Short-term optimal wind power generation capacity in liberalized electricity markets

ARTICLE IN PRESS

Energy Policy 35 (2007) 1257–1273 www.elsevier.com/locate/enpol

Short-term optimal wind power generation capacity in liberalized electricity markets Fernando Olsinaa,, Mark Ro¨scherb, Carlos Larissona, Francisco Garce´sa a

Instituto de Energı´a Ele´ctrica, Av. Lib. Gral. San Martı´n 1109(O) J5400ARL San Juan, Argentina b University of Aachen, Schinkelstrasse 6 D-52056 Aachen, Germany Available online 9 May 2006

Abstract Mainly because of environmental concerns and fuel price uncertainties, considerable amounts of wind-based generation capacity are being added to some deregulated power systems. The rapid wind development registered in some countries has essentially been driven by strong subsidizing programs. Since wind investments are commonly isolated from market signals, installed wind capacity can be higher than optimal, leading to distortions of the power prices with a consequent loss of social welfare. In this work, the influence of wind generation on power prices in the framework of a liberalized electricity market has been assessed by means of stochastic simulation techniques. The developed methodology allows investigating the maximal wind capacity that would be profitably deployed if wind investments were subject to market conditions only. For this purpose, stochastic variables determining power prices are accurately modeled. A test system resembling the size and characteristics of the German power system has been selected for this study. The expected value of the optimal, short-term wind capacity is evaluated for a considerable number of random realizations of power prices. The impact of dispersing the wind capacity over statistical independent wind sites has also been evaluated. The simulation results reveal that fuel prices, installation and financing costs of wind investments are very influential parameters on the maximal wind capacity that might be accommodated in a market-based manner. r 2006 Published by Elsevier Ltd. Keywords: Wind power capacity; Electricity markets; Stochastic simulation

1. Introduction Because of environmental concerns, particularly global warming, as well as uncertainties on the long-run prices of fossil fuels, many countries in the world are aggressively promoting wind power capacity development. Consequently, wind has become one of the fastest growing energy technologies around the world (EWEA, 2004). In some areas, wind power is now contributing substantially to meet the electrical demand. Such is particularly the case of Germany, Spain and Denmark with, respectively, 16.6 GW, 8.3 GW and 3.1 GW of installed wind power capacity by the end of 2004 (DEWI, 2005; GWEC, 2005). Further additions of considerable amounts of wind capacity in the near future are very likely. Indeed, the coming into force of the Kyoto Protocol will probably Corresponding author. Tel.: +54 264 4226444; fax: +54 264 4210299.

E-mail address: [email protected] (F. Olsina). 0301-4215/$ - see front matter r 2006 Published by Elsevier Ltd. doi:10.1016/j.enpol.2006.03.018

provide additional incentives for further increasing wind installed capacity, especially in EU countries. Some other countries might undertake wind-favoring mechanisms, if wind generation demonstrates to be cost-effective or to have a positive value in mitigating global-warming risks. In addition, investment costs of wind generating facilities have shown a substantial reduction over the last decade (Junginger et al., 2005), turning wind projects increasingly attractive for independent investors. Wind power generation has three prominent characteristics that differentiate from most conventional generation technologies. Generation proceeding from this primary source is fluctuating over a wide range of frequencies; it can only be controlled to a very limited extent, and its availability is affected by considerable uncertainty.1 Even 1 Short-term wind prediction errors rapidly increase for longer forecast horizons. Typical forecast errors (RMSE) of the power output for a single wind farm normalized to the installed wind capacity range from 12% for a

ARTICLE IN PRESS 1258

F. Olsina et al. / Energy Policy 35 (2007) 1257–1273

though the short-term, marginal costs of wind generation are commonly negligible, wind power does not work as a simple fuel saver, since it cannot easily be controlled and accurately predicted (Giebel, 2000). For this reason, the value of wind capacity for power systems in the framework of the vertically integrated monopoly industry has been extensively investigated in the past (Barnes et al., 1994, 1995; Wan and Parsons, 1993). Particularly, the capacity credit provided by a certain amount of installed wind capacity has called the attention of researchers for many years (Billinton and Bai, 2004; Milborrow, 1996; Milligan and Parsons, 1999; van Wijk et al., 1992; Kahn, 1979). More recently, the required capacity reserve necessary to reliably operate the power system for increasing wind penetrations had been extensively evaluated (Dany, 2000; Doherty and O’Malley, 2003). Hirst and Hild (2004) have also assessed the revenues’ net of balancing costs perceived by wind farms operating in a small utility under scenarios of increasing installed wind capacity. The work of Kennedy (2005) presents a methodology for evaluating the long-term social benefits (including avoided environmental costs) of adding wind generation capacity to a power system. The methodology is based on a nonchronological description of load and wind power injection combined with screening curves for characterizing the economies of the different conventional generating technologies. The stochastic behavior of conventional units, i.e. forced outages, is neglected from this analysis and the available generating capacity at each time is assumed to be deterministic. A drawback of this analysis is the assumption of an electricity market continuously operating under static, long-run equilibrium conditions. Under this assumption, it is presumed that the mix of generation technologies can immediately be adjusted to the least-cost generation system when increasing wind capacity is added to the system. A similar approach, although more detailed, has been implemented by Kra¨mer (2003) for determining the optimal technology mix of the German power system for different scenarios of wind capacity, fuel prices, emission caps, etc. Because of temporal constraints and some inherent time lags, e.g. construction of new power plants, the generation mix usually requires a prolonged time to achieve the long-run equilibrium. The rapid deployment of wind power capacity in a non-market-based manner might temporarily provoke a major deviation of the equilibrium. Therefore, the consequent loss of social welfare might outweigh some of the benefits associated to wind generation. Although the assumption of long-run equilibrium provides a well-defined benchmark, it has recently been demonstrated that liberalized power markets do not actually behave in this way. Indeed, large deviations from

the long-run economic equilibrium are likely to happen (Olsina et al., 2006). The rapid wind development registered in some countries has essentially been supported by means of strong subsidizing programs. Wind generation output is remunerated with substantially higher prices than the prevailing prices in the power markets. Furthermore, some incentive mechanisms, as the feed-in tariff currently used in Germany, completely remove market risks for wind investments. That explains why considerable amounts of wind capacity, mostly private-owned, are being added to some competitive power systems. Unlike conventional generating technologies, wind investments are commonly isolated from market signals. Therefore, the economic optimality of the installed wind power capacity cannot be warranted. If installed wind capacity exceeds for a considerable margin the optimal value, distortions of the power market with significant loss of social welfare might occur. The impact of increasing amounts of wind capacity on spot prices in the framework of liberalized power markets has not yet been assessed. The investigation presented in this article is essential to determine the economically optimal wind capacity that at a certain time can be accommodated in a market-based manner (i.e. in absence of subsidies) to an existing generation system. In this work, the influence of wind generation on the electricity market has been assessed by means of stochastic simulations of power prices. A Especial emphasis is given to the contribution of geographically dispersing the installed wind capacity over a number of statistically independent wind sites. This topic is of particular interest, as most of the new wind capacity is being installed within the interconnected European system (UCTE), which extends over a very wide region. This paper is organized as follows. In Section 2, the basic concepts referring the optimal installed wind capacity are briefly exposed. Section 3 presents several modeling techniques applied in this investigation. For its relevance as a leading country in wind capacity development, a test system resembling the size and technology mix of the German power system has been selected for this study. The main hypotheses, assumptions and simplifications are provided in this section. The stochastic behavior of the variables driving power prices, i.e. load demand, the existing generation system and wind resources are modeled in detail in the subsections of Section 3. In Section 4, the results of stochastic simulations for the base case as well as a sensitivity analysis to some relevant influencing variables are presented and analyzed. With a discussion of the implications of the results presented in Section 4, Section 5 concludes the present article.

(footnote continued) 3-h ahead prediction to 30% for a 36-h forecast. By combining various forecasting techniques, the day-ahead prediction error for the aggregate wind power output across a wide region is currently about 8–10%. As comparison, day-ahead load forecast errors lie around 0.5–2.5%.

