The strong effect of network resolution on electricity system models with high shares of wind and solar
Martha Maria Frysztacki, Jonas Hörsch, Veit Hagenmeyer, Tom Brown
HHighlights
The strong effect of network resolution on electricity system models with high shares of wind andsolar
Martha Maria Frysztacki,Jonas Hörsch,Veit Hagenmeyer,Tom Brown• Highly-renewable European power system is optimized at high spatial resolution• High-resolution capacity placement for wind and solar reduces costs by up to 10.5%• Models with low network resolution ignore congestion, underestimating costs by 19%• Costs underestimated most when grid expansion limited by, e.g., public acceptance• Grid reinforcements relieve congestion and lower system costs by up to 15% a r X i v : . [ phy s i c s . s o c - ph ] J a n he strong effect of network resolution on electricity system modelswith high shares of wind and solar ⋆ Martha Maria Frysztacki a,1 , Jonas Hörsch a,b , Veit Hagenmeyer a and Tom Brown a,b a Institute for Automation and Applied Informatics, Karlsruhe Institute of Technology, 76344 Eggenstein-Leopoldshafen, Germany b Frankfurt Institute for Advanced Studies, Ruth-Moufang-Straße 1, 60438 Frankfurt am Main, Germany
A R T I C L E I N F O
Keywords :energy system modellingspatial scale clusteringtransmission grid modellingresource resolution
A B S T R A C T
Energy system modellers typically choose a low spatial resolution for their models based on admin-istrative boundaries such as countries, which eases data collection and reduces computation times.However, a low spatial resolution can lead to sub-optimal investment decisions for wind and solargeneration. Ignoring power grid bottlenecks within regions tends to underestimate system costs, whilecombining locations with different wind and solar capacity factors in the same resource class tendsto overestimate costs. We investigate these two competing effects in a capacity expansion model forEurope’s power system with a high share of renewables, taking advantage of newly-available high-resolution datasets as well as computational advances. We vary the number of nodes, interpolatingbetween a 37-node model based on country and synchronous zone boundaries, and a 512-node modelbased on the location of electricity substations. If we focus on the effect of renewable resource reso-lution and ignore network restrictions, we find that a higher resolution allows the optimal solution toconcentrate wind and solar capacity at sites with better capacity factors and thus reduces system costsby up to 10.5% compared to a low resolution model. This results in a big swing from offshore to on-shore wind investment. However, if we introduce grid bottlenecks by raising the network resolution,costs increase by up to 19% as generation has to be sourced more locally at sites with worse capacityfactors. These effects are most pronounced in scenarios where grid expansion is limited, for example,by low local acceptance. We show that allowing grid expansion mitigates some of the effects of thelow grid resolution, and lowers overall costs by around %.
1. Introduction
Electricity systems with high shares of wind and solarphotovoltaic generation require a fundamentally different kindof modelling to conventional power systems with only dis-patchable generation [67]. While investments in conven-tional power plants can be dimensioned according to simpleheuristics like screening curves [11], the assessment of windand solar resources requires a high temporal and spatial res-olution to capture their weather-driven variability. The needto assess investments in generation, transmission and flexi-bility options over thousands of representative weather anddemand situations, as well as over thousands of potential lo-cations, means that balancing model accuracy against com-putational resources has become a critical challenge.The effects of temporal resolution have been well re-searched in the electricity system planning literature, includ-ing the need for at least hourly modelling resolution [67, 13],the consequences of clustering representative conditions [45],and the need to include extreme weather events [54]. On thespatial side, it has been recognized that integrating renew-able resources on a continental scale can smooth large-scaleweather variations, particularly from wind, and avoid theneed for temporal balancing [24, 43, 57, 9, 48, 59]. However,there has been little research on the effects of spatial resolu-tions on planning results. This is partly due to the fact thatcollecting high-resolution spatial data is challenging, as well ⋆ This document is the results of the research project funded byHelmholtz Association under grant no. VH-NG-1352.
ORCID (s): Corresponding author [email protected] (Martha Maria) as the fact that optimization at high-resolution over large ar-eas is computationally demanding.Choosing the spatial resolution based on administrativeboundaries such as country borders –which is a common ap-proach in the literature [24, 57, 32]– fails to account for thevariation of resources inside large countries like Germany.Aggregating low-yield sites together with high-yield sitestakes away the opportunity to optimize generation placement,which distorts investment decisions and drives up costs.On the other hand, aggregating diverse resources to sin-gle points tends to underestimate network-related costs, sincethe models are blind to network bottlenecks that might hin-der the welfare-enhancing integration of renewable resourceslocated far from demand centers. The effects of networkrestrictions are all the more important given the apparentlow public acceptance for new overhead transmission lines[21, 31] and the long planning and construction times fornew grid infrastructure [27].In the present contribution we disentangle these two com-peting effects of spatial resolution by running simulationsin a model of the future European electricity system. Weoptimize investments and operation of generation, storageand transmission jointly in a system with a high share ofrenewables under a 95% reduction in CO emissions com-pared to 1990, which is consistent with European targetsfor 2050 [26]. A recently-developed, high-resolution, open-source model of the European transmission network, PyPSA-Eur [38], is sequentially clustered from 512 nodes down to37 nodes in order to examine the effects on optimal invest-ments in generation, transmission and storage. Martha Maria et al.:
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Page 1 of 15 he strong effect of network resolution on energy system models
Previous work in the engineering literature has focusedon the effect of different network clustering algorithms [41]on the flows in single power flow simulations [12, 34], orused clustering algorithms that are dependent on specific dis-patch situations [19, 66, 63] and therefore unsuitable whenmaking large changes to generation and transmission capac-ities. In the planning literature that considers a high share ofrenewables in the future energy system, the effects of cluster-ing applied separately to wind, solar and demand were inves-tigated in [65], but neglected potential transmission line con-gestion within large regions. In [47] the previous study wasextended by including a synthesized grid and renewable pro-files, but it ignored the existing topology of the transmissiongrid. Effects of varying the resolution were not consideredin either of the studies. Recent work has examined regionalsolutions for the European power system, but did not takeinto account existing transmission lines, potential low pub-lic acceptance for grid reinforcement or the grid flow physics[71]. Other studies have examined transmission grid expan-sion at substation resolution, but either the temporal reso-lution was too low to account for wind and solar variability[25, 36], or only single countries were considered [50, 1, 36],or transmission expansion was not co-optimized with gen-eration and storage [25, 16, 62]. The competing effect ofclustering transmission lines versus variable resource siteson the share of renewables was also discussed in [22], butthe report did not provide an analysis of how strongly the re-spective clustering impacts modeling and planning results.The effects of model resolution on system planning resultswere considered for the United States in [46], where a cost-benefit was seen for higher wind and solar resolution, butthe resource resolution was not separated from the networkresolution, and only a small number of time slices were con-sidered to represent weather variations.Advances in solver algorithms and code optimization inthe modelling framework PyPSA [14], as well as hardwareimprovements, allow us to achieve what was previously notpossible in the literature: the co-optimization of transmis-sion, generation and storage at high temporal and spatialresolution across the whole of Europe, while taking into ac-count linearized grid physics, existing transmission lines andrealistic restrictions on grid reinforcement. In previous workby some of the authors large effects of spatial resolution oninvestment results were seen [37], but because the resourceand network resolution were changed in tandem, it was notpossible to analyse which effect dominates the results. In thepresent contribution we present a new study design that sep-arates the effects of resource and network resolution, and wedo indeed see substantial differences.
2. Methods
In this section we present an overview of the underlyingmodel and the study design, before providing more details onthe clustering methodology and the investment optimisation.A list of notation is provided in Table 2.
Figure 1:
PyPSA-Eur model of the European electricity sys-tem, including all existing and planned high-voltage alternatingcurrent (HVAC) and direct current (HVDC) lines.
