AckermannTh. (ed) Wind Power in Power Systems
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(4 % standard deviation). The standard deviation implies that the level of 126.75 MW
varies by 4 % between different years. The aim is now to select the subscription levels
in order to obtain a minimal expected power charge. If the subscribed level is too high
then too much will always be paid. However, selection of a level that is too low will be
too expensive as, there will be a need to pay extra for excess capacity. The optimal level
can be obtained in the following way.
Assume that the real measured power, which defines the charge, varies between
different years with a mean value m and a standard deviation . Assume also that the
subscription level has been selected to be a (see Figure 9.10).
This implies that in a year with a measured level x that is lower than or equal to the
subscribed level (i.e. x a) the cost is C( x) ¼ T
1
a, where T
1
is the cost for the subscrip-
tion per kilowatt per year. During a year when the measured power exceeds the
subscribed level (i.e. x > a), the cost becomes C(x) ¼ T
1
a þ T
2
(x a) instead, where
T
2
is the price for excess power per kilowatt per year. The expected cost E
C
that weighs
the probabilities for different years can be calculated as:
E
C
¼
Z
1
1
CðxÞfðxÞdx
¼
Z
a
1
T
1
afðxÞdx þ
Z
1
a
½T
1
a þ T
2
ðx aÞfðxÞdx
¼ T
1
a
Z
1
1
fðxÞdx þ T
2
Z
1
a
xfðxÞdx T
2
a
Z
1
a
fðxÞdx
¼ T
1
a þ T
2
Z
1
a
xfðxÞdx T
2
a½1 FðaÞ
¼ T
1
a þ T
2
m½1 FðaÞ þ
2
fðaÞ
T
2
a½1 FðaÞ
¼ðT
1
T
2
Þa þ T
2
m þ T
2
FðaÞða mÞþT
2
2
fðaÞ; ð9:20Þ
where
f(x) is the Gaussian probability function;
F(x) is the Gaussian distribution function.
0.08
0.06
0.04
0.02
0
110 115 120 125 130 135 140 145
Area = A
2
=1–F (a)
x > a
Area = A
1
= F (a)
x < a
a
m
f (x)
σσ
x
Figure 9.10 Gaussian approximation for the measured power, which defines the power charge
Wind Power in Power Systems 191
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Figure 9.11 shows how the expected cost E
C
depends on the selected subscription level.
As shown in Figure 9.11, a higher cost is obtained for too low or too high a
subscription level.
The minimum level is obtained when:
aðT
2
T
1
ÞA
2
¼ aT
1
A
1
:
This implies
ðT
2
T
1
Þ½1 FðaÞ ¼ T
1
FðaÞ;
which implies
FðaÞ¼
T
2
T
1
T
2
: ð9:21Þ
This solution can be explained in the following way:
1. Assume that the subscription level has been selected to be a. If this level is sligh tly
increased to a þ a, the cost for the years with a dimensioning power that is lower
than a will increase by aT
1
A
1
, where A
1
is the share of years when the dimensioning
power is lower than a (see Figure 9.10).
2. During years with a dimensioning power that is higher than a, the cost will instead
decrease by a(T
2
T
1
)A
2
, where A
2
is the share of years when the dimensioning
power is higher than a (see Figure 9.10).
3. The minimal cost is obtained when a small change of a does not affect the cost (i.e.
when a cost increase at point 1 equals the cost decrease at point 2), which is the same
as Equat ion (9.21).
Charges as in Table 9.4 make it possible to optimise subscription values. The resul t is
shown in Table 9.5
We now assume that there is wind power on the island of Gotland. This will decrease
the amount of power that is supplied from the mainland. The question is now whether
there is a new optimal subscription level and if the total expected cost would be reduced.
1250
1150
1050
1000
100 105 110 115 120 125 130 135 140 145 150
1100
1200
E
c
(a)
a
Figure 9.11 Expected cost E
C
, thousands of euros, as a function of selected subscription level a
192 The Value of Wind Power
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Figure 9.12 shows an example, where the load in Figure 9.10 is decreased by a certain
amount of wind power.
As shown in Figure 9.12, wind power will cause the dimensioning power to decrease.
