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Energy 133 (2017) 913e928

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Energy

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Optimal heterogeneity in a simplified highly renewable Europeanelectricity system

Emil H. Eriksen a, Leon J. Schwenk-Nebbe a, Bo Tranberg b, c, Tom Brown d,Martin Greiner b, *

a Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, 8000, Aarhus C, Denmarkb Department of Engineering, Aarhus University, Inge Lehmanns Gade 10, 8000, Aarhus C, Denmarkc Danske Commodities A/S, Vaerkmestergade 3, 8000, Aarhus C, Denmarkd Frankfurt Institute for Advanced Studies (FIAS), Johann Wolfgang Goethe Universit€at, Ruth-Moufang-Straße 1, 60438, Frankfurt am Main, Germany

a r t i c l e i n f o

Article history:Received 7 February 2017Received in revised form22 May 2017Accepted 28 May 2017Available online 31 May 2017

Keywords:Large-scale integration of renewablesSystem designRenewable energy networksWind power generationSolar power generationLevelised system cost of electricityEurope

* Corresponding author.E-mail addresses: emher@au.dk (E.H. Eriksen)

(L.J. Schwenk-Nebbe), bo@eng.au.dk (B. Tranberg),(T. Brown), greiner@eng.au.dk (M. Greiner).

http://dx.doi.org/10.1016/j.energy.2017.05.1700360-5442/© 2017 Elsevier Ltd. All rights reserved.

a b s t r a c t

The resource quality and the temporal generation pattern of variable renewable energy sources varysignificantly across Europe. In this paper spatial distributions of renewable assets are explored whichexploit this heterogeneity to lower the total system costs for a high level of renewable electricity inEurope. Several intuitive heuristic algorithms, optimal portfolio theory and a local search algorithm areused to find optimal distributions of renewable generation capacities that minimise the total costs ofbackup, transmission and renewable capacity simultaneously. Using current cost projections, an optimalheterogeneous distribution favours onshore wind, particularly in countries bordering the North Sea,which results in average electricity costs that are up to 11% lower than for a homogeneous referencedistribution of renewables proportional to each country's mean load. The reduction becomes even larger,namely 18%, once the transmission capacities are put to zero in the homogeneous reference distribution.Heuristic algorithms to distribute renewable capacity based on each country's wind and solar capacityfactors are shown to provide a satisfactory approximation to fully optimised renewable distributions,while maintaining the benefits of transparency and comprehensibility. The sensitivities of the results tochanging costs of solar generation and gas supply as well as to the possible cross-sectoral usage ofunavoidable curtailment energy are also examined.

© 2017 Elsevier Ltd. All rights reserved.

1. Introduction

The ambitious renewable energy targets set by European gov-ernments [1] imply that the share of renewables in electricitygeneration will increase significantly in the years to come. At pre-sent, the leading renewable technologies are wind, solar photo-voltaics (PV) and hydroelectricity, of which only wind and solar PVhave the potential for large scale expansion. The uneven distribu-tion of wind and solar resources across the continent raises thequestion of how best to exploit these heterogeneous resources. Ifwind and solar generation capacities are concentrated in thosecountries with the best resources, this may increase demand for

, leon@schwenk-nebbe.combrown@fias.uni-frankfurt.de

transmission and increase energy imbalances between countries; ifwind and solar generation are distributed homogeneously, then thebest renewable resources will not be fully used and total systemcosts may be higher than the heterogeneous optimum. In this pa-per, the consequences of heterogeneity for the whole electricitysystem, including backup generation and transmission, will bequantified.

Since wind and solar PV are both Variable Renewable EnergySources (VRES), backup generation is needed if the electrical de-mand is to be met at all times. Backup generation introducesadditional system costs, which depend on the mismatch betweenVRES generation and load. Using the degrees of freedom associatedwith the choice of the capacity distributions of VRES for eachcountry, it is possible to smooth out the aggregated temporalgeneration pattern or even shape it towards the load pattern. As aresult, the mismatch and thus the backup requirements is lowered.To decrease the dimensionality of the problem, renewable assetscan be assigned homogeneously, proportional to the mean load of

Table 2Capacity factors CFWn and CFSn for onshore wind and solar PV for the Europeancountries, derived from the EuroStat data [31e33]. The countries are sorted by theirrespective mean load ⟨L⟩ (in units of GW) over the 2000e2007 time series.*: esti-mated values, see text for details.

⟨L⟩ CFWn CFSn ⟨L⟩ CFWn CFSn ⟨L⟩ CFWn CFSn

DE 54.2 0.18 0.12 FI 9.0 0.20 0.08 RS 3.9 0.20* 0.14*

FR 51.1 0.22 0.12 CZ 6.6 0.19 0.11 IE 3.2 0.27 0.09*

GB 38.5 0.29 0.09 AT 5.8 0.20 0.11 BA 3.1 0.22* 0.14*

IT 34.5 0.20 0.14 GR 5.8 0.21 0.17 SK 3.1 0.21* 0.12ES 24.3 0.26 0.21 RO 5.4 0.21 0.14 HR 1.6 0.25 0.14*

SE 16.6 0.25 0.09 BG 5.1 0.22 0.14 LT 1.5 0.25 0.12*

PL 15.2 0.23 0.12* PT 4.8 0.28 0.18 EE 1.5 0.20 0.10*

NO 13.7 0.29 0.09* CH 4.8 0.20* 0.11* SI 1.4 0.20* 0.13NL 11.5 0.23 0.09 HU 4.4 0.22 0.13* LV 0.7 0.23 0.11*

BE 9.5 0.27 0.11 DK 3.9 0.31 0.10 LU 0.7 0.16 0.10

E.H. Eriksen et al. / Energy 133 (2017) 913e928914

each country, with a uniform wind-to-solar mixing factor. Thisapproach is demonstrated in Refs. [2,3], where optimal wind-to-solar mixes for Europe are found that minimise balancing andstorage costs. Further reductions in backup requirements arepossible by extending the transmission network to enable moreenergy exchange between the countries [4,5]. The implications fortotal system costs of different homogeneous renewable penentra-tions, wind-solar mixes and transmission levels were considered inRef. [6], where the cost-optimal design was found to consist of arenewable energy penetration of 50% and a wind fraction of 94%.Other relevant research on the advantages of grid extensions for theintegration of renewables, including reduced variability andsmaller forecast errors, can be found in Refs. [7e13].

In this paper the consequences of moving from a homogeneousspatial distribution of VRES and a uniform wind-to-solar mixingfactor to a cost-optimal placement of VRES capacities aroundEurope are explored. The distribution of VRES plants is determinedby at least two considerations. The first consideration is thegeographical variation of the VRES quality. The resource quality isquantified through the capacity factor (CF) defined as

CF ¼ average generationrated capacity

: (1)

The capacity factor is a number between 0 and 1, where 0meansno generation and 1 means maximum generation at all times. Ca-pacity factors for the European countries for onshore wind andsolar PV are calculated using (1) and listed in Table 2. The secondconsideration is the geographical variation of the temporal gener-ation pattern for a given VRES type. This effect is particularlyimportant for wind since Europe is large compared to the correla-tion length of wind of z 600 km [14e16], and wind thereforebenefits from smoothing effects across the continent.

With these points in mind, the optimal heterogeneous spatiallayouts of wind and solar PV across Europe is investigated andcompared to the homogeneous layouts. The main point of com-parison is the average cost of electricity, which is composed of theVRES, backup and transmission costs. Different approaches to copewith the resulting large number of degrees of freedom areconsidered. In the literature a common approach for heterogeneoussystems is to use linear programming to optimise generation andtransmission capacities simultaneously [17e20], but this has the

Table 1Nomenclature.