The study is performed in a model of the European elec-tricity system at the transmission level, PyPSA-Eur, which isfully described in a separate publication [38]. Here we givea brief outline of the input data.The PyPSA-Eur model shown in Figure 1 contains allexisting high-voltage alternating current (HVAC) and directcurrent (HVDC) lines in the European system, as well asthose planned by the European Network of Transmission Sys-tem Operators for Electricity (ENTSO-E) in the Ten YearNetwork Development Plan (TYNDP) [27]. The networktopology and electrical parameters are derived from the ENTSO-E interactive map [3, 73]. In total the network consists of4973 nodes, 5721 HVAC and 32 HVDC lines existing as of2018, as well as 279 HVAC and 29 HVDC planned lines.Historical hourly load data for each country are takenfrom the Open Power System Data project [5] and distributedto the nodes within each country according to population andgross domestic product data. Generation time series are pro-vided for the surrounding wind and solar plants based on his-torical wind and insolation data derived from the ERA5 re-analysis dataset [4] and the SARAH2 surface radiation dataset[55]. Renewable installation potentials are based on landcover maps, excluding for example nature reserves, cities orstreets.The model was partially validated in [38]. Further vali-dation against historical data was carried out in [29], whereit was shown that the model could reproduce curtailment ofwind and solar in Germany due to transmission bottlenecksin the years 2013-2018. The ability to reproduce historicalcongestion provides a strong check on the match between thetransmission network data and the availability of wind andsolar generation in the model.
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Case Short name Description 𝑠 and transmission nodes 𝑛 : 𝑠 = 𝑛 ∈ 𝑛 = 37 andincrease the number of generation and storage sites 𝑠 ∈ 𝑠 = 1024 and successivelyincrease the number of transmission nodes 𝑛 ∈ Table 1
Case descriptions. ( = {37} ∪ {⌊√ 𝑖 ⌋} 𝑖 =11 ,..., = {37 , , , , , , , , ) The nodes of the model are successively clustered in spaceinto a smaller number of representative nodes using the 𝑘 -means algorithm [35]. This groups close-by nodes together,so that, for example, multiple nodes representing a singlecity are merged into one node. Nodes from different coun-tries or different synchronous zones are not allowed to bemerged; to achieve this, the overall number of desired nodesis partitioned between the countries and synchronous zonesbefore the 𝑘 -means algorithm is applied in each partitionseparately. In total there are 37 ‘country-zones’ in the model,i.e. regions of countries belonging to separate synchronouszones.Figure 2, Case 1 shows the results for Ireland and theUnited Kingdom (where Northern Ireland is in a separatesynchronous zone to Great Britain). Once the nodes havebeen clustered, they are reconnected with transmission corri-dors representing the major transmission lines from the high-resolution model. Electricity demand, conventional genera-tion and storage options are also aggregated to the nearestnetwork node. More technical details on the clustering canbe found in subsection 2.5. An analysis of the effects of clus-tering on the network flows can be found in the Appendix,Section A.1. To separate the effects of the spatial resolution on the re-newable resources and the network, we consider three casesin which they are clustered differently. The three cases aresummarized in Table 1 and shown graphically in Figure 2 foreach case (rows) and for each level of clustering (columns).In
Case 1 the wind and solar sites are clustered to thesame resolution as the network. The number of clusters isvaried between 37, the number of country-zones, and 512,which represents the maximum resolution for which genera-tion, transmission and storage investment can be co-optimizedin reasonable time. The number of nodes is increased in half-powers of 2, so that nine different resolutions are considered: = {37} ∪ {⌊√ 𝑖 ⌋} 𝑖 =11 ,..., .In Case 2 network bottlenecks inside each country-zoneare removed so that there are only 37 transmission nodes, andonly the resolution of the wind and solar generation is varied.Inside each country-zone, all wind and solar generators areconnected to the central node. This allows the optimization to exploit the best wind and solar sites available.Finally in
Case 3 we fix a high resolution of renewablesites and vary the number of network nodes, in order to ex-plore the effects of network bottlenecks. Each renewable siteis connected to the nearest network node, where the trans-mission lines, electricity demand, conventional generatorsand storage are also connected.For each case we optimize investments and operationfor wind and solar power, as well as open cycle gas tur-bines, batteries, hydrogen storage and transmission. Flex-ibility from existing hydroelectric power plants is also takeninto account. The model is run with perfect foresight at a 3-hourly temporal resolution over a historical year of load andweather data from 2013, assuming a 95% reduction in CO emissions compared to 1990. The temporal resolution is 3-hourly to capture changes in solar generation and electricitydemand while allowing reasonable computation times. Thetechnology selection is also limited for computational rea-sons. More details on the investment optimization can befound in subsection 2.6.For each simulation we also vary the amount of newtransmission that can be built, in order to understand theeffect of possible grid reinforcements on the results. Themodel is allowed to optimize new transmission reinforce-ments to the grid as it was in 2018, up to a limit on the sumover new capacity multiplied by line length measured rela-tive to the grid capacity in 2018. For example, a transmis-sion expansion of 25% means that on top of 2018’s grid, newlines corresponding to a quarter of 2018’s grid can be addedto the network. The exact constraint is given in equation (17)in subsection 2.6. Before the clustering algorithm can be applied to the net-work, several simplifications are applied to the data.In order to avoid the difficulty of keeping track of dif-ferent voltage levels as the network is clustered, all lines aremapped to their electrical equivalents at 380 kV, the mostprevalent voltage in the European transmission system. Ifthe original reactance of the line 𝓁 𝑖,𝑗 was 𝑥 𝑖,𝑗 at its originalvoltage 𝑣 𝑖,𝑗 , the new equivalent reactance becomes 𝑥 ′ 𝑖,𝑗 = 𝑥 𝑖,𝑗 ( kV 𝑣 𝑖,𝑗 ) . (1)This guarantees that the per unit reactance is preserved after Martha Maria et al.:
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Page 3 of 15he strong effect of network resolution on energy system models generation site transmission node site connection transmission line
Figure 2:
