Optimal heterogeneity in a simplified highly renewable European electricity system
Emil H. Eriksen, Leon J. Schwenk-Nebbe, Bo Tranberg, Tom Brown, Martin Greiner
OOptimal heterogeneity in a simplified highly renewable European electricity 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, Denmark b Department of Engineering, Aarhus University, Inge Lehmanns Gade 10, 8000 Aarhus C, Denmark c Danske Commodities A/S, Vaerkmestergade 3, 8000 Aarhus C, Denmark d Frankfurt Institute for Advanced Studies (FIAS), Johann Wolfgang Goethe Universit¨at, Ruth-Moufang-Straße 1, 60438 Frankfurt amMain, Germany
Abstract
The resource quality and the temporal generation pattern of variable renewable energy sources vary significantly acrossEurope. In this paper spatial distributions of renewable assets are explored which exploit this heterogeneity to lowerthe total system costs for a high level of renewable electricity in Europe. Several intuitive heuristic algorithms, optimalportfolio theory and a local search algorithm are used to find optimal distributions of renewable generation capacities thatminimise the total costs of backup, transmission and renewable capacity simultaneously. Using current cost projections,an optimal heterogeneous distribution favours onshore wind, particularly in countries bordering the North Sea, whichresults in average electricity costs that are up to 11% lower than for a homogeneous reference distribution of renewablesproportional to each country’s mean load. The reduction becomes even larger, namely 18%, once the transmissioncapacities are put to zero in the homogeneous reference distribution. Heuristic algorithms to distribute renewablecapacity based on each country’s wind and solar capacity factors are shown to provide a satisfactory approximationto fully optimised renewable distributions, while maintaining the benefits of transparency and comprehensibility. Thesensitivities of the results to changing costs of solar generation and gas supply as well as to the possible cross-sectoralusage of unavoidable curtailment energy are also examined.
Keywords: large-scale integration of renewables, system design, renewable energy networks, wind power generation,solar power generation, levelised system cost of electricity, Europe
1. Introduction
The ambitious renewable energy targets set by Europeangovernments [1] imply that the share of renewables in elec-tricity generation will increase significantly in the yearsto come. At present, the leading renewable technologiesare wind, solar photovoltaics (PV) and hydroelectricity, ofwhich only wind and solar PV have the potential for largescale expansion. The uneven distribution of wind and solarresources across the continent raises the question of howbest to exploit these heterogeneous resources. If wind andsolar generation capacities are concentrated in those coun-tries with the best resources, this may increase demandfor transmission and increase energy imbalances betweencountries; if wind and solar generation are distributed ho-mogeneously, then the best renewable resources will not befully used and total system costs may be higher than theheterogeneous optimum. In this paper, the consequencesof heterogeneity for the whole electricity system, includingbackup generation and transmission, will be quantified.
Email addresses: [email protected] (Emil H. Eriksen), [email protected] (Leon J. Schwenk-Nebbe), [email protected] (Bo Tranberg), [email protected] (Tom Brown), [email protected] (Martin Greiner)
Since wind and solar PV are both Variable Renewable En-ergy Sources (VRES), backup generation is needed if theelectrical demand is to be met at all times. Backup gener-ation introduces additional system costs, which depend onthe mismatch between VRES generation and load. Usingthe degrees of freedom associated with the choice of the ca-pacity distributions of VRES for each country, it is possibleto smooth out the aggregated temporal generation patternor even shape it towards the load pattern. As a result, themismatch and thus the backup requirements is lowered.To decrease the dimensionality of the problem, renewableassets can be assigned homogeneously, proportional to themean load of each country, with a uniform wind-to-solarmixing factor. This approach is demonstrated in [2, 3],where optimal wind-to-solar mixes for Europe are foundthat minimise balancing and storage costs. Further re-ductions in backup requirements are possible by extend-ing the transmission network to enable more energy ex-change between the countries [4, 5]. The implications fortotal system costs of different homogeneous renewable pe-nentrations, wind-solar mixes and transmission levels wereconsidered in [6], where the cost-optimal design was foundto consist of a renewable energy penetration of 50% anda wind fraction of 94%. Other relevant research on theadvantages of grid extensions for the integration of renew-
Preprint submitted to Elsevier September 6, 2018 a r X i v : . [ phy s i c s . s o c - ph ] J un bles, including reduced variability and smaller forecasterrors, can be found in [7–13].In this paper the consequences of moving from a homoge-neous spatial distribution of VRES and a uniform wind-to-solar mixing factor to a cost-optimal placement of VREScapacities around Europe are explored. The distributionof VRES plants is determined by at least two considera-tions. The first consideration is the geographical variationof the VRES quality. The resource quality is quantifiedthrough the capacity factor (CF) defined asCF = average generationrated capacity . (1)The capacity factor is a number between 0 and 1, where0 means no generation and 1 means maximum generationat all times. Capacity factors for the European countriesfor onshore wind and solar PV are calculated using (1)and listed in Table 2. The second consideration is the geo-graphical variation of the temporal generation pattern fora given VRES type. This effect is particularly importantfor wind since Europe is large compared to the correlationlength of wind of ≈
600 km [14–16], and wind thereforebenefits from smoothing effects across the continent.With these points in mind, the optimal heterogeneous spa-tial layouts of wind and solar PV across Europe is inves-tigated and compared to the homogeneous layouts. Themain point of comparison is the average cost of electricity,which is composed of the VRES, backup and transmis-sion costs. Different approaches to cope with the resultinglarge number of degrees of freedom are considered. In theliterature a common approach for heterogeneous systemsis to use linear programming to optimise generation andtransmission capacities simultaneously [17–20], but thishas the drawback that only a selection of representativeweather conditions can be considered before computationtimes become infeasible. This makes the results suscepti-ble to over-tuning to the weather selection. Other groupshave used genetic algorithms to optimise generation, stor-age and transmission over a full year in Australia [21] andover three years in Europe, the Middle East and NorthAfrica [22]. In this paper a novel local search algorithmwas found to be most effective given the size and non-linearformulation of the optimisation problem, allowing 8 yearsof hourly weather to be considered.A downside of pure optimisation approaches is that oneloses an understanding of why particular solutions are op-timal. This makes it hard to justify investment strategiesto policy makers and to the public. To counter this down-side, more intuitive heuristic methods are developed hereto construct layouts based on knowledge of resource qual-ity, which are then compared to layouts obtained throughoptimisation. Distributions proportional to capacity fac-tors (similar to the approach in [11]) and distributionsbased on optimal portfolio theory that reduce risk, or stan-dard deviation, of the in-feed (similar to approaches in[23–26]) are considered and compared.
Table 1: Nomenclature
Name Description N Set of nodes n, m
Node index l Link index∆ n Mismatch (VRES generation minus load) α n Wind/solar mix γ n Renewable penetration G { W,S,B } n Generation of wind, solar or backup G Rn Total renewable generation L n Load P n Net power balance K { W,S,B } n Wind, solar or backup capacity K Tl Transmission capacity for link lE B Backup energy C n Curtailment B n Nodal balancing H PTDF matrix F l Power flow on link l CF { W,S } Wind/solar capacity factor (cid:104) x (cid:105) Average value of xq Quantile K Heterogeneity parameterThis paper is organised as follows: Section 2 discusses thegeneral modelling of the simplified European electricitysystem and the key infrastructure measures. Section 3describes the construction of heterogeneous layouts. InSection 4 the performance of the different layouts and theresulting renewable penetrations for individual Europeancountries are discussed. Section 5 contains an analysis ofthe sensitivity of the results to variations in componentcosts. We conclude the paper with a discussion on theresults and an outlook on future research.
