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Determining reserve requirements in DK1 area of Nord Pool using a probabilistic approach

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Determining reserve requirements in DK1 area of Nord Pool using a probabilistic approach

Abstract and Figures

Allocation of electricity reserves is the main tool for transmission system operators to guarantee a reliable and safe real-time operation of the power system. Traditionally, a deterministic criterion is used to establish the level of reserve. Alternative criteria are given in this paper by using a probabilistic framework where the reserve requirements are computed based on scenarios of wind power forecast error, load forecast errors and power plant outages. Our approach is first motivated by the increasing wind power penetration in power systems worldwide as well as the current market design of the DK1 area of Nord Pool, where reserves are scheduled prior to the closure of the day-ahead market. The risk of the solution under the resulting reserve schedule is controlled by two measures: the LOLP (Loss-of-Load Probability) and the CVaR (Conditional Value at Risk). Results show that during the case study period, the LOLP methodology produces more costly and less reliable reserve schedules, whereas the solution from the CVaR-method increases the safety of the overall system while decreasing the associated reserve costs, with respect to the method currently used by the Danish TSO (Transmission System Operator).
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Determining reserve requirements in DK1 area of Nord Pool using a
probabilistic approach
Javier Saez-Gallego,a, Juan M. Moralesa, Henrik Madsena, Tryggvi J´onssonb
aTechnical University of Denmark, Richard Petersens Plads , Building 322, 2800 Kgs. Lyngby, Denmark
bMeniga ehf. Kringlan 5 103 Reykjavik, Iceland
Abstract
Allocation of electricity reserves is the main tool for transmission system operators to guarantee a reliable
and safe real-time operation of the power system. Traditionally, a deterministic criterion is used to establish
the level of reserve. Alternative criteria are given in this paper by using a probabilistic framework where the
reserve requirements are computed based on scenarios of wind power forecast error, load forecast errors and
power plant outages. Our approach is first motivated by the increasing wind power penetration in power
systems worldwide as well as the current market design of the DK1 area of Nord Pool, where reserves are
scheduled prior to the closure of the day-ahead market. The risk of the solution under the resulting reserve
schedule is controlled by two measures: the Loss-of-Load Probability (LOLP) and the Conditional Value at
Risk (CVaR). Results show that during the case study period, the LOLP methodology produces more costly
and less reliable reserve schedules, whereas the solution from the CVaR method increases the safety of the
overall system while decreasing the associated reserve costs, with respect to the method currently used by
the Danish TSO.
Keywords: Reserve determination, wind power, probabilistic forecasts, loss-of-load probability,
conditional value-at-risk, Danish market
1. Introduction
Electricity is a commodity that must be supplied continuously at all times at certain frequency. When
this requirement is not fulfilled and there is shortage of electricity, industrial consumers can face the very
costly consequences of outages: their production being stopped or their systems collapsed. Households
will experience high discomfort and losses too. From a different point of view, service interruptions also
affect electricity producers as they are not able to sell the output of their power plants. Therefore, it is of
high importance that the demand is always covered. The main tool for transmission system operators to
avoid electricity interruptions is the allocation of operating reserves. In practice, scheduling reserves means
that the system is operating at less than full capacity and the extra capacity will only be used in case of
disturbances.
The term operating reserves is defined in this paper as “the real power capability that can be given or
taken in the operating time frame to assist in generation and load balance and frequency control” [1]. The
types of reserves are differentiated by three factors: first, the time frame when they have to be activated
ranging from few seconds to minutes; secondly, their activation mode, either automatically or manually;
finally, by the direction of the response, upwards or downwards. Members of the European Network for Sys-
tem Operators for Electricity (ENTSO-E) and more specifically, the Danish Transmission System Operator
(TSO), follow this classification criterion. Primary control is activated automatically within 15 seconds and
its purpose is to restore the balance after a deviation of ±0.2 mHz from the nominal frequency of 50Hz.
Corresponding author. Tel. +4545253369
Email addresses:jsga@dtu.dk (Javier Saez-Gallego), jmmgo@dtu.dk (Juan M. Morales), hmad@dtu.dk (Henrik Madsen),
tryggvij@gmail.com (Tryggvi J´onsson)
Preprint submitted to Energy June 16, 2014
Secondary control releases primary reserve and has to be automatically supplied within 15 minutes or 5
minutes if the unit is in operation. Manual reserve releases primary and secondary reserves and has to be
supplied within 15 minutes. In Denmark, this type of reserve is often provided by Combined Heat-and-Power
(CHP) plants and fast start units. The activated manual reserves are often referred as regulating power.
This paper deals with the total upward reserve requirements, namely the sum of primary, secondary and
manual reserves, neglecting the short-circuit power, reactive and voltage-control reserves. The result of the
proposed optimization models, namely the schedule of reserves, refers to the total MW of upward reserve
required. It is assumed that the reserve acts instantaneously to any generation deficit and no activation
times are considered.
Currently the provision of reserve capacity in the DK1 area of Nordpool obeys the following rules, which
can be found in the official documents issued by the Danish TSO [2]. The requirements for primary and
secondary reserve are ±27MW and ±90M W , respectively. The provision of tertiary or manual reserves
follows the recommendations in both the ENTSO-E Operation Handbook and the Nordic System Operation
Agreement [3], where it is stipulated that each TSO must procure the amount of tertiary reserves needed
to cover the outage of a dimensioning unit in the system (the so-called N1criterion), be it a domestic
transmission line, an international interconnection or a generating unit. The inspection of the historical
data reveals that this criterion roughly results in an amount of tertiary reserve in between 300 and 600 MW.
The methodologies discussed in this article are mainly targeted to the current structure of the Danish
electricity market, where reserve markets are settled independently of and before the day-ahead energy
market, implying that, at the moment of scheduling reserves, no information about which units will be
online is known. At the market closure, the TSO collects bids from producers willing to provide reserve
capacity, and selects them by a cost merit-order procedure. Most of the existing literature focuses on co-
optimizing the unit commitment and reserve requirements at the same time; such methods however cannot
be applied under the current design of the Danish electricity market.
This paper is also motivated by the increasing penetration of wind power production in Europe and, in
particular, in Denmark. As a matter of fact, the commission of the European Countries has set an ambitious
target such that the EU will reach 20 % share of energy from renewable sources by 2020, and in Denmark the
target is 30% [4]. As the share of electricity produced by renewables increases, several challenges must be
faced. Non-dispatchable electricity generation cannot ensure a certain production at all times, but instead
depends on meteorological factors. The stochastic nature of such factors inevitably leads to forecast errors
that will likely result in producers deviating from their contracted power, thus causing the system to be
imbalanced. Solutions call for methods capable of managing the uncertainty that wind power production
and other stochastic variables induce into the system.
The main contributions of this paper are the following:
1. A probabilistic framework to determine the total reserve requirements independently to the generation
power schedule in a power system with high penetration of wind production. The reserve levels in
Denmark are currently computed by deterministic rules such as allocating an amount of reserve equal
to the capacity of the largest unit online [2, 3]. Another example is the rule used in Spain and
Portugal, where the upward reserve is set equal to 2% of the forecast load plus the largest unit in
the system. These rules are designed for systems with very low penetration of renewable energy and
fairly predictable load, where the biggest largest need for reserve capacity arises from outages of large
generation units. With the increasing share of renewables (and decentralized production in general)
in the generation portfolio, renewables will naturally have a larger influence on the system imbalance.
Hence, the non-dispatchable and uncertain nature of these plants needs to be accounted for when
reserve power is scheduled [5]. Previous studies perform a co-optimization of the energy and reserve
markets, either in a deterministic manner [6] or in a probabilistic way [7, 8, 9, 10, 11, 12, 13]. However,
these methods cannot be applied directly to the DK1 area of Nord Pool since the reserve market and
the day-ahead energy market are cleared independently at different times and by different entities.
