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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 N−1criterion), 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

Ri≤Rmax

i∀i(1b)

RT=

M

X

i

Ri(1c)

LOLP = Z∞

RT

f(z)dz (1d)

LOLP ≤β(1e)

RT≥0∀i. (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 1−β=F(RT) being F(Z) = P(Z≤z)

the cumulative distribution function of Z(the required reserve). Finally, since βis a given parameter, then

the solution is RT∗=F−1(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(Z≤z) 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 RT∗has 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 = Pw∈S(zw−RT∗)πw,S={w∈W: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 Z“reserve 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

1−α

W

X

w=1

πwηw(3a)

where ξis, at the optimum, the α−Value 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 +VLOLLw∀w(3b)

RT=

J

X

j=1

Rj(3c)

rT

w=

G

X

g=1

rgw ∀w(3d)

0≤Rj≤IR

j∀j(3e)

0≤rgw ≤Ir

g∀g, w (3f)

Costw−ξ≤ηw∀w(3g)

rT

w≤RT∀w(3h)

zw−rT

w≤Lw∀w(3i)

0≤Lw, ηw∀w. (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 Costw−ξif C ostw≥ξ; 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 zw≤RT, implying that zw=rT

w; when

zw> RT, then Lwis equal to the diﬀerence between the reserve requirements and the deployed reserves,

namely, zw−rT

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 ×10−5is 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 ×10−4. 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, Yt∼bern(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

1−pt=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

xk−1e−x

st,(6)

with mean µt=kstand variance σ2=ks2

t. The canonical link for the gamma distribution is the inverse

link η= 1/µ [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.

References

[1] M. Milligan, P. Donohoo, D. Lew, E. Ela, B. Kirby, H. Holttinen, E. Lannoye, D. Flynn, M. O’Malley, N. Miller, et al.,

Operating reserves and wind power integration: an international comparison, in: proc. 9th International Workshop on

large-scale integration of wind power into power systems, 2010, pp. 18–29.

[2] Energinet.dk, Ancillary services to be delivered in denmark tender conditions (2011).

[3] Energinet.dk, Ancillary services strategy (2011).

[4] Energy strategy 2050 - from coal, oil and gas to green energy, Tech. Rep. February, The Danish Ministry of Climate and

Energy (2011).

[5] A. J. Conejo, M. Carri´on, J.M. Morales, Decision Making Under Uncertainty in Electricity Markets, International series

in operations research and management science, Springer US, 2010.

[6] N. Amjady, J. Aghaei, H. Shayanfar, Market clearing of joint energy and reserves auctions using augmented payment

minimization, Energy 34 (10) (2009) 1552–1559.

[7] F. Bouﬀard, F. Galiana, An electricity market with a probabilistic spinning reserve criterion, Power Systems, IEEE

Transactions on 19 (1) (2004) 300–307.

[8] M. A. Ortega-Vazquez, D. S. Kirschen, Estimating the spinning reserve requirements in systems with signiﬁcant wind

power generation penetration, Power Systems, IEEE Transactions on 24 (1) (2009) 114–124.

[9] J.M. Morales, A. Conejo, J. Perez-Ruiz, Economic valuation of reserves in power systems with high penetration of wind

power, Power Systems, IEEE Transactions on 24 (2) (2009) 900–910.

[10] A. Kalantari, J. F. Restrepo, F. D. Galiana, Security-constrained unit commitment with uncertain wind generation : The

loadability set approach, IEEE Transactions on Power Systems (2012) 1–10.

[11] J. Warrington, P. Goulart, S. Mariethoz, M. Morari, Robust reserve operation in power systems using aﬃne policies, in:

Decision and Control (CDC), 2012 IEEE 51st Annual Conference on, 2012, pp. 1111–1117.

[12] E. Karangelos, F. Bouﬀard, Towards full integration of demand-side resources in joint forward energy/reserve electricity

markets, Power Systems, IEEE Transactions on 27 (1) (2012) 280–289.

[13] F. Partovi, M. Nikzad, B. Mozafari, A. M. Ranjbar, A stochastic security approach to energy and spinning reserve

scheduling considering demand response program, Energy 36 (5) (2011) 3130 – 3137.

[14] M. Matos, R. Bessa, Setting the operating reserve using probabilistic wind power forecasts, Power Systems, IEEE Trans-

actions on 26 (2) (2011) 594–603.

[15] R. Doherty, M. O’Malley, A new approach to quantify reserve demand in systems with signiﬁcant installed wind capacity,

Power Systems, IEEE Transactions on 20 (2) (2005) 587–595.

[16] G. C. Pﬂug, Some remarks on the value-at-risk and the conditional value-at-risk, in: Probabilistic constrained optimization,

Springer, 2000, pp. 272–281.

[17] The Value of Lost Load (VoLL) for electricity in Great Britain, Tech. rep., London Economics (2013).

[18] E. Leahy, R. S. Tol, An estimate of the value of lost load for Ireland, Energy Policy 39 (3) (2011) 1514 – 1520.

[19] R. T. Rockafellar, S. Uryasev, Optimization of conditional value-at-risk, Journal of risk 2 (2000) 21–42.

[20] Energinet.dk.

URL www.energinet.dk

[21] J. Kloppenborg Møller, H. Nielsen Aalborg, H. Madsen, Time-adaptive quantile regression, Computational Statistics &

Data Analysis 52 (3) (2008) 1292–1303.

[22] H. A. Nielsen, H. Madsen, T. S. Nielsen, Using quantile regression to extend an existing wind power forecasting system

with probabilistic forecasts, Wind Energy 9 (1-2) (2006) 95–108.

[23] T. S. Nielsen, H. Madsen, H. A. Nielsen, G. Giebel, L. Landberg, Prediction of regional wind power, Tech. rep.

[24] H. Nielsen, K. Andersen, H. Madsen, Empirisk bestemt model for elforbruget i Østdanmark, Institut for Matematisk

Modellering, Danmarks Tekniske Universitet, Lyngby, 1998.

[25] P. Pinson, H. Madsen, H. A. Nielsen, G. Papaefthymiou, B. Kl¨ockl, From probabilistic forecasts to statistical scenarios of

short-term wind power production, Wind Energy 12 (1) (2009) 51–62.

14

[26] Nord Pool.

URL http://www.nordpoolspot.com

[27] J. Saez, Determination of optimal electricity reserve requirements, Master’s thesis (2012).

URL http://www2.imm.dtu.dk/pubdb/views/edoc_download.php/6467/pdf/imm6467.pdf

[28] H. Madsen, P. Thyregod, Introduction to General and Generalized Linear Models, CRC Press, 2011.

[29] Energinet.dk, Appendix 1 : Load-Frequency Control and Performance. UCTE Operation Handbook.

URL https://www.entsoe.eu/publications/system-operations- reports/operation-handbook/

[30] E. Shayesteh, A. Youseﬁ, M. P. Moghaddam, A probabilistic risk-based approach for spinning reserve provision using

day-ahead demand response program, Energy 35 (5) (2010) 1908 – 1915.

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.

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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.

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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.

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