2. Methods I: general modeling
Realistic time series describing the country-specific windand solar PV power generation and the load are the start-ing point of the advocated weather-driven modelling ofa simplified networked European electricity system. Theutilized data set has been released from the FraunhoferInstitute for Wind Energy and Energy System Technology(formerly ISET, now IWES) [27]. This data set covers theeight-year period from January 2000 to December 2007,has a temporal resolution of one hour and a spatial reso-lution of 50 ×
50 km over all of Europe. Fixed country-specific capacity layouts have been used to first convertthe weather data into onshore wind and solar power gen-eration, and then to aggregate the latter over each of the30 European countries; off-shore wind power generationis not considered. The country-specific load time series2ave been obtained from publicly available sources, ex-trapolated to cover missing data, and detrended from anannual growth of around 2% to their year 2007 values.For more specific details see [2, 27]. A good alternativedescription of the conversion modelling is given in [28–30].The obtained wind and solar PV power generation time se-ries have been rescaled to the capacity factors (CFs) from2014. The latter have been determined in accordance withequation (1) from the EuroStat data for the installed ca-pacities and the total generation for the year 2014 [31–33].The resulting CFs for each country and each technologyare listed in Table 2. For some of the countries (particu-larly smaller countries) no data was available or the cal-culated result was too uncertain because of too little orno installed capacity. For these countries the CFs are cal-culated as an average value from surrounding countries.These cases are marked by a star. Some countries withan already high installed capacity have a relatively lowcapacity factor compared to validated results from [28].For them the CFs have been raised by a small factor: 8%in Germany, 4% for wind in Spain, 4% for solar in Italyand 2% for wind in Great Britain. The final capacity fac-tors presented in Table 2 are in accordance with [29, 30],which presents a critical assessment of current and futurenational capacity factors. Capacity factors for wind arelikely to rise further in the future because of re-poweringof wind turbines with more efficient, modern turbines athigher hub heights [34]. The European electricity network is modelled as a simpli-fied 30-node model, where each node represents a country.For each node n the generation from VRES (see Table 1for a summary of nomenclature), G Rn ( t ) = G Wn ( t ) + G Sn ( t ) , (2)can be expressed through two parameters. The penetra-tion γ determines the amount of renewable energy gener-ated relative to the mean load of the node, (cid:104) G Rn (cid:105) = γ n (cid:104) L n (cid:105) , (3)while the mixing parameter α fixes the wind-to-solar ratio, (cid:104) G Wn (cid:105) = α n (cid:104) G Rn (cid:105) , (4) (cid:104) G Sn (cid:105) = (1 − α n ) (cid:104) G Rn (cid:105) . (5)Other forms of renewable power generation are neglectedin this simplistic modelling approach.The nodal difference between VRES generation and load∆ n ( t ) = G Rn ( t ) − L n ( t ) (6)is called the mismatch. To avoid power outages, the de-mand must be met at all times. Since storage is not con-sidered, any power deficits must be covered by backup generation. Dispatchable resources are not modelled ex-plicitly, but are considered as part of the backup gener-ation. If ∆ n ( t ) ≥
0, excess energy C n ( t ) must be cur-tailed, while if ∆ n ( t ) < G Bn ( t ) isneeded. Together the two terms form the nodal balancing B n ( t ) = C n ( t ) − G Bn ( t ). It is possible to lower the balancingneeds with transmission. Nodes with excess generation ex-port energy E n ( t ), allowing nodes with an energy deficit toimport energy I n ( t ) to (partly) cover their energy deficit.The nodal injection, E n ( t ) − I n ( t ), is denoted P n ( t ). Thisleads to the nodal balancing equation, G Rn ( t ) − L n ( t ) = B n ( t ) + P n ( t ) , (7)The vector of nodal injections is called the injection pat-tern, and fullfills (cid:80) n P n ( t ) = 0. The actual imports andexports, and thus the injection pattern, depend on the dis-patch of the nodal balancing. The synchronised balancingscheme, B n ( t ) = (cid:104) L n (cid:105) (cid:80) k (cid:104) L k (cid:105) (cid:88) m ∆ m ( t ) , (8)where all nodes are curtailing/generating backup synchron-ously (relative to (cid:104) L n (cid:105) ), fulfills two top priorities: it min-imises the total backup generation for each time step andit minimises the overall backup capacity [35]. This stylisedsynchronised balancing scheme has also been chosen inview of the layout optimisation, since the computationaltime for an update step is much smaller than for other dis-patch schemes, like for example the localised flow schemeused in two previous publications [4, 5].The injection pattern is fixed by Eqs. (7) and (8), anddetermines the power flows on the links l : F l ( t ) = (cid:88) n H ln P n ( t ) . (9)The linear relationship follows from the DC approxima-tion, which is known to be a good approximation for high-voltage flows. For the Power Transfer Distribution Factors H ln we have assumed unit susceptances [35], allowing itsconstruction from the Moore-Penrose pseudo inverse of theunderlying network Laplacian. Following [6], the energy system cost is calculated basedon a few key measures. Besides the cost of the VRES ca-pacities, K W and K S , costs for the backup system and thetransmission network are included. The backup systemcost is split into two components, the cost of backup ca-pacity K B and the cost of backup energy E B . The backupcapacity cost covers expenses related to construction andto keeping the power plants online while the backup en-ergy cost accounts for actual fuel costs. Expressed in unitsof the average annual load, the backup energy is given by E B = (cid:80) n (cid:80) t G Bn ( t ) (cid:80) m (cid:80) t L m ( t ) = (cid:80) n (cid:104) G Bn (cid:105) (cid:80) m (cid:104) L m (cid:105) . (10)3 able 2: Capacity factors CF Wn and CF Sn for onshore wind and solar PV for the European countries, derived from the EuroStat data [31–33].The countries are sorted by their respective mean load (cid:104) L (cid:105) (in units of GW) over the 2000-2007 time series. *: estimated values, see text fordetails. (cid:104) L (cid:105) CF Wn CF Sn (cid:104) L (cid:105) CF Wn CF Sn (cid:104) L (cid:105) CF Wn CF Sn DE FI RS ∗ ∗ FR CZ IE ∗ GB AT BA ∗ ∗ IT GR SK ∗ ES RO HR ∗ SE BG LT ∗ PL ∗ PT EE ∗ NO ∗ CH ∗ ∗ SI ∗ NL HU ∗ LV ∗ BE DK LU q n = (cid:90) K Bn p n ( G Bn ) dG Bn , (11)where p n ( G Bn ) is the time sampled distribution of backupgeneration and q n = 0 .
99. With this choice, the backupsystem will be able to fully cover the demand 99% of thetime. The remaining 1% is assumed to be covered by un-modelled balancing initiatives, e.g. demand side manage-ment. Given the nodal values K Bn , the overall backup ca-pacity K B = (cid:88) n K Bn (12)is calculated by summation.In analogy, the transmission capacity K Tl is defined so thatthe flow is met 99% of the time. Transmission can bepositive and negative, but since links are assumed bidirec-tional, only the magnitude (not the sign) of the flow is tobe considered. Hence q l = (cid:90) K Tl p l ( | F l | ) d | F l | , (13)where p l ( | F l | ) is the time sampled distribution of absoluteflows and q l = 0 .