Name Description

N Set of nodesn;m Node indexl Link indexDn Mismatch (VRES generation minus load)an Wind/solar mixgn Renewable penetration

GfW ;S;Bgn

Generation of wind, solar or backup

GRn Total renewable generation

Ln LoadPn Net power balance

K fW ;S;Bgn

Wind, solar or backup capacity

K Tl

Transmission capacity for link l

EB Backup energyCn CurtailmentBn Nodal balancingH PTDF matrixFl Power flow on link l

CFfW ;Sg Wind/solar capacity factor

⟨x⟩ Average value of xq QuantileK Heterogeneity parameter

drawback that only a selection of representative weather condi-tions can be considered before computation times become infea-sible. This makes the results susceptible to over-tuning to theweather selection. Other groups have used genetic algorithms tooptimise generation, storage and transmission over a full year inAustralia [21] and over three years in Europe, the Middle East andNorth Africa [22]. In this paper a novel local search algorithm wasfound to be most effective given the size and non-linear formula-tion of the optimisation problem, allowing 8 years of hourlyweather to be considered.

A downside of pure optimisation approaches is that one loses anunderstanding of why particular solutions are optimal. This makesit hard to justify investment strategies to policy makers and to thepublic. To counter this downside, more intuitive heuristic methodsare developed here to construct layouts based on knowledge ofresource quality, which are then compared to layouts obtainedthrough optimisation. Distributions proportional to capacity factors(similar to the approach in Ref. [11]) and distributions based onoptimal portfolio theory that reduce risk, or standard deviation, ofthe in-feed (similar to approaches in Refs. [23e26]) are consideredand compared.

This paper is organised as follows: Section 2 discusses thegeneral modelling of the simplified European electricity system andthe key infrastructure measures. Section 3 describes the construc-tion of heterogeneous layouts. In Section 4 the performance of thedifferent layouts and the resulting renewable penetrations for in-dividual European countries are discussed. Section 5 contains ananalysis of the sensitivity of the results to variations in componentcosts. We conclude the paper with a discussion on the results andan outlook on future research.

2. Methods I: general modelling

2.1. Renewable resource assessment

Realistic time series describing the country-specific wind andsolar PV power generation and the load are the starting point of theadvocated weather-driven modelling of a simplified networkedEuropean electricity system. The utilized data set has been releasedfrom the Fraunhofer Institute for Wind Energy and Energy SystemTechnology (formerly ISET, now IWES) [27]. This data set covers theeight-year period from January 2000 to December 2007, has atemporal resolution of one hour and a spatial resolution of 50� 50km2 over all of Europe. Fixed country-specific capacity layouts havebeen used to first convert the weather data into onshore wind andsolar power generation, and then to aggregate the latter over eachof the 30 European countries; off-shore wind power generation isnot considered. The country-specific load time series have been

E.H. Eriksen et al. / Energy 133 (2017) 913e928 915

obtained from publicly available sources, extrapolated to covermissing data, and detrended from an annual growth of around 2%to their year 2007 values. Formore specific details see Refs. [2,27]. Agood alternative description of the conversionmodelling is given inRefs. [28e30].

The obtained wind and solar PV power generation time serieshave been rescaled to the capacity factors (CFs) from 2014. Thelatter have been determined in accordance with equation (1) fromthe EuroStat data for the installed capacities and the total genera-tion for the year 2014 [31e33]. The resulting CFs for each countryand each technology are listed in Table 2. For some of the countries(particularly smaller countries) no data was available or thecalculated result was too uncertain because of too little or noinstalled capacity. For these countries the CFs are calculated as anaverage value from surrounding countries. These cases are markedby a star. Some countries with an already high installed capacityhave a relatively low capacity factor compared to validated resultsfrom Ref. [28]. For them the CFs have been raised by a small factor:8% in Germany, 4% for wind in Spain, 4% for solar in Italy and 2% forwind in Great Britain. The final capacity factors presented in Table 2are in accordancewith [29,30], which presents a critical assessmentof current and future national capacity factors. Capacity factors forwind are likely to rise further in the future because of re-poweringof wind turbines with more efficient, modern turbines at higherhub heights [34].

2.2. The electricity network

The European electricity network is modelled as a simplified 30-nodemodel, where each node represents a country. For each node nthe generation from VRES (see Table 1 for a summary ofnomenclature),

GRnðtÞ ¼ GW

n ðtÞ þ GSnðtÞ; (2)

can be expressed through two parameters. The penetration g de-termines the amount of renewable energy generated relative to themean load of the node,

DGRn

E¼ gn

DLnE; (3)

while the mixing parameter a fixes the wind-to-solar ratio,

DGWn

E¼ an

DGRn

E; (4)

DGSn

E¼ ð1� anÞ

DGRn

E: (5)

Other forms of renewable power generation are neglected inthis simplistic modelling approach.

The nodal difference between VRES generation and load

DnðtÞ ¼ GRnðtÞ � LnðtÞ (6)

is called themismatch. To avoid power outages, the demandmust bemet at all times. Since storage is not considered, any power deficitsmust be covered by backup generation. Dispatchable resources arenot modelled explicitly, but are considered as part of the backupgeneration. IfDnðtÞ � 0, excess energy CnðtÞmust be curtailed,whileif DnðtÞ<0 backup generation GB

nðtÞ is needed. Together the twoterms form the nodal balancing BnðtÞ ¼ CnðtÞ � GB

nðtÞ. It is possibleto lower the balancing needs with transmission. Nodes with excessgeneration export energy EnðtÞ, allowing nodes with an energydeficit to import energy InðtÞ to (partly) cover their energy deficit.

The nodal injection, EnðtÞ � InðtÞ, is denoted PnðtÞ. This leads to thenodal balancing equation,

GRnðtÞ � LnðtÞ ¼ BnðtÞ þ PnðtÞ ; (7)

The vector of nodal injections is called the injection pattern, andfullfills

PnPnðtÞ ¼ 0. The actual imports and exports, and thus the

injection pattern, depend on the dispatch of the nodal balancing.The synchronised balancing scheme,

BnðtÞ ¼ hLniPkhLki

Xm

DmðtÞ ; (8)

where all nodes are curtailing/generating backup synchronously(relative to hLni), fulfills two top priorities: it minimises the totalbackup generation for each time step and it minimises the overallbackup capacity [35]. This stylised synchronised balancing schemehas also been chosen in view of the layout optimisation, since thecomputational time for an update step is much smaller than forother dispatch schemes, like for example the localised flow schemeused in two previous publications [4,5].

The injection pattern is fixed by Eqs. (7) and (8), and determinesthe power flows on the links l:

FlðtÞ ¼Xn

HlnPnðtÞ : (9)

The linear relationship follows from the DC approximation,which is known to be a good approximation for high-voltage flows.For the Power Transfer Distribution Factors Hln we have assumedunit susceptances [35], allowing its construction from the Moore-Penrose pseudo inverse of the underlying network Laplacian.

