- Email: [email protected]
Optimal portfolio-theory-based allocation of wind power: Taking into account cross-border transmission-capacity constraints
Renewable Energy 36 (2011) 2374e2387
Contents lists available at ScienceDirect
Renewable Energy journal homepage: www.elsevier.com/locate/renene
Optimal portfolio-theory-based allocation of wind power: Taking into account cross-border transmission-capacity constraints Yannick Rombauts, Erik Delarue, William D’haeseleer* University of Leuven (K.U.Leuven) Energy Institute, TME Branch (Applied Mechanics and Energy Conversion), Celestijnenlaan 300A, Box 2421, B-3001 Leuven, Belgium
a r t i c l e i n f o
a b s t r a c t
Article history: Received 20 July 2010 Accepted 18 February 2011 Available online 12 March 2011
Allocating wind farms across different locations in different countries may reduce the variability of hourly wind power changes. Taking into account cross-border transmission-capacity constraints between countries can however decrease the effect of this diversification. A portfolio-theory-based model is developed that takes into account these cross-border transmission-capacity constraints when allocating wind power as efficient as possible across different locations. Three models are developed, looking to the cases where cross-border transmission-capacity constraints are equal to infinity, zero and a certain limited value, respectively. It is notably this last model that brings new perspectives in the allocation of wind power, based on portfolio theory modelling. Keeping cross-border transmission-capacity available for wind-power flows is an effective measure to limit hourly wind-power variations. 2011 Elsevier Ltd. All rights reserved.
Keywords: Portfolio theory Cross-border transmission Wind-power allocation
In March 2007, the council of the European Union committed Europe to strive to an energy-efficient, low carbon economy in order to compete against climate change and enhance the energy security [1]. Within this framework, one of the targets put forward is a 20% share of renewable energy resources (RES) in total energy consumption by 2020. In order to achieve this goal, as much as 34% of the electricity demand in Europe is expected to be met by RES, and of this total share, 47% might have to come from wind farms, both on- and offshore [2]. Many countries spend a large budget on research, development and demonstration (RD&D) of renewable energy sources [3], and ambitious targets for offshore wind-power developments are set ahead [4] (the use of feed-in tariffs and green certificates are envisaged to foster this development [5]). This poses significant challenges as wind is a variable and to some extent unpredictable energy resource. Measures have to be taken to allow for a secure integration and to lower the risk on possible difficulties as much as possible. Current studies dealing with the integration of wind energy into an electrical power system start from a certain expected amount of wind power installed in each location. The EU project Tradewind
[6] focuses in this way on the current flows that occur in and between the grids of the different member states. EWIS [7], coordinated by the European transmission operator ENTSO-E is the result of a similar research which on the one hand deals with the use of the current grid, and formulates on the other hand propositions for a harmonised policy to facilitate the integration of wind energy. Also in Wind Task 25 of the IEA [8] one starts from a certain amount of wind energy that is expected to be installed in the different locations considered. This paper will, in contrast to these studies, not start from a certain expected amount of wind power installed in every location. Rather, the objective is to make a first move to the optimal integration of wind turbine farms by installing them at the most optimal locations such that the production of wind power is smoothened as much as possible. Focussing only on short-term variability,1 geographical dispersion of wind turbines can help to flatten this variability as much as possible, as uncorrelated wind profiles balance each other out. Aggregating different wind-power profiles shows that the amplitudes of the variations decrease. The larger the geographical area considered, the larger this effect, further enhancing the possibility of increasing the amount of wind power installed into the system. Large-scale aggregation of wind can, however, only be done in an
Abbreviation: CBTCC, cross-border transmission-capacity constraints. * Corresponding author. Tel.: þ32 16 322511; fax: þ32 16 322985. E-mail addresses: [email protected] (Y. Rombauts), erik. [email protected] (E. Delarue), [email protected] (W. D’haeseleer).
1 Wind power, has an intermittent character, i.e., it is variable and to some extent unpredictable. This paper focuses on the short-term variability, meaning the hourly wind-power variations. The remainder of the electrical power system is not considered explicitly.
1. Introduction
0960-1481/$ e see front matter 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2011.02.010
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
Nomenclature
Index i,j u h k,r v v1 n
location time-interval hour country number of hours number of hourly differences number of countries
Sets Ik I K
subset of locations located in country k set of locations set of countries
Variables the share of total wind capacity that is attributed to xi location i the capacity factor of location i CPi the amount of wind capacity that has to be installed in Pk country k [MW] the total amount of wind capacity that has to be Ptot installed over all countries considered [MW] the vector of hourly wind-power differences before bi transmission of locations i [MW] the sample standard deviation of location i, based on s(bi) wind-power differences before transmission [MW] rij the correlation coefficient between the wind-power differences of locations i and j s(bi)s(bj)rij the covariance between location i and j [MW2] the hourly wind-power difference before transmission bu,i of location i [MW]
optimal way if a well interconnected grid exists. Just looking at aggregated wind-power profiles does not take into account the effect of cross-border transmission-capacity constraints (CBTCC) [9]. The aim of the paper is to define “efficient” wind-power allocations, i.e., allocations with minimum fluctuations (minimum variability) for a given output of wind power. Portfolio theory can offer a solution in order to correctly link the geographic diversification effect of wind-power variability with the effect of limited CBTCC. In the literature, portfolio theory is mostly applied to financial portfolios, with Markowitz as the founding father [10]. Markowitz stated that the risk of a diversified portfolio is lower than the sum of the risks of the individual assets. More specifically, standard portfolio theory calculates a Pareto efficient frontier of portfolios, weighing maximal return and minimal risk. Every point on this efficient frontier indicates an efficient portfolio, which means there is no portfolio that provides a higher return for the same risk or a lower risk for the same return. Which efficient portfolio is going to be chosen depends on the risk appetite of a particular individual. A person who is risk averse is going to pick a portfolio with a low risk and by consequence a lower return. On the contrary, a person willing to take a relatively high risk chooses a portfolio with high risk and a higher return. Nowadays, portfolio theory has been applied in various research fields beyond the financial sector. The energy sector is one of them [11e14]. In this paper, the focus is on applying portfolio theory on the wind-power sector and builds on the work of Roques et al. [15],
2375
bu,k
the hourly wind-power differences before transmission of country k [MW] E(b) the expected value of the hourly differences before transmission [MW] the hourly change in wind power in country k, after au,k transmission has taken place [MW] E(a) the expected value of the hourly differences after transmission [MW] the vector of hourly wind-power differences after ai transmission of locations i [MW] the vector of hourly wind-power differences after ak transmission of country k [MW] the variance of the hourly wind-power differences s(ak)2 after transmission of country k [MW2] the power transported in hour h, from countries k to ck,r,h country r [MW] capconstrk,r the capacity constraint put on the transmission line between the two countries k and r [MW] the fraction of wind power (expressed as a percentage frach,i of total installed capacity) in hour h at location i the amount of wind power produced in hour h in wh,i location i [MWh/h) ¼ [MW] the net amount of wind power produced (after ph,i balancing) in hour h in location i [MW] P ak Þ2 the variance of the hourly wind-power differences of sð k˛K the aggregated wind-power profile over all countries of set K [MW2] r(ak,ar) the correlation coefficient between the hourly windpower differences after transmission of country k and r the difference in net wind-power export in country k Yu,k between hour u and u þ 1 [MW] mu the mean hourly wind-power differences of all countries k [MW] Dmu,k the difference of bk,u and mu for each country [MW]
who developed a portfolio-theory-based model that maximises the average capacity factor and minimises normalised hourly windpower fluctuations, by allocating a share of wind power over Austria, France, Germany, Spain and Denmark. The minimisation was carried out for two cases. In a first case, they included all available wind data, whereas in the second case only peak demand hours are considered in order to maximise the wind-power contribution to the system reliability. Their data set consists of aggregated hourly wind-power-production data from the transmission-system operators of the different countries considered. In a further extension of the model, two types of constraints were added: constraints on the technical potential of wind power for each country and constraints on the cross-border transmissioncapacity. The latter was included as a proxy by taking the sum of the demand and the total transmission export capacity for each country, thereby defining an upper wind-power limit for the country considered. The main focus of the current work is on the inclusion of CBTCC. In contrast to the portfolio approach of Roques et al. [15], this work will model the cross-border capacities more explicitly so that these constraints can be taken into account and the electrical power flows stemming from wind-power production can be analysed.2 Furthermore, this work will apply portfolio theory on data of
2 CBTCC in this context is referred to as the capacity on cross-border connections, exclusively available for wind-power flows.
2376
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
Table 1 Illustration of the hourly wind-power differences before and after transmission.
Before After
Wind-power production in hour 1
Wind-power production in hour 2
Transmission in hour 2
Hourly difference
1000 1000
1500 1500
/ 200
500 300
different locations within one country instead of aggregated windpower data. Note that the methodology as presented in the paper considers a central planner approach. However, one could apply the developed algorithm in a deregulated market setting with different players, trying to minimise the variability of their own generation portfolios. Interconnections could then reflect possible contracts between different players. This is scope for future work. In the next section, the optimisation model is developed and the portfolio theory concepts are translated to the wind-power allocation problem. Three sub models are set up, putting CBTCC equal to infinity, to zero and to a certain specific value, respectively. This model is then applied to a case study, being discussed in Section 3. Conclusions are drawn in the final section. 2. Portfolio model description The fundamental concepts of classic portfolio theory are risk and return, which are clearly defined in the framework of financial theory. In order to apply portfolio theory to the wind-power problem, these financial concepts need appropriate equivalents that are related to wind-power generation. Comparable definitions as proposed by Roques et al. [15] are used. Return is defined as the “mean capacity factor over the different locations”, and risk is defined as the standard deviation of hourly wind-power differences, Pt Pt1 (with Pt the wind power at hour t). For this, either hourly wind-power differences before or after transmission, are used. In order to make this difference clear from the beginning, an illustration is shown in Table 1. The set up of the portfolio model consists of different adjacent countries, linked to each other through cross-border transmission lines. Each country has to install a specific amount of wind power by dividing it over different sites considered within that country, taking into account the wind speed profile characteristic for each particular site. The remainder of the electrical power system is not considered. As far as electric transmission is concerned, transmission lines within countries are assumed to have no congestion. CBTCC are taken into account in different steps. In a first model, cross-border transmission capacities are taken to be infinite, which is equivalent to looking at a single (combined) country with several wind farms. Secondly, cross-border transmission capacities are set to zero, effectively isolating each country. In this second model, all countries minimise their own risk.3 These two models can be seen as the two most extreme cases, and provide valuable insights. In the third portfolio model, the CBTCC can be chosen appropriately. 2.1. Model 1: unlimited cross-border transmission-capacity The optimisation model determines the optimal allocation of wind power (i.e., in terms of return and risk, as defined earlier) over the different locations in each country. It is assumed that each
3
Recall that throughout this work, the notion of risk is used to refer to the standard deviation of hourly wind-power differences.
country has an overall given amount of wind power that needs to be installed. In the first model, cross-border transmission capacities are infinite and therefore these constraints are non-binding, meaning that the aggregation of countries can be looked at as one single (combined) country (although accounting for the required amounts of wind power that need to be installed in each country). Firstly, the portfolio with the maximum return is calculated. This is done by means of the following objective function and constraints:
max
X
xi CPi
(1)
i˛I
Subject to:
X
xi ¼
i˛Ik
Pk Ptot
ck˛K and Ik ˛I
xi 0
(2)
(3)
The addition of Eq. (2) ensures that the right quantity of wind power is installed in a particular country. Moreover, the summation of Eq. (2) over the countries K, is equal to one. With this portfolio of maximum return, a certain risk can be associated. This risk (defined as a standard deviation) is equal to the square root of the variance s2:
s2max return ¼
XX i˛I
xi xj sðbi Þs bj rij
(4)
j˛I
The sample standard deviation si is defined as:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Pv1 u ¼ 1 bu;i EðbÞ sðbi Þ ¼ ðv 1Þ 1
(5)
Next, the portfolio with minimal risk is calculated. For this, the same constraints as in Eqs. (2) and (3) apply, only the objective function changes4:
min s2p ¼ min
XX i˛I
xi xj si sj rij
(6)
j˛I
Also for this portfolio, a certain return can be calculated as:
returnmin risk ¼
X
xi CPi
(7)
i˛I
The maximum return and the minimum risk portfolios constitute the extreme points of the efficient frontier. In addition, the other points of the efficient frontier have to be calculated. This is done in the same way as calculating the portfolio with minimal risk, except for an extra constraint added on the return5 as shown in Eq. (8), where this return is taken between the return of the portfolio with maximal return and the return of the portfolio with minimal risk. In other words, every return that is lying in between the returns of the two extreme points of the efficient frontier is taken as fixed, and risk is minimised.