In this specific case, the dimensioning power decreased by 4.3 MW, and the installed
amount of wind power was 43 MW. At the same time, the dimensioning hour is moved
within the week. The dimensioning hour could also be moved to another week with
lower wind speeds. On studying more wind situations, we found that the maximum net
load during one month was reduced by m
W
¼ 3:65 MW, corresponding to 43 MW of
wind power. However, the amount was different for different situations, and the
standard deviation was 3.1 MW.
In addition to this, one has to consider that the wind speeds differ between different
years. With 35 years of data from Visby airport on Gotland, it was found that there is an
additional standard deviation of 2.8 MW that has to be considered in order to arrive at a
correct estimation of how the maximum net load can vary in different situations.
The net level (power demand wind power production) for the measured power
levels, on which the charges are based, can now be described with a Gaussian distribu-
tion according to Figure 9.10. Assuming that there is no statistical dependence between
Table 9.5 Optimal subscription levels without wind power
Tariff
Yearly power High load power
Mean real measured power, m (MW) 126.75 126.75
Standard deviation, (MW) 5.1 5.1
Subscription cost per year, T
1
(e per kW) 3.4 8
Cost of excess power per year, T
2
(e per kW) 6.9 12
Minimum cost (millions of euros) 0.46 1.03
Subscription level, a (MW) 126.7 124.6
0
50
100
01234 657
Day
Week 1
MW
Figure 9.12 Net power consumption (actual load wind power.)
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the load and wind generation in January to February, the parameters for the Gaussian
distribution can be calculated as:
m
net
¼ m
load
m
w
¼ 123:1MW;
net
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
load
þ
2
loadw
þ
2
wyear
q
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
5:07
2
þ 3:1
2
þ 2:8
2
p
¼ 6:6MW:
An optimal subscription level can now be obtained by using Equation (9.21), and the
expected cost can be estimated by applying Equation (9.20). Table 9.6 presents the
results.
A comparison of Tables 9.6 and 9.5 shows that the subscription levels as well as the
expected costs decreas e depending on the installed amount of wind power. The cost
decreases by e0.04 million [(0:46 0:44) þ (1:03 1:01)] , which corresponds to (30 ke)/
(43,000) = e0.70 per kilowatt per year [(e30 000)/(43 000 kW)] of installed wind power
capacity, which lies in the range of 1 % of the total cost of wind power (i.e. it is a small
value).
It is important to note that the cost reduction is based on an optim ised subscription
level in the tariff. It was not part of this chapter to analyse whether the charges that are
reduced as a result of use of wind power also correspond to decreased costs in the
mainland grid.
9.5 Conclusions
This chapter has shown that the value of wind power in a power system can be divided
into the following parts:
.
operating cost value;
.
capacity credit;
.
control value;
.
loss reduction value;
.
grid investment value.
Table 9.6 Optimal subscription levels for wind power
Tariff
Yearly power High load power
Mean real measured power, m (MW) 123.1 123.1
Standard deviation, (MW) 6.6 6.6
Subscription cost per year, T
1
(e per kW) 3.4 8
Cost of excess power per year, T
2
(e per kW) 6.9 12
Minimum cost (millions of euros) 0.44 1.01
Subscription level, a (MW) 123.1 120.3
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In Section 9.3, it was shown how these values can be defined as ‘true values’ (i.e. the cost
decrease in an ideal system). Section 9.4 illustrated that payments to wind power owners
do not always correspond to these true values, at least not in the current deregulated
market in Sweden.
References
[1] IEEE (Institute for Electrical and Electronic Engineers) (1978) ‘Reliability Indices for Use in Bulk Power
Supply Adequacy Evaluation’, IEEE Transactions on Power Apparatus and systems PAS-97 (4) 1097–1103.
[2] Kirby, B., Hirst, E., Parsons, B., Porter K., Cadogan, J. (1997) ‘Electric Industry Restructuring, Ancillary
Services and the Potential Impact on Wind’, American Wind Energy Association, available at: http://
www.ornl.gov/ORNL/BTC/Restructuring/pub.htm.