2.3. Infrastructure measures

Following [6], the energy system cost is calculated based on afew keymeasures. Besides the cost of the VRES capacities,K W andK S , costs for the backup system and the transmission network areincluded. The backup system cost is split into two components, thecost of backup capacity K B and the cost of backup energy EB. Thebackup capacity cost covers expenses related to construction and tokeeping the power plants online while the backup energy costaccounts for actual fuel costs. Expressed in units of the averageannual load, the backup energy is given by

EB ¼P

nP

tGBnðtÞP

mP

tLmðtÞ¼

Pn

DGBn

EP

mhLmi: (10)

In principle, the backup capacity is fixed by a single extremeevent. However with this definition, the results will be highlycoupled to the particular data set used. To decrease the coupling,the 99% quantile is used rather than the maximum value,

qn ¼ZKB

n

0

pn�GBn

�dGB

n ; (11)

where pnðGBnÞ is the time sampled distribution of backup generation

and qn ¼ 0:99. With this choice, the backup system will be able tofully cover the demand 99% of the time. The remaining 1% isassumed to be covered by unmodelled balancing initiatives, e.g.demand side management. Given the nodal values K B

n, the overallbackup capacity

E.H. Eriksen et al. / Energy 133 (2017) 913e928916

K B ¼Xn

K Bn (12)

is calculated by summation.In analogy, the transmission capacity K T

l is defined so that theflow is met 99% of the time. Transmission can be positive andnegative, but since links are assumed bidirectional, only themagnitude (not the sign) of the flow is to be considered. Hence

ql ¼ZKT

l

0

plðjFljÞ djFlj ; (13)

where plðjFljÞ is the time sampled distribution of absolute flows andql ¼ 0:99. Since the link length varies, K T is not calculated directlyby summation, but instead as a weighted sum,

K T ¼Xl

K Tl dl; (14)

where dl denotes the length of link l. Link lengths are estimated asthe distance between the country capitals.

In this paper EB will be expressed in units of average annualload, K B in units of average hourly load and K T in units of averagehourly load � megametre.

2.4. Cost modelling

Cost assumptions for the elements of an electricity system varygreatly across the literature. In this study, the cost assumptionspublished by Ref. [6] have been adapted with a single modification.The cost of solar has been reduced by 50% in accordance with nearfuture solar PV panel price projections [36]. The resulting estimatesare listed in Table 3. In general, the cost assumptions are in the lowend for VRES which reflects the expectation that the cost of VRESwill go down in the future as the penetration increases. Backupgeneration is priced based on the cost of Combined Cycle GasTurbines (CCGTs).

From the VRES penetration, the mixing factor and the meanload, the mean generation of each node can be calculated. Dividingby the associated capacity factor, the capacity is obtained. Exceptfor transmission capacity, the present value of each element can becalculated directly as

V ¼ CapExþXTlifet¼1

OpExtð1þ rÞt ; (15)

where r is the rate of return assumed to be 4% per year. Thetransmission capacity cannot be translated directly into cost as thecost depends on the length and the type of the link. Link costs areassumed to be 400V per km perMW for AC links and 1500V per kmper MW for HVDC links. For HVDC links, an additional cost of150,000V per MW per converter station pair (one at each end) isadded [10,11,37]. The layout of AC and HVDC lines has been con-structed by Ref. [4] according to the existing European networkreported by ENTSO-E for the year 2011 [38] and new predicted lines

Table 3Cost assumptions for different assets separated into capital expenditures (CapEx) and fix

Asset CapEx [V/W] OpExfixed [V/kW

CCGT 0.90 4.5Solar PV 0.75 8.5Onshore wind 1.00 15.0

until 2014 [39,40]. It is shown in Fig. 10.To allow for comparison of different system layouts, the Lev-

elised Cost of Electricity (LCOE) is a convenient measure [6,41,42].The LCOE is the cost that every generated unit of energy consumedduring the lifetime of the project has to match the present value ofinvestment [43],

LCOEV ¼ VPTlifet¼1

LEU;tð1þrÞt

: (16)

Since the life time of the system elements differs, the LCOE isevaluated separately for each system element from each respectivepresent value. The LCOE for the complete system is calculated bysummation. Life times of 25 years for solar PV and onshorewind, 30years for CCGT plants and 40 years for transmission infrastructurewere assumed.

3. Methods II: heterogeneous layouts

The simplest way to distribute the renewable resources is toassign them homogeneously (relative to themean load of the node)so that gn ¼ gEU ¼ 1 and an ¼ aEU . This homogenous layout isdenoted as HOM. However this assignmentmight not be ideal sincethe capacity factors vary significantly between the nodes. Threeheuristic schemes and a straightforward optimisation for the con-struction of heterogeneous layouts will be presented in thefollowing four subsections. The naming of the distribution algo-rithms is summarised in Table 4.

3.1. Heuristic layout I: CF proportional (CFprop)

An intuitive first approach, called CFprop, is to assign resourcesproportional to the CF, or more general to the CF raised to anexponent b. For a wind-only layout, the nodal renewable penetra-tions gn are given by

gWn ¼�CFWn

�b�LEU

�P

m

�CFWm

�b�Lm

�gEU ; (17)

where gEU is the overall penetration assumed to be 1. An equivalentexpression for the solar-only layout is obtained by the substitutionW/S. Examples for b ¼ 1 are shown in Fig. 1a for the wind- andsolar-only layouts. In the layout illustrations, each bar represents acountry.

CFprop layouts for any value of a can be constructed as a linearcombination of the wind and solar only layouts with

gn ¼ aEUgWn þ ð1� aEUÞgSn (18)

and

an ¼ aEUgWn

aEUgWn þ ð1� aEUÞgSn: (19)

For practical reasons, it is not possible to realise extremely

ed/variable operational expenditures (OpEx) together with their expected life times.

/y] OpExvar [V/MWh] Life time [years]

56.0 300.0 250.0 25

Table 4Summary of the algorithms for distributing VRES.

Name Brief description

HOM Homogeneous distribution proportional to the mean of each country's loadCFprop Distribution proportional to a power ðCFÞb of the capacity factor CFCFmax Assignment to each country gn extremised within 1

K � gn � K depending on CFOPT Distribution using Optimal Portfolio TheoryGAS Distribution optimised using Greedy Axial Search algorithmGAS* As GAS, but with optimally constrained transmissionGASnoT As GAS but with no transmission between countries, so that each country is self-sufficient at all times

Fig. 1. Examples of heuristic (blue) wind-only and (yellow) solar-only layouts: (a) CFprop with b ¼ 1, (b) CFmax constrained to K ¼ 2, and (c) Pareto optimal OPT layouts obtainedwith K ¼ 2. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

E.H. Eriksen et al. / Energy 133 (2017) 913e928 917

heterogeneous layouts. On the one hand the geographical poten-tials for VRES installations in countries with good renewable re-sources may be a limiting factor. On the other hand countries withpoor renewable resources may not want to become too dependenton imports. To constrain heterogeneity, the heterogeneity param-eter K is introduced by requiring

1K� gn � K: (20)

With this definition, K¼ 1 corresponds to a homogeneous layoutwhile K ¼ ∞ represents unconstrained heterogeneity. For theCFprop layouts, each value of K translates into an a-dependent value

E.H. Eriksen et al. / Energy 133 (2017) 913e928918

of b. For a given value of a, the corresponding b value is found byincreasing b until the first country violates equation (20). At themixa ¼ 0:86 the values K ¼ 1;2;3 correspond to b ¼ 0:00;1:92;2:91,respectively.