X
xi CPi ¼ return
(8)
i˛I
By examining the calculation method of the risk of a particular portfolio into detail, a second calculation method can be derived no
4 The optimisation is performed towards minimum variance. The risk is equal to the square root of this variance. 5 Maximising return with a constraint on risk would yield the same results.
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
CBTCC apply. This second calculation method provides a better insight in the meaning of the risk definition. Considering two locations i and j for reasons of simplicity, risk as defined in Eq. (6) can be rewritten as:
2 s2p ¼ x2i sðbi Þ2 þx2j s bj þ2xi xj sðbi Þs bj rij 2 2 ¼ sðxi bi Þ2 þs xj bj þ2cov xi bi ; xj bj ¼ s xi bi þ xj bj
(9)
Minimising s(xibi þ xjbj)2 can be seen as minimising the variance of the hourly wind-power differences of aggregated6 wind-power profiles of both locations i and j, when a share xi of Ptot is installed in location i and a share xj of Ptot is installed in location j. Minimising the variance of the hourly wind-power differences of this single wind-power profile gives the same risk as minimising the objective function of Eq. (6). Note that Eq. (9) applies for a group of countries as a whole. However, for a country acting as a group of locations, the same equation can be applied. Note further that, in case all locations are considered, the following equation applies:
xi bi þ xj bj ¼ xi ai þ xj aj
(10)
This stems from the fact that no wind power is lost or created by transporting it between different countries. By consequence, summing the different wind-power profiles up before or after transmission till one aggregated wind-power profile makes no difference. Eq. (9) can also be written as:
2 2 s2p ¼ s xi bi þ xj bj ¼ s xi ai þ xj aj
(11)
2.2. Model 2: no cross-border transmission In the second model, the CBTCC are put equal to zero. Every country minimises its own risk. This way, a model is obtained for which countries cannot rely on neighbouring countries to balance any of their hourly wind power changes. Consequently, the global variance for this model is defined as the sum of each country’s individually optimised variance of hourly wind-power differences.7 This gives the following objective function when minimising risk8:
0 1 X XX @ min xi xj sðbi Þs bj rij A k˛K
(12)
i˛Ik j˛Ik
The remainder of the model is the same as the one described in Section 2.1. To summarise, the second model consists of Eqs. (1)e (4), (7), (8) and (12). Note that the return of Eq. (8) is still calculated and optimised for all countries together, instead of calculating it for each country separately. In this perspective, each country will look to the other countries in order to maximise the jointly obtained return. This makes that the global variance as calculated by Eq. (12) will be
lower than in the case where the return would be optimised per country separately. The reason for this can be found in the fact that more combinations for the allocation of shares giving a particular return can be obtained in the case returns are maximised jointly, making this constraint less strict. In other words, when calculating the efficient frontier, Eq. (8) is less strict than if this constraint would be imposed on each country separately. This will however have no effect on the portfolio of maximal return and the portfolio of minimal risk, because the portfolio with maximal return will still be composed of the locations with the highest return in each country, while the portfolio with minimal risk will still be the same most diversified portfolio.
2.3. Model 3: limited cross-border transmission-capacity In the previous sections, models for infinite and zero crossborder capacities have been developed, respectively. An intermediate way has to be found to evaluate cases with non-zero but finite cross-border capacities. Therefore, the objective function and the constraints of the previous models must be generalised. A different perspective on the objective function compared to the previous cases is taken. Instead of looking at the variances of the hourly wind-power differences before transmission, the variances of the hourly wind-power differences after transmission are considered. The minimisation of the objective function is comparable to Eq. (11) (considering in this case a country as a union of locations), but now each country separately takes cross-border transmission-capacity constraints into account. With this approach, no correlations between countries have to be considered in the minimisation, because the correlations are already implicitly incorporated through the cross-border transmissioncapacity constraints. One cannot transport more wind power from one country to another to flatten fluctuations, than allowed by available capacity on the transmission lines. This means that the risk of the different countries, defined as the variances of the hourly windpower differences after transmission, can be summed up. Further justification of this objective function is given in Section 2.4, in which the equivalence with the two previous models is demonstrated. This approach leads to the following objective function and constraints for calculating the minimal risk:
min
X
sðak Þ2
(13)
k˛K
Subject to:
ck;r;h capconstrk;r
ch˛1; .; v and
(14)
ck; r˛K with capconstrk;k ¼ 0 au;k ¼
X
wuþ1;i þ
0
@
X
cr;k;uþ1
r˛K
i˛Ik
X
wu;i þ
X
ck;r;uþ1
r˛K
X
cr;k;u
r˛K
i˛Ik 6 An aggregated wind-power profile can be obtained by taking the sum of the wind-power profiles of each location. Note that this aggregation can only be done if no congestion can take place on the transmission lines. 7 Performing one global risk minimisation (over all countries, with no interconnections) yields the same results as when minimising the risk for each country separately. This result stems from the fact that the risk of each country separately is independent from each other (meaning that no correlation coefficient has to be taken into account between the risks of two countries since cross-border transmission is put equal to zero in this second model). 8 See footnote 4.
2377
X
1 ck;r;u A
r˛K
cu˛1; .; v 1; ck˛K and Ik ˛I; with au;k ˛ak h
u
1
2
1
3
2
4
3
ð15Þ 5
4
Fig. 1. Representation of index h for hours and index u for hourly intervals.
2378
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
0.4 RETURN: Mean Capacity Factor Portfolio
3000
Wind power [kW]
2500 2000 1500 1000 500 0
0
5
10
15
20
25
0.39 0.38 0.37 0.36 0.35 0.34 0.33 550
30
Wind speed [m/s]
600 650 700 750 RISK: Portfolio Standard Deviation [MW]
800
Fig. 2. Power curve of the Vestas 112 3 MW wind turbine. Fig. 4. Efficient frontier of wind-power allocation portfolios, without CBTCC taken into account. Simulation is run with 192 hourly data.
wh;i ¼ frach;i $Ptot $xi ph;k ¼
X
wh;i þ
xi ¼
i˛Ik
Pk Ptot
ck;r;1 ¼ 0
cr;k;h
r˛K
i˛Ik
X
X
ch˛1; .; v X
(16)
ck;r;h ch˛1;.;v and ck˛K
(17)
r˛K
ck˛K and Ik ˛I
(18)
ck; r˛K
(19)
xi 0
(20)
ck;r;u 0
(21)
ph;k 0
(22)
B(1)
A(1) A(1)
As a clarification, index u and index h are represented in Fig. 1. In this figure, a time scale is displayed, where h refers to the different hours considered, whereas u refers to a time-interval. The addition of Eqs. (17), (18) and (22) ensures that a country cannot send more wind power to another country than it produces or receives from other countries, since the remainder of the electrical power system is not considered. Furthermore, as can be seen from Eq. (15), the hourly difference after transmission is calculated as the difference between on the one hand wind power produced (wuþ1), imported (cr,k,uþ1) and exported (ck,r,uþ1) in this hour (u þ 1) and on the other hand the wind power produced (wu), imported (cr,k,u) and exported (ck,r,u) in the previous hour (u). This way, previous hours are linked to each other correctly in order to be able to smooth out wind-power production as much as possible.
B(1)
B(2)
A(1)
B(1)
B(2)
B(2) A(2)
A(2)
B(3)
B(3)
C(1)
C(1)
C(2)
MODEL 1
A(2)
B(3)
C(1)
C(2)
MODEL 2 Fig. 3. Configuration of the different models.
C(2)
MODEL 3
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
2379
1
Location 7 C Location 6 C Location 5 B Location 4 B Location 3 B Location 2 A Location 1 A
0.9
Share of each location
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0.34 0.35 0.36 0.37 0.38 0.39 RETURN: Mean Capacity Factor Portfolio
Fig. 5. Wind-power allocation in different locations, for portfolios along the efficient frontier, without CBTCC taken into account. Simulation is run with 192 hourly data.
The objective function and constraints for calculating the portfolio with maximum return are the same as described in Section 2.1, because the return only depends on the shares allocated to each location. For calculating the efficient frontier, Eq. (8) is added to the constraints of the minimal risk calculation. 2.4. Equivalence of models
number of countries that is taken under consideration. This means that, for the portfolios lying on the efficient frontier for the third model, the variance is n times smaller than the variance of the same portfolio for model 1. Starting from Eq. (9) and taking into account Eq. (11), this can be shown as follows:
s
X
!2 bk
¼ s
k˛K
¼
Wind power [MW]
sðak Þ2 þ
s
X
!2 ak
8000 6000 4000 2000
50
100 Hours [h]
150
Fig. 6. Aggregated wind-power profile of portfolio with minimum risk and portfolio with maximum return. Each aggregated wind-power profile is composed of the windpower profiles of the three countries A, B and C. Simulation is run with 192 hourly data.
n1 X
n X
2sðak Þsðar Þrðak ; ar Þ
(23)
k ¼ 1 r ¼ kþ1
X
ðn 1Þ$n sðak Þ2 þ $2$sðak Þ2 2 k˛K X X X ¼ sðak Þ2 þðn 1Þ sðak Þ2 ¼ n sðak Þ2 ¼
k˛K
10000
ak
With cross-border transmission-capacity constraints set equal to infinity in the third model, it can be proven (Appendix A) that P ak Þ2 , results in s(ak) ¼ s(ar) and r(ak,ar) ¼ 1. This minimising sð k˛K results in:
12000
0
X k˛K
k˛K
Portfolio minimum risk Portfolio maximum return
!2
k˛K
It is possible to check upon the consistency of the third model, with the first and second models, when implementing a CBTCC value of infinity and zero respectively in model 3. 2.4.1. Equivalence of model 3 with model 1 and model 2 At first instance, CBTCC in model 3 are put equal to infinity. Comparing the results in this case with those of model 1 shows that the portfolio compositions and returns are the same, while the risk pffiffiffi of model 3 lies a factor of n times lower (expressed as the standard deviation of the hourly wind-power differences), with n the
X
k˛K
(24)
k˛K
Note that this difference does not affect the share allocation of each efficient portfolio because every risk is downscaled with a factor pffiffiffi of n. At second instance, all capconstrk,r in model 3 are put equal to zero. This way, the objective function optimises the variance of each country separately, without cross-border transmission taking place. Within the borders of a country, infinite wind-power transmission can occur between the different locations, as mentioned in Section 2. Taking this unlimited intra-country transmission into account, and combining it with Eq. (11) from Section 2.1 makes this case equivalent to model 2 where risk was calculated for each country separately. Both calculation methods are, as shown through Eq. (10) equal to each other.
3. Case study 3.1. Problem setting The model will be illustrated with a methodological case study. The conceptual problem set up consists of three adjacent countries
2380
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
60 Portfolio with minimum risk Portfolio with maximum return
50
Frequency
40 30 20 10 0 −2500
−2000
−1500
−1000
−500
0
500
1000
1500
2000
2500
Power difference [MW] Fig. 7. Histogram of the aggregated hourly wind-power differences of the portfolio with minimum risk and the portfolio with maximum return. Simulation is run with 192 hourly data.
3.2. Results 3.2.1. Model 1: unlimited cross-border transmission-capacity Running the portfolio model mentioned in Section 2.1 for 192 h of data, yields the efficient frontier represented in Fig. 4. Every
9
In future work, more distant wind profiles could be utilised. 10 The current set up can also be looked at as three geographic separated regions in the Netherlands (with and without electric connections as explained below). 11 All simulations are run on an Intel Pentium Dual Core 2.66 GHz processor with 3.0 GB of RAM, and solved using the Cplex 11.1 solver.
point on this efficient frontier indicates an efficient portfolio, which means there is no portfolio that provides a higher return for the same risk, or, a lower risk for the same return. As can be seen, the mean capacity factor (return), which is calculated with the 192 h of data used, following Eqs. (1), (7) and (8), lies in between 0.33 and 0.40. This range covers both the portfolios of minimal risk and maximal return. The risk of the portfolios is calculated according to Eqs. (4) and (6) and ranges from 550 MW to 775 MW. Fig. 5 presents the composition along the efficient frontier of Fig. 4 with increasing return: i.e., starting on the lower left-hand side of Fig. 4, and moving to the upper right-hand side. Each colour represents a certain percentage of wind power allocated to a particular location of one of the three countries. Consequently, the portfolio shown on the left-hand side of Fig. 5 is the portfolio with the least risk. As can be seen, this portfolio is the most diversified one, meaning each location has a certain amount of wind power installed because of the risk reduction attributed to the diversification effect. On the contrary, the right-hand side represents the portfolio with the highest return, but exposed to a higher risk. This
0.4 RETURN: Mean Capacity Factor Portfolio
(A, B, and C). Contrary to looking at each country as a single point of wind production [15], multiple wind-power producing locations are considered within each country. More specifically, countries A and C have two locations each, and country B has three locations. Each location is characterised by its own hourly wind profile. The actual wind profiles used, are based on data from the Dutch meteorological service (KNMI) [16] of seven locations situated in the Netherlands.9 These wind profiles are assigned randomly over the different locations in the three fictitious countries.10 All the locations are assumed to be onshore, and for the transformation of wind flows to power flows the Vestas 112 3 MW wind turbine is used [17]. The power curve of the Vestas 112 3 MW wind turbine is presented in Fig. 2. At first instance, 192 h of wind speed data (equal to eight days) are used. This is sufficient to gain insight in the developed methodology, and this problem can easily be solved by available CPU power.11 In Section 3.3, a more efficient ‘equivalent’ model formulation is provided (yielding identical results), and a more extended data set (covering hourly values of three years) is used there. The totally installed wind power amounts to 12,000 MW, of which 5000 MW needs to be installed in country A, 3000 MW in country B, and, finally, 4000 MW in country C. Recall that the remainder of the electrical power system is not considered. The three models are now applied to the case considered. As already mentioned in Section 2, no congestion takes place within countries. CBTCC are taken into account in different steps corresponding to the three developed models. In Fig. 3 the three models are represented from a country perspective. Recall that the objective in all models is to smooth out wind-power production as much as possible (i.e., to minimise hourly variations) for a given windpower output, or vice versa, taking cross-border transmissioncapacity constraints into account.