Clustering of network nodes (red, number 𝑛 ) and renewable sites (grey, number 𝑠 ) in each of the cases (rows) forIreland and the United Kingdom at different levels of clustering (columns). the equivalencing.The impedances and thermal ratings of all transformersare neglected, since they are small and cannot be consistentlyincluded with the mapping of all voltage levels to 380 kV.Univalent nodes, also known as dead-ends, are removedsequentially until no univalent nodes exist. That is, if node 𝑖 has no other neighbor than node 𝑗 , then node 𝑖 is merged tonode 𝑗 . We repeat the process until each node is multi-valentand update the merged node attributes and its attached assets(loads, generators and storage units) according to the rulesin Table 5.HVDC lines in series or parallel are simplified to a singleline 𝓁 using the rules in Tables 6 and 7. Capital costs perMW of capacity for HVDC lines 𝓁 𝑖,𝑗 with length 𝑙 𝓁 𝑖,𝑗 and afraction 𝑢 𝓁 𝑖,𝑗 ∈ [0 , underwater are given by 𝑐 𝑖,𝑗 = 1 . ⋅ 𝑙 𝑖,𝑗 ⋅ ( 𝑢 𝑖,𝑗 ⋅ 𝑐 marine + (1 − 𝑢 𝑖,𝑗 ) ⋅ 𝑐 ground ) , where 𝑐 marine is the capital cost for a submarine connection and 𝑐 ground for an underground connection. The factor of . accounts for indirect routing and height fluctuations. Different methods have been used to cluster networks inthe literature. We chose a version of 𝑘 -means clustering [35]based on the geographical location of the original substa-tions in the network, weighted by the average load and con-ventional capacity at the substations, since this representshow the topology of the network was historically planned toconnect major generators to major loads. It leaves the longtransmission lines between regions, which are expensive toupgrade and are more likely to encounter low local accep-tance, unaggregated, so that these lines can be optimized inthe model. Regions with a high density of nodes, for ex-ample around cities, are aggregated together, since the shortlines between these nodes are inexpensive to upgrade andrarely present bottlenecks. Geographical 𝑘 -means cluster-ing has the advantage over other clustering methods of not Martha Maria et al.:
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Page 4 of 15he strong effect of network resolution on energy system models making any assumptions about the future generation, storageand network capacity expansion.Other clustering methods applied in the literature are notsuitable for the co-optimization of supply and grid technolo-gies: these include clustering based on electrical distance[70, 12, 23, 2, 10] (which we do not use because we wantto optimize new grid reinforcements that alter electrical dis-tances), spectral partitioning of the graph Laplacian matrix[34] (avoided for same reason), an adaptation of 𝑘 -meanscalled 𝑘 -means ++ combined with a max- 𝑝 regions algorithmapplied to aggregate contiguous sites with similar wind, so-lar and electricity demand [65] (avoided since we want a co-herent clustering of all network nodes and assets), hierar-chical clustering based on a database of electricity demand,conventional generation and renewable profiles including asynthesized grid [47] (avoided for the same reason and be-cause we do not want to alter the topology of the existingtransmission grid), 𝑘 -means clustering based on renewableresources as well as economic, sociodemographic and ge-ographical features [20] (avoided because we need a clus-tering focused on network reduction), as well as clusteringbased on zonal Power Transfer Distribution Factors (PTDFs)[19, 53, 64] (avoided because they encode electrical param-eters that change with reinforcement), Available Tranfer Ca-pacities (ATCs) [63] (avoided because they depend on pre-defined dispatch patterns) and locational marginal prices (LMP)[66] (again avoided because they depend on pre-defined dis-patch patterns).We do not allow nodes in different countries or differ-ent synchronous zones to be clustered together, so that wecan still obtain country-specific results and so that all HVDCbetween synchronous zones are preserved during the aggre-gation. This results in a minimum number of 37 clusterednodes for the country-zones. First we partition the desiredtotal number 𝑛 of clusters between the 37 country-zones,then we apply the 𝑘 -means clustering algorithm within eachcountry-zone.In order to partition the 𝑛 nodes between the 37 country-zones, the following minimisation problem is solved argmin { 𝑛 𝑧 }∈ ℕ ∑ 𝑧 =1 ( 𝑛 𝑧 − 𝐿 𝑧 ∑ 𝑦 𝐿 𝑦 𝑛 ) , (2)where 𝐿 𝑧 is the total load in each country-zone 𝑧 . An ad-ditional constraint ensures that the number of clusters percountry-zone matches the desired number of clusters for thewhole network: ∑ 𝑧 𝑛 𝑧 = 𝑛 .Then the 𝑘 -means algorithm is applied to partition thenodes inside each country-zone into 𝑛 𝑧 clusters. The algo-rithm finds the partition that minimizes the sum of squareddistances from the mean position of each cluster 𝑥 𝑐 ∈ ℝ tothe positions 𝑥 𝑖 ∈ ℝ of its members 𝑖 ∈ 𝑁 𝑐 min { 𝑥 𝑐 ∈ ℝ } 𝑘 ∑ 𝑐 =1 ∑ 𝑖 ∈ 𝑁 𝑐 𝑤 𝑖 ⋅ ‖ 𝑥 𝑐 − 𝑥 𝑖 ‖ . (3)Each node is additionally assigned a normalised weighting 𝑤 𝑖 based on its nominal power for conventional generators and averaged load demand: 𝑤 𝑖 = ∑ 𝑠 conv . 𝐺 𝑖,𝑠 ∑ 𝑠 conv . ∑ 𝐵𝑖 =1 𝐺 𝑖,𝑠 + 𝑑 𝑖,𝑇 ∑ 𝐵𝑖 =1 𝑑 𝑖,𝑇 , ∀ 𝑖 (4)where 𝑑 𝑖,𝑇 corresponds to the averaged demand over the con-sidered time period 𝑇 . 𝑤 𝑖 is normalised according to ⌊ ⋅ 𝑤 𝑖 ‖ 𝑤 ‖ max ⌋ .The optimization is run with 𝑛 init = 10 different cen-troid seeds, a maximum number of iterations for a single runof max iter = 3 ⋅ and a relative tolerance with regards toinertia to declare convergence of 𝜀 = 10 −6 .Attributes of the nodes in 𝑁 𝑐 and their attached assetsare aggregated to the clustered node 𝑐 according to the rulesin Table 5.Lines connecting nodes 𝑁 𝑐 in cluster 𝑐 with nodes 𝑁 𝑑 in cluster 𝑐 , given by the set 𝑁 𝑐,𝑑 𝑁 𝑐,𝑑 = { 𝓁 𝑖,𝑗 , 𝑖 ∈ 𝑁 𝑐 , 𝑗 ∈ 𝑁 𝑑 } , ∀ 𝑐, 𝑑 (5)are aggregated to a single representative line. The lengthof the representative line is determined using the haversineformula (which computes the great-circle distance betweentwo points on a sphere) multiplied by a factor of . to takeindirect routing into account. The representative line inheritsthe attributes of the lines 𝑁 𝑐,𝑑 as described in Table 7. Ifany of the replaced lines in 𝑁 𝑐,𝑑 had the attribute that theircapacity was extendable, then the aggregated line inheritsthis extendability.An analysis of the effects of clustering on the networkflows can be found in the Appendix, Section A.1.For Case 1, generators are clustered to the same resolu-tion as the network. Times series containing hourly resolvedcapacity factors ̄𝑔 𝑖,𝑠,𝑡 ∈ [0 , for variable renewable genera-tion are aggregated using a weighted average ̄𝑔 𝑐,𝑠,𝑡 = 1 ∑ 𝑖 ∈ 𝑁 𝑐 𝑤 𝑖,𝑠 ∑ 𝑖 ∈ 𝑁 𝑐 𝑤 𝑖,𝑠 ⋅ ̄𝑔 𝑖,𝑠,𝑡 , ∀ 𝑐, 𝑠, 𝑡 (6)The resulting capacity factor ̄𝑔 𝑐,𝑠,𝑡 is in [0 , by definition.For renewables, the weighting 𝑤 𝑖,𝑠 is proportional to the max-imal yearly yield for technology 𝑠 at node 𝑖 , found by multi-plying the maximal installable capacity 𝐺 max 𝑖,𝑠 with the aver-age capacity factor. In the case of conventional technologiesthe weightings are distributed equally, i.e 𝑤 𝑖,𝑠 = 1 . Note thatthere is no relation between the weightings 𝑤 𝑖,𝑠 and the busweightings 𝑤 𝑖 of (4).For Case 2, the network is fixed at 37 nodes, and thewind and solar generators are merged in the aggregation step.Time series for VRE availability are aggregated according to(6) to their respective resolution.For Case 3, the network is clustered, but wind and solargenerators are not merged in the aggregation step. Their timeseries remain fixed at high resolution of 1024 nodes. Investments in generation, storage and transmission areoptimized in the PyPSA modelling framework [14], which