3.2. Heuristic layout II: extreme K-constrained (CFmax)

Although the overall capacity factor of a CFprop layout for b >0 is higher than the capacity factor of the homogeneous layout, it ispossible to achieve an even higher capacity factor without violatingthe constraints in equation (20). In the wind- and solar-only cases,the capacity factor is maximised by assigninggn ¼ K to the countrieswith the highest capacity factor and gn ¼ 1

K to the remainingcountries, except for a single in-between country which is fixed bythe constraintXn

gnhLni ¼ hLEUi: (21)

The wind- and solar-only cases of the CFmax layout constrainedby K ¼ 2 are shown in Fig. 1b. Similar to the CFprop layouts, theCFmax layouts for arbitrary aEU values can be constructed as linearcombinations (18) of the wind- and solar-only layouts.

Fig. 2. Scatter clouds for (blue) wind-only and (yellow) solar-only capacity layouts. Thediagram plots the overall capacity factor (22) vs. the standard deviation of the overallmismatch (24). The distribution (25) and the constraint (20) with K ¼ 2 have beenused for the Monte Carlo simulations. The white point in the upper left cloud cornersindicate Pareto optimal layouts; see also Fig. 1c. The line connecting the wind- andsolar-only Pareto optimal layouts results from the interpolation between these layoutsin (18). The black point marked on this line represents the OPT layout with minimumLCOE. For comparison, the three triangle points mark the (orange) optimal CFprop,(green) optimal CFmax and (blue) optimised GAS layouts for K ¼ 2. (For interpretationof the references to colour in this figure legend, the reader is referred to the webversion of this article.)

3.3. Heuristic layout III: OPT

The optimal portfolio theory (OPT) is well known in mathe-matical finance [44]. It discusses different assets obtained from thetradeoff between maximizing their return and minimising theirrisk. This concept has also been applied to find optimal deploymentof wind and solar energy resources in large-scale energy systems[24e26], where the overall capacity factor has been treated as thereturn and the variance of the renewable power generation as therisk. In modified form, we will use OPT to further explore VREScapacity layouts over Europe with low system cost of electricity.

The overall capacity factor of awind-only (gWn ) or solar-only (gSn)

layout is defined as

CFW=SEU ¼ hLEUi

K W=SEU

; (22)

where

K W=SEU ¼

Xn

gW=Sn

DLnE

CFW=Sn

(23)

represents the overall installed capacity. The overall capacity factoris a useful measure of return as this is of high importance for in-vestors of renewable generation capacity. Investors seek to mini-mise the overall capacity investment, which corresponds tomaximizing the overall capacity factor.

OPT's second measure is risk, for which we select the relativestandard deviation sD=hLEUi of the overall mismatch

DEUðtÞ ¼Xn

DnðtÞ (24)

based on the country-specific mismatches (6). The smaller the riskystandard deviation becomes the more likely is the reduced need fora backup infrastructure, which an investor tries to minimise [23].

A possible heterogeneous wind- or solar-only capacity layout issampled from a Monte Carlo procedure. The country-specificrenewable penetrations gn are randomly and independently drawnfrom a Beta distribution

pðgÞ ¼ Gðb1 þ b2ÞGðb1ÞGðb2Þ

�K

K2 � 1

�b1þb2�1

�g� 1

K

�b1�1ðK � gÞb2�1

(25)

defined on the compact support (20). GðbÞ is the Gamma function.The two shape parameters b1 and b2 are determined by requiringhgni ¼ 1 and by envoking the maximum entropy principle [45] tomaximal smear out the Beta distribution over the interval (20). ForK ¼ 2 the two parameters result in b1 ¼ 0:80, b2 ¼ 1:61, and forK ¼ 3 they are b1 ¼ 0:86, b2 ¼ 2:57. A capacity layout sampledwiththis procedure does not necessarily meet the requirement (21). Forsuch cases, all gn are uniformly rescaled upwards or downwardsuntil the requirement is fulfilled. During the rescaling some of thepenetration parameters hit the K constraint (20), and are thenfrozen for the remainder of the rescaling procedure.

The wind-only and solar-only portfolios for K ¼ 2 are shown inFig. 2 in blue and yellow, respectively, with the overall mismatchmeasure on the first axis and the overall capacity factor measure onthe second axis. Each of the portfolios consists of 100000 layouts.Due to the elongated shape of the portfolios there is no clearextended Pareto front in the upper left corners. The Pareto frontdefines a line, for which at the same time the standard deviation ofthe overall mismatch (risk) can not be reduced further for a fixedoverall capacity factor and the overall capacity factor (return) cannot be increased further for a fixed standard deviation of the overallmismatch. For both portfolios we identify a single point to char-acterise minimum risk and maximum return. This is done byextracting a subset of the points which are simultaneously a part ofthe top 200 capacity factors and bottom 200 standard deviations.For K ¼ 2 this leaves a sample of 28 layouts for wind and 67 layoutsfor solar to average and to calculate the respective new overallcapacity factor and new overall standard deviation. The resultingpoints are plotted in white on top of the portfolios. The layouts ofthese two Pareto optimal points are shown in Fig. 1c.

In order to find an optimal combined layout, we interpolatebetween the Pareto-optimal wind-only and solar-only layouts ac-cording to (18) and (19). This interpolation conserves the constraint

E.H. Eriksen et al. / Energy 133 (2017) 913e928 919

(21) and results in the line shown in Fig. 2. Apparently some of theinterpolated layouts are able to reduce the standard deviation ofthe global mismatch further. The interpolated layoutmarked with ablack dot comes with the mixing parameter aEU ¼ 0:87.

3.4. Optimised layouts

The full optimisation of the layouts is considered, with theobjective to minimise the LCOE with respect to the 60 variablesg1;…;gN;a1;…;aN for the N ¼ 30 countries. Given the highdimensionality of the search space, a number of optimisation al-gorithms were tested including the Nelder-Mead method [46],simulated annealing [47], genetic algorithms [48] and cuckoosearch [49]. It was found that the continuous enforcement of thenormalisation criterion (21) generally decreased the performanceof the tested algorithms, and for that reason a newhybrid algorithmwas developed to address this problem. While being a classicalgreedy algorithm in the sense that the locally optimal choice isalways taken, the renormalisation problem was circumvented bymoving only along the axial directions. The algorithm has beendenoted Greedy Axial Search (GAS).

When a solution is renormalised, all g values are scaled eitherup or down. Therefore, it is possible that some g values end upoutside the boundary (20). The g values are fixed at the boundaryand the rescaling is only applied to the remaining free g values. Ingeneral this approach is problematic since it can change the di-rection of the search. This is circumvented by holding the specific gvalue constant that is considered during the step up/down pro-cedure along a given axis. In this way only some g values are scaleddown/up and the feasibility of moving up/down along the consid-ered axis can be determined. This is the underlying principle ofGreedy Axial Search (GAS).

As any greedy algorithm, the GAS algorithmworks by taking thelocally optimal choice. Hence the feasibility for each direction isevaluated, but only the best choice is accepted. This process isrepeated until a convergence criterion is fulfilled. At this point thestep size is reduced and the iterative optimisation procedurerepeated until the step size drops below some tolerance. The algo-rithmstructure is sketched inAlgorithm1. The StepUp and StepDownsubroutines generate new solutions by stepping a solution (firstargument) up/down along axis i (second argument) with some stepsize (third argument) after which the solution is renormalised asdescribed above. Values of maxStepSize ¼ 1, minStepSize ¼ 5,10�4

and tolerance ¼ 10�4 were found to be appropriate.All optimised layouts have been obtained using the GAS routine.

These layouts will be denoted GAS layouts. Constraining the

transmission and thereby reducing the transmission capacity canlead to an overall lower LCOE. This is discussed in Section 4.4. Thelayouts resulting from this additional optimisation will be denotedGAS* layouts.