0.39 0.38 0.37 0.36 0.35 0.34 0.33 500
550
600
650
700
750
RISK: Portfolio Standard Deviation [MW] Fig. 8. Efficient frontier of portfolio with CBTCC equal to zero. Simulation is run with 192 hourly data.
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
1
Location 7 C Location 6 C Location 5 B Location 4 B Location 3 B Location 2 A Location 1 A
0.9 0.8 Share of each location
2381
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0.34
0.35
0.36
0.37
0.38
0.39
RETURN: Mean Capacity Factor Portfolio Fig. 9. Wind-power allocation in different locations, for portfolios along the efficient frontier, with CBTCC equal to zero. Simulation is run with 192 hourly data.
portfolio is obtained through maximising return, leading to the obvious result that the whole share in each country is attributed to the location with the highest capacity factor. This can be seen from the figure: the portfolio with maximum return consists solely of wind power on location 1 in country A, location 3 in country B and location 7 in country C. In case of infinite CBTCC, the different wind-power profiles of each location can easily be summed up till one aggregated windpower profile. The comparison of the aggregated wind-power profiles of the portfolio with minimal risk and the portfolio with maximal return is presented in Fig. 6. As can be seen from the figure, the aggregated wind-power profile of the portfolio with maximum return is for most of the time higher than for the portfolio with minimum risk. In addition, the portfolio with maximum return also shows larger hourly wind-power differences. This higher variability is also demonstrated the histograms presented in Fig. 7. From this figure, it is clear that the largest hourly differences for portfolio with minimum risk are much smaller than for the portfolio with maximum return. Furthermore, the hourly differences of the portfolio with minimum risk are grouped in a more
RETURN: Mean Capacity Factor Portfolio
0.4 0.39 0.38 0.37
3.2.2. Model 2: no cross-border transmission The results of this model are presented in Figs. 8 and 9. Fig. 8 represents the efficient frontier. All countries are taken into account, and risk is defined as in Eq. (12). Note that this risk cannot be compared with the risk calculated with Eq. (8) of model 1. For this, the risk obtained through model 1 has to be downscaled with pffiffiffi a factor of 3 because of the similarity between on the one hand model 3 and model 1, and on the other hand model 3 and model 2, pffiffiffi as shown in Section 2.4. The risk of model 3 is a factor n lower than that of model 1 (with n the number of countries, in this case n ¼ 3), while the risk of model 3 and model 2 is equal to each other. Using model 3 makes it by consequence possible to compare the efficient frontiers of model 1 and model 2. Fig. 9 represents the share allocation of the efficient frontier. Note that for country B, almost half of the portfolios on the efficient frontier are fully allocated to location 3. This result would not be possible if the return would have been maximised per country, instead of for all countries as a whole as mentioned in Section 2.3. In case of a separate optimisation, only the portfolio with the maximal return (situated on the right-hand side of Fig. 9) would have a full allocation of wind power to location 3 of country B.12 3.2.3. Model 3: limited cross-border transmission-capacity As already proven in Section 2.4, putting CBTCC equal to infinity and to zero in model 3 gives the same result as model 1 and 2 respectively, pffiffiffi except that the efficient frontier of the former lays a factor 3 lower with reference to the risk of model 1. The CBTCC are considered to be equal to 200 MW between A and B, 100 MW between A and C and 300 MW between B and C (and vice versa).13
0.36 0.35 0.34 0.33 300
condense way. Remark, however, that for the portfolio with maximum return, a lot of power differences are situated around zero. This is just a matter of coincidence because for the portfolio with maximum return, only the locations with the highest capacity factor are selected (in this case three), which had for several hours at the same time a stable wind-power profile.
350
400
450
500
RISK: Portfolio Standard Deviation [MW] Fig. 10. Efficient frontier of portfolio with CBTCC. Simulation is run with 192 hourly data.
12 Note that the same conclusions can be drawn as in Section 3.2.1 concerning the comparison between the portfolio with minimum risk and the portfolio with maximum return. 13 Note that the values for the CBTCC are chosen randomly.
2382
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
1
Location 7 C Location 6 C Location 5 B Location 4 B Location 3 B Location 2 A Location 1 A
0.9
Share of each location
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0.34
0.35
0.36
0.37
0.38
0.39
RETURN: Mean Capacity Factor Portfolio Fig. 11. Wind-power allocation in different locations, for portfolios along the efficient frontier, with CBTCC. Simulation is run with 192 hourly data.
0.4
RETURN: Mean Capacity Factor Portfolio
0.39
0.38
0.37
0.36
0.35 Infinite Transmission Capacity Limited Transmission Capacity No Transmission Capacity
0.34
0.33 300
350
400
450
500
550
600
650
700
750
RISK: Portfolio Standard Deviation [MW] Fig. 12. Efficient frontiers of the three considered cases.
The efficient frontier and wind-power allocation are presented in Figs. 10 and 11, respectively.14 In Fig. 12, a comparison of the efficient frontiers for the three considered CBTCC with model 3 is shown. Comparing this figure with Fig. 4 demonstrates once more that the efficient frontier of the former lays a factor O3 lower with reference to the risk of model 1. Fig. 12 shows that the case with CBTCC equal to infinity is the least risky, situated on the left-hand side because wind-power differences can be smoothed out as much as possible, whereas the case with CBTCC shows more risky portfolios and can, therefore, be found on the right-hand side in the figure. The case with CBTCC equal to 100 MW, 200 MW and 300 MW lies in between the
14
See footnote 12.
efficient frontier of the two other cases. As can be seen, there is not a big difference in risk between the case where CBTCC are put equal to infinity and the case where the CBTCC are equal to 100 MW, 200 MW and 300 MW. This means that the chosen cross-border transmission-capacity constraints are sufficiently large for balancing the hourly wind-power differences. The reason for this result can be seen from Fig. 13. In Fig. 13, a histogram of the hourly wind-power differences for each country is shown for a portfolio situated on the efficient frontier for the case where no transmission capacity is available. As can be seen from the histogram, the hourly differences are comparable to the chosen CBTCC, ensuring that these will often be not restrictive. The wind-power profile after transmission of country C, for the portfolio with maximum return, and the histogram of the hourly wind-power differences of these wind-power profiles are given in
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
120
Frequency
Figs. 14 and 15 respectively. From Figs. 14 and 15, it can be concluded that limited transmission capacity of 200 MW between A and B, 100 MW between A and C and 300 MW between B and C (and vice versa) gives more or less the same distribution of hourly wind-power differences compared to the infinity case. Furthermore, Fig. 15 clearly shows that cross-border transmission-capacity plays an important role in reducing the risk of a variable wind-power profile. Increasing CBTCC reduces the number of large hourly wind-power differences, making the distribution more condense. In the case of country C, increasing CBTCC from zero to a limited value significantly decreases the largest hourly difference from approximately 1300 MW to approximately 750 MW. This is also visible in Fig. 14 where up- and downward movements in the wind-power profiles occur in more steps when transmission capacity for wind-power flows is made available.
Country A Country B Country C
100 80 60 40 20 0 −1500
−1000 −500 0 500 1000 Hourly difference after transmission [MW]
2383
3.3. Increasing the use of hourly data
1500
The portfolio model optimizations as employed so far were solved for 192 h of data. This may be insufficient to evaluate practical cases, but it has permitted to evaluate the models on
Fig. 13. Frequency of the hourly wind-power differences after transmission when no cross-border transmission-capacity is available. 192 h of data are used.
Country C Wind power profile after transmission [MW]
5000 Infinite transmission capacity Limited transmission capacity No transmission capacity
4500 4000 3500 3000 2500 2000 1500 1000 500 0
0
20
40
60
80
100
120
140
160
180
Hours [h] Fig. 14. Wind-power profile after transmission of country C for infinite, limited and no transmission capacity. Simulation is run with 192 hourly data.
Country C 140 Infinite transmission capacity Limited transmission capacity No transmission capacity
120
Frequency
100 80 60 40 20 0 −2500
−2000
−1500
−1000
−500
0
500
1000
1500
2000
Hourly difference after transmission [MW] Fig. 15. Frequency of the hourly wind-power differences after transmission in country C for infinite, limited and no transmission capacity. Simulation is run with 192 hourly data.
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
0.49
Mean hourly difference after transmission, CBTCC = inf Mean hourly difference after transmission [MW]
200 Country A Country B Country C
150 100 50 0 −50
0
50 100 150 Number of hourly data used [h]
200
RETURN: Mean Capacity Factor Portfolio
2384
0.485 0.48 0.475 0.47 0.465 0.46 0.455 0.45 500
Mean hourly difference after transmission, CBTCC = zero Mean hourly difference after transmission [MW]
150
600
650
700
750
RISK: Portfolio Standard Deviation [MW]
Country A Country B Country C
100
Fig. 17. Efficient frontier of portfolio with CBTCC and zero-mean assumption. Simulation is run with 26,280 hourly data.
50
referred to as the zero-mean assumption. Previously, the formula for a variance was written as: 0
sðaÞ2 ¼ −50
0
50 100 150 Number of hourly data used [h]
200
Mean hourly difference after transmission, CBTCC = limited
sðaÞ2 ¼
Country A Country B Country C
150 100 50 0 −50
0
50 100 150 Number of hourly data used [h]
200
Fig. 16. Mean hourly wind-power difference after transmission for the cases with CBTCC equal to infinite, zero and a limited value.
a conceptual level. Towards larger data sets, however, these previously used formulations might encounter computational difficulties. A solution to this limitation can be found by adding to model 3 the assumption that the mean of the hourly differences after transmission is equal to zero.15 Physically, this means that the mean fluctuation of wind power comes closer to zero when considering long time frames, because the up and down movements of wind velocities translate themselves into up and down movements of wind power. Consequently, the mean of the hourly wind-power differences after transmission can be set to zero,
15 Since model 3 in fact covers both model 1 and model 2 (see Section 2.4), this model is the generic coordinating model.
Pv1
EðaÞÞ2 ¼ ðv 1Þ 1
u ¼ 1 ðau
Pv1
meanðaÞÞ2 ðv 1Þ 1
u ¼ 1 ðau
(25)
Taking the new assumption into account results in Eq. (26):
200 Mean hourly difference after transmission [MW]
550
Pv1
2 u ¼ 1 ðau Þ ðv 1Þ 1
(26)
The justification of this assumption can be understood from Fig. 16. In this figure, a plot is made of the mean hourly difference after transmission for each country for the cases where CBTCC are equal to infinity, zero and a certain value (as before). The portfolios with minimal risk are looked at.16 From this observation, we can conclude that the mean of the hourly differences after transmission is expected to be zero (certainly if a large number of hourly data is used). The efficient frontier and the allocation of shares of installed wind power across the efficient frontier for the model with CBTCC and zero-mean assumption are shown in Figs. 17 and 18. For these figures, 26,280 h of data (equal to three years) are used in the simulation. Note that, because 26,280 h of data are used, not only a different return and risk is obtained, but also a different share allocation across the efficient frontier compared to Figs. 10 and 11. By using more data, the return of all portfolios increases with approximately 9%. However, on the other hand, also a higher risk is obtained, which increased by approximately 160 MW. Furthermore, especially the share allocation in country B changes a lot, with more weight given to locations 4 and 5. This shows that the eight days selected before were no good long term representation, but rather an individual case. Consequently, using more data makes the results more robust, but at a penalty of increasing the calculation time.17
16 17
Same conclusions can be drawn for the other portfolios. Calculation time increased with a factor 4.