2. Optimal short-term wind power generation capacity Power economics shows that social welfare is maximized if resources are allocated optimally. If this condition is

ARTICLE IN PRESS F. Olsina et al. / Energy Policy 35 (2007) 1257–1273

achieved, the system is said to be in economic equilibrium. Under these circumstances, revenues perceived by each and every firm compensate exactly the total incurred costs and therefore the firm’s economic profit2 is zero. Thus, firms do not have any incentive to either invest in new capacity (since the market does not offer the possibility of gaining supernormal profits) or exit the business (since the costs are fully recovered). In deregulated markets, generation capacity is typically installed if it looks profitable. Additional generation capacity should be installed while the associated expected revenues pay off the total marginal incurred costs. In the absence of subsidies, this statement allows determining the maximal wind power capacity, Pmax W , that might be profitably deployed if the wind generator’s revenues were subject to market forces only. For PW ¼ Pmax W , the optimality condition is achieved, since the economic profit turns negative for any additional wind capacity unit. Under this condition, the total incurred costs of installing and operating a unit of wind-based capacity is equal to the revenues collected by selling the wind power output at market prices.3 This condition can be expressed in terms of the annual fixed investment cost of a wind capacity unit, FC measured in h/MW year, and the short-term, annual revenues, P(PW)4 in h/MW year: PðPW ¼ Pmax W Þ  FC ¼ 0, with FC ¼

r0  IC , 1  ð1 þ r0 ÞT a

where IC expressed in h/MW is the installation cost of a wind unit, Ta in year is the amortization period of the wind project and r0 in %/year is the discount rate or hurdle rate, defined by the actual cost of raising capital. For a given set of financial parameters and by means of stochastic simulations of the market revenues of wind generators for increasing values of PW, the maximal profitable wind capacity Pmax can be determined from the previous W equations.

1259

participants is usually required. Because of representing a well-defined benchmark, perfect competition is assumed in the framework of this study. Under this hypothesis, generators maximize their own profit by bidding their true marginal costs. Under this market setting, the competitive price is set at each time by the most expensive running generator, i.e. the marginal generator. In this work, only the wholesale energy market will be considered for determining hourly spot prices. The impact of wind generation on the market for ancillary services is neglected. Additionally, as congestions of the transmission network are very rare events for highly meshed systems, such is the case of Germany, only one price zone has been considered. For the purpose of the proposed investigation, a considerable number of yearly realizations of hourly load demand and power generation are necessary to perform Monte Carlo simulations of power prices. In order to obtain accurate statistical estimates, 1000 yearly realizations of the power system’s operation and the resulting hourly spot prices have been performed. This sample size ensures a maximum error of about 1% at probability levels as small as 0.01 within 99.7% confidence bounds. The following subsections provide a detailed description of the applied modeling techniques for describing the stochastic behavior of the driving variables, i.e. the load demand, the power supply of the conventional generation system and the aggregate wind generation.

3.1. Load demand modeling

The stochastic power market model presented here can be characterized as a fundamental or bottom-up model, since chronological hourly power prices are determined by simulating the mechanisms having direct influence on the short term, stochastic movements of supply and demand. Thus, explicit mathematical models for the aggregate supply and demand curves are provided. For this reason, some hypothesis on the degree of competition in the market and hence on the bid behavior of market

Load demand is a major driver of power prices. Aggregate power demand is assumed to be price-inelastic, which represents the typical inability of customers to adjust their consumption at a short notice.5 Thousand chronological time series of hourly power demand over a year have to be synthetically generated. Any consistent stochastic load model should account for some deterministic patterns, like the daily, weekly and yearly seasonality, as well as hourly stochastic fluctuations around the expected values. Unfortunately, the only publicly available data sets of the aggregate German load are not adequate for our purposes. The statistics provided by the UCTE (2005) only consists of 24-hourly load values recorded on the third Wednesday of each month over the period 1994–2004. Hourly load demand for a Saturday and a Sunday of each month are available only for year 2000. Therefore, the standard techniques for identifying an adequate stochastic model cannot be used in a straightforward manner and a preconditioning of the data sets was necessary.

2 Economic profit is defined as the difference between total revenues and total costs including the opportunity cost of capital. 3 As no fuel is required, O&M costs of wind facilities are commonly negligible. 4 Computed short-term, annual revenues are based on the existing generation system and the prevailing load conditions.

5 It is important to note that some degree of price-elasticity of electricity demand might induce a valuable correlation between wind power injections and power load. A better coincidence between wind resources and load pattern lessens to some extent the difficulties faced by this generation technology in the framework of the current planning and operation procedures.

3. Stochastic simulation of power markets

ARTICLE IN PRESS F. Olsina et al. / Energy Policy 35 (2007) 1257–1273

1260

In order to obtain the load expressed on a common year basis, the available time series were detrended using the growth rate of monthly energy consumptions. The basis year is year 2000. By doing this, estimates of the unconditional expected hourly loads, m^ L , as well as unconditional hourly standard deviations, s^ L , for each type of day and month were obtained. As only one typical daily load pattern for each month becomes available after this procedure, load data sets were interpolated and smoothed via Fourier series to account for slow changes within a month. Fig. 1a depicts the unconditional expected hourly power demand over the year after the conditioning process. From this figure, the annual and weekly seasonality can be clearly identified. In Fig. 1b, only the third week of December and June is plotted aiming at showing in detail the daily and weekly pattern of load demand.

considered. By virtue of Central Limit Theorem, the stochastic fluctuations of the aggregate load demand are commonly assumed to be Gaussian. A Gauss–Markov model has been proposed in the literature for describing the stochastic behavior of load fluctuations in the framework of reliability analysis of power systems (Breipohl et al., 1992; Breipohl and Lee, 1991). For a specific US utility, it was shown that on average this load model outperforms higher order, autorregresive models as well as other forecasting techniques. Because of the lack of enough chronological load observations, hourly fluctuations of the aggregated German load are assumed to follow a Gauss–Markov stochastic process given by the following sampling expression for the load Li within the ith time interval:

8 > m^ L þ s^ 2Li N i ð0; 1Þ > > < i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s^ 2Li Li ¼ > ^ Li þ r^ 2 ðLi1  m^ Li1 Þ þ s^ 2Li ð1  r^ 2 ÞN i ð0; 1Þ m > > : s^

for i ¼ 1; for i ¼ 2 . . . 8760;

Li1

Additionally, random deviations of the load from the hourly expected values, for example due to weather fluctuations, are also an important load feature to be 80

Mean hourly load [GW]

70 60 50 40

3.2. Modeling the generation system

30

(a)

20

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

80

Hourly load demand [GW]

3rdweek-Dec 70

3rdweek-Jun

60 50 40 30 20

(b)

where m^ Li is the estimated unconditional mean load at hour i, s^ 2Li the estimated unconditional variance for this time interval, r^ denotes the correlation coefficient between two consecutive hours (assumed to be fixed and equal to 0.9) and N i ð0; 1Þ identical independent Gaussian distributed random samples with zero mean and unit variance. Fig. 2a and b show, respectively, a sample realization of the load path for 1 year and the third week of December and June. By comparing Figs. 1 and 2, the necessity of considering the random load fluctuations when simulating the stochastic behavior of power prices it can be clearly recognized.

Mon

Tue

Wed

Thu

Fri

Sat

Sun

Fig. 1. (a,b) Expected hourly load patterns for the German system.

In this study, a test system, resembling the size and generation technology mix of the German generation system, has been modeled at the plant level. The thermal generation system encompasses four generating technologies and five fuels amounting 340 power plants and 81 568 MW of installed operating capacity.6 Parameters of the considered system are given in Table 1. Generating units are differentiated by technology, fuel, unit size and thermal efficiencies. For each plant, the heat input function has been linearized at nominal power and through the origin, which leads to consider constant marginal costs of generation. This simplification allows dispatching of generation units by the merit order or priority listing method. It is worth to note that the merit order is a simplified heuristic method to solve the unit commitment problem (UC). This dispatching methodology 6

In Germany, hydropower accounts for a small fraction of the total installed capacity. For simplicity, only thermal generating units have been considered.