The methodology in [14] is not suitable either as the Capacity Probability Table (COPT) refers to the
units that are online; this information is not available to the Danish TSO at the time of clearing the
reserve market.
2
2. A flexible scenario-based approach for modeling system uncertainty, which takes into account the
limited predictability of wind and load, and plausible equipment failures. Moreover, the distributions
from which the scenarios are generated are time-dependent, being the distributions of the scenarios of
forecasts errors of load and wind power production non-parametric and correlated. The authors of [9]
characterize the uncertainty in the system only by scenarios of wind power forecast errors. Load and
wind generation uncertainty is described in [8, 15] by independent Gaussian distributions and not in
a scenario framework. Other authors [7, 15] use outage probabilities as a constant parameter for each
unit and for each hour. The authors of [14] represent the forecast error distributions of the load and
wind generation by a set of quantiles, assuming both distributions are independent.
3. Equipment failures are modeled as the amount of MW that fail in the whole system due to the forced
outages of generating unit. This way we can model and simulate simultaneous outages. Furthermore,
the distribution of failures is dependent on time. Existing literature takes into account just one or two
simultaneous failures [7, 8, 15] or several [14].
4. Two different methods for controlling the risk of the resulting capacity reserve schedule. The first one
imposes a target on the probability of load shedding as in [7], while the second one is based on the
Conditional Value at Risk of the reserve cost distribution. The latter method minimizes the societal
costs, while penalizing high cost scenarios given a certain level of risk aversion.
The remaining of the paper is organized as follows. Section II presents two different optimization models
for reserve determination. Section III describes the methodology to generate scenarios of load forecast error,
wind power forecast error and equipment failures, which altogether constitute the input information to the
proposed reserve determination models. Section IV elaborates on the estimation of the cost of allocating
and deploying reserves. Section V discusses the results and comments on the implications of applying the
two reserve determination models to the Danish electricity market. Conclusions are summarized in Section
VI.
2. Modeling Framework
This section presents two formulations for determining the reserve requirements in DK1, both of them
solved using a scenario-based approach. The first limits the Loss-Of-Load Probability (LOLP), while the
second one minimizes the Conditional Value at Risk (CVaR) of the cost distribution of reserve allocation,
reserve deployment and load shedding. Both models are meant to be run to clear the reserve market and
can be used by the TSO to decide on how many MW of reserve should be scheduled. In Denmark, where
the study case in this paper is focused, the reserve market is cleared previous to and independently of the
day-ahead energy market. This implies that the unit commitment problem is not addressed at the time
when the reserve market closes and thus neither is it in this paper.
2.1. LOLP Formulation
The objective is to minimize the total cost of allocating reserves,
Minimize
Ri
M
X
i
λiRi,(1a)
3
where Riis a variable representing the total amount of reserve assigned to producer iand λiis the price
bid submitted to the reserve market by this producer. Mis the total number of bids. The objective (1a) is
subject to the following constraints
RiRmax
ii(1b)
RT=
M
X
i
Ri(1c)
LOLP = Z
RT
f(z)dz (1d)
LOLP β(1e)
RT0i. (1f)
The set of inequalities (1b) indicates that the amount of reserve provided by producer icannot be
greater than its bid quantity. The total reserve to be scheduled is defined in (1c) as the sum of the reserve
contribution from each producer. The probability density function of balancing requirements is represented
by f(z), and hence the integral from z=RTto z=is the probability of not scheduling enough reserves
to cover the demand, namely the loss-of-load probability, defined in (1d). It is constrained by a parameter
target β[0,1] in Equation (1e), which is to be specified by the transmission system operator. The smaller
βis, the more reserves are scheduled, as the LOLP is desired to be small. On the other hand, if βis equal
to 1, no reserves are allocated at all.
The optimal solution to problem (1a) can be found analytically, under the assumption that the objective
function (1a) is monotonically increasing with respect to the total scheduled reserve RT(i.e., reserve capacity
prices are non-negative) and because the LOLP is a decreasing function with respect to RT(note that f(z)
is a density function and therefore, always non-negative). Indeed, under the above assumption, greater
RTimplies greater costs, thus RTis pushed as low as possible until the relation LOLP = βis satisfied.
Therefore, at the optimum, it holds that β=R
RTf(z)dz or similarly 1β=F(RT) being F(Z) = P(Zz)
the cumulative distribution function of Z(the required reserve). Finally, since βis a given parameter, then
the solution is RT=F1(1 β).
In practice, f(z) can be difficult to estimate in a closed form; one way of dealing with this issue is to
describe the uncertainty by scenarios. Let zwbe the reserve required to cover balancing needs in scenario
wand πwthe associated probability of occurrence. Then the optimal solution to problem (1a) boils down
to the quantile 1 βof the scenarios. In other words, let ˆ
F(Z) = P(Zz) be the empirical cumulative
distribution function of the set of scenarios {zw}with ˆ
F: (−∞,)(0,1), then the analytical solution is
RT=inf{z(−∞,) : (1 β)ˆ
F(z)}.
Finally, we define the Expected Power Not Served (EPNS) as the expected amount of MW of balancing
power needed during one hour which cannot be covered by the scheduled reserves. It can be computed, once
the total scheduled reserve RThas been obtained, as
EPNS = Z
RT
zf (z)dz.(2)
In the case where the uncertainty of reserve requirements is characterized by scenarios, the EPNS can
be determined as EPNS = PwS(zwRT)πw,S={wW:zw> RT}.
2.2. Conditional Value at Risk (CVaR) Formulation
The following reserve determination model corresponds to a two-stage stochastic linear program where
each scenario is characterized by a realization of the stochastic variable Zreserve requirements”. Variable
RTrepresents the amount of MW that the TSO should buy at the reserve market. In the jargon of stochastic
programming, this variable is referred to as a first stage variable, or equivalently, as a here-and-now decision,
i.e., a decision that must be made before any plausible scenario zwof energy shortage is realized. This models
the fact that reserve capacity is to be scheduled before the scenarios of reserve requirement are realized.
4
For their part, the second stage variables, or recourse variables, rT
wand Lw, are relative to each scenario w,
and represent the deployed regulating power and the MW of shed load, respectively. Consequently, during
the real-time operation of the power system, once a certain scenario wof wind power production, load and
equipment failures realizes, reserve is activated rT
wor some load is shed (Lw). In such a way, the first stage of
our stochastic programming model represents the reserve availability market and the second stage represents
the reserve activation market. Finally, the probability of occurrence of each scenario is denoted by πw.
The objective function to be minimized is the CVaRαof the distribution of total cost. By definition,
the Value-at-Risk at the confidence level α(VaRα) of a probability distribution is its α-quantile, whereas
the CVaRαis the conditional expectation of the area below the VaRα. The CVaR is known to have better
properties than the VaR [16] and hence, it is used in this paper. Parameter α[0,1) represents the risk-
aversion of the TSO, i.e. the greater αis, the more conservative the solution will be in terms of costs. The
objective is to minimize the CVaRαof the distribution of the total cost:
Minimize
RT,Rg,rT
w,rgw ,Lw,ξ,ηw,C ostw
CVaRα=ξ+1
1α
W
X
w=1
πwηw(3a)
where ξis, at the optimum, the αValue at Risk (VaRα) and ηwis an auxiliary variable indicating the
positive difference between the VaR and the cost associated with scenario w. The cost of each scenario,
named Costw, is computed in (3b) as the sum of the cost of allocating and deploying reserve capacity plus
the cost incurred by involuntary load shedding. The objective (3a) is subject to the following constraints:
Costw=
J
X
j=1
λcap
jRj+
G
X
g=1
λbal
grgw +VLOLLww(3b)
RT=
J
X
j=1
Rj(3c)
rT
w=
G
X
g=1
rgw w(3d)
0RjIR
jj(3e)
0rgw Ir
gg, w (3f)
Costwξηww(3g)
rT
wRTw(3h)
zwrT
wLww(3i)
0Lw, ηww. (3j)
The first term in Equation (3b) represents the cost of allocating RTMW of reserve capacity. The TSO
has information about the marginal cost of allocating reserves at the closure time of the reserve market,
as it is given by the bids submitted by producers to the reserve market. These bids, however, are treated
confidentially and hence, were not available for the study case. Consequently, we estimate a cost function
for the supply of reserve capacity from the historical series of clearing prices in the Danish reserve market.