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
2385
1
Location 7 C Location 6 C Location 5 B Location 4 B Location 3 B Location 2 A Location 1 A
0.9
Share of each location
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0.455
0.46
0.465
0.47
0.475
0.48
0.485
RETURN: Mean Capacity Factor Portfolio Fig. 18. Wind-power allocation in different locations, for portfolios along the efficient frontier, with CBTCC and zero-mean assumption. Simulation is run with 26,280 hourly data.
4. Summary and conclusions In this paper, a new portfolio-theory-based model for wind-power allocation is developed. This model optimises wind-power allocation towards lowest risk (defined as the standard deviation of hourly wind-power differences) with a given return (defined as wind generation), or, vice versa. The model takes into account cross-border transmission-capacity constraints. Starting from two models that calculate an efficient frontier (based on correlations) for the case of infinite and zero transmission capacity, a new portfolio perspective for balancing wind-power variations is considered in the third new model. Instead of looking at the standard deviation of the hourly differences before transmission has taken place, the perspective changes to hourly wind-power differences after transmission. As a result, efficient frontiers for different CBTCC can be compared in an easy way, including the case where CBTCC are put equal to infinity or zero. The availability of infinite transmission capacity between each country results in an efficient frontier with the least risk for each amount of return, and by consequence a more flattened profile. The case where transmission capacity is set equal to zero gives the opposite result. This case should be avoided in practice because it has a negative effect on the integration of wind power in an electric power system. Each country has to balance its own wind-power variations with the technology available in its electric power system, which can come at a higher cost if a massive amount of wind power is installed in one country. Increasing transmission capacity from zero to a certain value will move the efficient frontier to the left and consequently lower risk. It can be concluded that cross-border transmissioncapacity plays an important role in reducing the risk of a variable wind-power profile. Increasing cross-border transmission-capacity increases the effect of diversification compared to the case with zero transmission capacities. However, in order to find the right transmission capacity, one has to balance on the one hand between risk and return (as defined in this paper), and on the other hand between the benefits and costs of the transmission capacity. Finally, in this paper, wind-power profiles of different locations within one country are considered, instead of aggregated windpower profiles per country, and transmission within countries is assumed to have no congestion.
“Balancing wind energy in the grid: an overall, techno-economic and coordinated approach”, granted by the Belgian State e “FOD Wetenschapsbeleid”. E. Delarue is a post-doctoral researcher of the Research Foundation e Flanders (FWO). Appendix A This section proves that when minimising the sum of the variances of the hourly wind-power differences after transmission per country in the case of infinite transmission capacity, the standard deviations of the hourly differences of each country are equal to each other and the correlation between the hourly differences is equal to one, considering b as given:
min
X
sðak Þ2 /ck; r˛K : sðak Þ ¼ sðar Þ and rkr ¼ 1
Writing the minimisation function, using the definition of s(ak)2:
The authors gratefully acknowledge partial financial support for the research reported in this paper by the SD/EN/02A project
12 v X au;k C B C B u¼1 C Bau;k @ v1 A 0
X
sðak Þ2 ¼
v X X
(A.2)
ðv 1Þ 1
k˛K u ¼ 1
k˛K
By definition, au,k can also be expressed as:
au;k ¼ bu;k Yu;k
(A.3)
with Yu,k equal to the hourly difference in net wind-power export in country k between hour u and u þ 1:
Yu;k ¼
X
ck;r;uþ1
r˛K
X
cr;k;uþ1
r˛K
X r˛K
ck;r;u þ
X
cr;k;u
(A.4)
r˛K
Eq. (A.4) can be understood from the fact that P P bu;k ¼ wuþ1;i wu;i , and comparing Eq. (15) with Eq. (A.3),18 i˛Ik
i˛Ik
As the sum of all net wind-power exports over all countries is equal to zero, from (A.4) it follows that:
cu˛1; .; v : Acknowledgement
(A.1)
k˛K
X
Yu;k ¼ 0
(A.5)
k˛K
18 Note that when calculating the hourly wind-power differences after transmission, the transmission in previous periods is taken into account.
2386
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
Furthermore, bu,k can be rewritten as a the difference between the average mu and the divergence from this average Dmu,k:
P bu;k ¼
bu;k
Dmu;k ¼ mu Dmu;k
k˛K
n
Regrouping and again applying Eqs. (A.5) and (A.7), and using PP PP xij ¼ xij , yields: i
j
j
i
¼ nm2u þ
(A.6)
2 P v m n u u¼1
For these divergences, the following holds:
X
Dmu;k ¼ 0 cu˛1; .; v
(A.7)
þ
Using Eq. (A.2) and substituting Eqs. (A.3) and (A.6) gives: v X
6 6 6mu Dmu;k Yu;k u¼1 6 4 X
sðak Þ2 ¼
X
7 7 7 5
v2
3 2
v P
P6 6 u¼1 6mu Dmu;k Yu;k k˛K 4
mu Dmu;k Yu;k 7 v1
2
mu Dmu;k Yu;k v1
Pv
u¼1
2mu Dmu;k 2mu Yu;k 2mu u ¼ 1
mu Dmu;k Yu;k
¼ nm2u þ
Dmu;k þ Yu;k 2 þ
k˛K
v P
2
P
u¼1
!
mu
v P u¼1 2
ðv 1Þ
! Yu;k
6 P6 6 6 k˛K 4
u¼1
!2
mu
þ
k˛K
Dmu;k þ Yu;k
#
cu˛1; .; v
v1
(A.9)
P Since n$½mu ð vu ¼ 1 mu =vÞ2 in Eq. (A.9) is variable independent (i.e. independent of the transmission), Eq. (A.9) is minimised by putting
Pv
u¼1
Dmu;k þ Yu;k
2
Dmu;k þ Yu;k
2 #
v1
Pv
u¼1
Dmu;k þ Yu;k
cu˛1; .; v and ck˛K
v1
(A.10)
v P u¼1
!2
Dmu;k
þ
This condition is satisfied if:
Dmu;k þ Yu;k ¼ cte ¼ dk cu˛1; .; v
(A.11)
Furthermore, because of Eqs. (A.5) and (A.7):
/
X
dk ¼ 0
(A.12)
k˛K
v P u¼1
ðv 1Þ2 !
Dmu;k þ Yu;k
2
!2 Yu;k
2
v P u¼1
2 !3 ! v P 7 6 Dmu;k mu 2 Yu;k 7 2mu 62mu 7 X6 u¼1 u¼1 u¼1 7 6 þ 7 6 v1 ðv 1Þ2 7 k˛K 6 5 4 v P
" X
equal to zero, or in other words:
7 5
With mu independent of k and by regrouping and taking into account Eqs. (A.5) and (A.7), this can be further written as:
v P
ðv 1Þ2
k˛K
32
þ 2Dmu;k Yu;k v1 v v 3 P P mu Dmu;k Yu;k mu Dmu;k Yu;k 7 7 þ 2Dmu;k u ¼ 1 þ 2Yu;k u ¼ 1 5 v1 v1
2
þ
v1
Dmu;k þ Yu;k ¼ v P
mu
#
u¼1
" X
cu˛1; .; v
v 2P
P6 6 2 2 þ 6u ¼ 1 4 6mu þ Dm2u;k þ Yu;k k˛K 4
¼
7 7 5
2
Pv
(A.8)
2
v P
!
mu
v P u¼1 2
!
Dmu;k
ðv 1Þ
v P u¼1
!
Dmu;k
v1
2mu
!3 Yu;k 7 7 7 u¼1 7 7 v1 7 5 v P
3 3 2 2 v v v v v v P P P P P P D m D D D m D 2 m 2 m m 2 m Y 2Y 2Y m 2Y Y u;k u u;k u;k u;k u;k 7 u u;k u;k u;k u;k 7 6 u;k X6 7 X6 7 6 u¼1 u¼1 u¼1 u¼1 u¼1 u¼1 þ 7þ 7 6 6 5 5 4 4 v1 v1 v1 v1 v1 v1 k˛K
k˛K
¼ n$ mu
Looking at the term on the right-hand side of Eq. (A.8) and making temporary abstraction of the summation over time-intervals u and the division by v 2, this term can be written out as follows:
v u¼1 mu
v1 ðv 1Þ2 # " P 2 P v v u¼1 Dmu;k þ u¼1 Yu;k X k˛K
"
v X
k˛K u¼1
k˛K
3
P
# " Pv Pv X 2 Dmu;k þ Yu;k u¼1 Dmu;k 2 Dmu;k þ Yu;k u¼1 Yu;k þ v1 v1
mu Dmu;k Yu;k 72 v1
Dmu;k þ Yu;k 2 þ
k˛K
k˛K
2
X
2nmu
k˛K
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
Taking Eq. (A.11) into account, the variance of the hourly windpower differences after transmission in country k can be written as:
2
v P
6 6 u¼1 6mu Dmu;k Yu;k 4 v P sðak Þ2 ¼ u¼1 0
12
v1
7 7 5
rkr ¼
0
v2 v P
mu
12
B C B C u¼1 dk þ dk C Bmu @ A v1 ¼
v P u¼1
0
v P
v2 12
mu C B C B u¼1 C Bmu @ v1 A ¼
v P u¼1
v2
¼ sðmu Þ2 (A.13) Since Eq. (A.13) is independent of K, it holds that ck,r ˛ K: s(ak) ¼ s (ar). Furthermore, following Eqs. (A.3) and (A.11), the hourly differences after transmission become:
au;k ¼ bu;k Yu;k ¼ mu Dmu;k Yu;k ¼ mu dk
(A.16)
References
v P u¼1
Covðmu dk ; mu dr Þ sðak Þ2 ¼ 1 ¼ sðak Þsðar Þ sðak Þ2
ck˛K
v2
v P mu dk C B C B u¼1 C Bmu dk @ v1 A
¼
Calculating next the correlation between country k and r taking into account Eq. (A.15) gives:
3 2
mu Dmu;k Yu;k 7
2387
ck˛K
(A.14)
From Eqs. (A.14) and (A.13), the covariance can be calculated for country k and r:
Covðmu dk ; mu dr Þ ¼ Covðmu ; mu Þ ¼ sðmu Þ2 ¼ sðak Þ2
(A.15)
[1] EC The EU climate and energy package. Last consulted on 05/05/2010. Available from: http://ec.europa.eu/environment/climat/climate_action.htm. [2] Ecofys. Promotion and growth of renewable energy sources and systems. Available from: http://www.res-progress.eu/; 2008. [3] Ragwitz M, Miola A. Evidence from RD&D spending for renewable energy sources in the EU. Renewable Energy 2005;30(11):1635e47. [4] Breton SP, Moe G. Status, plans and technologies for offshore wind turbines in Europe and North America. Renewable Energy 2008;34(3):646e54. [5] Ringel M. Fostering the use of renewable energies in the European Union: the race between feed-in-tariffs and green certificates. Renewable Energy 2005;31(1):1e17. [6] Tradewind project. Integrating wind. Developing Europe’s power market for the large-scale integration of wind power. Available from: http://www.tradewind.eu/; 2009. [7] European Wind Integration Study (EWIS). Towards a successful integration of large scale wind power into European electricity grids. Available from: http:// www.wind-integration.eu/. [8] IEA. IEA Wind Task 25: design and operation of power systems with large amounts of wind power. Available from: http://www.ieawind.org/AnnexXXV. html; 2006. [9] EWEA. Large scale integration of wind energy in the European power supply: analysis, issues and recommendations. Available from: http://www.ewea.org/ ; 2005. [10] Markowitz H. Portfolio selection. The Journal of Finance 1952;7:77e91. [11] Awerbuch S. Portfolio-based electricity generation planning: policy implications for renewables and energy security. Mitigation and Adaptations Strategies for Global Change 2006;11:693e710. [12] Awerbuch S, Berger M. Applying portfolio theory to EU Electricity planning and policy making. IEA/EET Working Paper; 2003. [13] Bar-Lev D, Katz S. A portfolio approach to fossil fuel procurement in the electric utility industry. The Journal of Finance 1976;31:933e47. [14] Delarue E, De Jonghe C, Belmans R, D’haeseleer W. Applying portfolio theory to the electricity sector: energy versus power. Energy Economics 2011;33(1):12e23. [15] Roques F, Hiroux C, Saguan M. Optimal wind power deployment in Europe e a portfolio approach. Energy Policy 2010;38(7):3245e56. [16] Koninklijk Nederlands Meteorologisch Instituut. K. Hourly wind speed of seven locations in the Netherlands: Vlakte van de Raan, Huibertgat, Nieuwe Beerta, Goeree, Volkel, Twenthe, Lelystad. Last consulted on 14/12/2009. Available from: www.knmi.nl. [17] Vestas Last consulted on 12/12/2009. Available from: www.vestas.com.