4. Results

The optimal heuristic layouts CFprop, CFmax, OPT as well as theoptimised layouts GAS will be discussed in the next three sub-sections, first for K ¼ 1, then for K ¼ 2, and finally for K ¼ 3. Thefourth subsection focuses on the transmission capacities.

4.1. K ¼ 1 layouts

By construction, the layouts CFprop, CFmax and OPT becomeidentical and homogeneous for K ¼ 1. Due to Eq. (20), theirrespective renewable penetrations are gn ¼ 1. Moreover, accordingto Eq. (19) their renewable mixes an ¼ aEU also turn out to be in-dependent of the country index. For these strictly homogeneouslayouts Fig. 3 shows the dependence of the key infrastructuremeasures on aEU as the blue curves. For the backup energy andbackup capacity, the optimal mixing parameters are located aroundaEU ¼ 0:85, which is slightly larger than the values found byRefs. [2,3]. For the transmission capacity, the minimum occursaround aEU ¼ 0:45. The main measure of interest, the LCOE, has aminimum at aEU ¼ 0:90. The high cost at aEU ¼ 0 is caused by acombination of high backup energy/capacity costs and the fact thatthe CF of solar is generally lower than for onshore wind. The cost ofproducing one unit of energy is thus higher for solar than foronshore wind even though the specific CapEx is lower for solar.

The homogeneous layout producing the minimum LCOE ataEU ¼ 0:90 is denoted as the ‘HOM’ layout. It is illustrated in Fig. 4a.Its total LCOE amounts to 59.7 V/MWh. The componentwise LCOEcorresponding to thewind, solar, backup and transmission parts arelisted in the third column of Table 5 and graphed as the second barin Fig. 5. Wind power dominates the overall LCOE. Its contributionamounts to 61%, and is followed by 21% from backup,10% from solarand 8% from transmission.

Contrary to the HOM layout, the K ¼ 1 GAS layout is no longerstrictly homogeneous. Of course, all renewable penetrations arestill equal to gn ¼ 1, but as a result of the optimisation the wind-solar mixing parameters become heterogeneous. This is illus-trated in Fig. 4b. Two-thirds of the countries are wind-only withan ¼ 1. The remaining countries have a significant share of solar.For some of those this was to be expected. Spain, Greece, Italy,Romania and Serbia have very large solar capacity factors. See againTable 2. However, other solar-rich countries, like Portugal, Bulgaria,Bosnia and Croatia, are not amongst them. Instead, Germany is alsoassigned a significant share of solar, although its solar capacityfactor is only average. By taking a closer inspection of Table 2 wediscover the following empirical finding for the K ¼ 1 GAS layout:all countries with an ¼ 1 come with a ratio between their solar andwind capacity factor which is smaller than CFSn=CF

Wn <0:65. The

countries with an <1 have a larger ratio CFSn=CFWn � 0:65, except for

the three smallest countries Estonia, Latvia and Luxembourg.Compared to the HOM layout, the a-heterogeneity of the K ¼ 1

GAS layout is able to reduce the total LCOE by 3%. This is mostly aconsequence of the reduced combined component costs for windand solar power. Note, that the overall mixing parameteraEU ¼ P

nanhLni=hLEUi has also slightly reduced from 0.90 (HOM) to

0.84 (GAS). See the fourth column of Table 5 and the third bar ofFig. 5. The costs for backup and transmission have not changedmuch; which is also apparent from the rightmost panel of Fig. 3.

All K ¼ 1 layouts discussed so far include the transmissioninfrastructure. It is also interesting to compare them to an

Fig. 3. Overview of the infrastructure measures: (a) the backup energy EB (in units of average annual European load), (b) the backup capacity K B (in units of average hourlyEuropean load), (c) the transmission capacity K T (in units of average hourly European load times megametre) and (d) the associated LCOE as a function of aEU . The CFprop andCFmax layouts are shown as solid and dashed lines respectively. The dependence of the OPT layouts on aEU is not shown; only the interpolations leading to a LCOE minimum areplotted as asterisks. The GAS layouts are plotted as dots. The blue diamond represents the GASnoT layout. Different constraints are shown: K ¼ 1 (blue), 2 (yellow) and 3 (green). (Forinterpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 4. Comparison of (a) the optimal homogeneous layout HOM with the optimised (b) GAS and (c) GASnoT layouts constrained by K ¼ 1.

E.H. Eriksen et al. / Energy 133 (2017) 913e928920

optimised layout without transmission. No exports and importswould then be possible and the injection pattern PnðtÞ would

always be zero. No transmission investment would be needed andthe respective componentwise LCOE would be zero. However, the

Table 5Componentwise LCOE for the optimal CFprop, optimal CFmax, optimal OPT, optimised GAS and optimised GAS* layouts for K ¼ 1 (left), 2 (middle) and 3 (right). Note that theK ¼ 1 layouts CFprop, CFmax and OPT are identical and denoted as HOM. The K ¼ 1 layout GASnoT without transmission is listed as reference. All costs are given in V/MWh.

K ¼ 1 K ¼ 2 K ¼ 3

GASnoT HOM GAS GAS* CFprop CFmax OPT GAS GAS� CFprop CFmax OPT GAS GAS*

aEU 0.86 0.90 0.84 0.84 0.86 0.87 0.87 0.83 0.83 0.85 0.86 0.87 0.82 0.82

LCOE(K W ) 35.0 36.4 33.4 33.4 33.1 31.9 33.6 30.7 30.7 31.9 30.0 32.5 29.1 29.2

LCOE(K S) 7.8 5.8 7.3 7.4 7.1 6.6 6.7 6.7 6.7 7.0 6.5 6.7 6.7 6.7

LCOE(K B) 6.8 4.3 4.4 4.5 4.2 4.2 4.2 4.2 4.3 4.2 4.4 4.1 4.2 4.3

LCOE(EB) 14.9 8.3 8.0 8.8 7.7 7.7 7.6 7.5 8.4 7.6 8.0 7.4 7.5 8.2

LCOE(K T ) 0.0 4.9 4.7 2.6 5.3 6.8 5.9 6.2 3.7 5.9 8.0 6.5 7.1 4.6

LCOE(total) 64.5 59.7 57.8 56.6 57.4 57.2 57.9 55.3 53.8 56.6 56.8 57.4 54.5 53.0

E.H. Eriksen et al. / Energy 133 (2017) 913e928 921

countries then have to balance their mismatches all by themselves,and this in turn requires more backup infrastructure with higherrespective componentwise LCOE. For the GAS layout without thetransmission infrastructure, which for clarity we denote as GASnoT,the total LCOE turns out to be 64.5 V/MWh. Compared to the HOMlayout, the combined LCOE components for wind and solar powergeneration are almost the same, but the increase of the LCOEcomponents for the backup power generation and capacity issignificantly larger than the disappearance of the transmissioncomponent. See again Fig. 3, Table 5 and Fig. 5. The total LCOE of theGASnoT layout is 8% and 11.5% larger than for the HOM and GASlayout respectively. This clearly demonstrates the benefit of trans-mission [4,6].

The GAS and GASnoT layouts are obtained from two indepen-dent optimisation efforts. This explains why the two layouts areactually quite different in the distribution of the wind and solarresources. Fig. 4c illustrates the resulting wind-solar mixing pa-rameters for the GASnoT layout. Contrary to the more extreme GASlayout, themajority of the countries comeswith amix below an ¼ 1and well above 0. Only the most northern countries turn out to bewind-only. However, on average the mixing parameter aEU ¼ 0:86for the GASnoT layout is again close to aEU ¼ 0:84 for the GASlayout.