99. Since the link length varies, K T isnot calculated directly by summation, but instead as aweighted sum, K T = (cid:88) l K Tl d l , (14)where d l denotes the length of link l . Link lengths areestimated as the distance between the country capitals.In this paper E B will be expressed in units of averageannual load, K B in units of average hourly load and K T in units of average hourly load × megametre. Table 3: Cost assumptions for different assets separated into capitalexpenditures (CapEx) and fixed/variable operational expenditures(OpEx) together with their expected life times.
Asset CapEx OpEx fixed
OpEx var
Life time [ e /W] [ e /kW/y] [ e /MWh] [years]CCGT 0.90 4.5 56.0 30Solar PV 0.75 8.5 0.0 25Onshore wind 1.00 15.0 0.0 25 Cost assumptions for the elements of an electricity systemvary greatly across the literature. In this study, the costassumptions published by [6] have been adapted with asingle modification. The cost of solar has been reducedby 50% in accordance with near future solar PV panelprice projections [36]. The resulting estimates are listedin Table 3. In general, the cost assumptions are in thelow end for VRES which reflects the expectation that thecost of VRES will go down in the future as the penetrationincreases. Backup generation is priced based on the costof Combined Cycle Gas Turbines (CCGTs).From the VRES penetration, the mixing factor and themean load, the mean generation of each node can be cal-culated. Dividing by the associated capacity factor, thecapacity is obtained. Except for transmission capacity, thepresent value of each element can be calculated directly as V = CapEx + T life (cid:88) t =1 OpEx t (1 + r ) t , (15)where r is the rate of return assumed to be 4% per year.The transmission capacity cannot be translated directlyinto cost as the cost depends on the length and the typeof the link. Link costs are assumed to be 400 e per km perMW for AC links and 1,500 e per km per MW for HVDClinks. For HVDC links, an additional cost of 150,000 e per MW per converter station pair (one at each end) isadded [10, 11, 37]. The layout of AC and HVDC lines has4een constructed by [4] according to the existing Europeannetwork reported by ENTSO-E for the year 2011 [38] andnew predicted lines until 2014 [39, 40]. It is shown inFigure 10.To allow for comparison of different system layouts, theLevelised Cost of Electricity (LCOE) is a convenient mea-sure [6, 41, 42]. The LCOE is the cost that every generatedunit of energy consumed during the lifetime of the projecthas to match the present value of investment [43],LCOE V = V (cid:80) T life t =1 L EU,t (1+ r ) t . (16)Since the life time of the system elements differs, the LCOEis evaluated separately for each system element from eachrespective present value. The LCOE for the complete sys-tem is calculated by summation. Life times of 25 years forsolar PV and onshore wind, 30 years for CCGT plants and40 years for transmission infrastructure were assumed.
3. Methods II: heterogeneous layouts
The simplest way to distribute the renewable resources isto assign them homogeneously (relative to the mean loadof the node) so that γ n = γ EU = 1 and α n = α EU . Thishomogenous layout is denoted as HOM. However this as-signment might not be ideal since the capacity factors varysignificantly between the nodes. Three heuristic schemesand a straightforward optimisation for the construction ofheterogeneous layouts will be presented in the followingfour subsections. The naming of the distribution algo-rithms is summarised in Table 4. An intuitive first approach, called CFprop, is to assignresources proportional to the CF, or more general to theCF raised to an exponent β . For a wind-only layout, thenodal renewable penetrations γ n are given by γ Wn = (cid:0) CF Wn (cid:1) β (cid:104) L EU (cid:105) (cid:80) m (cid:0) CF Wm (cid:1) β (cid:104) L m (cid:105) γ EU , (17)where γ EU is the overall penetration assumed to be 1. Anequivalent expression for the solar-only layout is obtainedby the substitution W → S . Examples for β = 1 are shownin Figure 1a for the wind- and solar-only layouts. In thelayout illustrations, each bar represents a country.CFprop layouts for any value of α can be constructed as alinear combination of the wind and solar only layouts with γ n = α EU γ Wn + (1 − α EU ) γ Sn (18)and α n = α EU γ Wn α EU γ Wn + (1 − α EU ) γ Sn . (19)For practical reasons, it is not possible to realise extremelyheterogeneous layouts. On the one hand the geograph-ical potentials for VRES installations in countries with good renewable resources may be a limiting factor. Onthe other hand countries with poor renewable resourcesmay not want to become too dependent on imports. Toconstrain heterogeneity, the heterogeneity parameter K isintroduced by requiring1K ≤ γ n ≤ K . (20)With this definition, K = 1 corresponds to a homogeneouslayout while K = ∞ represents unconstrained heterogene-ity. For the CFprop layouts, each value of K translatesinto an α -dependent value of β . For a given value of α ,the corresponding β value is found by increasing β until thefirst country violates equation (20). At the mix α = 0 . K = 1 , , β = 0 . , . , . Although the overall capacity factor of a CFprop layoutfor β > γ n = K to the countrieswith the highest capacity factor and γ n = to the re-maining countries, except for a single in-between countrywhich is fixed by the constraint (cid:88) n γ n (cid:104) L n (cid:105) = (cid:104) L EU (cid:105) . (21)The wind- and solar-only cases of the CFmax layout con-strained by K = 2 are shown in Figure 1b. Similar tothe CFprop layouts, the CFmax layouts for arbitrary α EU values can be constructed as linear combinations (18) ofthe wind- and solar-only layouts. The optimal portfolio theory (OPT) is well known in math-ematical finance [44]. It discusses different assets obtainedfrom the tradeoff between maximizing their return andminimizing their risk. This concept has also been ap-plied to find optimal deployment of wind and solar en-ergy resources in large-scale energy systems [24–26], wherethe overall capacity factor has been treated as the returnand the variance of the renewable power generation as therisk. In modified form, we will use OPT to further exploreVRES capacity layouts over Europe with low system costof electricity.The overall capacity factor of a wind-only ( γ Wn ) or solar-only ( γ Sn ) layout is defined as CF W/SEU = (cid:104) L EU (cid:105)K W/SEU , (22)where K W/SEU = (cid:88) n γ W/Sn (cid:104) L n (cid:105) CF W/Sn (23)5 E F R G B I T E S S
E P L N O N L B E F I C Z A T G R R O B G P T C
H HU D K R S I E B A S K H R L T EE S I L V L U γ n (a) CFprop ( β = 1) D E F R G B I T E S S
E P L N O N L B E F I C Z A T G R R O B G P T C
H HU D K R S I E B A S K H R L T EE S I L V L U . . . . . γ n (b) CFmax ( K = 2) D E F R G B I T E S S
E P L N O N L B E F I C Z A T G R R O B G P T C
H HU D K R S I E B A S K H R L T EE S I L V L U . . . . . γ n (c) OPT ( K = 2) Figure 1: Examples of heuristic (blue) wind-only and (yellow) solar-only layouts: (a) CFprop with β = 1, (b) CFmax constrained to K = 2, and (c) Pareto optimal OPT layouts obtained with K = 2. represents the overall installed capacity. The overall ca-pacity factor is a useful measure of return as this is of highimportance for investors of renewable generation capacity.Investors seek to minimise the overall capacity investment,which corresponds to maximising the overall capacity fac-tor.OPT’s second measure is risk, for which we select the rela-tive standard deviation σ ∆ / (cid:104) L EU (cid:105) of the overall mismatch∆ EU ( t ) = (cid:88) n ∆ n ( t ) (24)based on the country-specific mismatches (6). The smallerthe risky standard deviation becomes the more likely isthe reduced need for a backup infrastructure, which aninvestor tries to minimise [23].A possible heterogeneous wind- or solar-only capacity lay-out is sampled from a Monte Carlo procedure. The country-specific renewable penetrations γ n are randomly and inde-pendently drawn from a Beta distribution p ( γ ) = Γ( β + β )Γ( β )Γ( β ) (cid:18) KK − (cid:19) β + β − (cid:18) γ − K (cid:19) β − ( K − γ ) β − (25) Figure 2: Scatter clouds for (blue) wind-only and (yellow) solar-only capacity layouts. The diagram plots the overall capacity factor(22) vs. the standard deviation of the overall mismatch (24). Thedistribution (25) and the constraint (20) with K = 2 have beenused for the Monte Carlo simulations. The white point in the upperleft cloud corners indicate Pareto optimal layouts; see also Figure1c. The line connecting the wind- and solar-only Pareto optimallayouts results from the interpolation between these layouts in (18).The black point marked on this line represents the OPT layout withminimum LCOE. For comparison, the three triangle points markthe (orange) optimal CFprop, (green) optimal CFmax and (blue)optimised GAS layouts for K = 2. defined on the compact support (20). Γ( β ) is the Gammafunction. The two shape parameters β and β are deter-mined by requiring (cid:104) γ n (cid:105) = 1 and by envoking the maxi-mum entropy principle [45] to maximal smear out the Betadistribution over the interval (20). For K = 2 the two pa-rameters result in β = 0 . β = 1 .