Naturally, this function must be monotonically increasing. The estimation of the parameters of this function
is discussed in Section 3. In order to keep formulation (3) linear, the marginal cost of reserve capacity is
further approximated by a stepwise function consisting of Jintervals of length IR
jeach, as indicated in (3e),
and associated values λcap
j, which result from evaluating the estimated reserve cost function at the midpoint
of each interval. The term PJ
j=1 λcap
jRjrepresents thus the total cost of allocating RTMW of reserves.
Furthermore, the total allocated reserves are given by (3c). Note that the formulation would remain equal
5
if the real bids were used instead of the estimated cost function. One could interpret λcap
jand IR
jas the bid
that producer jsubmit to the reserve market and Rjas the MW of reserve capacity provided this producer.
The second term of (3b) represents the reserve deployment cost. This cost is unknown at the time of
clearing the reserve market and therefore, has to be estimated by as well. The estimation procedure is
discussed in Section 3. Similarly as before, λbal
gcan be seen as the cost of deploying rgw MW of reserve in
interval gand scenario w. The length of the intervals is Ir
g, as stated in (3f), having a total of Gintervals.
The total deployed reserve in scenario wis then given by (3d).
The third term of (3b) represents the cost of involuntary load curtailment. The parameter “Value of
Lost Load” VLOL expresses the societal cost of shedding 1 MWh of load. Often, the VLOL is interpreted as
the maximum price of upward regulation that is permitted to bid in the market, which in Denmark is 37 500
DKK/MWh or roughly 5 000 e/MWh. In Great Britain, the VLOL is estimated to be from 1 400 £/MWh
to 39 000 £/MWh depending on the type of consumer and the time of the year [17]. A study performed on
the Irish power system indicates that, on average, the VLOL is 12.9e/KWh [18]. In this paper, a sensitivity
analysis is performed to study how the parameter VLOL affects the solution.
Constraint (3g) is used to linearly define the CVaRαas in [19]. Variable ηwis equal to zero if Costw< ξ,
and equal to Costwξif C ostwξ; in other words, ηwaccounts for the difference between the cost in
each scenario and the VaRαwhen such a difference is positive. Equation (3h) indicates that the deployed
reserve cannot be greater than the scheduled reserves. Equation (3i) is used to define the shed load Lw(or
similarly, the lack of reserve). At the optimum, Lwis equal to zero if zwRT, implying that zw=rT
w; when
zw> RT, then Lwis equal to the difference between the reserve requirements and the deployed reserves,
namely, zwrT
w. In this case, the deployed reserve is equal to the scheduled capacity reserve rT
w=RT.
Once the CVaR problem has been solved, one can calculate the EPNS by multiplying the lacking reserve
from each scenario L
wat the optimum by its probability of occurrence πw, namely EPNS = PW
w=1 πwL
w.
3. Cost Functions
This section elaborates on the estimation of the cost of allocating reserves and the cost of providing
regulating power.
In practice, the bids that producers submit to the reserve market, that are used to define (3b), are
available to the Danish TSO at the closure of the reserve market. Nevertheless, this information is not
available to us for the case study presented in Section 5. Consequently, in order to adjust the optimization
models to the available data and test the efficiency of such, the bids of producers are substituted by a cost
function, being gR(z) the cost in e/MW of allocating of zMW of upward reserve. This function is built
from the series of clearing prices in the Danish reserve market, which is publicly available in [20]. The price
per MW of reserve capacity is assumed to be quadratic for simplicity, in particular, of the form gR(z) = az2.
The coefficient a= 1.25 ×105is estimated using least-square method.
A staircase linear approximation of gR(z) is then used in order to maintain formulation (3) linear. The
reason for the choice of a staircase function is that, due to market rules, the aggregated bidding curve is also
a staircase function. The feasible region of RTis split into intervals of length Ir
j= 30 MW j, ranging from
0 to an upper bound of RTchosen to be 1890MW. For every interval, we compute the estimated marginal
cost λcap
jat the mid point of the interval and set it to the height of each stair. Figure 1 shows on the left the
data points and the estimated curve of prices in Euro per MW of allocated reserve. The data appears very
homoscedastic, for example, the variability around RT= 300 MW is much lower than around RT= 450
MW. Nevertheless, the curve is not intended to capture all the variability of the data but to represent a
plausible aggregated bidding curve in the reserve market.
The cost of deploying reserves is a necessary input to Equation (3b) and must be estimated in practice, as
it is unknown at the time of clearing the reserve market. We denote the marginal cost of deploying zMW of
reserve by gr(z). In order to compute this cost, we approximate the clearing prices of the regulating market
by a quadratic term plus an intercept, gr(z) = µ+bz2. The parameters are estimated using the least-squares
method and data relative to the clearing price of the regulating market in DK1 collected from [20]. The
regulating power traded versus the market price is displayed in dots in Figure 1. The resulting estimates of
6
the parameters are µ= 48.2 and b= 6 ×104. In order to maintain the optimization problem (3) linear,
gr(z) is linearized as a stair-case function, which is shown in the right plot in Figure 1. More complex
functions could possibly be estimated, for example using time and other external factors as explanatory
variables. This implementation is left for future work.
Lastly, it should be noted the difference in scale between the settlement prices of the two markets. On
average, the price of allocating reserve is approximately 40 times lower than deploying them. Allocating
reserve is cheaper as no energy is actually deployed but only the capacity is allocated.
[Figure 1 goes here.]
4. Scenarios of Reserve Requirements
The total reserve capacity that should be scheduled and allocated in advance is mainly affected by three
factors or uncertainty sources: the forecast error of wind power production, the forecast error of electricity
demand and the forced outages of power plants, namely failures of the plants that cause their production
to stop. They are all taken into account in this paper.
Suppose that wind power production is the only source of uncertainty. We assume that wind power
producers bid their expected production in the day-ahead market. If the actual wind power production is
greater than what was expected, then there will be extra power to sell and hence a reduction in power supply
(down-reserves) will be required to maintain the system balance; if, on the other hand, the realized wind is
lower than the expected value, upward reserves will be required. In other words, if the forecasts were perfect
and the errors equal to zero, no reserve would be needed. Likewise, as the forecast errors increase, more
reserves are required to account for the possible mismatches between supply and demand. Similarly with the
power load: it is assumed that the amount of power traded in the day-ahead energy market is equal to the
expected power load demand, therefore positive errors imply upward reserve requirements while negative
errors imply downward reserve requirements. The predicted outages of power plants lead directly to upward
reserve requirements.
The probability distributions of the forecast errors and the power plant outages can be combined into one
by convolving them, resulting in a function which will represent the probability distribution of the combined
balancing requirements f(z). In this paper, we draw scenarios from each individual distribution and sum
them up to produce scenarios characterizing the total reserve requirements in the DK1 area of Nord Pool. A
scenario-based approach is chosen because the convolution of the probability distributions of the individual
stochastic variables does not have a closed form and can be highly complex. The remaining of this section
elaborates on how the individual scenarios are obtained.