4.2. K ¼ 2 layouts

More heterogeneity is introduced once K is chosen to be larger

Fig. 5. Componentwise LCOE for the optimal CFprop, CFmax, OPT, GAS and GAS*layouts for K ¼ 1 (left), 2 (middle) and 3 (right). The K ¼ 1 layout GASnoT withouttransmission is shown as reference.

than one. Fig. 6aec illustrate the optimal heuristic CFprop, CFmaxand OPT layouts for K ¼ 2. Their respective aEU values are0.86e0.87 (see Table 5), and have been fixed by minimising theLCOE (see Fig. 3d). The general aEU-dependence of the otherinfrastructure measures are illustrated in Fig. 3aec. The backupenergies required for the three layouts are quasi identical, and nodifference is seen to the K ¼ 1 HOM layout. Also the backup ca-pacities are almost identical for the three layouts, and are slightlyless than for the K ¼ 1 HOM layout. Differences are observed for thetransmission capacities. The CFprop layout comes with the smallesttransmission capacities, followed by the OPT layout. The CFmaxlayout has the largest transmission capacities because its hetero-geneity is the largest. All K ¼ 2 layouts are found to have largertransmission capacities than the respective K ¼ 1 layouts.

The total LCOE of the three heuristic K ¼ 2 layouts are inbetw-een 57.2e57.9V/MWh. See columns 6e8 in Table 5 and bars 5e7 inFig. 5. This is very close to the value 57.8 V/MWh found for the K ¼1 GAS layout. In this respect, the larger heterogeneity of the K ¼ 2layouts do not represent a clear cost advantage when compared tothe K ¼ 1 GAS layout, which is homogeneous in the renewablepenetration parameters gn. The situation changes once the opti-mised K ¼ 2 GAS layout is considered, which is exemplified inFig. 6d. It exploits the wind resources over Europe in a more effi-cient way and reduces the wind component in the LCOE; consultcolumn 9 of Table 5 and bar 8 in Fig. 5. This reduces the total LCOEto 55.3 V/MWh.

The overall renewable penetration of the K ¼ 2 GAS layout isgEU ¼ 1; consult again Equation (21). However, the individualrenewable penetration parameters now scatter within 0:5 � gn � 2.As can be seen in Fig. 6d, their distribution is extremely heteroge-nous. For half of the countries they are either gn ¼ 2 or gn ¼ 0:5, andfor the other countries just somewhere in-between. A more carefullinspection reveals an approximate heuristic law, which expressesthe renewable penetration parameters

gn ¼

8>>>>>>><>>>>>>>:

1=K�CFeffn � CF1

��K � 1

K

�CFeffn � CF1CF2 � CF1

þ 1K

�CF1 � CFeffn � CF2

�

K�CFeffn � CF2

�(26)

as a continuous and piece-wise linear function of an effective ca-pacity factor

CFeffn ¼ aCFWn þ ð1� aÞCFSn : (27)

A least-square fit is shown in Fig. 7.The overall mixing parameter aEU ¼ 0:83 of the K ¼ 2 GAS

layout is almost the same as for the K ¼ 1 GAS layout. Both layouts

Fig. 6. Comparison of different layouts constrained by K ¼ 2: (a) CFprop, (b) CFmax, (c) OPT and (d) GAS.

E.H. Eriksen et al. / Energy 133 (2017) 913e928922

also have in common that 20 out of the 30 countries come withan ¼ 1. The five largest of the an <1 countries with a non-zero solarcomponent are also identical.

It is worth to take again a quick look at Fig. 2. It shows that forK ¼ 2 the optimal CFprop, CFmax and OPT layouts have more orless the same close-to-minimum standard deviation of the overall

mismatch (24) as the optimised GAS layout. This indicates that aminimised mismatch standard deviation serves as a good measureto determine an optimal infrastructure [23]. However, it is still arough measure, since it does not allow to finetune the minimum-cost infrastructure.

Fig. 7. Renewable penetration parameters gn from the K ¼ 2 GAS layout as a function of the effective capacity factor CFeffn defined in Equation (27). The continuous and piecewiselinear green function represents the heuristic law (26) with least-square-fitted parameters a ¼ 0:596, CF1 ¼ 0:173 and CF2 ¼ 0:211. (For interpretation of the references to colour inthis figure legend, the reader is referred to the web version of this article.)

E.H. Eriksen et al. / Energy 133 (2017) 913e928 923

4.3. K ¼ 3 layouts

For K ¼ 3 the GAS algorithm has more freedom to optimise theheterogeneous layout and to reduce the overall LCOE, see (20). Theresulting layout is depicted in Fig. 8. It has some similarity to theK ¼ 2 GAS layout, but of course the K ¼ 3 GAS layout is even moreextreme. Its overall wind-solar mixing parameter aEU ¼ 0:82 isalmost the same as for the K ¼ 2 counterpart. The overall costreduction turns out to be small. As can be seen in Table 5, the totalLCOE for the K ¼ 2 and K ¼ 3 GAS layouts are 55.3 and 54.5V/MWh,respectively. This small cost reduction is mainly caused by theopportunity to allocatemorewind resources to the sites with a veryhigh capacity factor, and it is weakened to some extend by slightlyincreased costs for the transmission component; compare column14 with column 9 in Table 5.

Bulk results for the optimal heuristic K ¼ 3 layouts CFprop,CFmax and OPT are also listed in Table 5 and Fig. 5. Their layouts arealso found to be wind-dominated, with nearly the same aEU valuesas for the respective GAS layout. The LCOE for these three heuristiclayouts are larger than for the K ¼ 3 GAS layout. This of course wasto be expected. However, their LCOE also turn out to be slightlylarger than for the less heterogeneous K ¼ 2 GAS layout.

Another reason that the GAS optimisation might have been

Fig. 8. GAS layout con

better than the heuristic layouts is that the GAS algorithm sees notjust the capacity factors at each site, like the heuristic layouts, butalso the geographical variation of the temporal generation pattern,which the GAS algorithm can exploit to shape the VRES generationpattern towards the load. However if this was the reason, thebackup generation costs would have decreased from the heuristicto the GAS layout, which they do not. This suggests that the GASoptimisation's success really lies with the free exploitation of ca-pacity factors.

4.4. Transmission capacities

So far, only the total contribution of the transmission capacitiesto the overall LCOE have been discussed for various system layoutsin Table 5 and Fig. 5. Its geographic distribution has not yet beenspecified. This will be done in this Subsection, but not right away. Atfirst we will investigate a procedure which further reduces theoverall LCOE by reducing the transmission capacities to someextend.

The transmission capacities defined in Equation (13) have beenderived from unconstrained power flows. They are determined bythe most extreme flow events, which typically occur betweencountries with a large energy deficit and others with a large excess.

strained by K ¼ 3.

E.H. Eriksen et al. / Energy 133 (2017) 913e928924

These events are not expected to overlapwith other extreme eventswhen all countries face a large energy deficit. The latter determinethe required backup capacities. Consequently, it can be expectedthat a modest reduction of the total transmission capacities willnot, or at least not much, affect the total backup capacities and thetotal backup energy, and will lower the overall LCOE.