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Page 5 of 15he strong effect of network resolution on energy system models minimises the total system costs. The objective function is min 𝐺 𝑖,𝑠 , 𝐹 𝓁 ,𝑔 𝑖,𝑠,𝑡 , 𝑓 𝓁 ,𝑡 [ 𝐵 ∑ 𝑖 =1 𝑆 ∑ 𝑠 =1 ( 𝑐 𝑖,𝑠 𝐺 𝑖,𝑠 + 𝑇 ∑ 𝑡 =1 𝑤 𝑡 𝑜 𝑖,𝑠 𝑔 𝑖,𝑠,𝑡 ) + 𝐿 ∑ 𝓁 =1 𝑐 𝓁 𝐹 𝓁 ] , consisting of the annualised fixed costs 𝑐 𝑖,𝑠 for capacities 𝐺 𝑖,𝑠 at each node 𝑖 and storage/generation technology 𝑠 , the dis-patch 𝑔 𝑖,𝑠,𝑡 of the unit at time 𝑡 and associated variable costs 𝑜 𝑖,𝑠 multiplied by a weight factor 𝑤 𝑡 corresponding to thetemporal resolution of the system, and the line capacities 𝐹 𝓁 for each line 𝓁 including both high voltage alternating cur-rent and direct current lines and their annualised fixed costs 𝑐 𝓁 . The time period 𝑇 runs over a full year at a 3-hourlyresolution, so each time period 𝑡 is weighted with 𝑤 𝑡 = 3 .Investment cost assumptions are provided in Table 3, basedon projections for the year 2030 [7, 61, 17, 72]. 2030 is cho-sen for the cost projections since this is the earliest possi-ble time that such a system transformation might be feasible,and because it results in conservative cost assumptions com-pared to projections for a later date. The only CO -emittinggenerators are the open cycle gas turbines with natural gaswith specific emissions 0.187 tCO /MWh th and fuel cost21.6 €/MWh th . Investment costs are annualized with a dis-count rate of 7%. Lifetimes, efficiencies and operation andmaintenance costs can be found in the GitHub repository [8].The dispatch of conventional generators 𝑔 𝑖,𝑠,𝑡 is constrainedby their capacity 𝐺 𝑖,𝑠 ≤ 𝑔 𝑖,𝑠,𝑡 ≤ 𝐺 𝑖,𝑠 ∀ 𝑖, 𝑡, 𝑠 ∈ 𝐶𝐺 (7)The maximum producible power of renewable genera-tors depends on the weather conditions, which is expressedas an availability ̄𝑔 𝑖,𝑠,𝑡 per unit of its capacity: ≤ 𝑔 𝑖,𝑠,𝑡 ≤ ̄𝑔 𝑖,𝑠,𝑡 𝐺 𝑖,𝑠 ∀ 𝑖, 𝑡, 𝑠 ∈ 𝑅𝐸 (8)The installable renewable capacity 𝐺 𝑖,𝑠 is constrained byland eligibility for placing e.g. wind turbines or solar panelsin each node and for each renewable technology. The land re-strictions are derived using the Geospatial Land Availabilityfor Energy Systems (GLAES) tool [58] and are always finitefor renewable carriers: 𝐺 𝑖,𝑠 ≤ 𝐺 max 𝑖,𝑠 < ∞ ∀ 𝑖, 𝑠 ∈ 𝑅𝐸 (9)There is no capacity constraint for conventional genera-tors: 𝐺 𝑖,𝑠 < ∞ ∀ 𝑖, 𝑠 ∈ 𝐶𝐺 (10)The energy levels 𝑒 𝑖,𝑠,𝑡 of all storage units have to be con-sistent between all hours and are limited by the storage en-ergy capacity 𝐸 𝑖,𝑠 𝑒 𝑖,𝑠,𝑡 = 𝜂 𝑤 𝑡 𝑒 𝑖,𝑠,𝑡 −1 + 𝜂 𝑤 𝑡 [ 𝑔 𝑖,𝑠,𝑡 ] − − 𝜂 −12 𝑤 𝑡 [ 𝑔 𝑖,𝑠,𝑡 ] + + 𝑤 𝑡 𝑔 inflow 𝑖,𝑠,𝑡 − 𝑤 𝑡 𝑔 spillage 𝑖,𝑠,𝑡 ≤ 𝑒 𝑖,𝑠,𝑡 ≤ 𝐸 𝑖,𝑠 ∀ 𝑖, 𝑠, 𝑡 (11) Positive and negative parts of a value are denoted as [ ⋅ ] + =max( ⋅ , , [ ⋅ ] − = − min( ⋅ , . The storage units can havea standing loss 𝜂 , a charging efficiency 𝜂 , a dischargingefficiency 𝜂 , inflow (e.g. river inflow in a reservoir) andspillage. The energy level is assumed to be cyclic, i.e. 𝑒 𝑖,𝑠,𝑡 =0 = 𝑒 𝑖,𝑠,𝑡 = 𝑇 .CO emissions are limited by a cap CAP 𝐶𝑂 , implementedusing the specific emissions 𝑒 𝑠 in CO -tonne-per-MWh ofthe fuel 𝑠 and the efficiency 𝜂 𝑖,𝑠 of the generator: ∑ 𝑖,𝑠,𝑡 𝜂 𝑖,𝑠 𝑤 𝑡 ⋅ 𝑔 𝑖,𝑠,𝑡 ⋅ 𝑒 𝑠 ≤ CAP 𝐶𝑂 ↔ 𝜇 𝐶𝑂 (12)In all simulations this cap was set at a reduction of 95% ofthe electricity sector emissions from 1990.The (perfectly inelastic) electricity demand 𝑑 𝑖,𝑡 at eachnode 𝑖 must be met at each time 𝑡 by either local generatorsand storage or by the flow 𝑓 𝓁 ,𝑡 from a transmission line 𝓁 ∑ 𝑠 𝑔 𝑖,𝑠,𝑡 − 𝑑 𝑖,𝑡 = ∑ 𝓁 𝐾 𝑖, 𝓁 𝑓 𝓁 ,𝑡 ∀ 𝑖, 𝑡 (13)where 𝐾 𝑖, 𝓁 is the incidence matrix of the network. Thisequation is Kirchhoff’s Current Law (KCL) expressed in termsof the active power.In the present paper the linear load flow is used, whichhas been shown to be a good approximation for a well-compensatedtransmission network [69, 16]. To guarantee the physicalityof the network flows, in addition to KCL, Kirchhoff’s Volt-age Law (KVL) must be enforced in each connected network.KVL states that the voltage differences around any closedcycle in the network must sum to zero. If each independentcycle 𝑐 is expressed as a directed combination of lines 𝓁 bya matrix 𝐶 𝓁 ,𝑐 then KVL becomes the constraint ∑ 𝓁 𝐶 𝓁 ,𝑐 𝑥 𝓁 𝑓 𝓁 ,𝑡 = 0 ∀ 𝑐, 𝑡 (14)where 𝑥 𝓁 is the series inductive reactance of line 𝓁 . It wasfound in [40] that expressing the linear load flow equations inthis way with cycle constraints is computationally more ef-ficient than angle- or PTDF-based formulations. Note thatpoint-to-point HVDC lines have no cycles, so there is noconstraint on their flow beyond KCL.The flows are also constrained by the line capacities 𝐹 𝓁 | 𝑓 𝓁 ,𝑡 | ≤ 𝑏 𝐵 ⋅ 𝐹 𝓁 ∀ 𝓁 , 𝑡 (15)Although the capacities 𝐹 𝓁 are subject to optimisation, nonew grid topologies are considered beyond those planned inthe TYNDP 2018 [27]. The factor 𝑏 𝐵 = 0 . leaves a bufferof 30% of the line capacities to account for 𝑛 − 1 line outagesand reactive power flows. The choice of 70% for 𝑏 𝐵 is stan-dard in the grid modelling literature [68, 18, 30, 16] and isalso the target fraction of cross-border capacity that shouldbe available for cross-border trading in the European Union(EU) by 2025, as set in the 2019 EU Electricity Market Reg-ulation [6].Since line capacities 𝐹 𝓁 can be continuously expanded torepresent the addition of new circuits, the impedances 𝑥 𝓁 of Martha Maria et al.:
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Page 6 of 15he strong effect of network resolution on energy system models
37 45 64 90 128 181 256 362 512 number of clusters c o s t s [ y e a r ] Case 1
37 45 64 90 128 181 256 362 512 1024 number of clusters
Case 2 transmission expansion [%] number of clusters
Case 3
Figure 3:
Total annual system costs as a function of the number of clusters for Cases 1, 2 and 3. the lines would also decrease. In principle this would intro-duce a bilinear coupling in equation (14) between the 𝑥 𝓁 andthe 𝑓 𝓁 ,𝑡 . To keep the optimisation problem linear and there-fore computationally fast, 𝑥 𝓁 is left fixed in each optimisa-tion problem, updated and then the optimisation problem isrun, in up to 4 iterations to ensure convergence, followingthe methodology of [33, 51].In order to investigate the effects of transmission expan-sion, each line capacity 𝐹 𝓁 can be extended beyond the ca-pacity in 2018, 𝐹 𝓁 ≥ 𝐹 𝓁 , up to a a line volume cap CAP trans ,which is then varied in different simulations: ∑ 𝓁 𝑙 𝓁 ⋅ ( 𝐹 𝓁 − 𝐹 𝓁 ) ≤ CAP trans ↔ 𝜇 trans (16)The caps are defined in relation to 2018’s line capacities 𝐹 𝓁 , i.e.CAP trans = 𝑥 ⋅ ∑ 𝓁 𝑙 𝓁 ⋅ 𝐹 𝓁 (17)where 𝑥 is varied between zero and 50%.Since there is a cap on the transmission expansion, theline costs 𝑐 𝓁 can be set to zero. For the results, costs areadded after the simulation based on the assumptions in Table3. The optimised model returns the spatially-resolved ca-pacity for each technology 𝐺 𝑖,𝑠 as well as the amount oftransmission expansion of each included line 𝐹 𝓁 . Addition-ally, the results also provide dispatch time series for eachof the generators 𝑔 𝑖,𝑠,𝑡 and electricity flows 𝑓 𝓁 ,𝑡 for includedlines that obey the constraints described above in subsection2.6.