61, and for K = 3they are β = 0 . β = 2 .
57. A capacity layout sampledwith this procedure does not necessarily meet the require-ment (21). For such cases, all γ n are uniformly rescaledupwards or downwards until the requirement is fulfilled.During the rescaling some of the penetration parametershit the K constraint (20), and are then frozen for the re-mainder of the rescaling procedure.The wind-only and solar-only portfolios for K = 2 areshown in Figure 2 in blue and yellow, respectively, with theoverall mismatch measure on the first axis and the overallcapacity factor measure on the second axis. Each of theportfolios consists of 100000 layouts. Due to the elongatedshape of the portfolios there is no clear extended Paretofront in the upper left corners. The Pareto front defines aline, for which at the same time the standard deviation ofthe overall mismatch (risk) can not be reduced further for afixed overall capacity factor and the overall capacity factor(return) can not be increased further for a fixed standarddeviation of the overall mismatch. For both portfolios weidentify a single point to characterise minimum risk andmaximum return. This is done by extracting a subset ofthe points which are simultaneously a part of the top 200capacity factors and bottom 200 standard deviations. For K = 2 this leaves a sample of 28 layouts for wind and 67layouts for solar to average and to calculate the respectivenew overall capacity factor and new overall standard de-viation. The resulting points are plotted in white on top6f the portfolios. The layouts of these two Pareto optimalpoints are shown in Figure 1c.In order to find an optimal combined layout, we interpo-late between the Pareto-optimal wind-only and solar-onlylayouts according to (18) and (19). This interpolation con-serves the constraint (21) and results in the line shown inFigure 2. Apparently some of the interpolated layouts areable to reduce the standard deviation of the global mis-match further. The interpolated layout marked with ablack dot comes with the mixing parameter α EU = 0 . The full optimisation of the layouts is considered, withthe objective to minimise the LCOE with respect to the60 variables γ , ..., γ N , α , ..., α N for the N = 30 coun-tries. Given the high dimensionality of the search space, anumber of optimisation algorithms were tested includingthe Nelder-Mead method [46], simulated annealing [47],genetic algorithms [48] and cuckoo search [49]. It wasfound that the continuous enforcement of the normalisa-tion criterion (21) generally decreased the performance ofthe tested algorithms, and for that reason a new hybridalgorithm was developed to address this problem. Whilebeing a classical greedy algorithm in the sense that thelocally optimal choice is always taken, the renormalisationproblem was circumvented by moving only along the axialdirections. The algorithm has been denoted Greedy AxialSearch (GAS).When a solution is renormalised, all γ values are scaledeither up or down. Therefore, it is possible that some γ values end up outside the boundary (20). The γ valuesare fixed at the boundary and the rescaling is only appliedto the remaining free γ values. In general this approach isproblematic since it can change the direction of the search.This is circumvented by holding the specific γ value con-stant that is considered during the step up/down proce-dure along a given axis. In this way only some γ valuesare scaled down/up and the feasibility of moving up/downalong the considered axis can be determined. This is theunderlying principle of Greedy Axial Search (GAS).As any greedy algorithm, the GAS algorithm works bytaking the locally optimal choice. Hence the feasibilityfor each direction is evaluated, but only the best choice isaccepted. This process is repeated until a convergencecriterion is fulfilled. At this point the step size is re-duced and the iterative optimisation procedure repeateduntil the step size drops below some tolerance. The algo-rithm structure is sketched in Algorithm 1. The StepUp and
StepDown subroutines generate new solutions by step-ping a solution (first argument) up/down along axis i (sec-ond argument) with some step size (third argument) afterwhich the solution is renormalised as described above. Val-ues of maxStepSize = 1, minStepSize = 5 · − and tolerance = 10 − were found to be appropriate.All optimised layouts have been obtained using the GASroutine. These layouts will be denoted GAS layouts. Con-straining the transmission and thereby reducing the trans- Table 4: Summary of the algorithms for distributing VRES.
Name Brief descriptionHOM Homogeneous distribution proportional tothe mean of each country’s loadCFprop Distribution proportional to a power( CF ) β of the capacity factor CF CFmax Assignment to each country γ n extremisedwithin K ≤ γ n ≤ K depending on CF OPT Distribution using Optimal Portfolio The-oryGAS Distribution optimised using Greedy AxialSearch algorithmGAS* As GAS, but with optimally constrainedtransmissionGASnoT As GAS but with no transmission betweencountries, so that each country is self-sufficient at all timesmission capacity can lead to an overall lower LCOE. Thisis discussed in Section 4.4. The layouts resulting from thisadditional optimisation will be denoted GAS* layouts.
4. Results
The optimal heuristic layouts CFprop, CFmax, OPT aswell as the optimized layouts GAS will be discussed in thenext three subsections, first for K = 1, then for K = 2,and finally for K = 3. The fourth subsection focuses onthe transmission capacities. K = 1 layouts By construction, the layouts CFprop, CFmax and OPTbecome identical and homogeneous for K = 1. Due to Eq.(20), their respective renewable penetrations are γ n = 1.Moreover, according to Eq. (19) their renewable mixes α n = α EU also turn out to be independent of the countryindex. For these strictly homogeneous layouts Figure 3shows the dependence of the key infrastructure measureson α EU as the blue curves. For the backup energy andbackup capacity, the optimal mixing parameters are lo-cated around α EU = 0 .
85, which is slightly larger thanthe values found by [2, 3]. For the transmission capacity,the minimum occurs around α EU = 0 .