The synchronised balancing scheme (8) presented in Section 2.2is based on unconstrained power flows. In order to include con-strained power flows, a generalisation is needed:

minB

Pn

ðBnðtÞÞ2hLni

s:t:PnPnðtÞ ¼ 0

s:t: �K conTl � FlðtÞ ¼

Xn

HlnPnðtÞ � K conTl :

(28)

The objective is to minimise the expression in the first line,taking into account the two constraints of the second and third line.K conT

l denotes the constrained transmission capacity of line l. Inthe limit of unconstrained flows, where the second constraint canbe discarded, the objective (28) can be rewritten asminB

PnðB2nðtÞ=hLni � lPnÞ with the method of Lagrange multipliers

and leads to the solution (8). For the following we will downscalethe unconstrained transmission capacities from (13) by a uniformscaling parameter z to obtain the constrained transmissioncapacities

K conTl ¼ zK T

l : (29)

Fig. 9 illustrates the dependence of the LCOE on the trans-mission constraints by taking the unconstrained transmission ca-pacities of the K ¼ 2 GAS layout and scaling them down by theuniform factor z. At first, as z decreases, the LCOE also decreases. Aminimum is found at z ¼ 0:60. For the K ¼ 1 and K ¼ 3 GAS layoutsthe minimum is found at the optimal values z ¼ 0:55 and 0.65,respectively. If the transmission capacities are downscaled furtherthe LCOE starts to increase again due to increasing requirements forbackup energy and backup capacity.

Table 5 lists also the modified GAS layouts resulting from theoptimal scaling parameters. For clarity, we denote them as GAS*

layouts. Compared to theGAS layouts, the transmission contribution

Fig. 9. Non-VRES components of the LCOE as a function of the scaling parameter z. Thedashed line indicates the minimum leading to the lowest LCOE. The calculations wereperformed using the K ¼ 2 GAS layout at z ¼ 1. The VRES part, which does not dependon z, is not shown; it consists of 30.7V/MWh for wind and 6.7V/MWh for solar.

to the total LCOE is reduced and the backup contributions areslightly increased. The wind and solar components of the GAS andGAS* layouts are of course identical. Compared to the K ¼ 1 GASlayout, the total LCOE of the K ¼ 1 GAS* layout is reduced by 1.2V/MWh in absolute units and by 2.1% in relative units. For K ¼ 2 andK ¼ 3 the reductions are 2.7% and 2.8%, respectively. The reductionsare also illustrated in Fig. 5.

The geographic distribution of the transmission capacities forthe K ¼ 2 GAS* layout is shown in Fig. 10. The transmission ca-pacities are not homogeneously distributed across the network. Byfar the strongest links are attached to Spain and Great Britain,which are the two largest countries with severe renewable excessgeneration. Links to their second neighbours with big deficits inrenewable power generation, in particular Germany and Italy, alsoturn out to be quite strong. The more expensive HVDC transmissionlines are utilized less extensively.

5. Sensitivity analysis

5.1. Reduced solar cost

For the optimised GAS layouts as well as for the heuristicCFprop, CFmax and OPT layouts the optimal mixing parameter aEUminimising the overall costs is located in the wind dominated re-gion. This is a consequence of the substantially higher costs of solargeneration compared to wind. The future price development ofsolar photovoltaic systems is rather uncertain. To analyse thesensitivity to future price drops in solar cost, we calculate opti-mised layouts for solar cost reductions of 25%, 50% and 75%. Costreductions could come from improved production processes, oralternatively from increasing capacity factors. Based on data fromRef. [28], the capacity factor can be increased by up to 40% byapplying dual axis tracking compared to the fixed position instal-lation assumed in Table 2, which may offset the higher capital costsof such systems. In addition, studies on increasing the energyconversion efficiency are still being conducted. A recent studysuggests a huge decrease in the total system cost of PVs in a farfuture system [34].

The resulting K ¼ 2 GAS portfolios are visualized in Fig. 11. Notsurprisingly we find that a decrease in solar cost leads to acontinuously increase in totally installed solar capacity. This in-crease is not found to be equal at all nodes. The main solar elec-tricity supplier, Spain, initially increases its solar capacity, but for

Fig. 10. Geographic distribution of the transmission capacities for the K ¼ 2 GAS*

layout. AC links are shown in black while HVDC links are shown in red. Link capacitiesare indicated relative to the highest capacity, which is 68 GW between France andSpain. (For interpretation of the references to colour in this figure legend, the reader isreferred to the web version of this article.)

Fig. 11. GAS optimised layouts constrained by K ¼ 2 for a solar cost reduction of 0%, 25%, 50% and 75%, from left to right.

E.H. Eriksen et al. / Energy 133 (2017) 913e928 925

the more extreme price reductions decreases it again. It seemsmore efficient to shift the production to other sites. Spain is theclear leader in terms of solar generation for large solar costs.However, in the case of 50% solar cost reductions Germany almostproduces equal amounts as Spain. This might not appear to beintuitive from the figure as the renewable penetration of Germanyis always smaller than for Spain, but the mean load of Germany ismore than twice as large as the one of Spain. In the 75% scenarioGermany passes Spain and becomes the main producer of solarpower. In this most extreme scenario almost all countries deploysolar resources.

We illustrate the change in the associated LCOE due to the costreductions in the cases of K ¼ 1, 2 and 3 in Fig. 12. For all cases ofheterogeneity the associated total European LCOE drops steadily fordecreasing solar costs. For a reduction of the solar cost by 25% theoptimalmix is shifted from above aEU ¼ 0:8 to below this point, andthe LCOE values drop by around 2V. As the solar cost is reduced by50% the optimal mix drops further and lies between 0.6 and 0.7. ForK ¼ 2 the LCOE is reduced by almost 5V compared to the referencescenario. When reducing the cost of solar by 75%, solar becomesmuch cheaper than wind, and the optimal mix is shifted belowaEU ¼ 0:5, indicating a dominant share of solar. Compared to thereference scenario, the LCOE dropped by around 9V for the caseK ¼ 2. We have to be aware that such large cost reductions for solar

Fig. 12. LCOE of the GAS optimised layouts when the solar cost is reduced by 25%(triangle), 50% (square) and 75% (diamond). The 0% scenario (circle) is included as areference. Different constraints are shown: K ¼ 1 (blue), 2 (yellow) and 3 (green). (Forinterpretation of the references to colour in this figure legend, the reader is referred tothe web version of this article.)

photovoltaic systems might not be plausible. A cost reduction ismostly to be expected from material and production costs but notfrom installation costs.

5.2. Increased backup cost

The future price developments of fossil fuels, which are likely toincrease, will affect the cost of electricity. An increase in the cost ofgas used by the CCGT generators leads to an increase in the variableoperational expenses associated with backup generation. In prin-ciple this will also affect the structure of the optimised layouts, butwe expect the structural change to be very small. As Fig. 3a reveals,the mixing parameters aEU ¼ 0:82� 0:84 of the optimised GASlayouts also produce the minimum of the backup energy. Conse-quently, the structure of the layouts will more or less not change,but of course their LCOEwill increase as the gas price increases. Thisincrease is linear. For the K ¼ 2 GAS layout an increase in backupfuel price to 150% leads to a LCOE of 59.0V/MWh, which is an in-crease of 6.8%. An increase to 200% of the gas price results in a LCOEof 62.8V/MWh, which equals an increase of 13.6%.

