aggregator, with a possible opportunity cost if activated
reserve would have increased his revenue). Finally, a mod-
erately risk-averse decision maker will observe that scenar-
ios generated from separate forecasts create less penalties
due to their sharper distribution and more realistic vari-
ability: an aggregator bidding AS and energy will increase
his Conditional Value -at-Risk, and a system operator will
decrease his lower operational costs for most days of the
year. In conclusion, the assessment of scenario generation
methodologies in this paper informs the decision-makers
on the optimal way to generate short-term scenarios of ag-
gregated RES production according to their risk aversion
and to the contribution of each source in the aggregation.
9. Acknowledgments
The authors wish to thank ENGIE GREEN, HESPUL
and HYDRONEXT for providing historical production data
for the various power plants considered in the present work.
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