4.1. Scenario Generation of Wind Power Production and Load Forecast Errors
In this subsection both the generation of scenarios of wind power production and load forecast errors
are discussed. Scenarios from both stochastic variables are generated together to account for correlation
between them.
Regarding the wind power production in DK1, point quantile forecast have been issued using a conditional
parametric model, i.e., a linear model in which the parameters are replaced by a smooth unknown functions
of one or more explanatory variables. The explanatory variables are on-line and off-line power measurements
from wind turbines and numerical weather prediction of wind speed and wind direction. The functions are
estimated adaptively. The errors are modeled as a sum of non-linear smooth functions of variables forecast by
the meteorological model or variables derived from such forecasts. Further information about the employed
modeling approach can be found in [21, 22, 23].
The load in DK1 area has been modeled as a function of the temperature, the wind, and the solar
radiation. The annual trend is modeled by a cubic B-spline basis with orthogonal columns. The daily
variations are modeled as a combination of different sinusoids, one referring to each time of the day. The
reader is referred to [24] for a detailed description of the methodology used in this paper to model the
electricity demand in DK1.
7
The scenarios of wind production and load are generated in pairs in order to account for their mutual
correlation. Each scenario is composed by two variables and is built in three steps as in [25]: first by a
sample of a multivariate Gaussian distribution where the covariance matrix is estimated recursively as new
observations are collected; then, by applying the inverse probit function of such sample, and finally by using
the estimated inverse cumulative function of the desired variables.
Figure 2 shows the distribution of the scenarios of forecast errors of wind power production and load in
gray and white respectively, during the 15th Dec 2011 from 13:00 to 13:59. Note that the distribution of
the forecast error of wind power is wider, indicating that, in general, wind power production has a greater
impact on reserve requirements than the load. On average, forecast error scenarios of wind exhibit five times
more variance than the load scenarios. Finally, note that both distributions are centered around zero.
[Figure 2 goes here.]
4.2. Scenario Generation of Power Plant Outages
The modeling of individual power plant outages requires historical data and specific information on each
power plant which might not always be available to the TSO. Secondly, it requires computing an individual
model for each unit, thus increasing complexity significantly. Thirdly, it requires information about which
units will be on/off during the operation horizon, which is not available at the clearing of the Danish reserve
market. An alternative approach taken in this paper is to model the total amount of MW that fail in the
entire system by aggregating all the units into one. The predicted MW failed in the entire system depend on
time and on the load. Historical data of power plant outages can be found at the Urgent Market Messages
service of Nord Pool [26]. The left plot in Figure 3 shows the forced outages in MW during 2011. The area
inside the box corresponds to the MW failed in May, also zoomed in the right plot. In the course of 2011,
there was 92% of the hours where 0 MW failed; during the rest of the hours, either an outage of a single
unit, a partly outage or simultaneous outages occurred.
[Figure 3 goes here.]
The procedure proposed in this paper to model power plant outages is comprised of two steps. In the
first step, we model the presence or absence of an outage. In the second step, we model the amount of MW
failed, conditioned on the fact that a failure occurred. In [27] we explored alternative methodologies based
on Hidden Markov Models that were proven to perform worse at predicting future outages.
The variable modeled in the first step Ytis defined as
Yt=(1 if failure occurs at time t
0 otherwise. (4)
It is natural to assume that Ytfollows a Bernoulli distribution, Ytbern(pt), and therefore, it is
appropriate to model Ytas a Generalized Linear Model [28]. The link function chosen is the logit function.
The explanatory variables are the hour of the day, the day of the week and the month, all represented through
sinusoidal curves. Sinusoidal terms of the form k(1)cos(2πhourt
24 ), k(2)cos(2πdayt
7) and k(3)cos(2πmontht
12 ) with
k(1) = 1...24, k(2) = 1...7, and k(3) = 1...12, are considered, also using the sin function. Only the most
relevant were kept using a likelihood ratio test as in [28]. The final model is
ηt=log pt
1pt=µ+α1cos 2πdayt
7+α2sin 2πdayt
7+α3cos 2πmontht
12 +
α4cos 52πmontht
12 +α5sin 2πmontht
12 +α6sin 22πmontht
12 +
α7sin 32πmontht
12 +α8sin 42πmontht
12 +α9sin 52πmontht
12 .(5)
8
The reduced model indicates that the hour of the day is not significant when predicting the probability of
an outage. The day of the week and the month are both significant variables. The parameters of the model
are optimized using train data and updated everyday including data from the previous 24 hours during the
test period.
The second stage of the model accounts for the amount of failed MW at time t,Xt, conditioned on the
fact that a failure has occurred. Note that the more energy is demanded, the more power plants are online
and more generators are subject to fail, meaning that the load ntwill affect our predictions of Xt. The
histogram of (Xt
nt|Yt= 1) depicted in Figure 4 clearly resembles the density of a Gamma distribution. Thus,
we assume that (Xt
nt|Yt= 1) Gamma(st, k), where kis the shape parameter, common for all observations,
and stthe scale parameter at time t.
[Figure 4 goes here.]
The probability density function of a Gamma distribution is defined as
f(x) = 1
Γ(k)sk
t
xk1ex
st,(6)
with mean µt=kstand variance σ2=ks2
t. The canonical link for the gamma distribution is the inverse
link η= 1[28]. As in the previous binary model, the explanatory variables are several sinusoidal curves.
Several approximate χ2-distribution tests were performed to disregard irrelevant terms. The final model
only including the significant terms is
ηt=1
µt
=µ+α1cos 2πhourt
7+α2cos 22πhourt
7+α3sin πhourt
12 +
α4sin 22πhourt
12 +α5sin 3πhourt
12 +α6cos 2πdayt
12 +
α7cos 22πdayt
12 +α8sin 2πdayt
12 +α9sin 32πdayt
12 +
α10cos 2πmontht
12 +α11sin 22πmontht
12 +α12sin 32πmontht
12 .(7)
When predicting the ratio Xt/nt, the hour, the week day and the month are statistically significant.
Scenarios are generated in an iterative process. Every day at 9:00 am, the parameters of both models are
updated including data from the previous day. At this time, 5 000 scenarios for each hour of the next day are
generated, i.e., with lead time ranging from 16 to 40 hours. Each scenario corresponds to an independent
simulation of a Bernoulli multiplied by a Gamma simulated value and by the predicted load nt.
5. Results and Discussion
The performance of the proposed reserve determination models is assessed by comparing the reserve
capacity scheduled by the models and the reserve capacity actually deployed in the DK1 area of Nord
Pool. The latter is calculated as the sum of the activated secondary reserve, regulating power produced
in DK1 and the regulating power exchanged through the interconnections with neighboring areas. Data
pertaining to the activated secondary reserve and the total volume of regulating power can be downloaded
from [26] and [20], respectively. The regulating power exchanged through the interconnections with Germany,
Norway, Sweden and East Denmark is estimated using data from [20]. More specifically, it is computed by
subtracting the total power scheduled for each interconnector in the day-ahead market from the actual power
that eventually flows through it. Primary regulation was not considered due to unavailability of the data.
However, conclusions would be barely affected by considering the primary regulation as it is comparatively
9
very low. In the remainder of the paper, a shortage event is defined as an hour when the scheduled reserves
are lower than the actual reserve deployed. In practice, a shortage event does not necessarily imply that
load is shed. In the Nordic market, producers who are scheduled to provide a certain capacity in the
reserve market must place an offer of regulating power of the corresponding size in the regulating market,
45 minutes before operational time. However, other players who are not committed to provide reserves in
the reserve market may still bid in the regulating market and thereby provide regulating power. This case
is not considered in this paper.