3. Results
Figure 3 presents the total annual system costs for eachcase. To obtain a better understanding of the system com-position, Figure 4 breaks down the total costs into individ-ual components when there is no grid expansion. In Figure5 we present total system costs for different grid expansionscenarios for 256 clusters in the simultaneous case (Case 1). An example map of investments can be found in Figure 6 fora 25% grid expansion (a similar level to ENTSO-E’s TYNDP[27]).
If the resource and network resolutions increase in tan-dem according to Case 1 without grid expansion, the totalannual system costs in Figure 3 rise gently with the increas-ing number of nodes, reaching a maximum of billion eu-ros per year at 512 nodes, which is 9% more expensive thanthe solution with 37 nodes. This corresponds to an averagesystem cost of 87 €/MWh. If some transmission expansionis allowed, costs are lower, and there is almost no change intotal system costs as the number of nodes is varied.However, the fact that costs are flat does not mean thatthe solutions are similar: a large shift from offshore windat low resolution to onshore wind at high resolution can beobserved in the left column of Figure 4 (Case 1). This is anindication that spatial resolution can have a very strong effecton energy modelling results. To understand what causes thiseffect, we must examine Cases 2 and 3.
In Case 2 we use the lowest network resolution of 37nodes, corresponding to one-node-per-country-zone, and in-vestigate the effect of changing the number of wind and solarsites on the results. As the resolution increases, total costswithout grid expansion in Figure 3 drop by . % from to billion euro per year. Although the slope of the costcurve appears constant, note that the 𝑥 -axis is logarithmic,so that the rate of cost decrease slows as the number of sitesincreases.The cost reduction is driven by strong changes in the in-vestment between generation technologies, particularly theratio between offshore and onshore wind (see Figure 4). Atlow spatial resolution, good and bad onshore sites are mixedtogether, diluting onshore capacity factors and making on-shore a less attractive investment. Figure 9 in the Appendixshows how the capacity factors for wind and solar vary acrossthe continent. While offshore is spatially concentrated and Martha Maria et al.:
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Page 7 of 15he strong effect of network resolution on energy system models c o s t s [ y e a r ] Case 1 Case 2 t y p e = g e n . Case 3
37 45 64 90 128 181 256 362 512 number of clusters
37 45 64 90 128 181 256 362 512 number of clusters
37 45 64 90 128 181 256 362 512 number of clusters t y p e = f l e x . onshore windoffshore windsolarrun of riveropen cycle gas turbinepumped hydro storagebattery storagehydrogen storagetransmission lines Figure 4:
Breakdown of the annual system costs for generation (top) and flexibility options (bottom) as a function of the numberof clusters for Cases 1, 2 and 3 when there is no grid expansion. transmission expansion [%] c o s t s [ y e a r ] Case 1: simultaneous clustering onshore windoffshore windsolaropen cycle gas turbinerun of riverpumped hydro storagebattery storagehydrogen storagetransmission lines
Figure 5:
Costs as a function of the transmission expansionlevel for 256 nodes in Case 1. solar capacity factors are relatively evenly spread in eachcountry-zone, onshore wind is stronger near coastlines. Athigh spatial resolution the model can choose to put onshorewind only at the best sites (within land restrictions), increas-ing average capacity factors and thus lower the per-MWh-cost. (The increasing average capacity factors are plotted inFigure 11 in the Appendix.) As a result, onshore wind in-vestments more than double from to billion euros peryear, while offshore investments drop % from to billion per year and solar by . The biggest effect on thetechnology mix is when going from 37 to around clus-ters; beyond that the changes are smaller. In Case 3 we fix a high resolution of wind and solargenerators (1024 sites) and vary the resolution of the trans-mission network to gauge the impact of transmission bot-tlenecks. With 37 network nodes many bottlenecks are notvisible, so costs are lower, but as the resolution increases to512 nodes it drives up the costs by %. Note that becausethe 𝑥 -axis is logarithmic, the highest rate of cost increase iswhen the number of nodes is small.As can be seen from the breakdown in Figure 4, the ris- ing transmission investments from the higher resolution onlyhave a small contribution to the result. Instead, rising costsare driven by generation and storage. Unlike Cases 1 and2, the ratio between the generation technologies does notchange dramatically with the number of clusters, but the ca-pacities for onshore wind, solar, batteries and hydrogen stor-age all rise.The transmission bottlenecks limit the transfer of powerfrom the best sites to the load, forcing the model to buildonshore wind and solar more locally at sites with lower ca-pacity factors. Average capacity factors of onshore wind andsolar sink by % and % respectively with no grid expan-sion (see Figure 11 in the Appendix), meaning that morecapacity is needed for the same energy yield. Curtailmentis generally low in the optimal solution (around 3% of avail-able wind and solar energy) and has less of an effect on costs(see Figure 12 in the Appendix).Investment in battery and hydrogen storage rises with thenumber of network nodes since the storage is used to balancelocal wind and solar variations in order to avoid overloadingthe grid bottlenecks. Separating the effects of resource resolution from net-work resolution reveals that the apparent stability of totalsystem costs in Case 1 in Figure 3 as the number of clusterschanges, as reported in [37], is deceptive. In fact, the sinkingcosts from the higher resource resolution are counter-actedby the rising costs from network bottlenecks. With no gridexpansion, the system cost of network bottlenecks is doublethe benefit of the higher resource resolution.While these two effects offset each other at the level oftotal system costs, they have very different effects on thetechnology mix. Resource resolution leads to much strongerinvestment in onshore wind, once good sites are revealed.Network bottlenecks have only a weak effect on the ratio ofgeneration technologies, but lead to lower average capacityfactors and drive up storage requirements.