ARTICLE IN PRESS F. Olsina et al. / Energy Policy 35 (2007) 1257–1273

as well as mean-time-to-failure (MTTF) and mean-time-torepair (MTTR) for each considered technology are given in Table 2. Under the Markov hypothesis, operation (TTF) and reparation (TTR) times of generating units can be simulated by taking i.i.d. random samples from an exponential distribution with parameters l and m respectively (Billinton and Allan, 1996):     1 1 1 1 TTF ¼ ln U½0; 1 ; TTR ¼ ln U½0; 1 , l l m m

Hourly load demand [GW]

80 70 60 50 40 30 20 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

(a)

Hourly load demand [GW]

80

3rdweek-Jun 3rdweek-Dec

70 60 50 40 30 20

(b)

1261

Mon

Tue

Wed

Thu

Fri

Sat

Sun

Fig. 2. (a,b) Stochastic hourly realizations of the German load demand.

assumes constant unit’s marginal costs and total operating flexibility for generating units. In fact, operating constraints of generating units influencing the optimal UC, like unit’s startup costs, minimum downtime–uptime and minimum unit output are not considered.7 Marginal costs of generation can therefore be computed by dividing the corresponding fuel price by the unit’s thermal efficiency. Fuel prices for the different generating technologies are taken constant over the year. The range of marginal costs of generation is given for each technology in Table 1. For this generation system, Fig. 3 illustrates the aggregate supply curve, which results of plotting the cumulated power capacity against the sorted marginal cost of generation. A major source of price fluctuations is the capacity that at a certain time is available to meet the load. The available capacity stochastically fluctuates from hour to hour as consequence of random outages of the individual units. For this reason, the stochastic behavior of the thermal units has to be modeled. A two-state Markov model is deemed to be enough for describing random outages of thermal power plants. Typical forced outage rates (FOR)

where l ¼ MTTF1 in h1 is referred as the failure rate and m ¼ MTTR1 measured in h1 is the repair rate, and U½0; 1 are uniform i.i.d. samples over the interval [0,1]. Both reliability parameters can be related to the outage probabilities (FOR) by the following expression: FOR ¼

l . lþm

Fig. 4a shows seven simulated paths over a period exceeding 1 year for some individual units of different technologies. With A it is denoted that the time interval within the considered generating units are available. On the contrary, U denotes the period of technical unavailability due to forced outages. In Fig. 4b, the simulated total available capacity over 1 year is plotted along with the total installed capacity (dotted line). As it can be noted, the total available capacity fluctuates randomly over a quite wide range. Although unlikely, at some hours the available generating capacity is lower than the peak load value. At those times, the risk of having the system unable to fully satisfy the power demand, which is referred in the reliability literature as loss of load probability (LOLP), is greatly increased. Under circumstances of capacity deficit, there is no intersection of the aggregate supply and demand curves, and therefore, the power market cannot be cleared. In these cases, prices would rise to a very high value to reflect the scarcity and a price cap has to be administratively set. The price cap is commonly determined by regulators and theoretically, it should set equal to the cost of being curtailed, which is referred in the literature as value of lost load (VOLL). For the base-case simulations, the VOLL is set to be equal to 10 000 h/MWh. 3.3. Modeling wind power generation 3.3.1. Statistical description of wind speeds The occurrence frequencies of wind speeds are usually very well fitted by the Weibull distribution function, whose probability density function (PDF), wðvÞ, and cumulative probability function, W ðvÞ, are respectively defined for the interval v 2 ½0; 1Þ as a

wðvÞ ¼ aba va1 eðv=bÞ , a

W ðvÞ ¼ 1  eðv=bÞ , 7

For this linearization, Cumperayot (2004) finds an average overestimation of 6.33% when simulating System Marginal Costs for the German power system.

where a is the shape parameter and b [s/m] the scale parameter, with a, b40. For some specific wind sites, it is

ARTICLE IN PRESS F. Olsina et al. / Energy Policy 35 (2007) 1257–1273

1262

Table 1 Model data for the German conventional generation system Technology

Number of units

Avg. unit size (MW)

Installed capacity (GW)

Fuel prices (h/GJ)

Efficiency (%)

Marginal costs (h/MWh)

Nuclear Lignite Hard coal CCGT Oil steam Gas turbine Gas steam Oil turbine

19 66 124 23 6 29 36 37

1112 294 201 174 400 54 175 51

21.121 19.402 24.879 4.013 2.398 1.570 6.295 1.890

1.07 1.40 2.06 4.65 3.62 4.65 4.65 7.92

38.6–31.4 43.1–27.0 45.1–25.0 59.4–40.0 37.3–35.6 38.1–25.0 37.3–28.0 30.5–25.0

9.98–12.27 11.7–18.68 16.4–29.6 29.64–41.86 34.86–36.6 44–66.98 44.88–66.98 93.5–114

s2v ¼ 8:43 m2 =s2 . From the previous equations, the parameters of the Weibull distribution can be determined by solving numerically for a and b. The Weibull parameters are found to be 6.4919 and 2.0937 respectively. In Fig. 5a the Weibull probability density function of wind velocities is plotted for the found values for a and b. Inset shows some statistical parameters of the probability distribution, i.e. mean, variance, skewness and kurtosis.

120

Marginal cost [C/ MWh]

100 80 60 40 20 0 0

20

40 60 Aggregated Supply [GW]

80

Fig. 3. Aggregate supply curve for the German power system.

Table 2 Typical reliability parameters for conventional generating units Technology

FOR

MTTF (h)

MTTR (h)

Nuclear Lignite Hard coal CCGT Oil-fired steam Gas turbine Gas-fired steam Oil turbine

0.042 0.044 0.055 0.036 0.071 0.031 0.071 0.031

1100 900 900 1100 1000 600 1000 600

48.23 41.42 52.38 41.08 76.43 19.20 76.43 19.20

usual that only some statistical measures of the wind resources, like the mean and the total variance, are available. The mean mv and the variance s2v for the Weibull distribution are defined in term of the Gamma function as   mv ¼ bG 1 þ a1 ,      s2v ¼ b2 G 1 þ 2a1  G2 1 þ a1 . In this study, a homogeneous wind field over all considered wind sites has been assumed. The mean wind speed is given as mv ¼ 5:75 m=s and the total variance

3.3.2. Power spectrum of wind speed fluctuations In addition to the statistical characterization of wind speeds, the description of wind velocity fluctuations in the frequency domain also provides relevant information. Indeed, wind velocities exhibit a strong auto-correlation structure between successive time intervals. By virtue of the Wiener–Khintchine theorem, the power spectral density (PSD) function, denoted as S f 0 ðoÞ, of a stochastic process describing the temporal sequence of wind speeds f 0 ðtÞ, is related to the auto-correlation function, Rf 0 ðtÞ, by Fourier Transform pair8: Z 1 1 Sf 0 ðoÞ ¼ Rf ðtÞ eiot dt; 2p 1 0 Z 1 Rf 0 ðoÞ ¼ S f 0 ðtÞ eiot dt; 1

where o ¼ 2pf is measured in rad/s, f being the frequency in Hz. In a famous article, Van der Hoven (1957) presented for the first time a composite picture of the spectrum of wind speed fluctuations over a wide frequency band ranging from 0.0007 h1 (period of about 2 months) to 900 h1 (period of 4 s). He recognized the existence of two very well differentiated spectrum zones separated by a wide zone centered at frequencies of about 1 h1 with negligible spectral amplitudes. Numerous studies carried out in many different meteorological stations confirmed further that the two spectral wind regimes and the spectral gap characterize consistently wind speed fluctuations. 8

The PSD function is an alternative description of the temporal autodependence of a time series. It is worth to note that the total variance s2f ¼ Rðt ¼ 0Þ is equal to the area under the PSD function. 0

ARTICLE IN PRESS F. Olsina et al. / Energy Policy 35 (2007) 1257–1273

0.14

A

Weibull probability density

U A U A U A U A U A U A

0.12

0.04 0.02

0

5

10 15 20 Wind speed [m/s]

(a)