The presented case study has been performed on data spanning three years. The training period of the
scenario-generating models goes from the 1st January 2009 at 00:00 CET to the 30st June 2011 at 23:00
CET. The test period covers week 36 of year 2011 and weeks 3, 17 and 22 of 2012, all of them randomly
selected.
Recall that scenarios are generated every day at 9:00 am with a lead time of 16 to 40 hours, namely
the next operational day. The optimization models are run using the same lead time, as if they were to be
solved at the clearing of the reserve market. The solutions to the LOLP and CVaR models are compared
with the actual deployed reserves in DK1 during the four testing weeks.
It is worth stressing that the proposed models for reserve determination focus on the total reserve
requirements, which are triggered by unexpected fluctuations in the load and in the wind power production,
and by outages of power plants. We do not distinguish, therefore, between primary, secondary and tertiary
reserve. In the case of the LOLP-formulation, it is up to the TSO to decide how to split the total reserve
requirements into the different types of reserve that may be considered. Likewise, the CVaR-formulation
can be easily tuned to represent the three types of reserves through the estimated cost functions. Indeed, if
it is much easier for plants to participate in the tertiary reserve market, because providing tertiary reserve is
cheaper than providing primary and secondary reserve, then the distinction between these should be made
through the supply cost function for reserve, with the tertiary reserve being cheaper than the secondary and
primary ones. If, on the contrary, it is much easier for the plants to participate in the tertiary reserve market
for reasons that cannot be translated into costs, then the required primary and secondary reserve should be
treated as input information in the CVaR-method and subtracted from the total reserve requirements.
5.1. LOLP-model Results
This subsection shows the results of applying the LOLP model introduced in Section 2.1. The model was
run during the four testing weeks and the simulation results are included in Figure 5. The actual deployed
reserves in DK1 and the total scheduled reserves by Energinet.dk are also shown in the figure. The shaded
areas in the background represent the solution to the LOLP model for different values of β. Weeks are
separated by vertical lines. The amount of scheduled reserve varies substantially depending on the level of
uncertainty of the wind, the load and the power plant outages. As an example, on the Thursday of the
fourth week, the prediction of load and wind power production happened to be wrong, and up to 1 100 MW
of regulating power were needed. In this special case, both the LOLP solution and Energinet.dk’s reserve
scheduling criterion led to a shortage event.
[Figure 5 goes here.]
The reliability plot in Figure 6 shows i.e. the desired or expected LOLP (parameter β), against the
observed LOLP, namely, the number of shortage events divided by the time span. The whole data set was
used to compute this reliability plot. The ideal case is illustrated by the dashed line where both quantities,
expected and observed, are equal. The actual performance of the model is represented by the continuous
line, which is fairly close to the ideal one, indicating that the expected probability of reserve shortage is well
adjusted to the observed one.
[Figure 6 goes here.]
It is worth mentioning that the value of the parameter βis directly connected to the reliability level of
the underlying power system. With this in mind, a simple rule for the TSO to decide on an appropriate
10
value for this parameter would read as follows: divide the number of hours in a year where shedding load is
tolerated to happen by the total number of hours in the year. This simple rule would roughly indicate the
probability that, in each hour, the need for upward balancing power exceeds the scheduled reserves. The
UCTE suggests that enough reserve should be scheduled to manage energy deviations in 99,9% of all hours
during the year [29]. For example, if the TSO tolerates that there are around 96 hours during a year where
some load might not be covered, then the resulting LOLP would be equal to 0.01. The case presented in
[14] is performed using LOLP={1,0.5,0.1}. The authors in [30] consider five scenarios of demand, where
the LOLP is in between 0.005 and 0.016. Note, however, that the LOLP does not account for how many
MW of load are shed or the cost of such load shedding events.
Next, we perform a sensitivity analysis to asses how changes in βaffect the solution. We choose several
plausible values of βwhich are displayed in the first column of Table 1, and then compute the optimal
reserve schedule for each of them. The four testing weeks are considered and the results shown are averaged
by the number of days. The second column shows the numbers of shortage events. The cost of allocating the
reserve given by the LOLP model is displayed in the third column. It is computed using the cost function
gR(z), presented in Section 3. The fourth column shows the cost of deploying the actual reserve computed
using the function gr(z), which estimation is discussed in Section 3. Lastly, the MW shed, or in other words,
the number of MW of actual deployed reserve exceeding the LOLP solution, is presented in the fifth column.
Note that a decrease in the parameter βimplies that the solution becomes more conservative and hence,
more reserve will be scheduled. For this reason, as βdecreases, the number of shortage events decreases
too, at the expense of increasing the allocation and deploy cost. On the other hand, the amount of MW not
covered by the scheduled reserves, collated in the fourth column, decreases as βdiminishes.
βShortage events Alloc. cost in e×103Deploy. cost in e×103MW not covered
0.2 5.214 1.194 95.643 862.056
0.15 4.107 2.077 110.966 668.879
0.089 2.678 4.528 134.826 423.645
0.07 1.928 6.132 143.782 346.910
0.05 1.464 9.150 153.939 270.271
0.01 0.428 33.032 187.791 85.464
Table 1: The first column shows several values of parameter βwhich is an input to the LOLP model. The second column
presents the number of shortage events on average per day. The third column includes the cost of allocating the amount of
reserves given by the LOLP model on average per day. The fourth column displays the cost of deploying the actual reserve
requirements. The number of MW not covered by the scheduled reserve on average per day when using the LOLP solution are
shown on the fifth column.
Next, we compare the solution to the LOLP model with the solution given by Energinet.dk. TSO’s
solution incurs 75 shortage events during the four testing weeks, or equivalently 2.67 shortage events per
day. The estimated total cost of allocating reserve is 4 355 e; the estimated deployment cost is 132350 e,
and the MW not covered 422.39.
For the same number of shortage events, the LOLP gives a worse solution that Energinet.dk’s solution:
the allocation costs are 3.97% higher, the deployment cost 1.18% higher and the MW shed increase in 1.15
MW per day. This means that during the four testing weeks, the LOLP methodology underperforms the
solution given by Energinet.dk in terms of reliability and economic efficiency. The main advantage that
the LOLP method brings is the analogy of the parameter βwith the probability of a shortage event to
occur, which is a very easy risk measure to interpret. On the other hand, the method has two drawbacks.
As discusses before, its solution does not depend on the cost of allocating reserves, namely on λior on
the estimated cost function g(z) (as long as it is increasingly monotonic). Neither it depends on the cost
of deploying reserves. The solution only depends on the parameter β, as the relation LOLP =βin the
optimization problem (1) will always be satisfied at the optimum, no matter what the cost is. The second
disadvantage is that load shedding costs are not taken into account. These drawbacks are overcome by the
CVaR model, for which results are presented in the next subsection.
11
5.2. CVaR-Method Results
In this section we discuss the results of the CVaR-based reserve determination method, which has been
presented in Section 2.2, and compare them with the deployed reserves that were actually needed in DK1
during the simulation horizon.
The CVaR model needs as input two parameters which, in practice, are to be determined by the TSO:
α, which controls the CVaR risk measure and represents the risk aversion of the TSO, and VLOL, which
accounts for the cost in eof shedding 1 MW of load. We performed a sensitivity analysis to determine
how changes in these parameters affect the level of procured reserve. The model was run for values of
α={0,0.25,0.5,0.75,0.9,0.95,0.99}and Vlol ={200,500,1 000,2 000,5 000}e/MW.