The increased backup costs can to some degree be counter-balanced by the sale of curtailment energy. So far we have assumedthat curtailed electricity is wasted renewable production. Sellingthe curtailment energy to other energy sectors like the heating andtransportation sector is a promising possibility. The resultingdecrease in LCOE depends on the selling price and the amount ofelectricity sold. Since we are discussing an all-European renewablepenetration of gEU ¼ 1 throughout this paper, the total amount ofcurtailment energy is identical to the backup energy. Assuming tosell 1=3 of it at a price of 80V/MWh, the LCOE of the K ¼ 2 GASlayout is reduced to 50.2V/MWh, which is a decrease of 9.2%. Notehowever, that the sale of curtailment energy might have a slightlybigger impact on the structural change of the optimised GAS lay-outs than increased backup costs. Since, again, the amount ofcurtailment energy is equal to the backup energy, Fig. 3a also il-lustrates the dependence of the curtailment energy on the mixingparameter aEU. For parameter values below aEU ¼ 0:82� 0:84 thecurtailment energy increases strongly. Consequently, when takingthe sale of curtailment energy into account, a proper layout opti-misation will shift to some degree towards smaller mixingparameters.

5.3. Interpolations towards more and less heterogeneity

As the heterogeneity parameter changes from K ¼ 1 to 2 and 3,the LCOE of the optimised GAS layouts has decreased further;

Fig. 14. LCOE of the layouts interpolated between the HOM and the GAS layouts forK ¼ 1 (blue), 2 (yellow) and 3 (green). (For interpretation of the references to colour inthis figure legend, the reader is referred to the web version of this article.)

E.H. Eriksen et al. / Energy 133 (2017) 913e928926

consult again Table 5 and Fig. 5. It is quite natural to ask how muchfurther the LCOEmight decrease as K gets even larger. The answer isshown in Fig. 13. The LCOE decreases continuously with increasingheterogeneity. However, the benefit of increased heterogeneitybecomes smaller and smaller. The increasing cost of transmissionleads to a point where it is almost no longer economic beneficial toincrease the heterogeneity. The LCOE of 54.5V/MWh for the K ¼ 3GAS layout is already very close to the asymptotic value of 54V/MWh for very large K.

On the contrary, it might be more politically correct to reducethe heterogeneity. If the optimised GAS layouts were to representthe minimum of a rather shallow cost landscape, then other, morehomogeneous layouts could be found in their vicinity withoutincreasing the LCOE too much. Unfortunately, the search space forthe exploration is high-dimensional, 60-dimensional to be moreprecise, as each of the 30 countries comes with its two variables gnand an. If for each variable we were to test two smaller and twobigger values around its GAS value, we would end up in testing 560

layout explorations. This is infeasible. Instead, we explore simpleone-parameter interpolations between the heterogenous GAS lay-outs and the homogeneous HOM layout:

gn ¼ ð1� sÞgHOMn þ sgGASn ;

an ¼ ð1� sÞaHOMn þ saGASn :(30)

The interpolation parameter is confined to 0 � s � 1. A value ofs ¼ 1 represents the GAS layout while s ¼ 0 reproduces the ho-mogeneous layout. Fig. 14 illustrates the LCOE of the interpolatedlayouts. The dependence on s turns out to be almost linear. It is onlyweakly convex. This might indicate that the cost landscape aroundthe GAS minimum is not flat, and that it might not be possible tofind more homogeneous layouts without increasing the LCOE toomuch.

6. Discussion and outlook

In this paper the heterogeneity of renewable resources indifferent countries has been explored, but the distribution of windand solar capacities within each country was fixed. Further het-erogeneity of renewables, particularly wind, could be exploited byfine-tuning the distribution of renewables within each country, orby using a finer-scale model of Europe that exposes the locationswith high capacity factors. In a recent paper [50] it was shown thatthe VRES costs in a heterogenous optimisation are up to 10% lowerwhen using a 362 node model of Europe compared to a one-node-per-country model with 37 nodes, because the better exploitation

Fig. 13. LCOE of the optimised GAS layouts as a function of the constraint parameter1 � K � 5.

of good sites offsets the increased exposure of grid bottleneckswithin each country.

Only three generation technologies were considered here: solarPV, onshore wind and natural gas. The inclusion of offshore windmight not improve system costs, given its high LCOE, but the LCOEmay be offset by the system benefit of its steadier feed-in profile. Inaddition, offshore offers other benefits compared to onshore windwhich are not accounted for by the cost optimisation, such ashigher rates of public acceptance. Given that offshore wind isgeographically concentrated along the coastlines of countries, afiner-resolution grid model would be advisable to fully assess theintegration of offshore wind.

Modelling hydroelectricity, which already supplies 17% ofEurope's electricity, would reduce the costs of backup energy andprovide extra flexibility to integrate the VRES. Similarly, the incor-poration of storage or the use of flexibility from the electrificationof transport and heating may allow VRES to be balanced morelocally, favouring homogeneous solutions.

Finally, while the cost reduction is a strong argument for aheterogeneous VRES layout, the realisation might be a politicalchallenge. Since the optimal placing of resources was derived froma system perspective, a realisation would require full collaborationfrom all countries. Countries with low capacity factors would nolonger be self sufficient, while countries with high shares of re-newables, such as the countries bordering the North Sea with goodwind sites, may encounter problems finding enough sites or withpublic acceptance.

An unequal distribution of renewable energy generation alsoraises the question of who should pay for the generation andtransmission assets. Current market conditions do not allowrenewable generators to recover their capital costs from theenergy-onlymarket, forcing countries to subsidise the expansion ofrenewables. A highly heterogeneous system would thereforerequire a system for countries to compensate each other for theirrenewable imbalances. Recent work on the allocation of networkflows to users in highly renewable networks [51,52] may providethe basis for an equitable distribution of such costs in a highlyheterogeneous system.

7. Conclusions

In this paper the cost-optimal spatial distribution of VRES in asimplified European electricity systemhas been investigated for thecase where the mean VRES generation equals the mean load(gEU ¼ 1). A heterogenous distribution of wind and solar capacitieshas been shown to result in an average electricity cost that is up to11% lower than a homogeneous distribution of renewables

E.H. Eriksen et al. / Energy 133 (2017) 913e928 927

proportional to each country's mean load. This is because thecapital costs of wind and solar dominate the total system costs, andallowing the system to build more VRES in countries with bettercapacity factors means that fewer wind turbines and solar panelsneed to be built in order to produce the same amount of energy.

If the heterogeneity parameter K, which controls the maximumand minimum levels of renewables generation in each countrycompared to its mean load, is gradually relaxed from K ¼ 1 (ho-mogeneous) to larger values (heterogenous) then there is a cleartrend of cost reduction, which is steepest for smaller values of K andflattens out above K ¼ 3. This has the important policy conse-quence that Europe can profit from the benefits of heterogeneitywithout allowing renewable imbalances between countries tobecome excessive.

The optimal mixing parameter between wind and solar isremarkably robust as the heterogeneity is increased, favouring ahigh proportion of wind of between 80% and 90% in the VRES mix.The mixing parameter is, however, sensitive to the relative capitalcosts of wind and solar, dropping to between 60% and 70% as solarcapital costs are decreased by 50% compared to the default costassumption.

While the best results in terms of low total system costs havebeen obtained here by explicit optimisation, heuristic methods forheterogeneously distributing wind and solar capacities, based forexample on capacity factors, produce results that have costs only afew percent higher than the optimal systems. Given the increasedcomprehensibility and transparency that heuristic methods pro-vide, this may be a price worth paying for policy makers.

Acknowledgments

Tom Brown is funded by the CoNDyNet project, which is sup-ported by the German Federal Ministry of Education and Researchunder grant no. 03SF0472C. The responsibility for the contents liessolely with the authors.

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