25

30

6 Mean spectral intensity [m2/s2]

Available generation capacity [GW]

γv2 = 3.13763 m4/s4

0.06

82

80

78

76

74

daily peak 5

σv2 = 9.42 m / s2

4 3 2 annual peak 1

72

0 10-5 (b)

70

(b)

γv1 = 0.57054 m3/s3

0.08

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000011000 Hours

μv = 5.75 m/s σv2 = 8.43 m2/s2

0.1

U

(a)

1263

10-4

10-3

10-2 10-1 100 101 Frequency [1/h]

102

103

104

Jan Feb March April May June July Aug Sept Oct Nov Dec

Fig. 5. (a,b) Statistical and spectral characterization of the wind resources. Fig. 4. (a,b) Random unit outages and a yearly stochastic realization of the hourly available capacity. 9

Fig. 5b illustrates the mean power spectral intensity of horizontal wind speed fluctuations at 10 m (33 ft) above the ground, computed from a 10-year time series of hourly records of wind speeds at Caribou, Maine, USA (Oort and Taylor, 1969). The spectrum covered fluctuation periods from about 2 years to 1 h. For the high-frequency range, the Kaimal spectrum, for which an analytical expression exists (Kaimal et al., 1972), has been adopted.10 The figure clearly reveals the spectral gap as well as the two spectral regions. On one hand, the low-frequency spectral region is explained by the passage of large synoptic scale pressure systems with period of 4–7 days. On the other hand, the high-frequency region is originated in turbulent wind speed fluctuations, i.e. gusts, with periods in the order of minutes and seconds.11 Very noticeable is the diurnal peak at 9 The mean spectral intensity measured in m2/s2 is related to the PSD as fSðf Þ, where f is the frequency in Hz and the power spectral density function S(f) expressed in m2/(s2Hz) or m2/s. 10 Many other analytical descriptions are available for the spectral modeling of wind turbulence, such as the von Karman and the Davenport turbulence models. 11 For this study, only hourly wind speeds time series are required. The variance corresponding to fluctuations with frequencies bellow 1 h1 is

1 1 h . Although smaller, the yearly frequencies of about 24 seasonality can also be clearly recognized as an annual 1 peak at frequencies of about 8760 h1 .

3.3.3. Stochastic simulation of hourly wind speeds Before the widespread use of simulation techniques, probabilistic evaluation of power systems, including unconventional generation sources, were performed in the framework of the equivalent load duration curve (ELDC)(Singh and Lago-Gonzalez, 1985). The initial efforts in applying stochastic simulation techniques for generating wind speed time series were based on statistical considerations only. Monte Carlo methods were applied to obtain independent samples of a given marginal probability distribution of wind speeds. Wind speed time series obtained with this methodology failed in considering the auto-correlation structure of wind speeds observed in consecutive time intervals. (footnote continued) equal to 8.43 m2/s2. The low contribution to the total variance corresponding to intrahour wind speed fluctuations will be further disregarded.

ARTICLE IN PRESS F. Olsina et al. / Energy Policy 35 (2007) 1257–1273

1264

Presently, most of the proposed wind speed simulations techniques are based on autoregressive moving average (ARMA) models (Torres et al., 2005; Kamal and Jafri, 1997; Daniel and Chen, 1991). The same approach is chosen by Billinton et al. (1996a, b) in the framework of Monte Carlo techniques for evaluating reliability of power systems when considering wind power generation. Although ARMA models seem to be suitable for simulating sequences of wind speeds at some particular location, some considerations must be taken into account. In order to apply an ARMA model, a transformation (e.g. the Box–Cox transformation) and standardization of the observed time series are commonly required to obtain stationary, Gaussian distributed time series. The digital simulation of stochastic processes and fields by means of the spectral representation is a very efficient alternative approach for generating sample wind speed time series (Shinozuka and Jan, 1972). A thorough review of the method for generating univariate, Gaussian stochastic processes according to a prescribed PSD (or autocorrelation function) can be found in Shinozuka and Deodatis (1991). In the following, only some introductory details of the simulation algorithm will be outlined. Every zero-mean, real-valued, univariate stochastic process, f o ðtÞ, can be represented in terms of two mutually orthogonal real processes, uðoÞ and vðoÞ, as follows: Z 1 ½cosðotÞ duðoÞ þ sinðotÞ dvðoÞ. f 0 ðtÞ ¼ 0

By adequately defining the orthogonal increments duðoÞ and dvðoÞ, the integral equation can be expressed in terms of an infinite series as: 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X S f 0 ðok ÞDo cosðok t þ Fk Þ, f 0 ðtÞ ¼ 2 k¼0

where Fk are independent random phase angles distributed uniformly over the interval ½0; 2p. Therefore, a sample realization f ðiÞ ðtÞ of the stochastic process f 0 ðtÞ can be simulated by the following truncated series as N ! 1: f ðiÞ ðtÞ ¼ 2

N 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X

S f 0 ðon ÞDo cosðon t þ f ðiÞ n Þ,

n¼0

where n ¼ 0; 1; 2; . . . ; N  1, on ¼ nDo; Do ¼ ou =N. In the previous equations, ou denotes an upper cutoff frequency beyond which the PSD S f 0 ðoÞ may be assumed to be zero and N is the number of frequency bands of width Do in which the PSD has been discretized. The sequence of random phase angles F0 ; F1 ; F2 ; . . . ; FN1 are replaced by ðiÞ ðiÞ ðiÞ their respective ith realizations fðiÞ 0 ; f1 ; f2 ; . . . ; fN1 . According to the sample theorem, the time interval Dt of the simulated stochastic process has to obey the condition that Dtp2p=ou in order to avoid aliasing. This approach is computationally very efficient, as it takes advantage of the

fast fourier transform (FFT). For this purpose, the simulation formula is rewritten as follows, where M, the number of time intervals of length Dt, must be selected an integer power of two: " # M1 X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðiÞ Sf 0 ðon ÞDo eif n einp2p=M Þ , f ðiÞ ðpDtÞ ¼ Re 2 n¼0

p ¼ 0; 1; 2; . . . ; M  1; M  2N;

M ¼ 2m ;

One relevant property of the Shinozuka method is that each and every sample function obtained with this method is ergodic in the mean value and auto-correlation function. Hence, the temporal average and the temporal autocorrelation function of every sample are identical to the corresponding targets. In order to guarantee the ergodicity with respect to the target mean and auto-correlation function, the following condition must additionally be set: Sf 0 ðon Þ n¼0 ¼ S f 0 ðo0 Þ ¼ 0. It is easy to demonstrate that sample functions generated with this algorithm are periodic with period T 0 ¼ 2p=Do. The temporal average and the temporal auto-correlation function of any sample function f ðiÞ ðtÞ are identical to the corresponding targets only when the length T of the simulated sample is equal to the period T 0 or when T ! 1. Yamazaki and Shinozuka (1988) proposed an iterative procedure (also known as spectral correction method) to generate stochastic univariate sample functions to match simultaneously a specified PSD and a prescribed skewed marginal probability density function. This algorithm presents, however, some limitations to match highly skewed non-Gaussian stochastic processes. Since wind speeds are non-Gaussian with significant skewness, i.e. commonly Weibull, an improved extension of Yamazaki–Shinozuka method proposed by Deodatis and Micaletti (2001) has been applied in this work to synthesize hourly wind speed time series. For the jth iteration, the method starts with the generation of an underlying stochastic process f jG ðtÞ according to a PSD, Sjf ðoÞ, which evolves as the iterations G proceed. For the first iteration, the target spectrum, T denoted as S f W ðoÞ, can be used to generate the underlying stochastic process. Deodatis and Micaletti (2001) demonstrated that only in the first iteration, f jG ðtÞ is Gaussian and the Gaussian property is lost in the further iterations. F jG denotes the empirical cumulative probability distribution function (CDF) of the underlying stochastic process at the jth iteration. The CDF is computed via the Kaplan–Meier estimate of the underlying sample function. By using the concept of translation process or memory-less transformation, the underlying stochastic process is mapped to a new stochastic process, f jW ðtÞ distributed according to the prescribed non-Gaussian distribution. By denoting as FW the Weibull cumulative distribution function, the mapping