The cost of allocating and deploying reserve capacity, the cost of shedding load, and the total cost, are
displayed on the upper-left, upper-right, lower-left and lower-right plot of Figure 7. The cost is shown on
the y-axis in e×103, while the risk parameter αis shown on the x-axis. Each line represents a cost of the
reserve schedule solution for a certain VLOL. All costs are averaged by the number of days in the test period.
As the TSO becomes more risk averse, i.e., as αincreases, the allocation and deployment costs increase,
because a larger amount of reserve is procured. The same occurs as VLOL increases, since shortage events
become more penalized and more reserves are scheduled to avoid them. On the other hand, the cost of
curtailing load, depicted in the left-lower plot, decreases as αincreases, but does not necessarily increase as
VLOL increases. The reason for this is that, even though the amount of curtailed load decreases as VLOL
increase, the product VLOL ×Lwrepresenting the cost of curtailing load in scenario wmay still increase.
The total cost shown in the down-right subplot of Figure 7 is computed by summing up the reserve
allocation, the reserve deployment and the load shedding costs. In general, the total cost increases as the
risk-aversion parameter αincreases. However, this is not always the case when VLOL ={1 000,2 000,5 000}
e/MW. The reason for this discrepancy is that the generated scenarios do not represent the potential need
for reserve capacity accurately enough. Adding more variables to the scenario representation of the reserve
requirements, increasing the amount of scenarios or adding more weeks to the test period could solve this
issue, in particular, they underestimate the amount of upward balancing power that may potentially be
required, as confirmed by the plot in Figure 6. Finally, one should notice that changes in the total cost are
mainly driven by changes in the VLOL , while changes in αhave smaller impact on the solution.
[Figure 7 goes here.]
The scheduled reserves when VLOL = 500 e/MW and VLOL = 5 000 e/MW, over time, are displayed
in shadowed areas in the upper and lower plot of Figure 8. The actual deployed reserve and the reserve
scheduled by Energinet.dk are drawn on top. Weeks are separated by vertical lines. It is interesting to
note how the reserve schedule given by the CVaRαbased reserve determination model changes as VLOL
changes. When minimizing the CVaR of the cost distribution of reserve allocation and deployment, and
load shedding costs, an increase in VLOL makes the load shedding costs have more weight in the total costs,
and hence the events of reserve shortage will be penalized to a larger extent. When VLOL is low, those
events are less relevant and the curves look more flat. Another reason for the flatness of the curves is the
linear approximation of the cost functions gR(z) and gr(z), both introduced in Section 3. The increase in
cost when increasing the reserve in one unit is much higher when jumping from one step of the stepwise
function to another, than when the function remains in the same step. The step lengths are defined in (3e)
and in (3f). A finer linearisation of such cost functions by reducing the step length would solve this issue.
[Figure 8 goes here.]
The number of interruption events and the amount of load that is involuntarily shed are further ana-
lyzed in Figure 9, in the left and right plots, respectively. As αand/or VLOL increase, both the number of
interruptions and the MW shed on average per day decrease. Under the assumption that only the producers
committed to the reserve market are allowed to participate in the regulating market, the Danish TSO’s solu-
tion incurs 75 shortage events during the four testing weeks. On average per day, the Danish TSO’s solution
incurs 2.67 shortage events, with an estimated total cost of allocation equal to 4 355 e, an estimated deploy-
ment cost of 13 2350 e, and amount of load shed of 422.39 MW. The CVaR-method produces cheaper results
12
in terms of total cost. The CVaR solution is from 3.38% cheaper with {α= 0.99, V LO L = 200 e/MW},
to 82.9% for {α= 0.99, V LOL = 5 000 e/MW}, compared to the solution given by the TSO. Note that the
CVaR method tends to schedule more reserves than the Danish TSO’s solution, while at the same time the
solution is cheaper, because shedding load is highly penalized by the coefficient VLOL.
[Figure 9 goes here.]
Figure 10 illustrates the so-called efficient frontier [5]. This plot can be used by the TSO to choose an
appropriate value for the risk-aversion parameter αaccording to its attitude towards risk. The efficiency
frontier shows the expected total cost per day, namely, the expected cost of allocating and deploying reserve
plus the load shedding cost per day, against the expected LOLP, that is, the expected probability of a load
shedding event. The numbers along the curve indicate the value of the risk parameter αused to obtain
such a point in the curve. The efficient frontier shown in Figure 10 has been determined for a VLOL equal
to 5 000e/MW. Needless to say, the efficient frontier would change for different values of VLOL , but the
interpretation of the resulting curves would remain similar.
The TSO can thus use this efficient frontier to resolve the trade-off between desired or expected LOLP
versus the expected total cost that such a level of reliability would entail. For example, the TSO can achieve
a LOLP of 0.001 with an expected total cost of approximately 3×105eper day. To this end, the TSO should
set the risk-aversion parameter αto 0.5. If, for instance, the LOLP is to be decreased down to 0.0002, the
expected total cost would raise up to 3.2×105eper day. In that case, the parameter αshould be set to 0.9.
[Figure 10 goes here.]
The main advantage that the CVaR method offers over the LOLP method is that the TSO is able to
input the cost of shedding load, VLOL, in the model and, therefore, the reserve dispatch solution is dependent
on it. On the contrary, the LOLP method depends only on the number of interruption events and their
associated cost is not accounted for. Another advantage of the CVaR method is that the reserve schedule
depends on the reserve costs, both the allocation and deployment cost. In a real set-up, the solution would
depend on the bids from the producers, while the solution of the LOLP method is independent of the reserve
costs. Lastly, both optimization models are able to reflect the risk aversion of the TSO through the risk
parameters βor α. However, the risk parameter αof the CVaR methodology does not have a straightforward
interpretation in real power systems, as compared to the parameter βfrom the LOLP formulation, which
has a direct physical interpretation.
6. Conclusion
In this paper we present two methods to determine the reserve requirements using a probabilistic ap-
proach, suited for a market structure where the reserves are scheduled independently of and before to the
day-ahead energy market. This is the case in the Nordic countries and, more specifically, in the DK1 area
of Nord Pool, under which the study case of this paper is framed. The first method ensures that the LOLP
is kept under a certain target. The second method considers the costs of allocating and deploying reserve
and of shedding load, and minimizes the CVaR of the total cost distribution at a given confidence level α.
Both approaches are based on scenarios of potential balancing requirements, induced by the forecast error
of the wind power production, the forecast error of the load, and the forced failures of the power plants in
the power system.
The performance of the proposed reserve determination models is assessed by comparing the resulting
optimal scheduled reserves with the Danish TSO’s solution approach and with the actual deployed reserves
during four testing weeks, in terms of costs and shortage events. The results from the case study show that
the LOLP method underperforms the Danish TSO’s solution in terms of costs, for the same shortage events.
By using a CVaR risk approach, the cost of allocating reserves is reduced from 3.38% to 82.9%, depending
on the value of the parameters of confidence level and value of lost load. The CVaR methodology provides
adequate levels of reserves.
13
Further studies should focus on the applicability of these methods to the Nordic reserve market, by
differentiating between types of reserves. This could be achieved by modeling the amount of MWh of each
type of reserve required at every hour and the cost of allocating and activating each of them. Also, further
improvements should be done on the modeling of the failed MW in the whole system. More specifically,
time-dependencies could be modeled, since a power plant is more likely to be off-line if the previous hour
was off-line too. This could be achieved by, for example, a non-homogeneous Hidden Markov Model, where
the transition probabilities between states depend on time and other external variables.
Acknowledgement
The work presented in this paper was partly supported through the iPower platform project DSF (Det
Strategiske Forskn- ingsr˚ad) and RTI (R˚adet for Teknologi og Innovation) and the OSR Nordic Project.