[3] Milligan, M., Porter, K., Parsons, B., Caldwell, J. (2002) ‘Wind Energy and Power System Operations:
A Survey of Current Research and Regulatory Actions’, The Electricity Journal (March 2002) 56–67.
[4] Milligan, M. R. (2002) ‘Modelling Utility-scale Wind Power Plants, Part 2: Capacity Credit’, NREL
Report TP-500-29701, available at http://www.nrel.gov/publications/.
[5] Munksgaard, J., Pedersen, M. R., Pedersen, J. R. (1995) ‘Economic Value of Wind Power’ Report 1,
Amternes of Kommunernes Forskningsinstitut (AKF), Coperhagen (in Danish).
[6] So
¨
der, L. (1997) ‘Capacity Credit of Wind Power’, Elforsk report 97:27, final report from Elforsk project
2061 (in Swedish); copy available from author.
[7] So
¨
der, L. (1992) ‘Reserve Margin Planning in a Wind–Hydro–Thermal Power System’, 92 WM 168-5
PWRS, presented at IEEE/PES Winter Meeting, New York, 26–30 January 1992; published in IEEE
Transactions on Power Systems 8(2) 564–571.
[8] So
¨
der, L. (1988) Benefit Assessment of Wind Power in Hydro–Thermal Power Systems, PhD thesis, Royal
Institute of Technology, Stockholm.
[9] So
¨
der, L. (1994) ‘Integration Study of Small Amounts of Wind Power in the Power System’, Electric
Power Systems, Royal Institute of Technology, Stockholm.
[10] So
¨
der, L. (1999a) ‘The Value of Wind Power for an Owner of a Local Distribution Network’, paper
presented at 15th International Conference on Electricity Distribution, 1–4 June, 1999, Nice, France; copy
available from author.
[11] So
¨
der, L. (1999b) ‘Wind Energy Impact on the Energy Reliability of a Hydro–Thermal Power System in a
Deregulated Market’, For Proceedings of 13th Power Systems Computation Conference, June 28–July 2,
1999, Trondheim, Norway; copy available from author.
[12] van Wijk, A. (1990) Wind Energy and Electricity Production, PhD thesis, Utrecht University, Utrecht, The
Netherlands.
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Part B
Power System
Integration
Experience
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10
Wind Power in the Danish Power
System
Peter Borre Eriksen and Carl Hilger
10.1 Introduction
The past decade has seen a massive expansion of wind power in Denmark (see Figure 10.1).
This trend is the result of targeted political efforts to rely increasingly on renewable energy
production by introducing extensive public subsidising schemes.
Denmark is divided into two power systems (Figure 10.1) with no electrical inter-
connection:
.
The power system in Eastern Denmark (Zealand), with Elkraft as the Transmission
System Operator (TSO). This system is part of the Nordi c synchronous power system
(Nordel).
.
The power system in Western Denmark (Jutland–Funen), with Eltra as the TSO. This
system is part of the European synchronous power system, the Union for the Coord-
ination of Transmission of Electricity [the Union pour la Coordination du Transport
d’Electricite
´
(UCTE)].
Figure 10.2 shows the wind power expansion that has taken place in Denmark since
1985, for both Western Denmark, consisting of Jutland–Funen (with Eltra as TSO) and
Eastern Denmark, consisting of Zealand (with Elkraft as TSO). Because there are far
better opportunities for placing wind power facilities in the Jutland–Funen area and
there is a higher number of suitable sites this part of the country shows the by far largest
increase in wind power. There is currently (in mid-2003) a total installed capacity of
approximately 3000 MW in Denmark, of which a total of 2400 MW is located in the
Eltra area.
Wind Power in Power Systems Edited by T. Ackermann
Ó 2005 John Wiley & Sons, Ltd ISBN: 0-470-85508-8 (HB)
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Eltra
Elkraft
SYSTEM
Figure 10.1 Location of Denmark and definition of the Eastern and Western Danish power
systems
0
400
800
1200
1600
2000
2400
2800
1977
1979
1981
1983
1985
1987
1989
1991
1993
1995
1997
1999
2001
2003
Power (MW)
Wind power: Eltra area
Wind power: Denmark
Year
Figure 10.2 Wind power capacity expansion in Denmark
200 The Danish Power System