ARTICLE IN PRESS F. Olsina et al. / Energy Policy 35 (2007) 1257–1273

and hence n h io j j f jW ðtÞ ¼ F 1 F f ðtÞ , W G G F 1 W

where denotes the inverse of the Weibull cumulative distribution function of wind speeds, for which an analytical expression exists. The power spectral density of the generated Weibull sample function after the j-th iteration can be computed as 2 Z T 1 j j iot S f ðoÞ ¼ f W ðtÞ e dt W 2pT 0 2 X j 1 1 M1 iopDt ¼ f W ðpDtÞ e , Do M p¼0 which can be readily calculated via the FFT algorithm. Because of the non-linear mapping, the non-Gaussian sample function f jW ðtÞ does no longer possess the target spectrum, i.e. S jf ðoÞaS TfW ðoÞ. Therefore, an iterative W scheme for correcting the underlying power spectrum j S fG ðoÞ is necessary by applying the following updating formula: " T #a Sf W ðoÞ jþ1 j S f ðoÞ ¼ Sf ðoÞ j , G G Sf ðoÞ W

where the exponent a is set lower than unity in order to achieve a more rapid convergence behavior.12 The described iterative process proceeds while the error between the spectrum of the simulated sample function and the target spectrum exceeds some specified tolerance. The convergence tolerance has been defined in terms of the root mean square error (RMSE). The stop criterion was set to be vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N 1 h i2 u1 X j RMSE ¼ t S f ðon Þ  S TfW ðon Þ p0:008, W N n¼0 which provides an accurate matching to the target spectrum with reasonable computation time. The speed of convergence depended largely on the set of random angles selected. In the case of a well-conditioned phase angle set, the tolerance criterion was achieved after 15 iterations. The skewness of the distribution of generated underlying sample function was found to be a very good, early indicator of bad-conditioned phase angles and hence of slow convergence behavior. In those cases, the reseeding of the algorithm with a new, better-conditioned set of random phase angles saved much computation time. 12

Deodatis and Micaletti (2001) propose for the exponent a value of a ¼ 0.3. From an extensive experimentation, in this case a value of a equal to 0.7 leads to the fastest convergence speed.

20 Simulated wind speed [m/s]

procedure can be mathematically expressed as: h i h i F W f jW ðtÞ ¼ F jG f jG ðtÞ ,

1265

15

10

5

0

0

100

200

300 400 500 Simulated hours

600

700

Fig. 6. Generated sample path of hourly wind speed at 10 m height.

The described algorithm has been applied to synthesize a database of 15 000 yearly realizations of hourly wind velocities. These time series will be used to determine the aggregated power output delivered by a number of wind farms located at different wind sites. Fig. 6 shows 730 h of a yearly sample of wind speeds generated with the spectral representation method. By means of the normal kernel function, the empirical PDF of the generated non-Gaussian sample wind speed has been computed and it is shown in Fig. 7a along with the target distribution. For the purpose of comparison with the values of the target distribution shown in Fig. 5a, statistical parameters of the empirical distribution, i.e. mean, variance, skewness and kurtosis, are provided as well. As it can be observed, the algorithm achieves a very good matching with the prescribed Weibull distribution. Fig. 7b illustrates additionally the computed PSD of a sample function, which shows a very good agreement with the wind target spectrum depicted in Fig. 5b. 3.3.4. Simulation of aggregated wind power supply In this study, the installed wind power capacity is varied between 1 and 20 GW. This capacity consists of a number of 1-MW wind turbines spread over 100 identical wind parks. Therefore, the number of considered wind turbines within a wind farm ranges from 10 to 200. In order to assess the influence of allocating the wind capacity among uncorrelated wind sites, simulation experiments with different amounts of wind capacity dispersed over 1, 2, 5 and 100 statistically independent wind locations have been conducted. A typical turbine’s power output curve with a nominal power of 1 MW delivered at 13 m/s (nominal wind speed) and a cut-in speed of 2.5 m/s and cut-off speed of 25 m/s have been adopted for this study. Fig. 8 illustrates the nonlinear conversion characteristics of the considered wind turbine. This output characteristics is used as a mapping function to convert the synthetic hourly wind speed time series in hourly power output time series. As wind speed

ARTICLE IN PRESS F. Olsina et al. / Energy Policy 35 (2007) 1257–1273

1266

0.25

Simulated Target

0.12 0.1

μv = 5.75012 m/s

0.08

σv2 = 8.43043 m2/s2 γv1 = 0.57101 m3/s3

0.06

γv2 = 3.13677 m4/s4

0.04

0.15

0.1

0.05

0.02 0 0

5

(a)

10 15 20 Wind speed [m/s]

25

30

Mean spectral intensity [m2/s2]

6

0

0

1

2

3 4 5 6 7 8 9 10 11 12 13 14 15 Number of unavailable wind turbines

Fig. 9. Binomial probability distribution of the unavailable number of wind turbines.

5

3

wind speed profile has been selected. The wind speed at a height z is related to the wind velocity measured at a height zbase as follows:

2

vðzÞ ¼ vðzbase Þ

4

1 0 10-5

10-4

(b)

10-3 10-2 10-1 Frequency [1/h]

100

101

Fig. 7. (a,b) Statistical and spectral characteristics of synthesized wind speed samples.

1200 1000 Power output [kW]

N = 200 wind turbines p = 2% failure probability

0.2 Outage probability

Weibull probability density

0.14

800 600

lnðz=z0 Þ , lnðzbase =z0 Þ

where z0 ¼ 0.75 m denotes the roughness length, which depends on the terrain characteristics, and the hub height z is assumed to be equal to 75 m. As far as only fluctuations of frequencies lower than 1 h1 are considered, full coherence in the power output of wind turbines of the wind farm can be assumed without loss of accuracy (Nanahara et al., 2004). Hence, the wind farm output can be readily computed by multiplying the generation output of a single wind turbine by the number of available turbines. The assumption of identical wind turbines allows the determination of the number of available wind turbines for each hour by means of taking random samples of a binomial probability distribution.13 The probability PrðnW Þ of having a number nW of unavailable turbines in a wind farm composed by N W identical machines can be analytically computed as follows:

400

W nW ð1  pÞN W nW , PrðnW Þ ¼ C N nW p

200

where p is the turbine’s outage probability, for which a typical value of 0.02 has been assumed. Fig. 9 shows the failure probability of increasing number of wind turbines for a wind farm containing 200 units. The number of available wind farms at any time (e.g. due to transformer and/or breakers’ outages) is considered in a similar manner. Forced outage rates for wind farms are taken to be 0.006. Finally, the generation output of the available wind farm is further corrected for array efficiency which is assumed to be equal to 0.9. The capacity factor delivered

0 0

5

10 15 20 Wind Speed [m/s]

25

30

Fig. 8. Power generation curve of a typical 1-MW wind turbine.

time series represent the prevailing wind conditions at 10 m height, they must be corrected for turbine’s hub height before this non-linear mapping. For this purpose, a standard logarithmic model for describing the vertical

13 A Markov, two-state reliability model for the turbine availability has been adopted.

ARTICLE IN PRESS F. Olsina et al. / Energy Policy 35 (2007) 1257–1273

Hourly powerprices [• / MWh]

120

18 16 14 12 10 8 6

10000

c / MWh

80 60 40 20 0

Jan Feb March April May June July Aug Sept Oct Nov Dec

2 0

Hours

(a) 0

100

200

300

(a)

400 Hours

500

600

700

10 Aggregated power output [GW]

Price spike 100

4

120 Hourly powerprices [• / MWh]

Aggregated power output [GW]

20

1267

9.5 9 8.5 8 7.5

80 60 40 20 0

7 (b) 6.5 6

Price spike 10000 c / MWh

100

0

100

200

300

400 Hours

500

600

700

Fig. 11. (a,b) Stochastic yearly realization of power prices.