Acknowledgements to Nord Pool, Energinet.dk and ENFOR A/S for the data provided.
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15
200 400 600
0
10
20
30
40
50
Euro/MW of reserve
MW
200 400 600
50
100
150
200
250
300
Euro/MWh of regulating power
MW
Figure 1: On the left, the settlement price and the allocated reserves in the reserve market. Dots represent data and the
staircase curve constitutes the estimated fit. On the right, the data relative to the settlement price of the regulating power
market is shown in dots while the fitted stair-wise curve is displayed on top.
16
Figure 2: The histogram of forecast error scenarios of wind power production and load during one specified hour shown are
shown in gray and white, respectively.
17
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
0
400
800
1200
Month
MW forced outages
01 04 07 10 13 16 19 22 25 28 31
0
400
800
1200
Day of the month
MW forced outages
Figure 3: Historical data of MW forced to fail in DK1. On the left, data relative to year 2011 is shown. The area inside the
rectangle is zoomed on the right plot and refers to May 2011.
18
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
0.04
0.09
0.13
0.17
0.21
0.26
Xt / nt
Frequency
Figure 4: Histogram of xt
nt|yt= 1, namely the amount of MW failed divided by the load at time t, knowing that a failure has
occurred. The curve represents the estimated Gamma distribution.
19
Time, day of the week
Total reserves
M T W T F S S M T W T F S S M T W T F S S M T W T F S S
0
200
400
600
800
1000
1200
0.2
0.1
0.05
0.01
β
Actual deployed reserves
TSO reserve schedule
Figure 5: The deployed reserves and the actual scheduled reserves by Energinet.dk are plotted as indicated in the text boxes.
The shaded areas in the background represent the solution of the LOLP model for different values of β. Weeks are separated
by vertical lines.
20
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
β
Prob DR > Rtot* (=LOLP)
Figure 6: Observed LOLP vs expected, namely parameter β
21
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14 x 104Allocation
α
Cost in Euro
0 0.2 0.4 0.6 0.8 1
2
3
4
5
6
7
8
9x 104Deployment
α
Cost in Euro
0 0.2 0.4 0.6 0.8 1
0
0.5
1
1.5
2x 105Shedding
α
Cost in Euro
0 0.2 0.4 0.6 0.8 1
0.5
1
1.5
2
2.5
3
3.5 x 105Total
α
Cost in Euro
200 500 1000 2000 5000
Vlol Euro/MW
Figure 7: Sensitivity analysis of the CVaR-based reserve determination model. The parameter αis displayed on the x-axis and
the cost in e×103on the y-axis. Each line represents the cost of the solution for a certain VLOL. The cost of allocating and
deploying reserve capacity, the cost of shedding load and the total cost are displayed on the upper-left, upper-right, lower-left
and lower-right plot, respectively. All costs are averaged by the number of days in the test period.
22
Vlol =500 Euro/MW
Time, day of the week
Total reserve
M T W T F S S M T W T F S S M T W T F S S M T W T F S S
0
200
400
600
800
1000
1200
Vlol =5000 Euro/MW
Time, day of the week
Total reserve
M T W T F S S M T W T F S S M T W T F S S M T W T F S S
0
200
400
600
800
1000
1200
1400
1600
0
0.25
0.5
0.75
0.9
0.95
0.99
α
Activated deployed reserves
TSO’s scheduled reserves
Activated deployed reserves
TSO’s scheduled reserves
Figure 8: The reserve schedule using the CVaR methodology is displayed for VLOL = 500 e/MW in the upper plot and for
VLOL = 5 000 e/MW in the lower plot. The shaded areas in the background represent the solution of the CVaR model for
different values of the risk aversion parameter α. The actual deployed reserve in DK1 and the reserve capacity scheduled by
Energinet.dk (the Danish TSO) are depicted on top.
23
0 0.2 0.4 0.6 0.8 1
0
50
100
150
200
250
α
MW shed
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
α
Interruption events
200
500
1000
2000
5000
Vlol Euro/MW
Figure 9: In both plots, the reserve schedule, computed by the CVaR-based method, is compared to the actual deployed reserve
in the DK1 area, for different values of VLOL and α. On the left, we depict the average number of shortage events per day,
and on the right, the average MW of load shedding.
24
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 10−3
2.9
3
3.1
3.2
3.3
3.4
3.5 x 105
Expected LOLP
Expected total cost
α=0.99
α=0.95
α=0.9
α=0.75
α=0.5
α=0.25
α=0
Figure 10: Efficiency frontier plot of the CVaR model. The expected total cost, computed as the sum of the expected cost of
allocating and deploying reserve, and the expected cost of shedding load for VLOL = 5 000 e, is shown in the the x-axis. In
the y-axis, the expected LOLP is displayed. The numbers along the curve indicate the value of the risk parameter αused to
obtain each of the solutions.
25
... In light of these problems, there are also many research efforts from academia. Some researches utilize the forecast methods, such as the density forecasts which consider the probability distributions of future observations [5], or the scenario forecasts which consider several typical future scenarios [6], to define reserve requirements in systems with high renewable penetrations, see [7] for a survey. In [8], a statistical clustering algorithm is proposed to enable dynamic reserve zone partition. ...
... Coefficients (C k , C k ) represent vectors of generation upward and downward re-dispatch prices 3 in scenario k. If one generator's downward reserve is deployed, it will pay back to the SO, so there is a negative sign before (C T k δg D k ) in (6). Coefficient C L denotes the vector of load shedding prices. ...
... The proposed pricing approach for energy and reserve is based on their marginal contributions to the expected system total cost presented in (6). Namely, consider any generator j, we first fix g(j), r U (j), r D (j) at their optimal values g(j) * , r U (j) * , r D (j) * and consider them as parameters instead of decision variables. ...
Preprint
Full-text available
We propose a scenario-oriented approach for energy-reserve joint procurement and pricing for electricity market. In this model, without the empirical reserve requirements, reserve is procured according to all possible contingencies and load/renewable generation fluctuations with the minimum expected system total cost. The innovative locational marginal pricing approach for loads, generations and reserve and the associated settlement process are proposed. We show that payments from loads, payments to generators and congestion rent will reach their balance in the basecase as well as in all scenarios, so that revenue adequacy can be guaranteed for the system operator.
... Probabilistic AS 3 1.1. CONTEXT sizing methodologies have been proposed to take into account the uncertainty of VRE production [11]. Stochastic unit commitment models propose to optimize the scheduling of dispatchable plants in order to accommodate VRE penetration while minimizing costs [12]. ...
... Reliable offers of AS suppose adequate power forecasting methods. For variable generation, assuming that the distribution of forecast errors depends only on historic performance does not capture the uncertainty inherent to the forecast model [11]. The nonlinearity of wind and solar generation induces that conditional distribution of forecast errors is easier to model with nonparametric approaches [12]. ...
... The nonlinearity of wind and solar generation induces that conditional distribution of forecast errors is easier to model with nonparametric approaches [12]. According to a review of probabilistic methods for reserve requirements [11], density forecasts can be applied to both wind and solar power, and give more reliability on reserve allocation problems than approaches based on historical forecast only. Kernel Density Estimation (KDE) is a density forecast method which figures among the top-ranked methods for wind and PV forecasting [13]. ...