Contents lists available at ScienceDirect
Renewable Energy journal homepage: www.elsevier.com/locate/renene
Optimal portfolio-theory-based allocation of wind power: Taking into account cross-border transmission-capacity constraints Yannick Rombauts, Erik Delarue, William D’haeseleer* University of Leuven (K.U.Leuven) Energy Institute, TME Branch (Applied Mechanics and Energy Conversion), Celestijnenlaan 300A, Box 2421, B-3001 Leuven, Belgium
a r t i c l e i n f o
a b s t r a c t
Article history: Received 20 July 2010 Accepted 18 February 2011 Available online 12 March 2011
Allocating wind farms across different locations in different countries may reduce the variability of hourly wind power changes. Taking into account cross-border transmission-capacity constraints between countries can however decrease the effect of this diversification. A portfolio-theory-based model is developed that takes into account these cross-border transmission-capacity constraints when allocating wind power as efficient as possible across different locations. Three models are developed, looking to the cases where cross-border transmission-capacity constraints are equal to infinity, zero and a certain limited value, respectively. It is notably this last model that brings new perspectives in the allocation of wind power, based on portfolio theory modelling. Keeping cross-border transmission-capacity available for wind-power flows is an effective measure to limit hourly wind-power variations. 2011 Elsevier Ltd. All rights reserved.
Keywords: Portfolio theory Cross-border transmission Wind-power allocation
In March 2007, the council of the European Union committed Europe to strive to an energy-efficient, low carbon economy in order to compete against climate change and enhance the energy security [1]. Within this framework, one of the targets put forward is a 20% share of renewable energy resources (RES) in total energy consumption by 2020. In order to achieve this goal, as much as 34% of the electricity demand in Europe is expected to be met by RES, and of this total share, 47% might have to come from wind farms, both on- and offshore [2]. Many countries spend a large budget on research, development and demonstration (RD&D) of renewable energy sources [3], and ambitious targets for offshore wind-power developments are set ahead [4] (the use of feed-in tariffs and green certificates are envisaged to foster this development [5]). This poses significant challenges as wind is a variable and to some extent unpredictable energy resource. Measures have to be taken to allow for a secure integration and to lower the risk on possible difficulties as much as possible. Current studies dealing with the integration of wind energy into an electrical power system start from a certain expected amount of wind power installed in each location. The EU project Tradewind
[6] focuses in this way on the current flows that occur in and between the grids of the different member states. EWIS [7], coordinated by the European transmission operator ENTSO-E is the result of a similar research which on the one hand deals with the use of the current grid, and formulates on the other hand propositions for a harmonised policy to facilitate the integration of wind energy. Also in Wind Task 25 of the IEA [8] one starts from a certain amount of wind energy that is expected to be installed in the different locations considered. This paper will, in contrast to these studies, not start from a certain expected amount of wind power installed in every location. Rather, the objective is to make a first move to the optimal integration of wind turbine farms by installing them at the most optimal locations such that the production of wind power is smoothened as much as possible. Focussing only on short-term variability,1 geographical dispersion of wind turbines can help to flatten this variability as much as possible, as uncorrelated wind profiles balance each other out. Aggregating different wind-power profiles shows that the amplitudes of the variations decrease. The larger the geographical area considered, the larger this effect, further enhancing the possibility of increasing the amount of wind power installed into the system. Large-scale aggregation of wind can, however, only be done in an
Abbreviation: CBTCC, cross-border transmission-capacity constraints. * Corresponding author. Tel.: þ32 16 322511; fax: þ32 16 322985. E-mail addresses: [email protected] (Y. Rombauts), erik. [email protected] (E. Delarue), [email protected] (W. D’haeseleer).
1 Wind power, has an intermittent character, i.e., it is variable and to some extent unpredictable. This paper focuses on the short-term variability, meaning the hourly wind-power variations. The remainder of the electrical power system is not considered explicitly.
1. Introduction
0960-1481/$ e see front matter 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2011.02.010
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
Nomenclature
Index i,j u h k,r v v1 n
location time-interval hour country number of hours number of hourly differences number of countries
Sets Ik I K
subset of locations located in country k set of locations set of countries
Variables the share of total wind capacity that is attributed to xi location i the capacity factor of location i CPi the amount of wind capacity that has to be installed in Pk country k [MW] the total amount of wind capacity that has to be Ptot installed over all countries considered [MW] the vector of hourly wind-power differences before bi transmission of locations i [MW] the sample standard deviation of location i, based on s(bi) wind-power differences before transmission [MW] rij the correlation coefficient between the wind-power differences of locations i and j s(bi)s(bj)rij the covariance between location i and j [MW2] the hourly wind-power difference before transmission bu,i of location i [MW]
optimal way if a well interconnected grid exists. Just looking at aggregated wind-power profiles does not take into account the effect of cross-border transmission-capacity constraints (CBTCC) [9]. The aim of the paper is to define “efficient” wind-power allocations, i.e., allocations with minimum fluctuations (minimum variability) for a given output of wind power. Portfolio theory can offer a solution in order to correctly link the geographic diversification effect of wind-power variability with the effect of limited CBTCC. In the literature, portfolio theory is mostly applied to financial portfolios, with Markowitz as the founding father [10]. Markowitz stated that the risk of a diversified portfolio is lower than the sum of the risks of the individual assets. More specifically, standard portfolio theory calculates a Pareto efficient frontier of portfolios, weighing maximal return and minimal risk. Every point on this efficient frontier indicates an efficient portfolio, which means there is no portfolio that provides a higher return for the same risk or a lower risk for the same return. Which efficient portfolio is going to be chosen depends on the risk appetite of a particular individual. A person who is risk averse is going to pick a portfolio with a low risk and by consequence a lower return. On the contrary, a person willing to take a relatively high risk chooses a portfolio with high risk and a higher return. Nowadays, portfolio theory has been applied in various research fields beyond the financial sector. The energy sector is one of them [11e14]. In this paper, the focus is on applying portfolio theory on the wind-power sector and builds on the work of Roques et al. [15],
2375
bu,k
the hourly wind-power differences before transmission of country k [MW] E(b) the expected value of the hourly differences before transmission [MW] the hourly change in wind power in country k, after au,k transmission has taken place [MW] E(a) the expected value of the hourly differences after transmission [MW] the vector of hourly wind-power differences after ai transmission of locations i [MW] the vector of hourly wind-power differences after ak transmission of country k [MW] the variance of the hourly wind-power differences s(ak)2 after transmission of country k [MW2] the power transported in hour h, from countries k to ck,r,h country r [MW] capconstrk,r the capacity constraint put on the transmission line between the two countries k and r [MW] the fraction of wind power (expressed as a percentage frach,i of total installed capacity) in hour h at location i the amount of wind power produced in hour h in wh,i location i [MWh/h) ¼ [MW] the net amount of wind power produced (after ph,i balancing) in hour h in location i [MW] P ak Þ2 the variance of the hourly wind-power differences of sð k˛K the aggregated wind-power profile over all countries of set K [MW2] r(ak,ar) the correlation coefficient between the hourly windpower differences after transmission of country k and r the difference in net wind-power export in country k Yu,k between hour u and u þ 1 [MW] mu the mean hourly wind-power differences of all countries k [MW] Dmu,k the difference of bk,u and mu for each country [MW]
who developed a portfolio-theory-based model that maximises the average capacity factor and minimises normalised hourly windpower fluctuations, by allocating a share of wind power over Austria, France, Germany, Spain and Denmark. The minimisation was carried out for two cases. In a first case, they included all available wind data, whereas in the second case only peak demand hours are considered in order to maximise the wind-power contribution to the system reliability. Their data set consists of aggregated hourly wind-power-production data from the transmission-system operators of the different countries considered. In a further extension of the model, two types of constraints were added: constraints on the technical potential of wind power for each country and constraints on the cross-border transmissioncapacity. The latter was included as a proxy by taking the sum of the demand and the total transmission export capacity for each country, thereby defining an upper wind-power limit for the country considered. The main focus of the current work is on the inclusion of CBTCC. In contrast to the portfolio approach of Roques et al. [15], this work will model the cross-border capacities more explicitly so that these constraints can be taken into account and the electrical power flows stemming from wind-power production can be analysed.2 Furthermore, this work will apply portfolio theory on data of
2 CBTCC in this context is referred to as the capacity on cross-border connections, exclusively available for wind-power flows.
2376
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
Table 1 Illustration of the hourly wind-power differences before and after transmission.
Before After
Wind-power production in hour 1
Wind-power production in hour 2
Transmission in hour 2
Hourly difference
1000 1000
1500 1500
/ 200
500 300
different locations within one country instead of aggregated windpower data. Note that the methodology as presented in the paper considers a central planner approach. However, one could apply the developed algorithm in a deregulated market setting with different players, trying to minimise the variability of their own generation portfolios. Interconnections could then reflect possible contracts between different players. This is scope for future work. In the next section, the optimisation model is developed and the portfolio theory concepts are translated to the wind-power allocation problem. Three sub models are set up, putting CBTCC equal to infinity, to zero and to a certain specific value, respectively. This model is then applied to a case study, being discussed in Section 3. Conclusions are drawn in the final section. 2. Portfolio model description The fundamental concepts of classic portfolio theory are risk and return, which are clearly defined in the framework of financial theory. In order to apply portfolio theory to the wind-power problem, these financial concepts need appropriate equivalents that are related to wind-power generation. Comparable definitions as proposed by Roques et al. [15] are used. Return is defined as the “mean capacity factor over the different locations”, and risk is defined as the standard deviation of hourly wind-power differences, Pt Pt1 (with Pt the wind power at hour t). For this, either hourly wind-power differences before or after transmission, are used. In order to make this difference clear from the beginning, an illustration is shown in Table 1. The set up of the portfolio model consists of different adjacent countries, linked to each other through cross-border transmission lines. Each country has to install a specific amount of wind power by dividing it over different sites considered within that country, taking into account the wind speed profile characteristic for each particular site. The remainder of the electrical power system is not considered. As far as electric transmission is concerned, transmission lines within countries are assumed to have no congestion. CBTCC are taken into account in different steps. In a first model, cross-border transmission capacities are taken to be infinite, which is equivalent to looking at a single (combined) country with several wind farms. Secondly, cross-border transmission capacities are set to zero, effectively isolating each country. In this second model, all countries minimise their own risk.3 These two models can be seen as the two most extreme cases, and provide valuable insights. In the third portfolio model, the CBTCC can be chosen appropriately. 2.1. Model 1: unlimited cross-border transmission-capacity The optimisation model determines the optimal allocation of wind power (i.e., in terms of return and risk, as defined earlier) over the different locations in each country. It is assumed that each
3
Recall that throughout this work, the notion of risk is used to refer to the standard deviation of hourly wind-power differences.
country has an overall given amount of wind power that needs to be installed. In the first model, cross-border transmission capacities are infinite and therefore these constraints are non-binding, meaning that the aggregation of countries can be looked at as one single (combined) country (although accounting for the required amounts of wind power that need to be installed in each country). Firstly, the portfolio with the maximum return is calculated. This is done by means of the following objective function and constraints:
max
X
xi CPi
(1)
i˛I
Subject to:
X
xi ¼
i˛Ik
Pk Ptot
ck˛K and Ik ˛I
xi 0
(2)
(3)
The addition of Eq. (2) ensures that the right quantity of wind power is installed in a particular country. Moreover, the summation of Eq. (2) over the countries K, is equal to one. With this portfolio of maximum return, a certain risk can be associated. This risk (defined as a standard deviation) is equal to the square root of the variance s2:
s2max return ¼
XX i˛I
xi xj sðbi Þs bj rij
(4)
j˛I
The sample standard deviation si is defined as:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Pv1 u ¼ 1 bu;i EðbÞ sðbi Þ ¼ ðv 1Þ 1
(5)
Next, the portfolio with minimal risk is calculated. For this, the same constraints as in Eqs. (2) and (3) apply, only the objective function changes4:
min s2p ¼ min
XX i˛I
xi xj si sj rij
(6)
j˛I
Also for this portfolio, a certain return can be calculated as:
returnmin risk ¼
X
xi CPi
(7)
i˛I
The maximum return and the minimum risk portfolios constitute the extreme points of the efficient frontier. In addition, the other points of the efficient frontier have to be calculated. This is done in the same way as calculating the portfolio with minimal risk, except for an extra constraint added on the return5 as shown in Eq. (8), where this return is taken between the return of the portfolio with maximal return and the return of the portfolio with minimal risk. In other words, every return that is lying in between the returns of the two extreme points of the efficient frontier is taken as fixed, and risk is minimised.