0

100

200

300

(b)

400 Hours

500

600

700

Fig. 10. (a,b) Random samples of the aggregate power output for 1 and 100 uncorrelated wind sites, respectively.

by a wind farm subjected to the considered wind resource is about 35%. With this methodology, hourly time series representing the aggregate wind power output of 100 wind farms encompassing a different number of wind turbines and dispersed over a number of uncorrelated wind sites has been generated. Figs. 10a and b illustrate sample realizations of the aggregated wind power generation for 20 GW of installed wind capacity. Fig. 10a shows the aggregate power output when wind capacity is concentrated in only one wind location (full correlation). On the other hand, Fig. 10b case depicts a sample of the injected power of 20 GW of installed wind power capacity dispersed over 100 uncorrelated wind sites. 4. Simulation results 4.1. Influence on power prices Fig. 11a provides a sample simulation of hourly power prices for 10 GW of installed wind capacity dispersed over 100 statistically independent wind sites. From this figure,

the yearly seasonality of power prices can be easily recognized. Prices follow approximately the yearly demand pattern, the summer being the period of lower load demand and hence of the lowest market prices. On the contrary, winter loads are considerably higher and therefore power prices consistently situate on a higher level. In the figure, price spikes due to unexpectedly high demand along with tight or even insufficient available power capacity can be distinguished. In Fig. 11b, a detailed portion of the price sample containing a price spike event is given. In this figure, the daily and the weekly seasonality of power prices as well as the random price fluctuations are illustrated. Because of the negligible marginal costs of wind generation, wind power generation alters the dispatching merit order by entering first in the priority list. Therefore, the aggregate supply curve is simply shifted toward right and the more expensive conventional generation is replaced. Consequently, market prices for energy are decreased at times when wind power is available. The empirical probability density functions of power prices computed after 1000 stochastic simulations of the power market for 1 and 20 GW of installed wind capacity are illustrated in Figs. 12. Fig. 12a shows the case of full spatial correlation. Alternatively, Fig. 12b depicts the case of wind capacity dispersed over 100 uncorrelated wind sites. Inset, the expected power prices for the several conditions are provided. A massive addition of wind

ARTICLE IN PRESS F. Olsina et al. / Energy Policy 35 (2007) 1257–1273

45 1 GW wind installed capacity 20 GW wind installed capacity

0.2

0.15 Expected price for 1 GW: 38 C /MWh

0.1

Expected price for 20 GW: 23.6 C /MWh

0.05

0 10

20

Probability density function of power prices

25 30 35 40 Power prices [ c / MWh]

45

50

1 wind site 100 uncorrelated wind sites

40 35 30 25 20 15

15

(a)

(b)

Expected power prices [•/MWh]

0.25

0

2

4 6 8 10 12 14 16 Installed wind power capacity [GW]

18

20

Fig. 13. Behavior of the expected power prices for increasing wind power capacity.

0.25 1 GW wind installed capacity 20 GW wind installed capacity

0.2

0.14 1 wind site

0.15 Expected price for 1 GW: 37.6 C /MWh

0.1

Expected price for 20 GW: 18.9 C /MWh

0.05

0 10

15

20

25 30 35 40 Power prices [ c / MWh]

45

50

Fig. 12. (a,b) Probability density functions of market prices for electricity.

capacity results in a dramatic reduction of power prices. Although the qualitative influence of wind generation on the power prices looks similar in both cases, a noticeable impact on the expected electricity prices was found. This difference might have a major impact on the investment activity in conventional units. In fact, a scenario of average electricity prices of about 19 h/MWh might turn unprofitable almost any investment in conventional power generation. In Fig. 13, the non-linear decline of the expected power prices for increasing wind generating capacity is shown. The impact of wind generation on the expected market prices depends not only on the amount of wind capacity but also on its dispersion among uncorrelated wind sites. Although contributing substantially to the price expectation, the tail of the probability distribution corresponding to the price spikes is not shown in Fig. 12. Fig. 14 shows the probability of encountering the generation system unable to meet the total load demand for increasing amounts of installed wind capacity.14 Thus, Fig. 14 14 Conventional generating capacity remains unchanged in LOLP computations.

Loss of load probability [in %]

Probability density function of power prices

1268

0.12 0.1

2 uncorrelated wind sites 5 uncorrelated wind sites 100 uncorrelated wind sites

0.08 0.06 0.04 0.02 0

1 2 3 4 5 7.5 10 12.5 15 17.5 Installed wind power capacity [GW]

20

Fig. 14. Loss of load probability (LOLP) for increasing levels of wind penetration.

represents the occurrence probability of price spikes reaching the price cap. This probability reduces rapidly after adding the first MWs of wind capacity. Because the marginal contribution of wind as firm capacity declines as more wind capacity is installed, a saturation effect takes place for high penetration levels. Nevertheless, by dispersing the wind capacity even over a small number of uncorrelated wind sites, this saturation level can be noticeably reduced. Other perspective of the value provided by wind capacity as firm capacity is the reduction of the peak load. Figs. 15a and b depict the load duration curves (LDC) and the equivalent load duration curves (ELDC) remaining after adding 20 GW of wind capacity, for both cases: full wind speed correlation and dispersion of wind capacity over 100 independent locations. In the first case, the reduction of the peak load is negligible. For the case of statistical independence of wind resources, the firm capacity delivered by wind amounts 33% of the wind installed capacity.

ARTICLE IN PRESS F. Olsina et al. / Energy Policy 35 (2007) 1257–1273

100 Expected internal rate of return [in %]

12

Equivalent load [GW]

ELDC, 20 GW, 1 wind site LDC 80

60

40

20

0

1269

Peak LDC: 83.247 GW Peak ELDC: 82.731 GW 0

0.25

(a)

0.5 Normalised time

0.75

1

1 wind site 100 uncorrelated wind sites

10 8 6 4 2 0 2

4

6 8 10 12 14 16 Installed wind capacity [GW]

18

20

100 Fig. 16. Profitability of wind investments for increasing wind power capacity.

Equivalent load [GW]

ELDC, 20 GW, 100 wind sites LDC 80

60

specified level or hurdle rate. The expected IRR denoted as E½r0 , is estimated as follows:

40

E½r0  ¼ Peak LDC:83.247 GW Peak ELDC: 76.566 GW

20 0 (b)

0.25

0.5 Normalised time

0.75

1

Fig. 15. (a,b) Equivalent load duration curves for 20 GW of installed wind capacity.

4.2. Optimal wind power capacity The maximal wind power capacity that would profitably be deployed under full market conditions can be assessed by applying NPV methods. As it was shown in Section 2, this can be done by comparing the revenues perceived by a wind capacity unit, resulting in selling its production at the prevailing market prices, with its investment fixed costs. Thousand yearly realizations of power prices are used to compute short-term, annual revenues Pi ðPW Þ perceived by a wind investment. For an installed wind capacity PW, the ith realization of the profitability delivered by a wind project is computed by solving numerically the following equation for ri0 : Pi ðPW Þ 

ri0  IC ¼ 0; 1  ð1 þ ri0 ÞT a

i ¼ 1 . . . 1000,

where ri0 is the discount factor that brings the wind project to the breakeven condition. This figure is a convenient measure of the project profitability and it is referred in the finance literature as internal rate of return (IRR). Investments are profitable if the project’s IRR, exceeds some

1000 1 X ri . 1000 i¼1 0

For the base case, specific investment costs for wind facilities are assumed to be 1000 h/kW. The capital cost is set to 8%/year and the amortization period for wind investments is assumed to be 20 years. The influence of these adopted values on the optimal wind installed capacity is further investigated by means of a sensitivity analysis. In Fig. 16, the behavior of expected IRR for increasing amounts of installed wind capacity is illustrated. For the considered set of parameters, the wind capacity that could be accommodated under market conditions is about 3 GW. It should be noted that a moderate reduction of the costs of accessing to capital resources for financing wind projects could lead to a considerable higher amount of wind capacity. In addition, by dispersing the wind capacity over uncorrelated wind sites, considerably more wind power can be economically deployed, particularly if capital costs are lower. One important feature of stochastic simulations is that it allows computing not only the expected value of the investment profitability but also the probability distribution of exceeding some specified value. Figs. 17a and b depict for one and hundred independent wind sites, respectively the empirical probability density functions of the IRR for different amount of wind capacity. Note that the high dispersion of the annual profitability involves a large cumulated probability of not exceeding the hurdle rate. 4.3. Sensitivity analysis In this subsection, the sensitivity of the simulation results to a change in some relevant input parameters is assessed.