Grid expansion does not affect the main qualitative fea-tures of the different Cases, but it does have the overall ef-
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Page 8 of 15he strong effect of network resolution on energy system models
Figure 6:
Example of investments with 25% grid expansionand 256 nodes in Case 1. fect of lowering total system costs. In Case 1, the total cost-benefit of grid expansion is highest at around 15% for a 50%increase in grid capacity, with the marginal benefit still in-creasing, but it is subject to diminishing returns (see Ap-pendix Figure 14 for a comparison of the marginal benefitto the cost of transmission). The first 9% of additional gridcapacity brings total cost savings of up to 8%, but for each ex-tra increment of grid expansion, the benefit is weaker. Thereis more benefit from grid expansion at a higher number ofnodes, since the higher network resolution reveals more crit-ical bottlenecks in the transmission system.The total savings from 25% and 50% grid expansion arearound 33 and 40 billion euros per year respectively. In a2018 study ENTSO-E examined scenarios with up to 75%renewable electricity in Europe in 2040 with and withoutplanned TYNDP grid expansions (corresponding to around25% grid expansion), given fixed demand and a fixed gener-ation fleet. They found that the grid reinforcements broughta cost benefit of 43 billion euros per year. This is higher thanour value, despite their study’s lower level of renewable elec-tricity, because in our simulations the generation and storagefleet can be re-optimised to accommodate the lower level ofgrid capacity.The breakdown of system cost as the grid is expandedfor a fixed number of clusters (256), plotted in Figure 5, re-veals how costs are reduced. Although the investment intransmission lines rises, generation and storage costs reducefaster as investment shifts from solar and onshore wind tooffshore wind. Offshore wind reduces costs because of itshigh capacity factors and more regular generation pattern intime. It can be transported around the continent more eas-ily with more transmission, and benefits from the smooth-ing effects over a large, continental area that grid expan-sion enables. The map of investments in Figure 6 showshow offshore wind is balanced by new transmission around the North Sea, which smooths out weather systems that rollacross the continent from the Atlantic. Further transmissionreinforcements bring energy inland from the coastlines toload centers. With more transmission, there is less invest-ment in battery and hydrogen storage, as a result of the betterbalancing of weather-driven variability in space.Turning to Case 3, we see that grid expansion mitigatesthe effect of network resolution by allowing bottlenecks to bealleviated. For a 50% increase in transmission capacity, totalcosts rise by only 3% from 90 nodes up to 512 nodes. Thedistribution of investments between technologies also barelychanges in this range (see Appendix Figure 10). This meansthat a grid resolution of around 90 nodes can give accept-able solutions for grid expansion scenarios if computationalresources are limited, as long as the wind and solar resolu-tion is high enough (as in Case 2, 181 generation sites wouldsuffice). Without grid expansion, a higher grid resolution isneeded to capture the effects of bottlenecks and achieve re-liable results.
Besides the poor availability of data at high resolution,one of the main motivations for clustering the network isto reduce the number of variables and thus the computationtime of the optimisation. In Appendix Figure 15 the memoryand solving time requirements for each Case are displayed asa function of the number of clusters. Both memory and solv-ing time become limiting factors in Cases 1 and 3, with ran-dom access memory (RAM) usage peaking at around 55 GBand solving time at around 2 days for 500 clusters. Beyondthis number of clusters no consistent convergence in the so-lutions was seen.Case 2, where the network resolution is left low and theresource resolution is increased, shows five times lower mem-ory consumption and eight times faster solving times com-pared to Cases 1 and 3 for the same number of clusters. It istherefore the network resolution rather than the resource res-olution that drives up computational requirements, which itdoes by introducing many new variables and possible spatialtrade-offs into the optimisation. Since Case 2 proved rela-tively reliable for estimating the ratio between technologies,if not their total capacity, it may prove attractive to increasethe resource resolution rather than the network resolution ifcomputational resources are limited.
Further results on curtailment, average capacity factors,the distribution of technologies between countries, maps, net-work flows and shadow prices can be found in the Appendix,as well as a discussion of the limitations of the model.
4. Discussion
From these investigations we can draw several conclu-sions. Modellers need to take account of spatial resolution,since it can have a strong effect on modelling results. In ourco-optimization of generation, storage and network capac-ities, higher network resolution can drive up total system
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Page 9 of 15he strong effect of network resolution on energy system models costs by as much as 19%. Higher costs are driven by thenetwork bottlenecks revealed at higher resolution that limitaccess to wind and solar sites with high capacity factors.On the other hand, resource resolution affects the balance oftechnologies by revealing more advantageous onshore windsites. In both cases the system costs are driven more by theuseable generation resources than investments in the grid orstorage.If grid expansion can be assumed, a grid resolution of90 nodes for Europe is sufficient to capture costs and tech-nology investments as long as the solar and onshore windresolution is at least around 181 nodes. If grid expansionis not possible, a higher spatial resolution for the grid is re-quired for reliable results on technology choices. Since gridexpansion is likely to be limited in the future by low publicacceptance, more attention will have to be paid to the compu-tational challenge of optimizing investments at high spatialgranularity.
5. Data availability
Please contact the Lead Contact, Martha M. Frysztacki([email protected]), for information related to thedata and code described in the following Material and Meth-ods section.
No materials were used in this study.
All the code and input data from PyPSA-Eur are openlyavailable online on GitHub and Zenodo [8, 39]. All modeloutput data is available on Zenodo under a Creative Com-mons Attribution Licence [28].
6. Glossary
All notation is listed in Table 2.
7. Acknowledgements
We thank Martin Greiner, Fabian Neumann, Lina Re-ichenberg, Mirko Schäfer, David Schlachtberger, Kais Sialaand Lisa Zeyen for helpful discussions, suggestions and com-ments. MF, JH and TB acknowledge funding from the HelmholtzAssociation under grant no. VH-NG-1352. The responsibil-ity for the contents lies with the authors.
8. Declaration of Interests
The authors declare that they have no competing finan-cial interests.
A. Appendix
A.1. Preservation of flow patterns with clustering
To understand how well the 𝑘 -means clustering preservesflow patterns, we took a fixed dispatch pattern for the assets
37 45 64 90 128 181 256 362 512 1024 number of clusters P e a r s o n ' s r Figure 7:
Pearson’s correlation coefficient of mapped flows(blue). Note that the x-axis is non-linear, therefore we mark alinear fit to the data (red). F l o w s f r o m a gg r e g a t e d p o w e r i n j e c t i o n . . . . . . . . . . . . . F l o w s f r o m a gg r e g a t e d p o w e r i n j e c t i o n . . . . . . . Figure 8:
Kernel Density Estimation (KDE) of aggregatedflows from a high resolution network grid with 1024 nodes onthe 𝑥 -axis and a low resolution grid with 45 nodes (left) and362 nodes (right) on the 𝑦 -axis. 0.25, 0.5 and 0.75 quantilesof the distribution are displayed as purple isolines around theKDE. in Europe at high resolution and examined how the networkflows changed as the network was clustered.The fixed dispatch was determined by solving the lin-earised optimal power flow problem for a 1024-node repre-sentation of today’s European electricity system. The assetdispatch was then mapped into the clustered networks, anda regular linearised power flow was solved in the clusterednetwork. Martha Maria et al.:
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Page 10 of 15he strong effect of network resolution on energy system models
If lines 𝓁 ∈ 𝑁 𝑐,𝑑 in the 1024-node network were mappedto a single representative line 𝓁 𝑐,𝑑 in the clustered network,the summed flows from the original network ̂𝑓 𝑐,𝑑,𝑡 = ∑ 𝓁 ∈ 𝑁 𝑐,𝑑 𝑓 𝓁 ,𝑡 (‘microscopic flows’) were then compared to the flow 𝑓 𝑐,𝑑,𝑡 in line 𝓁 𝑐,𝑑 of the clustered network (‘macroscopic flows’).Figure 7 shows the Pearson correlation coefficient be-tween the flows 𝑓 𝑐,𝑑,𝑡 of aggregated lines 𝓁 𝑐,𝑑 in the lowerresolution network and the summed flows ̂𝑓 𝑐,𝑑,𝑡 of all linesin 𝑁 𝑐,𝑑 in the full resolution network. Red is a linear fitthrough the points. The distortion from linearity is due toa non-linear scale in the 𝑥 -axis. Even at 37 nodes the cor-relation between the flows is good (Pearson correlation co-efficient above 0.90) and shows an improving trend until atfull 1024-node resolution the flows are once again perfectlyequal.Example density plots of the ̂𝑓 𝑐,𝑑,𝑡 against the 𝑓 𝑐,𝑑,𝑡 forall lines and all times are plotted for different clustering lev-els in Figure 8. The match between the flows is better forhigher resolution networks, with a near-diagonal line alreadyfor 362 nodes.For a more probabilistic approach, we perform a kerneldensity estimation (KDE) by applying a fast Fourier transfor-mation of aggregated flows of the higher resolved networkversus the flows of the low resolution network. Aggregatedflows ̂𝑓 𝑐,𝑑,𝑡 are considered an estimator for the flow 𝑓 𝑐,𝑑,𝑡 in the representative lower resolution network. The result-ing density functions from the KDE are displayed in Figure8. For the low resolution network, the probability distribu-tion has two different modes, while a higher resolution net-work approaches a Gaussian distribution. The variance ofthe probability density function for a low resolution networkis higher than for a high resolution network, as each of thequantile isolines are broader. A.2. Maps of capacity factors for wind and solar
Figures 9a, 9b, 9c present average capacity factors overthe weather year 2013 for solar, wind on- and off-shore re-spectively, i.e. ̄𝑔 𝑛,𝑠 = ⟨ ̄𝑔 𝑛,𝑠,𝑡 ⟩ 𝑡 ∀ 𝑛 , where 𝑠 ∈ {solar , wind onshore , wind of fshore} . The ca-pacity factors are shown in the Voronoi cells around each ofthe 1024 node of the original network, i.e. the set of pointsclosest to each node.The graphics show that capacity factors for solar are de-creasing from South to North while those for wind are in-creasing towards the North and Baltic Sea. The average ca-pacity factors are spatially correlated, but as they are aggre-gated over larger and larger areas using the weighted averagefrom the clustering approach in equation (6), they decline asbad sites are mixed with good sites. This is reflected in Fig-ure 11, which shows how the average capacity factors pertechnology for the generation fleet optimized over the wholeof Europe change with the clustering. A.3. Breakdowns for multiple transmissionexpansion scenarios
Figure 10 shows an extension of the cost breakdowns inFigure 4 from the scenario with no transmission to scenarioswith 12.5% and 50% grid expansion. The general trends arethe same as for the scenario without grid expansion, but gridexpansion generally allows more wind capacity to be built,resulting in lower investment in solar, batteries and hydrogenstorage, as was seen in Figure 5.