X
xi CPi ¼ return
(8)
i˛I
By examining the calculation method of the risk of a particular portfolio into detail, a second calculation method can be derived no
4 The optimisation is performed towards minimum variance. The risk is equal to the square root of this variance. 5 Maximising return with a constraint on risk would yield the same results.
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
CBTCC apply. This second calculation method provides a better insight in the meaning of the risk definition. Considering two locations i and j for reasons of simplicity, risk as defined in Eq. (6) can be rewritten as:
2 s2p ¼ x2i sðbi Þ2 þx2j s bj þ2xi xj sðbi Þs bj rij 2 2 ¼ sðxi bi Þ2 þs xj bj þ2cov xi bi ; xj bj ¼ s xi bi þ xj bj
(9)
Minimising s(xibi þ xjbj)2 can be seen as minimising the variance of the hourly wind-power differences of aggregated6 wind-power profiles of both locations i and j, when a share xi of Ptot is installed in location i and a share xj of Ptot is installed in location j. Minimising the variance of the hourly wind-power differences of this single wind-power profile gives the same risk as minimising the objective function of Eq. (6). Note that Eq. (9) applies for a group of countries as a whole. However, for a country acting as a group of locations, the same equation can be applied. Note further that, in case all locations are considered, the following equation applies:
xi bi þ xj bj ¼ xi ai þ xj aj
(10)
This stems from the fact that no wind power is lost or created by transporting it between different countries. By consequence, summing the different wind-power profiles up before or after transmission till one aggregated wind-power profile makes no difference. Eq. (9) can also be written as:
2 2 s2p ¼ s xi bi þ xj bj ¼ s xi ai þ xj aj
(11)
2.2. Model 2: no cross-border transmission In the second model, the CBTCC are put equal to zero. Every country minimises its own risk. This way, a model is obtained for which countries cannot rely on neighbouring countries to balance any of their hourly wind power changes. Consequently, the global variance for this model is defined as the sum of each country’s individually optimised variance of hourly wind-power differences.7 This gives the following objective function when minimising risk8:
0 1 X XX @ min xi xj sðbi Þs bj rij A k˛K
(12)
i˛Ik j˛Ik
The remainder of the model is the same as the one described in Section 2.1. To summarise, the second model consists of Eqs. (1)e (4), (7), (8) and (12). Note that the return of Eq. (8) is still calculated and optimised for all countries together, instead of calculating it for each country separately. In this perspective, each country will look to the other countries in order to maximise the jointly obtained return. This makes that the global variance as calculated by Eq. (12) will be
lower than in the case where the return would be optimised per country separately. The reason for this can be found in the fact that more combinations for the allocation of shares giving a particular return can be obtained in the case returns are maximised jointly, making this constraint less strict. In other words, when calculating the efficient frontier, Eq. (8) is less strict than if this constraint would be imposed on each country separately. This will however have no effect on the portfolio of maximal return and the portfolio of minimal risk, because the portfolio with maximal return will still be composed of the locations with the highest return in each country, while the portfolio with minimal risk will still be the same most diversified portfolio.
2.3. Model 3: limited cross-border transmission-capacity In the previous sections, models for infinite and zero crossborder capacities have been developed, respectively. An intermediate way has to be found to evaluate cases with non-zero but finite cross-border capacities. Therefore, the objective function and the constraints of the previous models must be generalised. A different perspective on the objective function compared to the previous cases is taken. Instead of looking at the variances of the hourly wind-power differences before transmission, the variances of the hourly wind-power differences after transmission are considered. The minimisation of the objective function is comparable to Eq. (11) (considering in this case a country as a union of locations), but now each country separately takes cross-border transmission-capacity constraints into account. With this approach, no correlations between countries have to be considered in the minimisation, because the correlations are already implicitly incorporated through the cross-border transmissioncapacity constraints. One cannot transport more wind power from one country to another to flatten fluctuations, than allowed by available capacity on the transmission lines. This means that the risk of the different countries, defined as the variances of the hourly windpower differences after transmission, can be summed up. Further justification of this objective function is given in Section 2.4, in which the equivalence with the two previous models is demonstrated. This approach leads to the following objective function and constraints for calculating the minimal risk:
min
X
sðak Þ2
(13)
k˛K
Subject to:
ck;r;h capconstrk;r
ch˛1; .; v and
(14)
ck; r˛K with capconstrk;k ¼ 0 au;k ¼
X
wuþ1;i þ
0
@
X
cr;k;uþ1
r˛K
i˛Ik
X
wu;i þ
X
ck;r;uþ1
r˛K
X
cr;k;u
r˛K
i˛Ik 6 An aggregated wind-power profile can be obtained by taking the sum of the wind-power profiles of each location. Note that this aggregation can only be done if no congestion can take place on the transmission lines. 7 Performing one global risk minimisation (over all countries, with no interconnections) yields the same results as when minimising the risk for each country separately. This result stems from the fact that the risk of each country separately is independent from each other (meaning that no correlation coefficient has to be taken into account between the risks of two countries since cross-border transmission is put equal to zero in this second model). 8 See footnote 4.
2377
X
1 ck;r;u A
r˛K
cu˛1; .; v 1; ck˛K and Ik ˛I; with au;k ˛ak h
u
1
2
1
3
2
4
3
ð15Þ 5
4
Fig. 1. Representation of index h for hours and index u for hourly intervals.
2378
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
0.4 RETURN: Mean Capacity Factor Portfolio
3000
Wind power [kW]
2500 2000 1500 1000 500 0
0
5
10
15
20
25
0.39 0.38 0.37 0.36 0.35 0.34 0.33 550
30
Wind speed [m/s]
600 650 700 750 RISK: Portfolio Standard Deviation [MW]
800
Fig. 2. Power curve of the Vestas 112 3 MW wind turbine. Fig. 4. Efficient frontier of wind-power allocation portfolios, without CBTCC taken into account. Simulation is run with 192 hourly data.
wh;i ¼ frach;i $Ptot $xi ph;k ¼
X
wh;i þ
xi ¼
i˛Ik
Pk Ptot
ck;r;1 ¼ 0
cr;k;h
r˛K
i˛Ik
X
X
ch˛1; .; v X
(16)
ck;r;h ch˛1;.;v and ck˛K
(17)
r˛K
ck˛K and Ik ˛I
(18)
ck; r˛K
(19)
xi 0
(20)
ck;r;u 0
(21)
ph;k 0
(22)
B(1)
A(1) A(1)
As a clarification, index u and index h are represented in Fig. 1. In this figure, a time scale is displayed, where h refers to the different hours considered, whereas u refers to a time-interval. The addition of Eqs. (17), (18) and (22) ensures that a country cannot send more wind power to another country than it produces or receives from other countries, since the remainder of the electrical power system is not considered. Furthermore, as can be seen from Eq. (15), the hourly difference after transmission is calculated as the difference between on the one hand wind power produced (wuþ1), imported (cr,k,uþ1) and exported (ck,r,uþ1) in this hour (u þ 1) and on the other hand the wind power produced (wu), imported (cr,k,u) and exported (ck,r,u) in the previous hour (u). This way, previous hours are linked to each other correctly in order to be able to smooth out wind-power production as much as possible.
B(1)
B(2)
A(1)
B(1)
B(2)
B(2) A(2)
A(2)
B(3)
B(3)
C(1)
C(1)
C(2)
MODEL 1
A(2)
B(3)
C(1)
C(2)
MODEL 2 Fig. 3. Configuration of the different models.
C(2)
MODEL 3
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
2379
1
Location 7 C Location 6 C Location 5 B Location 4 B Location 3 B Location 2 A Location 1 A
0.9
Share of each location
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0.34 0.35 0.36 0.37 0.38 0.39 RETURN: Mean Capacity Factor Portfolio
Fig. 5. Wind-power allocation in different locations, for portfolios along the efficient frontier, without CBTCC taken into account. Simulation is run with 192 hourly data.
The objective function and constraints for calculating the portfolio with maximum return are the same as described in Section 2.1, because the return only depends on the shares allocated to each location. For calculating the efficient frontier, Eq. (8) is added to the constraints of the minimal risk calculation. 2.4. Equivalence of models
number of countries that is taken under consideration. This means that, for the portfolios lying on the efficient frontier for the third model, the variance is n times smaller than the variance of the same portfolio for model 1. Starting from Eq. (9) and taking into account Eq. (11), this can be shown as follows:
s
X
!2 bk
¼ s
k˛K
¼
Wind power [MW]
sðak Þ2 þ
s
X
!2 ak
8000 6000 4000 2000
50
100 Hours [h]
150
Fig. 6. Aggregated wind-power profile of portfolio with minimum risk and portfolio with maximum return. Each aggregated wind-power profile is composed of the windpower profiles of the three countries A, B and C. Simulation is run with 192 hourly data.
n1 X
n X
2sðak Þsðar Þrðak ; ar Þ
(23)
k ¼ 1 r ¼ kþ1
X
ðn 1Þ$n sðak Þ2 þ $2$sðak Þ2 2 k˛K X X X ¼ sðak Þ2 þðn 1Þ sðak Þ2 ¼ n sðak Þ2 ¼
k˛K
10000
ak
With cross-border transmission-capacity constraints set equal to infinity in the third model, it can be proven (Appendix A) that P ak Þ2 , results in s(ak) ¼ s(ar) and r(ak,ar) ¼ 1. This minimising sð k˛K results in:
12000
0
X k˛K
k˛K
Portfolio minimum risk Portfolio maximum return
!2
k˛K
It is possible to check upon the consistency of the third model, with the first and second models, when implementing a CBTCC value of infinity and zero respectively in model 3. 2.4.1. Equivalence of model 3 with model 1 and model 2 At first instance, CBTCC in model 3 are put equal to infinity. Comparing the results in this case with those of model 1 shows that the portfolio compositions and returns are the same, while the risk pffiffiffi of model 3 lies a factor of n times lower (expressed as the standard deviation of the hourly wind-power differences), with n the
X
k˛K
(24)
k˛K
Note that this difference does not affect the share allocation of each efficient portfolio because every risk is downscaled with a factor pffiffiffi of n. At second instance, all capconstrk,r in model 3 are put equal to zero. This way, the objective function optimises the variance of each country separately, without cross-border transmission taking place. Within the borders of a country, infinite wind-power transmission can occur between the different locations, as mentioned in Section 2. Taking this unlimited intra-country transmission into account, and combining it with Eq. (11) from Section 2.1 makes this case equivalent to model 2 where risk was calculated for each country separately. Both calculation methods are, as shown through Eq. (10) equal to each other.
3. Case study 3.1. Problem setting The model will be illustrated with a methodological case study. The conceptual problem set up consists of three adjacent countries
2380
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
60 Portfolio with minimum risk Portfolio with maximum return
50
Frequency
40 30 20 10 0 −2500
−2000
−1500
−1000
−500
0
500
1000
1500
2000
2500
Power difference [MW] Fig. 7. Histogram of the aggregated hourly wind-power differences of the portfolio with minimum risk and the portfolio with maximum return. Simulation is run with 192 hourly data.
3.2. Results 3.2.1. Model 1: unlimited cross-border transmission-capacity Running the portfolio model mentioned in Section 2.1 for 192 h of data, yields the efficient frontier represented in Fig. 4. Every
9
In future work, more distant wind profiles could be utilised. 10 The current set up can also be looked at as three geographic separated regions in the Netherlands (with and without electric connections as explained below). 11 All simulations are run on an Intel Pentium Dual Core 2.66 GHz processor with 3.0 GB of RAM, and solved using the Cplex 11.1 solver.
point on this efficient frontier indicates an efficient portfolio, which means there is no portfolio that provides a higher return for the same risk, or, a lower risk for the same return. As can be seen, the mean capacity factor (return), which is calculated with the 192 h of data used, following Eqs. (1), (7) and (8), lies in between 0.33 and 0.40. This range covers both the portfolios of minimal risk and maximal return. The risk of the portfolios is calculated according to Eqs. (4) and (6) and ranges from 550 MW to 775 MW. Fig. 5 presents the composition along the efficient frontier of Fig. 4 with increasing return: i.e., starting on the lower left-hand side of Fig. 4, and moving to the upper right-hand side. Each colour represents a certain percentage of wind power allocated to a particular location of one of the three countries. Consequently, the portfolio shown on the left-hand side of Fig. 5 is the portfolio with the least risk. As can be seen, this portfolio is the most diversified one, meaning each location has a certain amount of wind power installed because of the risk reduction attributed to the diversification effect. On the contrary, the right-hand side represents the portfolio with the highest return, but exposed to a higher risk. This
0.4 RETURN: Mean Capacity Factor Portfolio
(A, B, and C). Contrary to looking at each country as a single point of wind production [15], multiple wind-power producing locations are considered within each country. More specifically, countries A and C have two locations each, and country B has three locations. Each location is characterised by its own hourly wind profile. The actual wind profiles used, are based on data from the Dutch meteorological service (KNMI) [16] of seven locations situated in the Netherlands.9 These wind profiles are assigned randomly over the different locations in the three fictitious countries.10 All the locations are assumed to be onshore, and for the transformation of wind flows to power flows the Vestas 112 3 MW wind turbine is used [17]. The power curve of the Vestas 112 3 MW wind turbine is presented in Fig. 2. At first instance, 192 h of wind speed data (equal to eight days) are used. This is sufficient to gain insight in the developed methodology, and this problem can easily be solved by available CPU power.11 In Section 3.3, a more efficient ‘equivalent’ model formulation is provided (yielding identical results), and a more extended data set (covering hourly values of three years) is used there. The totally installed wind power amounts to 12,000 MW, of which 5000 MW needs to be installed in country A, 3000 MW in country B, and, finally, 4000 MW in country C. Recall that the remainder of the electrical power system is not considered. The three models are now applied to the case considered. As already mentioned in Section 2, no congestion takes place within countries. CBTCC are taken into account in different steps corresponding to the three developed models. In Fig. 3 the three models are represented from a country perspective. Recall that the objective in all models is to smooth out wind-power production as much as possible (i.e., to minimise hourly variations) for a given windpower output, or vice versa, taking cross-border transmissioncapacity constraints into account.