ARTICLE IN PRESS 45 1 GW 3 GW 5 GW 10 GW 15 GW 20 GW

Probability density

35 30 25 20 15 10 5 0 0

0.05

(a)

0.1 0.15 0.2 Internal rate of return [in %]

0.25

40

Probability density

30 25 20 15 10 5 0

(b)

0

0.05

0.1 0.15 0.2 Internal rate of return [in%]

20 1 wind site 2 uncorrelated wind sites 5 uncorrelated wind sites 100 uncorrelated wind sites

15

10

5

0 600

700

800 900 1000 1100 1200 1300 1400 Wind power installation costs [ C /kW]

Fig. 18. Sensitivity analysis to changes in the installation costs of wind turbines.

1 GW 3 GW 5 GW 10 GW 15 GW 20 GW

35

0.25

Fig. 17. (a,b) Probability distributions of the IRR for increasing wind capacity.

For each assessed parameter, simulative investigations were performed for the case of dispersing the installed wind capacity over a different number of statistical independent wind sites. The sensitivity analysis is univariate in the sense that only one parameter is changed each time. The remaining parameters are fixed and set equal to the values corresponding to the base case. The optimal installed wind power capacity is computed as the mathematical expectation after 1000 stochastic realization of the market revenues of a wind unit. The first investigation concerns the impact of a reduction of the installation costs on the maximal wind capacity that might be installed in a market-based manner. As it is shown in Fig. 18, this parameter has a major influence on the results. Particularly, the wind capacity that might profitably be accommodated increases dramatically if wind investment costs fall bellow 800 h/kW. This result is very relevant, as installation costs have been reduced rapidly in the last decade and currently, investments costs reported for some wind projects are below this figure. In addition, wind technology is yet considered an immature technology because the learning rate remains high. Hence, some room

Optimal installed wind power capacity [GW]

40

Optimal installed windpower capacity [GW]

F. Olsina et al. / Energy Policy 35 (2007) 1257–1273

1270

4.5 4 3.5

1 wind site 2 uncorrelated wind sites 5 uncorrelated wind sites 100 uncorrelated wind sites

3 2.5 2 1.5 1 0.5 0 10

15 20 Amortization period [years]

25

Fig. 19. Sensitivity analysis to changes in the amortization period of wind investments.

for further cost reductions for wind facilities is expected. Additionally, installation costs exceeding 1400 h/kW turn unprofitable any wind power capacity. That is an important result for off-shore wind projects, which have significantly higher installation costs than similar on-shore counterparts. It is worth to note that by dispersing the wind capacity over uncorrelated sites has only some initial effect in increasing the optimal wind capacity. After spreading the capacity over five locations with uncorrelated wind resources, no further apparent improvement is achieved. The second simulation experiment is regarding the influence of the investment amortization period on the economically deployable wind power capacity. Unlike the previous case, a considerable extension of the amortization does not lead to high profitable penetration. That is because the amortization period has only a relative minor effect in the profitability of wind investments. In addition

ARTICLE IN PRESS 20 18 16

1 wind site 2 uncorrelated wind sites 5 uncorrelated wind sites 100 uncorrelated wind sites

14 12 10 8 6 4 2 0

60

80 100 120 140 160 Modification of the fuel costs for conventional power plants [in %]

180

Optimal installed wind power capacity [GW]

Optimal installed wind power capacity [GW]

F. Olsina et al. / Energy Policy 35 (2007) 1257–1273

5

1271

1 wind site 2 uncorrelated wind sites 5 uncorrelated wind sites 100 uncorrelated wind sites

4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

0

2

4 6 8 10 12 14 16 18 Value of lost load [Thousand c / MWh]

20

Fig. 21. Sensitivity analysis to changes in the price cap. Fig. 20. Sensitivity analysis to price escalation of fossil fuels.

to this, a saturation effect can be recognized from Fig. 19, particularly when considering a single wind location. A sensitivity analysis of the maximal installable wind power capacity for different scenarios of fuel prices has also been conducted. Price variations, in both directions, of fossil fuels required for the conventional generation system (coal, lignite, gas and oil) were investigated. This investigation is relevant, as an increase of fuel prices in the long-run is deemed to be likely. Although somewhat unrealistic, for the sake of simplicity all considered fuel prices are equally increased or reduced in percentage term. Fuel prices are equally modified from a decline of 50% to an increase of 180% regarding the base case values given in Table 1. The base case level of fuel prices is denoted as 100%. The results of this simulation experiment are illustrated in Fig. 20. The impact of an escalation of fuel prices on the maximal profitable wind capacity is highly non-linear. The magnitude of the influence of this parameter is only comparable with the reduction of the installation costs of wind turbines. Consequently, the combined scenario of a moderate reduction of the investment cost with a modest raise of fuel prices can turn profitable large amounts of wind capacity even without subsidizing programs. Note in this case at that the diversity of wind resources plays a more significant role than the exhibited in Fig. 18. Finally, the last sensitivity analysis is carried out on the price cap set by regulatory authorities. Results of these investigations are given in Fig. 21. Even though the influence of this parameter is minor compared with fuel prices and the investment cost, it is important to note that if the price cap is set too low, e.g. below 2500 h/MWh, no wind capacity is profitable under market conditions. That is because despite the fact that price spikes are unlikely events, they are still necessary to fully recover the plant

investment fixed costs. Similar to the amortization costs, the simulations reveal a saturation effect, even stronger, for high values of the price cap. 5. Conclusions Wind power is progressively becoming part of the mainstream generation. Penetration goals are constantly set higher as clean and sustainable generation is encouraged. A high penetration of this generation technology implies a major change in the market conditions. In this work, the influence of wind power generation on power prices in liberalized electricity markets has been assessed by means of various simulation techniques. The developed methodology allowed investigating the optimal wind capacity that would be profitably deployed, if wind investments were only subjected to the prevailing conditions in power markets. The market simulation approach used in this investigation is based on modeling explicitly the variables that determine the price formation in power markets, i.e. the supply and demand curve. For this purpose, the load characteristics observed in the German power system have been adopted for this study. The stochastic fluctuations have been assumed to follow a Gauss–Markov stochastic process, what is deemed to be accurate for the purpose of this investigation. Additionally, the German conventional generation system as well as the stochastic behavior of thermal units has been considered for the supply side. Under the assumption of perfect competition, short-term demand inelasticity and neglecting intertemporal constraints, price simulations can be carried out by means of a priority listing algorithm, which is computationally very efficient for large-scale systems. Simulations of power prices reflect stochastic characteristics of prices observed in actual power markets, such as daily, weekly and yearly

ARTICLE IN PRESS 1272

F. Olsina et al. / Energy Policy 35 (2007) 1257–1273

patterns as well as some circumstantial, unexpected price spikes. The wind resources are characterized by the PSD of wind speed fluctuations and the marginal PDF of wind velocities. The simulation of stochastic processes by means of the spectral representation method constitutes one of the best modeling alternatives, as it allows synthesizing time series with a prescribed target power spectrum. Additionally, an iterative method makes possible simulation of nonGaussian processes with a specified target marginal probability Weibull distribution. One relevant topic investigated is the influence of increasing the spatial diversity of wind resources on the overall behavior of the power system. In this work, the extreme case of adding capacity in uncorrelated wind sites has been investigated. The overall conclusion over all the wide range of simulation results is that the marginal contribution of a new statistical independent site is very low after the first five uncorrelated wind sites are already exploited. The investigation of the influence of the wind on the market prices reveals that the addition of considerable wind power capacity results in a significant reduction of the power prices, particularly for the case of some independent wind resources. That might dramatically impact the profitability of investments in conventional power plants, which are still necessary to provide the higher requirements imposed on reserve and regulating power. Finally, from the simulation experiments, it can be concluded that the profitable deployment of high wind power capacity in a market-based manner might be possible only for scenarios of high escalation of fossil fuel prices and a drastic reduction of either, the installation or financing costs of wind investment projects. For the evaluated test system and typical values for current wind projects (IC ¼ 800 h/ kW, r ¼ 8% year, T a ¼ 20 year), the economically optimal wind power capacity that should be installed is about 7.12 GW (see Fig. 18), representing a penetration of 8.7% in the generation mix. The presented methodology allows regulatory authorities and policy makers to determine an objective benchmark for wind capacity based on economic considerations only. The installation of wind capacity beyond the economic optimal values results in inefficiencies in the resource allocation with the consequent loss of social economic welfare. The internalization in market prices of externalities associated to conventional generation, as it is expected in the future, would have a major impact on the optimal wind capacity that should be placed, and therefore, on the maximal penetration of wind. Presumably, as long as market prices account for external costs of emissions, wind power becomes a more attractive alternative. The quantification of the amount of wind capacity that maximizes the social welfare under consideration of the avoided external costs is a topic subject of ongoing investigations.