Thesis
Full-text available
As variable renewable energy plants penetrate significantly the electricity generation mix, they are expected to contribute to the supply of reserve power, albeit the high uncertainty levels on their production. A solution to reduce the uncertainty consists in aggregating renewable plants dispersed over several climates to obtain a smoother production profile and operate them within a Virtual Power Plant control system. In this thesis, a series of probabilistic forecasting models are proposed to assess the capacity of a variable renewable Virtual Power Plant to provide ancillary services with maximum reliability: these models are adapted decision-tree regression models, recurrent and convolutional neural networks, as well as distributions dedicated to extremely low quantiles. The combination of energy sources (Photovoltaics, Wind, Run-of-river Hydro) is considered in detail. Optimal strategies for the joint offer of energy and ancillary services by a variable renewable Virtual Power Plant are later defined, based on production forecasts and market uncertainties. Offer strategies explore several modelling options:dependence between renewable production and prices via a copula, controlled rate of reserve underfullfilment with a chance-constraint optimization, and finally offer of multiple ancillary services thanks to a Lagrangian formulation.
... In Ref. [16], a probabilistic approach has been used to achieve the reserve under uncertainty of wind power generation, electrical demand, and power generation of conventional units. Besides, energy and reserve market are cleared independently, that the reserve market is cleared before energy market in Ref. [16]. ...
... In Ref. [16], a probabilistic approach has been used to achieve the reserve under uncertainty of wind power generation, electrical demand, and power generation of conventional units. Besides, energy and reserve market are cleared independently, that the reserve market is cleared before energy market in Ref. [16]. In Ref. [17], the Transmission System Operator (TSO) is responsible for apportioning dynamic reserves in the power system. ...
... Scenarios of wind power generation in scenarios 2, 3 and 4 in IEEE-RTS [16]. the power system. ...
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In this paper, a new mechanism is proposed to apportion expected reserve costs between electricity market agents in the power system. The uncertainties of generation units, transmission lines, wind power generation and electrical loads are considered in this model. Hence, a Stochastic Unit Commitment (SUC) is used to apply the uncertainty of stochastic variables in the simultaneous energy and reserve market-clearing problem. Moreover, electrical customers can participate in the electricity market based on their desired strategies. In this paper, a novel method is proposed to allocate reserve costs between GenCos, TransCos, electrical customers and wind farm owners. Consequently, market agents are responsible for paying a portion of the allocated expected reserve costs based on the economic metrics that are defined for the first time in this paper. Finally, two cases including a 3-bus test system and IEEE-RTS are utilized to illustrate the performance of the proposed mechanism to share the expected reserve costs.
... For reserve requirement selection, parametric and non-parametric probabilistic forecasting techniques are incorporated in (I) by characterizing the underlying probability distribution of future situations [7]- [10]. Scenariobased approaches have also been proposed in [11], [12]. In addition, on the deliverability of reserve, a statistical clustering method is proposed in [13] that partitions the network into reserve zones. ...
... (V) : minimize {g,r U ,r D ,δg U k ,δg D k ,δd k ,θ,θ k } F V (·), subject to (6), (7), (8), (11), (12), (13) Λ : ...
Preprint
Full-text available
We consider some crucial problems related to the secure and reliable operation of power systems with high renewable penetrations: how much reserve should we procure, how should reserve resources distribute among different locations, and how should we price reserve and charge uncertainty sources. These issues have so far been largely addressed empirically. In this paper, we first develop a scenario-oriented energy-reserve co-optimization model, which directly connects reserve procurement with possible outages and load/renewable power fluctuations without the need for empirical reserve requirements. Accordingly, reserve can be optimally procured system-wide to handle all possible future uncertainties with the minimum expected system total cost. Based on the proposed model, marginal pricing approaches are developed for energy and reserve, respectively. Locational uniform pricing is established for energy, and the similar property is also established for the combination of reserve and re-dispatch. In addition, properties of cost recovery for generators and revenue adequacy for the system operator are also proven.
... There are many methods reported in the market studies and in the articles for solving the SRR problem. These techniques can be abstracted; Experimental method [10], Deterministic method [11], Probabilistic method [12], Deterministicprobabilistic combined method [13] and Cost-benefit method [14]. The experimental method is the simplest method used to determine the amount of reserve. ...
... Traditionally, a deterministic criterion is used to establish the level of reserves needed. With increased wind power, the uncertainties related to ramping of wind power production become more important than the sizes of conventional power plants; consequently, (Saez-Gallego et al., 2014) uses the uncertainty of future wind power production to determine the spinning reserve. In that study, the reserve requirements are computed on the basis of scenarios of wind power forecast errors, load forecast errors, and power plant outages. ...
Article
Long-term power-system planning and operation, build on expectations concerning future electricity demand and future transmission/generation capacities. This paper reviews current methodologies for forecasting long-term hourly electricity demand on an aggregate scale (regional or nationally), for 10–50 years ahead. We discuss the challenges of these methodologies in a future energy system featuring more renewable energy sources and tighter coupling between the power sector and the building and transport sectors. Finally, we conclude with some recommendations on aspects to be taken into account regarding long-term load forecasts in a changing power system.
... There are many methods reported in the market studies and in the articles for solving the SRR problem. These techniques can be abstracted; Experimental method [10], Deterministic method [11], Probabilistic method [12], Deterministicprobabilistic combined method [13] and Cost-benefit method [14]. The experimental method is the simplest method used to determine the amount of reserve. ...
Conference Paper
Full-text available
Spinning reserve is one of the significant sources that are used to define the all quantity of generation actual from all units synchronized on the power system, minus the available load and losses being supplied. The reserve is used by system operators as a response to unforeseen events such as sudden load changes and outages. By keeping the generation levels at high amount of reserve, it is possible to protect the power system against outages in the contingencies, thus the possibility of applying to load shedding process decreases. However, in this situation supplying reserve cost becomes high. In the Classic Unit Commitment formulations, deterministic standard, which determine the capacity of the largest synchronous generator, are used in order to adjust Spinning Reserve requirements. In this work, reserve requirements are adjusted according to the deterministic and probabilistic criteria. Socioeconomic parameters have been taken into consideration while achieving the optimum amount of reserves. The unit commitment for the most appropriate reserve cost was obtained using the benders decomposition method. As a result, in the event of sudden load changes and unforeseen events in the power system, the power required by the units is commitment at the most reasonable cost.
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This paper provides a high-level international comparison of methods and key results from both operating practice and integration analysis, based on an informal International Energy Agency Task 25: Large-scale Wind Integration.
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The determination of additional operating reserves in power systems with high wind penetration is attracting a significant amount of attention and research. Wind integration analysis over the past several years has shown that the level of operating reserve that is induced by wind is not a constant function of the installed capacity. Observations and analysis of actual wind plant operating data has shown that wind does not change its output fast enough to be considered as a contingency event. However, the variability that wind adds to the system does require the activation or deactivation of additional operating reserves. This paper provides a high-level international comparison of methods and key results from both operating practice and integration analysis, based on the work in International Energy Agency IEA WIND Task 25 on Large-scale Wind Integration. The paper concludes with an assessment of the common themes and important differences, along with recent emerging trends.
Chapter
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This paper presents a new concept for predicting the total wind power production in a larger region based on a combination of on-line measurements of power production from selected wind farms, power measure- ments for all wind turbines in the area and numerical weather predictions of wind speed and wind direction. The models are implemented in the Zephyr/WPPT system - an on-line software system for calculating short-term pre- dictions of wind power currently being developed by IMM and Risø in coorporation with Elsam, Eltra, Elkraft and SEAS - the major electrical utilities with respect to wind power in Denmark. Zephyr/WPPT employs statistical models to describe the relationship between power production and the numerical weather predictions. The statistical models belong to the class of conditional parametric models - a model class particular useful for estimating non-linear relationships on-line. The estimation is furthermore made adaptively in order to allow for slow changes in the system e.g. caused by the annual variations of the climate. Keywords: Forecasting Methods, Models (Mathematical), Adaptive Estimation, Statistics.
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