0.39 0.38 0.37 0.36 0.35 0.34 0.33 500
550
600
650
700
750
RISK: Portfolio Standard Deviation [MW] Fig. 8. Efficient frontier of portfolio with CBTCC equal to zero. Simulation is run with 192 hourly data.
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
1
Location 7 C Location 6 C Location 5 B Location 4 B Location 3 B Location 2 A Location 1 A
0.9 0.8 Share of each location
2381
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0.34
0.35
0.36
0.37
0.38
0.39
RETURN: Mean Capacity Factor Portfolio Fig. 9. Wind-power allocation in different locations, for portfolios along the efficient frontier, with CBTCC equal to zero. Simulation is run with 192 hourly data.
portfolio is obtained through maximising return, leading to the obvious result that the whole share in each country is attributed to the location with the highest capacity factor. This can be seen from the figure: the portfolio with maximum return consists solely of wind power on location 1 in country A, location 3 in country B and location 7 in country C. In case of infinite CBTCC, the different wind-power profiles of each location can easily be summed up till one aggregated windpower profile. The comparison of the aggregated wind-power profiles of the portfolio with minimal risk and the portfolio with maximal return is presented in Fig. 6. As can be seen from the figure, the aggregated wind-power profile of the portfolio with maximum return is for most of the time higher than for the portfolio with minimum risk. In addition, the portfolio with maximum return also shows larger hourly wind-power differences. This higher variability is also demonstrated the histograms presented in Fig. 7. From this figure, it is clear that the largest hourly differences for portfolio with minimum risk are much smaller than for the portfolio with maximum return. Furthermore, the hourly differences of the portfolio with minimum risk are grouped in a more
RETURN: Mean Capacity Factor Portfolio
0.4 0.39 0.38 0.37
3.2.2. Model 2: no cross-border transmission The results of this model are presented in Figs. 8 and 9. Fig. 8 represents the efficient frontier. All countries are taken into account, and risk is defined as in Eq. (12). Note that this risk cannot be compared with the risk calculated with Eq. (8) of model 1. For this, the risk obtained through model 1 has to be downscaled with pffiffiffi a factor of 3 because of the similarity between on the one hand model 3 and model 1, and on the other hand model 3 and model 2, pffiffiffi as shown in Section 2.4. The risk of model 3 is a factor n lower than that of model 1 (with n the number of countries, in this case n ¼ 3), while the risk of model 3 and model 2 is equal to each other. Using model 3 makes it by consequence possible to compare the efficient frontiers of model 1 and model 2. Fig. 9 represents the share allocation of the efficient frontier. Note that for country B, almost half of the portfolios on the efficient frontier are fully allocated to location 3. This result would not be possible if the return would have been maximised per country, instead of for all countries as a whole as mentioned in Section 2.3. In case of a separate optimisation, only the portfolio with the maximal return (situated on the right-hand side of Fig. 9) would have a full allocation of wind power to location 3 of country B.12 3.2.3. Model 3: limited cross-border transmission-capacity As already proven in Section 2.4, putting CBTCC equal to infinity and to zero in model 3 gives the same result as model 1 and 2 respectively, pffiffiffi except that the efficient frontier of the former lays a factor 3 lower with reference to the risk of model 1. The CBTCC are considered to be equal to 200 MW between A and B, 100 MW between A and C and 300 MW between B and C (and vice versa).13
0.36 0.35 0.34 0.33 300
condense way. Remark, however, that for the portfolio with maximum return, a lot of power differences are situated around zero. This is just a matter of coincidence because for the portfolio with maximum return, only the locations with the highest capacity factor are selected (in this case three), which had for several hours at the same time a stable wind-power profile.
350
400
450
500
RISK: Portfolio Standard Deviation [MW] Fig. 10. Efficient frontier of portfolio with CBTCC. Simulation is run with 192 hourly data.
12 Note that the same conclusions can be drawn as in Section 3.2.1 concerning the comparison between the portfolio with minimum risk and the portfolio with maximum return. 13 Note that the values for the CBTCC are chosen randomly.
2382
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
1
Location 7 C Location 6 C Location 5 B Location 4 B Location 3 B Location 2 A Location 1 A
0.9
Share of each location
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0.34
0.35
0.36
0.37
0.38
0.39
RETURN: Mean Capacity Factor Portfolio Fig. 11. Wind-power allocation in different locations, for portfolios along the efficient frontier, with CBTCC. Simulation is run with 192 hourly data.
0.4
RETURN: Mean Capacity Factor Portfolio
0.39
0.38
0.37
0.36
0.35 Infinite Transmission Capacity Limited Transmission Capacity No Transmission Capacity
0.34
0.33 300
350
400
450
500
550
600
650
700
750
RISK: Portfolio Standard Deviation [MW] Fig. 12. Efficient frontiers of the three considered cases.
The efficient frontier and wind-power allocation are presented in Figs. 10 and 11, respectively.14 In Fig. 12, a comparison of the efficient frontiers for the three considered CBTCC with model 3 is shown. Comparing this figure with Fig. 4 demonstrates once more that the efficient frontier of the former lays a factor O3 lower with reference to the risk of model 1. Fig. 12 shows that the case with CBTCC equal to infinity is the least risky, situated on the left-hand side because wind-power differences can be smoothed out as much as possible, whereas the case with CBTCC shows more risky portfolios and can, therefore, be found on the right-hand side in the figure. The case with CBTCC equal to 100 MW, 200 MW and 300 MW lies in between the
14
See footnote 12.
efficient frontier of the two other cases. As can be seen, there is not a big difference in risk between the case where CBTCC are put equal to infinity and the case where the CBTCC are equal to 100 MW, 200 MW and 300 MW. This means that the chosen cross-border transmission-capacity constraints are sufficiently large for balancing the hourly wind-power differences. The reason for this result can be seen from Fig. 13. In Fig. 13, a histogram of the hourly wind-power differences for each country is shown for a portfolio situated on the efficient frontier for the case where no transmission capacity is available. As can be seen from the histogram, the hourly differences are comparable to the chosen CBTCC, ensuring that these will often be not restrictive. The wind-power profile after transmission of country C, for the portfolio with maximum return, and the histogram of the hourly wind-power differences of these wind-power profiles are given in
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
120
Frequency
Figs. 14 and 15 respectively. From Figs. 14 and 15, it can be concluded that limited transmission capacity of 200 MW between A and B, 100 MW between A and C and 300 MW between B and C (and vice versa) gives more or less the same distribution of hourly wind-power differences compared to the infinity case. Furthermore, Fig. 15 clearly shows that cross-border transmission-capacity plays an important role in reducing the risk of a variable wind-power profile. Increasing CBTCC reduces the number of large hourly wind-power differences, making the distribution more condense. In the case of country C, increasing CBTCC from zero to a limited value significantly decreases the largest hourly difference from approximately 1300 MW to approximately 750 MW. This is also visible in Fig. 14 where up- and downward movements in the wind-power profiles occur in more steps when transmission capacity for wind-power flows is made available.
Country A Country B Country C
100 80 60 40 20 0 −1500
−1000 −500 0 500 1000 Hourly difference after transmission [MW]
2383
3.3. Increasing the use of hourly data
1500
The portfolio model optimizations as employed so far were solved for 192 h of data. This may be insufficient to evaluate practical cases, but it has permitted to evaluate the models on
Fig. 13. Frequency of the hourly wind-power differences after transmission when no cross-border transmission-capacity is available. 192 h of data are used.
Country C Wind power profile after transmission [MW]
5000 Infinite transmission capacity Limited transmission capacity No transmission capacity
4500 4000 3500 3000 2500 2000 1500 1000 500 0
0
20
40
60
80
100
120
140
160
180
Hours [h] Fig. 14. Wind-power profile after transmission of country C for infinite, limited and no transmission capacity. Simulation is run with 192 hourly data.
Country C 140 Infinite transmission capacity Limited transmission capacity No transmission capacity
120
Frequency
100 80 60 40 20 0 −2500
−2000
−1500
−1000
−500
0
500
1000
1500
2000
Hourly difference after transmission [MW] Fig. 15. Frequency of the hourly wind-power differences after transmission in country C for infinite, limited and no transmission capacity. Simulation is run with 192 hourly data.
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
0.49
Mean hourly difference after transmission, CBTCC = inf Mean hourly difference after transmission [MW]
200 Country A Country B Country C
150 100 50 0 −50
0
50 100 150 Number of hourly data used [h]
200
RETURN: Mean Capacity Factor Portfolio
2384
0.485 0.48 0.475 0.47 0.465 0.46 0.455 0.45 500
Mean hourly difference after transmission, CBTCC = zero Mean hourly difference after transmission [MW]
150
600
650
700
750
RISK: Portfolio Standard Deviation [MW]
Country A Country B Country C
100
Fig. 17. Efficient frontier of portfolio with CBTCC and zero-mean assumption. Simulation is run with 26,280 hourly data.
50
referred to as the zero-mean assumption. Previously, the formula for a variance was written as: 0
sðaÞ2 ¼ −50
0
50 100 150 Number of hourly data used [h]
200
Mean hourly difference after transmission, CBTCC = limited
sðaÞ2 ¼
Country A Country B Country C
150 100 50 0 −50
0
50 100 150 Number of hourly data used [h]
200
Fig. 16. Mean hourly wind-power difference after transmission for the cases with CBTCC equal to infinite, zero and a limited value.
a conceptual level. Towards larger data sets, however, these previously used formulations might encounter computational difficulties. A solution to this limitation can be found by adding to model 3 the assumption that the mean of the hourly differences after transmission is equal to zero.15 Physically, this means that the mean fluctuation of wind power comes closer to zero when considering long time frames, because the up and down movements of wind velocities translate themselves into up and down movements of wind power. Consequently, the mean of the hourly wind-power differences after transmission can be set to zero,
15 Since model 3 in fact covers both model 1 and model 2 (see Section 2.4), this model is the generic coordinating model.
Pv1
EðaÞÞ2 ¼ ðv 1Þ 1
u ¼ 1 ðau
Pv1
meanðaÞÞ2 ðv 1Þ 1
u ¼ 1 ðau
(25)
Taking the new assumption into account results in Eq. (26):
200 Mean hourly difference after transmission [MW]
550
Pv1
2 u ¼ 1 ðau Þ ðv 1Þ 1
(26)
The justification of this assumption can be understood from Fig. 16. In this figure, a plot is made of the mean hourly difference after transmission for each country for the cases where CBTCC are equal to infinity, zero and a certain value (as before). The portfolios with minimal risk are looked at.16 From this observation, we can conclude that the mean of the hourly differences after transmission is expected to be zero (certainly if a large number of hourly data is used). The efficient frontier and the allocation of shares of installed wind power across the efficient frontier for the model with CBTCC and zero-mean assumption are shown in Figs. 17 and 18. For these figures, 26,280 h of data (equal to three years) are used in the simulation. Note that, because 26,280 h of data are used, not only a different return and risk is obtained, but also a different share allocation across the efficient frontier compared to Figs. 10 and 11. By using more data, the return of all portfolios increases with approximately 9%. However, on the other hand, also a higher risk is obtained, which increased by approximately 160 MW. Furthermore, especially the share allocation in country B changes a lot, with more weight given to locations 4 and 5. This shows that the eight days selected before were no good long term representation, but rather an individual case. Consequently, using more data makes the results more robust, but at a penalty of increasing the calculation time.17
16 17
Same conclusions can be drawn for the other portfolios. Calculation time increased with a factor 4.