Acknowledgments The authors thank Dr. Pariya Cumperayot for kindly providing data of the German conventional generation system. The financial support from the National Council and the National Agency for Science and Technology (CONICET and ANPCyT, Argentina) is gratefully acknowledged. References Barnes, P.R., Van Dyke, J.W., Tesche, F.M., Zaininger, H.W., 1994. The integration of renewables energy sources into electric distribution systems. ORNL 6775/V1. Barnes, P.R., Dykas, W.P., Kirby, B.J., 1995. The integration of renewables energy sources into electric transmission systems. ORNL 6827. Billinton, R., Allan, R., 1996. Reliability Evaluation of Power Systems. Plenum Press, New York. Billinton, R., Bai, G., 2004. Generating capacity adequacy associated with wind energy. IEEE Transactions on Energy Conversion 19 (3), 641–646. Billinton, R., Chen, H., Ghajar, R., 1996a. Time-series models for reliability evaluation of power systems including wind energy. Microelectronics and Reliability 36 (9), 1253–1261. Billinton, R., Chen, H., Ghajar, R., 1996b. A sequential simulation technique for adequacy evaluation of generating systems including wind energy. IEEE Transactions on Energy Conversion 11 (4), 728–734. Breipohl, A.M., Lee, F.N., 1991. A Stochastic load model for use in operating reserve evaluation. In: Third International Conference on Probabilistic Methods Applied to Electric Power Systems (PMAPS) 3–5 July, pp. 123–126. Breipohl, A.M., Lee, F.N., Zhai, D., Adapa, R., 1992. A Gauss–Markov load model for the application in risk evaluation and production simulation. Transactions on Power Systems 7 (4), 1493–1499. Cumperayot, P., 2004. Effects of modeling accuracy on system marginal costs simulations in deregulated electricity markets. Ph.D. Dissertation, RWTH Aachen University, ABEV Series (97), Klingberg Verlag, Aachen, Germany, ISBN 3-934318-50-9. Daniel, A.R., Chen, A.A., 1991. Stochastic simulation and forecasting of hourly average wind speed sequences in Jamaica. Solar Energy 46 (1), 1–11. Dany, G., 2000. Kraftwerksreserve in elektrischen Verbundsystemen mit hohem Windenergieanteil. Ph.D. Dissertation, RWTH Aachen University, ABEV Series (71), Klingberg Verlag, Aachen, Germany, ISBN 3-934318-10-X. Deodatis, G., Micaletti, R.C., 2001. Simulation of highly skewed nonGaussian stochastic processes. Journal of Engineering Mechanics (ASCE) 127 (12), 1284–1295. DEWI, 2005. Status der Windenergienutzung in Deutscland, Deutsche Windenergie-Institut, Wilhelmshaven, Germany, Stand 31.12.2004. Available at: www.dewi.de Doherty, R., O’Malley, M., 2003. Quantifying reserve demand due to increasing wind power penetration. In: IEEE Bologna Power Technology Conference, June. Italy, 23–26. EWEA, 2004. The current status of the wind industry. European Wind Energy Association, Brussels, Belgium. Available at: www.ewea.org Giebel, G., 2000. On the benefits of distributed generation of wind energy in Europe. PhD Thesis, Carl von Ossietzky Universita¨t Oldenburg. GWEC, 2005. Global wind power continues expansion. Global Wind Energy Council, Brussels, Belgium. Available at: www.gwec.net Hirst, E., Hild, J., 2004. The value of wind energy as a function of wind capacity. The Electricity Journal 17 (6), 11–20. Junginger, M., Faaij, A., Turkenburg, W.C., 2005. Global experience curves for wind farms. Energy Policy 33, 133–150.

ARTICLE IN PRESS F. Olsina et al. / Energy Policy 35 (2007) 1257–1273 Kahn, E., 1979. The reliability of distributed wind generators. Electric Power Systems Research 2, 1–14. Kaimal, J.C., Wyngaard, J.C., Izumi, Y., Cote´, O.R., 1972. Spectral characteristics of surface-layer turbulence. Quarterly Journal of Royal Meteorological Society 98, 563–589. Kamal, L., Jafri, Y.Z., 1997. Time series models to simulate and forecast hourly averaged wind speed in Quetta, Pakistan. Solar Energy 61 (1), 23–32. Kennedy, S., 2005. Wind power planning: assessing long-termcosts and beneifits. Energy Policy 33, 1661–1675. Kra¨mer, M., 2003. Modellanalyse zur Optimierung der Stromerzegung bei hoher Einspeisung von Windenergie. Ph.D. Disseration, (vol. 492), Bremer Energie Institut, VDI Verlag, Du¨sseldorf, Germany, ISBN 318-349206-7. Milborrow, D., 1996. Capacity credits—clarifying the issues. In: Proceedings of the BWEA Conference on Wind Energy, pp. 215–219. Milligan, M., Parsons, B., 1999. A comparison study of capacity credits algorithms for wind power plants. Wind Engineering 23 (3), 159–166. Nanahara, T., Asari, M., Maejima, T., Sato, T., Yagamuchi, K., Shibata, M., 2004. Smoothing effects of distributed wind turbines. Part I. Coherence and smoothing effects at a wind farm. Wind Engineering 7, 61–74. Olsina, F., Garce´s, F., Haubrich, H.-J., 2006. Modeling long-term dynamics of electricity markets. Energy Policy 34 (12), 1411–1433. Oort, A., Taylor, A., 1969. On the kinetic energy spectrum near the ground. Montly Weather Review 97 (9), 623–636.

1273

Singh, C., Lago-Gonzalez, A., 1985. Reliability modeling of generation systems including unconventional energy sources. IEEE Transactions on Power Apparatus and Systems, PAS 103, 569–575. Shinozuka, M., Jan, C.-M., 1972. Digital simulation of random processes and its applications. Journal of Sound and Vibration 25 (1), 111–128. Shinozuka, M., Deodatis, G., 1991. Simulation of stochastic processes by spectral representation. Applied Mechanics Review 44 (4), 191–203. Torres, J.L., Garcı´ a, A., De Blas, M., De Francisco, A., 2005. Forecast of hourly wind speed with ARMA models in Navarre (Spain). Solar Energy 9 (1), 65–77. UCTE, 2005. Union for the Co-ordination of Transmission of Electricity. Statistics available at www.ucte.org Van der Hoven, I., 1957. Power spectrum of horizontal wind speed in frequency range from 0.0007 to 900 cycles per hour. Journal of Meteorology 14 (2), 160–164. van Wijk, A.J.M., Halberg, N., Turkenburg, W.C., 1992. Capacity credit of wind power in the Netherlands. Electric Power Systems Research 23, 189–200. Wan, Y., Parsons, B., 1993. Factors relevant to utility integration of intermittent renewable technologies. NREL/TP-463-4953, National Renewable Laboratory. Yamazaki, M., Shinozuka, M., 1988. Digital generation ofstochastic fields. Journal of Engineering Mechanics, American Society of Civil Engineers (ASCE) 114 (7), 1183–1197.