A.4. Average capacity factors per technology
To understand how the model exploits the best avail-able resource sites per node, we examine a time-averagedtechnology-specific capacity factor ̄𝑔 𝑠 . The capacity factoris weighted by how much capacity 𝐺 𝑛,𝑠 of technology 𝑠 wasbuilt at each node 𝑛 with time-averaged capacity factor ̄𝑔 𝑛,𝑠 = ⟨ ̄𝑔 𝑛,𝑠,𝑡 ⟩ 𝑡 . ̄𝑔 𝑠 ∶= ∑ 𝑛 ̄𝑔 𝑛,𝑠 ⋅ 𝐺 𝑛,𝑠 ∑ 𝑛 𝐺 𝑛,𝑠 . We present this technology-specific capacity factor inFigure 11 for all three cases with the no-expansion trans-mission scenario, i.e. where 𝐹 𝓁 = 𝐹 𝓁 .As the number of clusters increases, Case 2 has a largervariety of sites per node to choose where capacity should beinstalled optimally and is not restricted by transmission con-straints beyond country-zones. Therefore, the more sites areavailable, the higher the weighted capacity factor is becauseit is not mixed with lower capacity factor sites in equation(6). The highest resolution of Case 2 is also the lowest reso-lution of Case 3: many resource sites and only one node percountry-zone. As the number of nodes in Case 3 increaseswhile the same sites are available, transmission bottlenecksforce the model to build more capacity in locations of worsecapacity factors. Therefore, the capacity factors drop again.For Case 1, where resource resolution and network resolu-tion change in tandem, the resource resolution dominatesand we see increasing capacity factors like in Case 2. A.5. Curtailment per technology
Curtailment is the amount of energy that is available intheory but cannot be injected into the grid because of trans-mission constraints or a lack of demand: ̄𝑔 𝑛,𝑠,𝑡 ⋅ 𝐺 𝑛,𝑠 − 𝑔 𝑛,𝑠,𝑡 Figure 12 shows total curtailment per technology in allCases. Curtailment in all situations is low (less than oftotal demand). Curtailment increases with higher networkresolution in both the Cases and that incorporate trans-mission constraints, while it is gently decreasing with re-source resolution in Case where there are only transmis-sion constraints at the boundaries of country-zones. A.6. Breakdowns by country
Figures 4 and 10 show the breakdown of total costs bytechnology for the whole of Europe. However, it could be
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Page 11 of 15he strong effect of network resolution on energy system models that for each technology, the spatial distribution is unstable,moving from country to country with the clustering changes.For a better understanding of the spatial distribution ofinstalled capacity, we examine the total installed renewablecapacity per country in all Cases in Figure 13 with no trans-mission expansion. The general trend is that the total in-stalled capacity per country is relatively stable with clus-ter resolution. In Case 2 capacity decreases with resolution,since the exploitation of better resource sites means that lesscapacity is needed for a given energy yield. The oppositeeffect is seen in Case 3, while Case 1 reveals a mix of theeffects of Case 2 and 3.
A.7. Shadow price of line volume constraint
The shadow price 𝜇 trans of the transmission expansionconstraint in equation (16) corresponds to the system costbenefit of an incremental MWkm of line volume. Read an-other way, it is the line cost required to obtain the same so-lution with the constraint removed (i.e. lifting the constraintinto the objective function as a Lagrangian relaxation).We present the resulting shadow prices in Figure 14, wherethey are compared with the annuity for underground and over-head lines. Using the cost of underground cables, the cost-optimal solution would give a grid expansion of 18-25% athigh resolution. For overhead transmission, the cost opti-mum would be over 50%. A.8. Computational times and memoryconsumption
The main motivation for clustering and network reduc-tion techniques are to reduce computation times. Both com-putation time and memory consumption are shown as a func-tion of the number of nodes or sites 𝑛 in Figure 15. Whilethe memory rises linearly with 𝑛 , the solving time is slightlyfaster (the interior point method has weakly polynomial worst-case complexity [44, 42]). The solving time in cases and is longer than days with a memory consumption of GBRAM at peak. Case needs less memory at peak and is faster in solving due to lacking transmission constraints. A.9. Capacity factors within each cluster regionfor wind and solar
In this subsection we analyse the homogeneity of time-average capacity factors for wind and solar within each clus-ter region as the number of clusters changes. Duration curvesof the capacity factors in each of the 0.3 ◦ × ◦ weather pix-els of the original ERA5 reanalysis dataset [4] for the Euro-pean area (‘cutout’) are plotted in blue in Figure 16. In ad-dition, the duration curves for the pixels in each cluster areplotted in orange, with the median for each cluster in red.This reveals how much the capacity factors of wind and so-lar vary within each cluster region, compared to the wholeof Europe. Table 4 presents the average standard deviationwith each cluster region for each technology and resolution.For a high resolution of 1024 clusters, we observe thatthe median values (red dots) for solar lie very close to therepresentative values of Europe (black line) with a relatively small average standard deviation of . ⋅ −3 inside eachcluster region (scattering of the orange dots). In the case ofonshore wind, the high capacity factors are underestimatedby the median value, while intermediate and low capacityfactors are represented with a minor difference between me-dian and representative European value. For onshore wind,the average standard deviation of the capacity factors withineach region is larger than for solar by one magnitude ( (10 −2 ) ,represented by the scattering of orange dots). The largestvariance can be observed in offshore regions, where the av-erage standard deviation is . ⋅ −2 , twice as large as foronshore regions, and the low capacity factors are overesti-mated by their representative median values.In the case of 256 clusters, the standard deviation perregion (scattered orange dots) doubles compared to a reso-lution of sites for solar and increases by ∼ 50% for on-shore and offshore wind. However, the median values (reddots) per site do not change much compared to the higher res-olution case. Only at very low resolutions or, in the extreme,one site representing one country-zone, the median values(red dots) do not agree with the European curve (black line),and the capacity values per site (orange scattered dots) covera wide range of values (for example . for wind onshore,or .