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
2385
1
Location 7 C Location 6 C Location 5 B Location 4 B Location 3 B Location 2 A Location 1 A
0.9
Share of each location
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0.455
0.46
0.465
0.47
0.475
0.48
0.485
RETURN: Mean Capacity Factor Portfolio Fig. 18. Wind-power allocation in different locations, for portfolios along the efficient frontier, with CBTCC and zero-mean assumption. Simulation is run with 26,280 hourly data.
4. Summary and conclusions In this paper, a new portfolio-theory-based model for wind-power allocation is developed. This model optimises wind-power allocation towards lowest risk (defined as the standard deviation of hourly wind-power differences) with a given return (defined as wind generation), or, vice versa. The model takes into account cross-border transmission-capacity constraints. Starting from two models that calculate an efficient frontier (based on correlations) for the case of infinite and zero transmission capacity, a new portfolio perspective for balancing wind-power variations is considered in the third new model. Instead of looking at the standard deviation of the hourly differences before transmission has taken place, the perspective changes to hourly wind-power differences after transmission. As a result, efficient frontiers for different CBTCC can be compared in an easy way, including the case where CBTCC are put equal to infinity or zero. The availability of infinite transmission capacity between each country results in an efficient frontier with the least risk for each amount of return, and by consequence a more flattened profile. The case where transmission capacity is set equal to zero gives the opposite result. This case should be avoided in practice because it has a negative effect on the integration of wind power in an electric power system. Each country has to balance its own wind-power variations with the technology available in its electric power system, which can come at a higher cost if a massive amount of wind power is installed in one country. Increasing transmission capacity from zero to a certain value will move the efficient frontier to the left and consequently lower risk. It can be concluded that cross-border transmissioncapacity plays an important role in reducing the risk of a variable wind-power profile. Increasing cross-border transmission-capacity increases the effect of diversification compared to the case with zero transmission capacities. However, in order to find the right transmission capacity, one has to balance on the one hand between risk and return (as defined in this paper), and on the other hand between the benefits and costs of the transmission capacity. Finally, in this paper, wind-power profiles of different locations within one country are considered, instead of aggregated windpower profiles per country, and transmission within countries is assumed to have no congestion.
“Balancing wind energy in the grid: an overall, techno-economic and coordinated approach”, granted by the Belgian State e “FOD Wetenschapsbeleid”. E. Delarue is a post-doctoral researcher of the Research Foundation e Flanders (FWO). Appendix A This section proves that when minimising the sum of the variances of the hourly wind-power differences after transmission per country in the case of infinite transmission capacity, the standard deviations of the hourly differences of each country are equal to each other and the correlation between the hourly differences is equal to one, considering b as given:
min
X
sðak Þ2 /ck; r˛K : sðak Þ ¼ sðar Þ and rkr ¼ 1
Writing the minimisation function, using the definition of s(ak)2:
The authors gratefully acknowledge partial financial support for the research reported in this paper by the SD/EN/02A project
12 v X au;k C B C B u¼1 C Bau;k @ v1 A 0
X
sðak Þ2 ¼
v X X
(A.2)
ðv 1Þ 1
k˛K u ¼ 1
k˛K
By definition, au,k can also be expressed as:
au;k ¼ bu;k Yu;k
(A.3)
with Yu,k equal to the hourly difference in net wind-power export in country k between hour u and u þ 1:
Yu;k ¼
X
ck;r;uþ1
r˛K
X
cr;k;uþ1
r˛K
X r˛K
ck;r;u þ
X
cr;k;u
(A.4)
r˛K
Eq. (A.4) can be understood from the fact that P P bu;k ¼ wuþ1;i wu;i , and comparing Eq. (15) with Eq. (A.3),18 i˛Ik
i˛Ik
As the sum of all net wind-power exports over all countries is equal to zero, from (A.4) it follows that:
cu˛1; .; v : Acknowledgement
(A.1)
k˛K
X
Yu;k ¼ 0
(A.5)
k˛K
18 Note that when calculating the hourly wind-power differences after transmission, the transmission in previous periods is taken into account.
2386
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
Furthermore, bu,k can be rewritten as a the difference between the average mu and the divergence from this average Dmu,k:
P bu;k ¼
bu;k
Dmu;k ¼ mu Dmu;k
k˛K
n
Regrouping and again applying Eqs. (A.5) and (A.7), and using PP PP xij ¼ xij , yields: i
j
j
i
¼ nm2u þ
(A.6)
2 P v m n u u¼1
For these divergences, the following holds:
X
Dmu;k ¼ 0 cu˛1; .; v
(A.7)
þ
Using Eq. (A.2) and substituting Eqs. (A.3) and (A.6) gives: v X
6 6 6mu Dmu;k Yu;k u¼1 6 4 X
sðak Þ2 ¼
X
7 7 7 5
v2
3 2
v P
P6 6 u¼1 6mu Dmu;k Yu;k k˛K 4
mu Dmu;k Yu;k 7 v1
2
mu Dmu;k Yu;k v1
Pv
u¼1
2mu Dmu;k 2mu Yu;k 2mu u ¼ 1
mu Dmu;k Yu;k
¼ nm2u þ
Dmu;k þ Yu;k 2 þ
k˛K
v P
2
P
u¼1
!
mu
v P u¼1 2
ðv 1Þ
! Yu;k
6 P6 6 6 k˛K 4
u¼1
!2
mu
þ
k˛K
Dmu;k þ Yu;k
#
cu˛1; .; v
v1
(A.9)
P Since n$½mu ð vu ¼ 1 mu =vÞ2 in Eq. (A.9) is variable independent (i.e. independent of the transmission), Eq. (A.9) is minimised by putting
Pv
u¼1
Dmu;k þ Yu;k
2
Dmu;k þ Yu;k
2 #
v1
Pv
u¼1
Dmu;k þ Yu;k
cu˛1; .; v and ck˛K
v1
(A.10)
v P u¼1
!2
Dmu;k
þ
This condition is satisfied if:
Dmu;k þ Yu;k ¼ cte ¼ dk cu˛1; .; v
(A.11)
Furthermore, because of Eqs. (A.5) and (A.7):
/
X
dk ¼ 0
(A.12)
k˛K
v P u¼1
ðv 1Þ2 !
Dmu;k þ Yu;k
2
!2 Yu;k
2
v P u¼1
2 !3 ! v P 7 6 Dmu;k mu 2 Yu;k 7 2mu 62mu 7 X6 u¼1 u¼1 u¼1 7 6 þ 7 6 v1 ðv 1Þ2 7 k˛K 6 5 4 v P
" X
equal to zero, or in other words:
7 5
With mu independent of k and by regrouping and taking into account Eqs. (A.5) and (A.7), this can be further written as:
v P
ðv 1Þ2
k˛K
32
þ 2Dmu;k Yu;k v1 v v 3 P P mu Dmu;k Yu;k mu Dmu;k Yu;k 7 7 þ 2Dmu;k u ¼ 1 þ 2Yu;k u ¼ 1 5 v1 v1
2
þ
v1
Dmu;k þ Yu;k ¼ v P
mu
#
u¼1
" X
cu˛1; .; v
v 2P
P6 6 2 2 þ 6u ¼ 1 4 6mu þ Dm2u;k þ Yu;k k˛K 4
¼
7 7 5
2
Pv
(A.8)
2
v P
!
mu
v P u¼1 2
!
Dmu;k
ðv 1Þ
v P u¼1
!
Dmu;k
v1
2mu
!3 Yu;k 7 7 7 u¼1 7 7 v1 7 5 v P
3 3 2 2 v v v v v v P P P P P P D m D D D m D 2 m 2 m m 2 m Y 2Y 2Y m 2Y Y u;k u u;k u;k u;k u;k 7 u u;k u;k u;k u;k 7 6 u;k X6 7 X6 7 6 u¼1 u¼1 u¼1 u¼1 u¼1 u¼1 þ 7þ 7 6 6 5 5 4 4 v1 v1 v1 v1 v1 v1 k˛K
k˛K
¼ n$ mu
Looking at the term on the right-hand side of Eq. (A.8) and making temporary abstraction of the summation over time-intervals u and the division by v 2, this term can be written out as follows:
v u¼1 mu
v1 ðv 1Þ2 # " P 2 P v v u¼1 Dmu;k þ u¼1 Yu;k X k˛K
"
v X
k˛K u¼1
k˛K
3
P
# " Pv Pv X 2 Dmu;k þ Yu;k u¼1 Dmu;k 2 Dmu;k þ Yu;k u¼1 Yu;k þ v1 v1
mu Dmu;k Yu;k 72 v1
Dmu;k þ Yu;k 2 þ
k˛K
k˛K
2
X
2nmu
k˛K
Y. Rombauts et al. / Renewable Energy 36 (2011) 2374e2387
Taking Eq. (A.11) into account, the variance of the hourly windpower differences after transmission in country k can be written as:
2
v P
6 6 u¼1 6mu Dmu;k Yu;k 4 v P sðak Þ2 ¼ u¼1 0
12
v1
7 7 5
rkr ¼
0
v2 v P
mu
12
B C B C u¼1 dk þ dk C Bmu @ A v1 ¼
v P u¼1
0
v P
v2 12
mu C B C B u¼1 C Bmu @ v1 A ¼
v P u¼1
v2
¼ sðmu Þ2 (A.13) Since Eq. (A.13) is independent of K, it holds that ck,r ˛ K: s(ak) ¼ s (ar). Furthermore, following Eqs. (A.3) and (A.11), the hourly differences after transmission become:
au;k ¼ bu;k Yu;k ¼ mu Dmu;k Yu;k ¼ mu dk
(A.16)
References
v P u¼1
Covðmu dk ; mu dr Þ sðak Þ2 ¼ 1 ¼ sðak Þsðar Þ sðak Þ2
ck˛K
v2
v P mu dk C B C B u¼1 C Bmu dk @ v1 A
¼
Calculating next the correlation between country k and r taking into account Eq. (A.15) gives:
3 2
mu Dmu;k Yu;k 7
2387
ck˛K
(A.14)
From Eqs. (A.14) and (A.13), the covariance can be calculated for country k and r:
Covðmu dk ; mu dr Þ ¼ Covðmu ; mu Þ ¼ sðmu Þ2 ¼ sðak Þ2
(A.15)
[1] EC The EU climate and energy package. Last consulted on 05/05/2010. Available from: http://ec.europa.eu/environment/climat/climate_action.htm. [2] Ecofys. Promotion and growth of renewable energy sources and systems. Available from: http://www.res-progress.eu/; 2008. [3] Ragwitz M, Miola A. Evidence from RD&D spending for renewable energy sources in the EU. Renewable Energy 2005;30(11):1635e47. [4] Breton SP, Moe G. Status, plans and technologies for offshore wind turbines in Europe and North America. Renewable Energy 2008;34(3):646e54. [5] Ringel M. Fostering the use of renewable energies in the European Union: the race between feed-in-tariffs and green certificates. Renewable Energy 2005;31(1):1e17. [6] Tradewind project. Integrating wind. Developing Europe’s power market for the large-scale integration of wind power. Available from: http://www.tradewind.eu/; 2009. [7] European Wind Integration Study (EWIS). Towards a successful integration of large scale wind power into European electricity grids. Available from: http:// www.wind-integration.eu/. [8] IEA. IEA Wind Task 25: design and operation of power systems with large amounts of wind power. Available from: http://www.ieawind.org/AnnexXXV. html; 2006. [9] EWEA. Large scale integration of wind energy in the European power supply: analysis, issues and recommendations. Available from: http://www.ewea.org/ ; 2005. [10] Markowitz H. Portfolio selection. The Journal of Finance 1952;7:77e91. [11] Awerbuch S. Portfolio-based electricity generation planning: policy implications for renewables and energy security. Mitigation and Adaptations Strategies for Global Change 2006;11:693e710. [12] Awerbuch S, Berger M. Applying portfolio theory to EU Electricity planning and policy making. IEA/EET Working Paper; 2003. [13] Bar-Lev D, Katz S. A portfolio approach to fossil fuel procurement in the electric utility industry. The Journal of Finance 1976;31:933e47. [14] Delarue E, De Jonghe C, Belmans R, D’haeseleer W. Applying portfolio theory to the electricity sector: energy versus power. Energy Economics 2011;33(1):12e23. [15] Roques F, Hiroux C, Saguan M. Optimal wind power deployment in Europe e a portfolio approach. Energy Policy 2010;38(7):3245e56. [16] Koninklijk Nederlands Meteorologisch Instituut. K. Hourly wind speed of seven locations in the Netherlands: Vlakte van de Raan, Huibertgat, Nieuwe Beerta, Goeree, Volkel, Twenthe, Lelystad. Last consulted on 14/12/2009. Available from: www.knmi.nl. [17] Vestas Last consulted on 12/12/2009. Available from: www.vestas.com.