45. The main mea-sure of interest, the LCOE, has a minimum at α EU = 0 . α EU = 0 is caused by a combination ofhigh backup energy/capacity costs and the fact that theCF of solar is generally lower than for onshore wind. Thecost of producing one unit of energy is thus higher for solarthan for onshore wind even though the specific CapEx islower for solar.The homogeneous layout producing the minimum LCOEat α EU = 0 .
90 is denoted as the ’HOM’ layout. It isillustrated in Figure 4a. Its total LCOE amounts to 59.7 e /MWh. The componentwise LCOE corresponding to the7 lgorithm 1 Pseudo code for the greedy axial search (GAS) routine. The
Evaluate function calculates the associatedcost of each new solution, and all new solutions are thereupon sorted by the
Sort function in ascending order. function
GreedyAxialSearch best ← solution selected randomly from within the solution space deltaCost ← ∞ stepSize ← maxStepSize while stepSize > minStepSize dowhile deltaCost > tolerance dofor index i = 1 to 2N do trailSolutions[i] ← StepUp( best , i , stepSize ) trailSolutions[i+2N] ← StepDown( best , i , stepSize )Evaluate( trailSolutions )Sort( trailSolutions ) deltaCost ← cost of best minus cost of trailSolutions[1] if deltaCost > then best ← trailSolutions[1]stepSize ← stepSize /2 return best . . . . . . α EU . . . . . E B . . . . . . α EU . . . . . . . . K B . . . . . . α EU . . . . . . . K T . . . . . . α EU L C O E [ € / M W h ] CFprop CFmax OPT GAS GASnoT K = 1 K = 2 K = 3 Figure 3: Overview of the infrastructure measures: (a) the backup energy E B (in units of average annual European load), (b) the backupcapacity K B (in units of average hourly European load), (c) the transmission capacity K T (in units of average hourly European load timesmegametre) and (d) the associated LCOE as a function of α EU . The CFprop and CFmax layouts are shown as solid and dashed linesrespectively. The dependence of the OPT layouts on α EU is not shown; only the interpolations leading to a LCOE minimum are plotted asasterisks. The GAS layouts are plotted as dots. The blue diamond represents the GASnoT layout. Different constraints are shown: K = 1(blue), 2 (yellow) and 3 (green). wind, solar, backup and transmission parts are listed in thethird column of Table 5 and graphed as the second bar inFigure 5. Wind power dominates the overall LCOE. Itscontribution amounts to 61%, and is followed by 21% frombackup, 10% from solar and 8% from transmission.Contrary to the HOM layout, the K = 1 GAS layout isno longer strictly homogeneous. Of course, all renewablepenetrations are still equal to γ n = 1, but as a result ofthe optimisation the wind-solar mixing parameters becomeheterogeneous. This is illustrated in Figure 4b. Two-thirds of the countries are wind-only with α n = 1. Theremaining countries have a significant share of solar. Forsome of those this was to be expected. Spain, Greece,Italy, Romania and Serbia have very large solar capacityfactors. See again Table 2. However, other solar-rich coun-tries, like Portugal, Bulgaria, Bosnia and Croatia, are notamongst them. Instead, Germany is also assigned a sig- nificant share of solar, although its solar capacity factor isonly average. By taking a closer inspection of Table 2 wediscover the following empirical finding for the K = 1 GASlayout: all countries with α n = 1 come with a ratio be-tween their solar and wind capacity factor which is smallerthan CF Sn / CF Wn < .
65. The countries with α n < Sn / CF Wn ≥ .
65, except for the three small-est countries Estonia, Latvia and Luxembourg.Compared to the HOM layout, the α -heterogeneity of the K = 1 GAS layout is able to reduce the total LCOE by3%. This is mostly a consequence of the reduced combinedcomponent costs for wind and solar power. Note, that theoverall mixing parameter α EU = (cid:80) n α n (cid:104) L n (cid:105) / (cid:104) L EU (cid:105) hasalso slightly reduced from 0.90 (HOM) to 0.84 (GAS). Seethe fourth column of Table 5 and the third bar of Figure 5.The costs for backup and transmission have not changedmuch; which is also apparent from the rightmost panel of8 able 5: Componentwise LCOE for the optimal CFprop, optimal CFmax, optimal OPT, optimised GAS and optimised GAS* layouts for K = 1 (left), 2 (middle) and 3 (right). Note that the K = 1 layouts CFprop, CFmax and OPT are identical and denoted as HOM. The K = 1 layout GASnoT without transmission is listed as reference. All costs are given in e /MWh. K = 1 K = 2 K = 3 GASnoT HOM GAS GAS ∗ CFprop CFmax OPT GAS GAS ∗ CFprop CFmax OPT GAS GAS ∗ α EU 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 29.2LCOE( 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 6.7LCOE( 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 4.3LCOE( E B ) 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 8.2LCOE( 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 4.6LCOE(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 D E F R G B I T E S S
E P L N O N L B E F I C Z A T G R R O B G P T C
H HU D K R S I E B A S K H R L T EE S I L V L U . . . . . . γ n (a) HOM ( K = 1) D E F R G B I T E S S
E P L N O N L B E F I C Z A T G R R O B G P T C
H HU D K R S I E B A S K H R L T EE S I L V L U . . . . . . γ n (b) GAS ( K = 1) D E F R G B I T E S S
E P L N O N L B E F I C Z A T G R R O B G P T C
H HU D K R S I E B A S K H R L T EE S I L V L U . . . . . . γ n (c) GASnoT ( K = 1) Figure 4: Comparison of (a) the optimal homogeneous layout HOMwith the optimised (b) GAS and (c) GASnoT layouts constrained by K = 1. Figure 3.All K = 1 layouts discussed so far include the transmissioninfrastructure. It is also interesting to compare them toan optimised layout without transmission. No exports andimports would then be possible and the injection pattern P n ( t ) would always be zero. No transmission investmentwould be needed and the respective componentwise LCOEwould be zero. However, the countries then have to bal-ance their mismatches all by themselves, and this in turnrequires more backup infrastructure with higher respectivecomponentwise LCOE. For the GAS layout without thetransmission infrastructure, which for clarity we denote as G A Sn o T H O M G A S G A S * C F p r o p C F m a x O P T G A S G A S * C F p r o p C F m a x O P T G A S G A S * L C O E [ € / M W h ] K = 1 K = 2 K = 3 K W K S K B E B K T Figure 5: Componentwise LCOE for the optimal CFprop, CFmax,OPT, GAS and GAS* layouts for K = 1 (left), 2 (middle) and 3(right). The K = 1 layout GASnoT without transmission is shownas reference. GASnoT, the total LCOE turns out to be 64.5 e /MWh.Compared to the HOM layout, the combined LCOE com-ponents for wind and solar power generation are almostthe same, but the increase of the LCOE components forthe backup power generation and capacity is significantlylarger than the disappearance of the transmission compo-nent. See again Figure 3, Table 5 and Figure 5. The totalLCOE of the GASnoT layout is 8% and 11.5% larger thanfor the HOM and GAS layout respectively. This clearlydemonstrates the benefit of transmission [4, 6].The GAS and GASnoT layouts are obtained from two in-dependent optimisation efforts. This explains why the twolayouts are actually quite different in the distribution ofthe wind and solar resources. Figure 4c illustrates theresulting wind-solar mixing parameters for the GASnoTlayout. Contrary to the more extreme GAS layout, themajority of the countries comes with a mix below α n = 1and well above 0. Only the most northern countries turnout to be wind-only. However, on average the mixing pa-rameter α EU = 0 .
86 for the GASnoT layout is again close9 E F R G B I T E S S
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H HU D K R S I E B A S K H R L T EE S I L V L U . . . . . . . . . γ n (a) CFprop ( K = 2) D E F R G B I T E S S
E P L N O N L B E F I C Z A T G R R O B G P T C
H HU D K R S I E B A S K H R L T EE S I L V L U . . . . . γ n (b) CFmax ( K = 2) D E F R G B I T E S S
E P L N O N L B E F I C Z A T G R R O B G P T C
H HU D K R S I E B A S K H R L T EE S I L V L U . . . . . γ n (c) OPT ( K = 2). D E F R G B I T E S S
E P L N O N L B E F I C Z A T G R R O B G P T C
H HU D K R S I E B A S K H R L T EE S I L V L U . . . . . γ n (d) GAS ( K = 2) Figure 6: Comparison of different layouts constrained by K = 2: (a)CFprop, (b) CFmax, (c) OPT and (d) GAS. to α EU = 0 .
84 for the GAS layout. K = 2 layouts More heterogeneity is introduced once K is chosen to belarger than one. Figures 6a-c illustrate the optimal heuris-tic CFprop, CFmax and OPT layouts for K = 2. Theirrespective α EU values are 0.86 – 0.87 (see Table 5), andhave been fixed by minimising the LCOE (see Figure 3d).The general α EU -dependence of the other infrastructuremeasures are illustrated in Figure 3a-c. The backup en-ergies required for the three layouts are quasi identical,and no difference is seen to the K = 1 HOM layout. Alsothe backup capacities are almost identical for the threelayouts, and are slightly less than for the K = 1 HOM lay-out. Differences are observed for the transmission capaci-ties. The CFprop layout comes with the smallest transmis-sion capacities, followed by the OPT layout. The CFmaxlayout has the largest transmission capacities because itsheterogeneity is the largest. All K = 2 layouts are foundto have larger transmission capacities than the respective K = 1 layouts.The total LCOE of the three heuristic K = 2 layouts areinbetween 57.2 – 57.9 e /MWh. See columns 6-8 in Table5 and bars 5-7 in Figure 5. This is very close to the value aCF wn + (1 − a ) CF sn γ n ATFI NL BAFR NOBEGBPLBG GR PTCH HRROCZ HURSDE IESE DKITSI ESLU SKEE LV LT
Figure 7: Renewable penetration parameters γ n from the K = 2GAS layout as a function of the effective capacity factor CF effn de-fined in Equation (27). The continuous and piecewise linear greenfunction represents the heuristic law (26) with least-square-fitted pa-rameters a = 0 . CF = 0 .
173 and CF = 0 . e /MWh found for the K = 1 GAS layout. In thisrespect, the larger heterogeneity of the K = 2 layouts donot represent a clear cost advantage when compared to the K = 1 GAS layout, which is homogeneous in the renewablepenetration parameters γ n . The situation changes oncethe optimised K = 2 GAS layout is considered, which isexemplified in Figure 6d. It exploits the wind resourcesover Europe in a more efficient way and reduces the windcomponent in the LCOE; consult column 9 of Table 5 andbar 8 in Figure 5. This reduces the total LCOE to 55.3 e /MWh.The overall renewable penetration of the K = 2 GAS lay-out is γ EU = 1; consult again Equation (21). However, theindividual renewable penetration parameters now scatterwithin 0 . ≤ γ n ≤
2. As can be seen in Figure 6d, theirdistribution is extremely heterogenous. For half of thecountries they are either γ n = 2 or γ n = 0 .
5, and for theother countries just somewhere in-between. A more care-full inspection reveals an approximate heuristic law, whichexpresses the renewable penetration parameters γ n = /K ( CF eff n ≤ CF )( K − K ) CF eff n − CF CF − CF + K ( CF ≤ CF eff n ≤ CF ) K ( CF eff n ≥ CF ) (26)as a continuous and piece-wise linear function of an effec-tive capacity factor CF eff n = aCF Wn + (1 − a ) CF Sn . (27)A least-square fit is shown in Figure 7.The overall mixing parameter α EU = 0 .
83 of the K = 2GAS layout is almost the same as for the K = 1 GASlayout. Both layouts also have in common that 20 out ofthe 30 countries come with α n = 1. The five largest of the α n < E F R G B I T E S S
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H HU D K R S I E B A S K H R L T EE S I L V L U . . . . . . . γ n GAS (K = 3)
Figure 8: GAS layout constrained by K = 3. It is worth to take again a quick look at Figure 2. Itshows that for K = 2 the optimal CFprop, CFmax andOPT layouts have more or less the same close-to-minimumstandard deviation of the overall mismatch (24) as theoptimised GAS layout. This indicates that a minimisedmismatch standard deviation serves as a good measure todetermine an optimal infrastructure [23]. However, it isstill a rough measure, since it does not allow to finetunethe minimum-cost infrastructure. K = 3 layouts For K = 3 the GAS algorithm has more freedom to op-timise the heterogeneous layout and to reduce the overallLCOE, see (20). The resulting layout is depicted in Figure8. It has some similarity to the K = 2 GAS layout, but ofcourse the K = 3 GAS layout is even more extreme. Itsoverall wind-solar mixing parameter α EU = 0 .
82 is almostthe same as for the K = 2 counterpart. The overall costreduction turns out to be small. As can be seen in Table5, the total LCOE for the K = 2 and K = 3 GAS layoutsare 55.3 and 54.5 e /MWh, respectively. This small costreduction is mainly caused by the opportunity to allocatemore wind resources to the sites with a very high capac-ity factor, and it is weakened to some extend by slightlyincreased costs for the transmission component; comparecolumn 14 with column 9 in Table 5.Bulk results for the optimal heuristic K = 3 layouts CF-prop, CFmax and OPT are also listed in Table 5 and Fig-ure 5. Their layouts are also found to be wind-dominated,with nearly the same α EU values as for the respective GASlayout. The LCOE for these three heuristic layouts arelarger than for the K = 3 GAS layout. This of course wasto be expected. However, their LCOE also turn out to beslightly larger than for the less heterogeneous K = 2 GASlayout.Another reason that the GAS optimisation might havebeen better than the heuristic layouts is that the GASalgorithm sees not just the capacity factors at each site,like the heuristic layouts, but also the geographical vari-ation of the temporal generation pattern, which the GASalgorithm can exploit to shape the VRES generation pat-tern towards the load. However if this was the reason, thebackup generation costs would have decreased from theheuristic to the GAS layout, which they do not. This sug- gests that the GAS optimisation’s success really lies withthe free exploitation of capacity factors. So far, only the total contribution of the transmission ca-pacities to the overall LCOE have been discussed for vari-ous system layouts in Table 5 and Figure 5. Its geographicdistribution has not yet been specified. This will be donein this Subsection, but not right away. At first we willinvestigate a procedure which further reduces the overallLCOE by reducing the transmission capacities to some ex-tend.The transmission capacities defined in Equation (13) havebeen derived from unconstrained power flows. They aredetermined by the most extreme flow events, which typ-ically occur between countries with a large energy deficitand others with a large excess. These events are not ex-pected to overlap with other extreme events when all coun-tries face a large energy deficit. The latter determine therequired backup capacities. Consequently, it can be ex-pected that a modest reduction of the total transmissioncapacities will not, or at least not much, affect the totalbackup capacities and the total backup energy, and willlower the overall LCOE.The synchronised balancing scheme (8) presented in Sec-tion 2.2 is based on unconstrained power flows. In order toinclude constrained power flows, a generalisation is needed:min B (cid:88) n ( B n ( t )) (cid:104) L n (cid:105) s.t. (cid:88) n P n ( t ) = 0s.t. − K conTl ≤ F l ( t ) = (cid:88) n H ln P n ( t ) ≤ K conTl . (28)The objective is to minimise the expression in the firstline, taking into account the two constraints of the secondand third line. K conTl denotes the constrained transmis-sion capacity of line l . In the limit of unconstrained flows,where the second constraint can be discarded, the objec-tive (28) can be rewritten as min B (cid:80) n ( B n ( t ) / (cid:104) L n (cid:105) − λP n )with the method of Lagrange multipliers and leads to thesolution (8). For the following we will downscale the un-constrained transmission capacities from (13) by a uniformscaling parameter ζ to obtain the constrained transmissioncapacities K conTl = ζ K Tl . (29)Figure 9 illustrates the dependence of the LCOE on thetransmission constraints by taking the unconstrained trans-mission capacities of the K = 2 GAS layout and scalingthem down by the uniform factor ζ . At first, as ζ de-creases, the LCOE also decreases. A minimum is foundat ζ = 0 .
60. For the K = 1 and K = 3 GAS layouts theminimum is found at the optimal values ζ = 0 .
55 and 0 . . . . . . . ζ L C O E [ € / M W h ] Minimum K T E B K B Figure 9: Non-VRES components of the LCOE as a function ofthe scaling parameter ζ . The dashed line indicates the minimumleading to the lowest LCOE. The calculations were performed usingthe K = 2 GAS layout at ζ = 1. The VRES part, which does notdepend on ζ , is not shown; it consists of 30.7 e /MWh for wind and6.7 e /MWh for solar. further the LCOE starts to increase again due to increas-ing requirements for backup energy and backup capacity.Table 5 lists also the modified GAS layouts resulting fromthe optimal scaling parameters. For clarity, we denotethem as GAS ∗ layouts. Compared to the GAS layouts, thetransmission contribution to the total LCOE is reducedand the backup contributions are slightly increased. Thewind and solar components of the GAS and GAS ∗ layoutsare of course identical. Compared to the K = 1 GAS lay-out, the total LCOE of the K = 1 GAS ∗ layout is reducedby 1.2 e /MWh in absolute units and by 2.1% in relativeunits. For K = 2 and K = 3 the reductions are 2.7% and2.8%, respectively. The reductions are also illustrated inFigure 5.The geographic distribution of the transmission capaci-ties for the K = 2 GAS ∗ layout is shown in Figure 10.The transmission capacities are not homogeneously dis-tributed across the network. By far the strongest linksare attached to Spain and Great Britain, which are thetwo largest countries with severe renewable excess gener-ation. Links to their second neighbours with big deficitsin renewable power generation, in particular Germany andItaly, also turn out to be quite strong. The more expensiveHVDC transmission lines are utilised less extensively.
5. Sensitivity analysis
For the optimised GAS layouts as well as for the heuris-tic CFprop, CFmax and OPT layouts the optimal mixingparameter α EU minimising the overall costs is located inthe wind dominated region. This is a consequence of thesubstantially higher costs of solar generation compared BEFR BGBA HRDE HU FIDKNLPT NO LVLTLU ROPLCH GR EEIT CZATIE ES RSSKSI SEGB − .
5% 2 . −
5% 10% 25% 50% 100%
Figure 10: Geographic distribution of the transmission capacities forthe K = 2 GAS ∗ layout. AC links are shown in black while HVDClinks are shown in red. Link capacities are indicated relative to thehighest capacity, which is 68 GW between France and Spain. to wind. The future price development of solar photo-voltaic systems is rather uncertain. To analyse the sensi-tivity to future price drops in solar cost, we calculate op-timised layouts for solar cost reductions of 25%, 50% and75%. Cost reductions could come from improved produc-tion processes, or alternatively from increasing capacityfactors. Based on data from [28], the capacity factor canbe increased by up to 40% by applying dual axis track-ing compared to the fixed position installation assumed inTable 2, which may offset the higher capital costs of suchsystems. In addition, studies on increasing the energy con-version efficiency are still being conducted. A recent studysuggests a huge decrease in the total system cost of PVsin a far future system [34].The resulting K = 2 GAS portfolios are visualized in Fig-ure 11. Not surprisingly we find that a decrease in solarcost leads to a continuously increase in totally installed so-lar capacity. This increase is not found to be equal at allnodes. The main solar electricity supplier, Spain, initiallyincreases its solar capacity, but for the more extreme pricereductions decreases it again. It seems more efficient toshift the production to other sites. Spain is the clear leaderin 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 appearto be intuitive from the figure as the renewable penetra-tion of Germany is always smaller than for Spain, but themean load of Germany is more than twice as large as theone of Spain. In the 75% scenario Germany passes Spainand becomes the main producer of solar power. In thismost extreme scenario almost all countries deploy solarresources.We illustrate the change in the associated LCOE due tothe cost reductions in the cases of K = 1, 2 and 3 in Fig-ure 12. For all cases of heterogeneity the associated totalEuropean LCOE drops steadily for decreasing solar costs.For a reduction of the solar cost by 25% the optimal mix12 E F R G B I T E S S
E P L N O N L B E F I C Z A T G R R O B G P T C
H HU D K R S I E B A S K H R L T EE S I L V L U . . . . . γ n Figure 11: GAS optimised layouts constrained by K = 2 for a solar cost reduction of 0%, 25%, 50% and 75%, from left to right. . . . . . . α EU L C O E [ € / M W h ] Figure 12: LCOE of the GAS optimised layouts when the solar costis reduced by 25% (triangle), 50% (square) and 75% (diamond). The0% scenario (circle) is included as a reference. Different constraintsare shown: K = 1 (blue), 2 (yellow) and 3 (green). is shifted from above α EU = 0 . e . As the solar costis reduced by 50% the optimal mix drops further and liesbetween 0.6 and 0.7. For K = 2 the LCOE is reducedby almost 5 e compared to the reference scenario. Whenreducing the cost of solar by 75%, solar becomes muchcheaper than wind, and the optimal mix is shifted below α EU = 0 .
5, indicating a dominant share of solar. Com-pared to the reference scenario, the LCOE dropped byaround 9 e for the case K = 2. We have to be aware thatsuch large cost reductions for solar photovoltaic systemsmight not be plausible. A cost reduction is mostly to beexpected from material and production costs but not frominstallation costs. The future price developments of fossil fuels, which arelikely to increase, will affect the cost of electricity. Anincrease in the cost of gas used by the CCGT generators leads to an increase in the variable operational expensesassociated with backup generation. In principle this willalso affect the structure of the optimised layouts, but weexpect the structural change to be very small. As Figure3a reveals, the mixing parameters α EU = 0 . K = 2 GAS layout an increase in backupfuel price to 150% leads to a LCOE of 59.0 e /MWh, whichis an increase of 6.8%. An increase to 200% of the gasprice results in a LCOE of 62.8 e /MWh, which equals anincrease of 13.6%.The increased backup costs can to some degree be coun-terbalanced by the sale of curtailment energy. So far wehave assumed that curtailed electricity is wasted renew-able production. Selling the curtailment energy to otherenergy sectors like the heating and transportation sec-tor is a promising possibility. The resulting decrease inLCOE depends on the selling price and the amount ofelectricity sold. Since we are discussing an all-Europeanrenewable penetration of γ EU = 1 throughout this paper,the total amount of curtailment energy is identical to thebackup energy. Assuming to sell 1 / e /MWh, the LCOE of the K = 2 GAS layout is re-duced to 50.2 e /MWh, which is a decrease of 9.2%. Notehowever, that the sale of curtailment energy might have aslightly bigger impact on the structural change of the op-timised GAS layouts than increased backup costs. Since,again, the amount of curtailment energy is equal to thebackup energy, Figure 3a also illustrates the dependence ofthe curtailment energy on the mixing parameter α EU . Forparameter values below α EU = 0 . K L C O E [ € / M W h ] GAS layouts
Figure 13: LCOE of the optimised GAS layouts as a function of theconstraint parameter 1 ≤ K ≤ As the heterogeneity parameter changes from K = 1 to 2and 3, the LCOE of the optimised GAS layouts has de-creased further; consult again Table 5 and Figure 5. It isquite natural to ask how much further the LCOE mightdecrease as K gets even larger. The answer is shown inFigure 13. The LCOE decreases continuously with increas-ing heterogeneity. However, the benefit of increased het-erogeneity becomes smaller and smaller. The increasingcost of transmission leads to a point where it is almost nolonger economic beneficial to increase the heterogeneity.The LCOE of 54.5 e /MWh for the K = 3 GAS layout isalready very close to the asymptotic value of 54 e /MWhfor very large K .On the contrary, it might be more politically correct toreduce the heterogeneity. If the optimised GAS layoutswere to represent the minimum of a rather shallow costlandscape, then other, more homogeneous layouts could befound in their vicinity without increasing the LCOE toomuch. Unfortunately, the search space for the explorationis high-dimensional, 60-dimensional to be more precise, aseach of the 30 countries comes with its two variables γ n and α n . If for each variable we were to test two smaller and twobigger values around its GAS value, we would end up intesting 5 layout explorations. This is infeasible. Instead,we explore simple one-parameter interpolations betweenthe heterogenous GAS layouts and the homogeneous HOMlayout: γ n = (1 − σ ) γ HOM n + σγ GAS n ,α n = (1 − σ ) α HOM n + σα GAS n . (30)The interpolation parameter is confined to 0 ≤ σ ≤
1. Avalue of σ = 1 represents the GAS layout while σ = 0reproduces the homogeneous layout. Figure 14 illustratesthe LCOE of the interpolated layouts. The dependence on σ turns out to be almost linear. It is only weakly convex.This might indicate that the cost landscape around theGAS minimum is not flat, and that it might not be possible . . . . . . σ L C O E [ € / M W h ] K = 1 K = 2 K = 3 Figure 14: LCOE of the layouts interpolated between the HOM andthe GAS layouts for K = 1 (blue), 2 (yellow) and 3 (green). to find more homogeneous layouts without increasing theLCOE too much.
6. Discussion and outlook
In this paper the heterogeneity of renewable resources indifferent countries has been explored, but the distribu-tion of wind and solar capacities within each country wasfixed. Further heterogeneity of renewables, particularlywind, could be exploited by fine-tuning the distribution ofrenewables within each country, or by using a finer-scalemodel of Europe that exposes the locations with high ca-pacity factors. In a recent paper [50] it was shown that theVRES costs in a heterogenous optimisation are up to 10%lower when using a 362 node model of Europe comparedto a one-node-per-country model with 37 nodes, becausethe better exploitation of good sites offsets the increasedexposure of grid bottlenecks within each country.Only three generation technologies were considered here:solar PV, onshore wind and natural gas. The inclusion ofoffshore wind might not improve system costs, given itshigh LCOE, but the LCOE may be offset by the systembenefit of its steadier feed-in profile. In addition, offshoreoffers other benefits compared to onshore wind which arenot accounted for by the cost optimisation, such as higherrates of public acceptance. Given that offshore wind isgeographically concentrated along the coastlines of coun-tries, a finer-resolution grid model would be advisable tofully assess the integration of offshore wind.Modelling hydroelectricity, which already supplies 17% ofEurope’s electricity, would reduce the costs of backup en-ergy and provide extra flexibility to integrate the VRES.Similarly, the incorporation of storage or the use of flexi-bility from the electrification of transport and heating mayalow VRES to be balanced more locally, favouring homo-geneous solutions.Finally, while the cost reduction is a strong argument fora heterogeneous VRES layout, the realisation might be apolitical challenge. Since the optimal placing of resourceswas derived from a system perspective, a realisation wouldrequire full collaboration from all countries. Countries14ith low capacity factors would no longer be self suffi-cient, while countries with high shares of renewables, suchas the countries bordering the North Sea with good windsites, may encounter problems finding enough sites or withpublic acceptance.An unequal distribution of renewable energy generationalso raises the question of who should pay for the gener-ation and transmission assets. Current market conditionsdo not allow renewable generators to recover their capi-tal costs from the energy-only market, forcing countriesto subsidise the expansion of renewables. A highly het-erogeneous system would therefore require a system forcountries to compensate each other for their renewable im-balances. Recent work on the allocation of network flowsto users in highly renewable networks [51, 52] may providethe basis for an equitable distribution of such costs in ahighly heterogeneous system.
7. Conclusions
In this paper the cost-optimal spatial distribution of VRESin a simplified European electricity system has been in-vestigated for the case where the mean VRES generationequals the mean load ( γ EU = 1). A heterogenous distribu-tion of wind and solar capacities has been shown to resultin an average electricity cost that is up to 11% lower thana homogeneous distribution of renewables proportional toeach country’s mean load. This is because the capital costsof wind and solar dominate the total system costs, and al-lowing the system to build more VRES in countries withbetter capacity factors means that fewer wind turbines andsolar panels need to be built in order to produce the sameamount of energy.If the heterogeneity parameter K , which controls the maxi-mum and minimum levels of renewables generation in eachcountry compared to its mean load, is gradually relaxedfrom K = 1 (homogeneous) to larger values (heteroge-nous) then there is a clear trend of cost reduction, whichis steepest for smaller values of K and flattens out above K = 3. This has the important policy consequence thatEurope can profit from the benefits of heterogeneity with-out allowing renewable imbalances between countries tobecome excessive.The optimal mixing parameter between wind and solaris remarkably robust as the heterogeneity is increased,favouring a high proportion of wind of between 80% and90% in the VRES mix. The mixing parameter is, how-ever, sensitive to the relative capital costs of wind andsolar, dropping to between 60% and 70% as solar capitalcosts are decreased by 50% compared to the default costassumption.While the best results in terms of low total system costshave been obtained here by explicit optimisation, heuristicmethods for heterogeneously distributing wind and solarcapacities, based for example on capacity factors, produceresults that have costs only a few percent higher than theoptimal systems. Given the increased comprehensibility and transparency that heuristic methods provide, this maybe a price worth paying for policy makers. Acknowledgments
Tom Brown is funded by the CoNDyNet project, which issupported by the German Federal Ministry of Educationand Research under grant no. 03SF0472C. The responsi-bility for the contents lies solely with the authors.
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