11 − 0 . . for solar). At 37 nodes, the average standarddeviation is three times larger for solar compared to a reso-lution of 1024 sites and twice as large for onshore wind.From this analysis we can conclude that a resource res-olution of at least several hundred nodes is required to ade-quately capture the resource variation within Europe, with ahigher resolution required for wind than for solar. A.10. Limitations of this study
The need to solve the models at high spatial resolutionand 3-hourly temporal resolution in reasonable time meansthat compromises have been made elsewhere: the conven-tional generation technologies are limited to hydroelectricityand gas turbines, the storage is limited to batteries and hydro-gen storage, only a single weather year is modelled, and an-cillary services, grid losses, discretisation of new grid capac-ities, distribution grids and forecast error are not modelled.This allows us to focus on the main interactions betweenwind, solar and the transmission grid; the effects of the otherfactors are expected to be small [13] since wind and solar in-vestment dominates system costs. If it were cost-effective tobuild dispatchable low-carbon generators like nuclear or fos-sil generators with carbon capture and sequestration, then theeffects of resource and network resolution would be damp-ened, since there would be less wind and solar investment.Some of the quantitative conclusions may depend on thetechnology assumptions, such as the relative cost of solar PV,onshore wind and offshore wind. However, investigations ofthe sensitivities of similar models to generation costs [60]and of the near-optimal space of solutions [52] have shownthat a large share of wind in low-cost scenarios for Europe isrobust across many scenarios because of the seasonal match-ing of wind to demand in Europe. It is the interactions be-tween wind and the transmission grid that drive the results
Martha Maria et al.:
Preprint submitted to Elsevier
Page 12 of 15he strong effect of network resolution on energy system models in this paper.The results may also change as additional energy sec-tors are coupled to the power sector, such as building heat-ing, transport and non-electric industry demand. While ex-tra flexibility from these sectors might offer an alternative togrid expansion, grid expansion is still expected to be cost-effective [15], while the effects of resource resolution on theoptimal solution remain the same.In the present paper different market structures to today’sare assumed, namely nodal pricing to manage grid conges-tion, and a high CO price to obtain a 95% CO reductioncompared to 1990 levels.We weighted the distribution of wind and solar insideeach nodal region (Voronoi cell) proportional to the instal-lable capacity and capacity factor at each weather grid cell[38]. This means good and bad sites are not mixed evenly,but skewed slightly towards good sites. This effect disap-pears at high resolution, where the capacity factor is moreuniform inside each Voronoi cell.Another approach would be to keep a low one-node-per-country network resolution and then have multiple resourceclasses defined not by region, like our Case 2, but by ca-pacity factor [59, 56, 49] (e.g. a good class with sites withfull load hours above 2000, a medium class between 1500and 2000, and a bad class below 1500). This would also bebeneficial but would not be compatible with the increasinggrid resolution, since the generators in each class would bespread non-contiguously over the country. CRediT authorship contribution statement
Martha Maria Frysztacki:
Conceptualization, Method-ology, Software, Formal Analysis, Data Curation, Writing -Original Draft, Writing - Review & Editing, Visualization.
Jonas Hörsch:
Conceptualization, Methodology, Software,Data Curation, Writing - Review & Editing, Visualization.
Veit Hagenmeyer:
Writing - Review & Editing, FundingAcquisition.
Tom Brown:
Conceptualization, Methodol-ogy, Writing - Original Draft, Writing - Review & Editing,Project Administration, Funding Acquistion.
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Table 2
Notationsymbol meaninggeneral abbreviations 𝑠 technology type 𝑡 time point 𝑖, 𝑗 nodes in high resolution network 𝑐, 𝑑 clustered nodes 𝓁 𝑖,𝑗 high resolution line connecting nodes 𝑖 and 𝑗 𝓁 𝑐,𝑑 aggregated representative line con-necting clusters 𝑐 and 𝑑𝑁 𝑐 set of high resolution nodes in cluster 𝑐𝑁 𝑐,𝑑 set of high resolution lines betweenclusters 𝑐 and 𝑑𝑅𝐸 set of renewable generator technolo-gies 𝐶𝐺 set of conventional generator andstorage technologiesline attributes 𝑥 𝑖,𝑗 reactance of line 𝓁 𝑖,𝑗 𝑣 𝑖,𝑗 voltage of line 𝓁 𝑖,𝑗 𝑐 𝑖,𝑗 capital costs for line 𝓁 𝑖,𝑗 𝑙 𝑖,𝑗 length of line 𝓁 𝑖,𝑗 𝐹 𝑖,𝑗 capacity of line 𝓁 𝑖,𝑗 𝑓 𝑖,𝑗,𝑡 flow of line 𝓁 𝑖,𝑗 at time 𝑡𝑐 marine∕ground capital costs for a submarine/ under-ground connectionnodal and technology attributes 𝑥 𝑖 coordinates of node 𝑖 in ℝ 𝑤 𝑖 nodal weighting 𝑒 𝑠 CO emissions of technology 𝑠𝑤 𝑖,𝑠 nodal technology weighting 𝑐 𝑖,𝑠 annualised fixed costs 𝐺 𝑖,𝑠 (optimal) capacity of technology 𝑠 atnode 𝑖𝐺 max 𝑖,𝑠 maximal installable capacity of tech-nology 𝑠 at node 𝑖𝑜 𝑖,𝑠 variable costs of technology 𝑠 at node 𝑖𝐸 𝑖,𝑠 storage energy efficiency 𝜂 𝑖,𝑠 storage losses or efficiencies at node 𝑖 for technology 𝑠𝑤 𝑡 time weighting 𝑑 𝑖,𝑡 demand per node 𝑖 and time 𝑡̄𝑔 𝑖,𝑠,𝑡 capacity factor for RE ∈ [0 , 𝑔 𝑖,𝑠,𝑡 dispatch in node 𝑖 of technology 𝑠 attime 𝑡𝑒 𝑖,𝑠,𝑡 energy level of technology 𝑠 in node 𝑖 at time 𝑡 graph related attributes 𝐾 𝑖, 𝓁 incidence matrix 𝐶 𝓁 ,𝑐 Cycle matrix, here, 𝑐 represents acylce Table 3
Technology investment costs with
1$ = 0 . €.asset cost unitonshore wind 1110 €/kWoffshore wind 1640 €/kW(AC/DC grid connection separate)solar PV utility 425 €/kWsolar PV rooftop 725 €/kWopen cycle gas turbine 400 €/kWrun of river 3000 €/kWpumped hydro storage 2000 €/kWhydro storage 2000 €/kWbattery storage 192 $/kWhbattery power conversion 411 $/kW el hydrogen storage 11.3 $/kWhhydrogen power conversion 689 €/kW el HVAC overhead transmission 400 €/(MWkm)HVAC underground transmission 1342 €/(MWkm)HVAC subsea transmission 2685 €/(MWkm)HVDC underground transmission 1000 €/(MWkm)HVDC subsea transmission 2000 €/(MWkm)n clusters solar wind onshore wind offshore . ⋅ −3 . ⋅ −2 . ⋅ −2
512 2 . ⋅ −3 . ⋅ −2 . ⋅ −2
362 3 . ⋅ −3 . ⋅ −2 . ⋅ −2
256 3 . ⋅ −3 . ⋅ −2 . ⋅ −2
181 4 . ⋅ −3 . ⋅ −2 . ⋅ −2
128 4 . ⋅ −3 . ⋅ −2 . ⋅ −2
90 5 . ⋅ −3 . ⋅ −2 . ⋅ −2
64 6 . ⋅ −3 . ⋅ −2 . ⋅ −2
45 6 . ⋅ −3 . ⋅ −2 . ⋅ −2
37 6 . ⋅ −3 . ⋅ −2 . ⋅ −2 Table 4 average standard deviation of the capacity factor (per unit)per region for a network resolution of , and sites.Martha Maria et al.: Preprint submitted to Elsevier
Page 16 of 15he strong effect of network resolution on energy system models
Table 5
Aggregation rules for attributes of nodes and attached assetsattribute aggregated attribute mapping values or unitslatitude & longitude 𝑥 𝑐 | 𝑁 𝑐 | ∑ 𝑖 ∈ 𝑁 𝑐 𝑥 𝑖 ℝ (optimal) power capacity 𝐺 𝑐,𝑠 ∑ 𝑖 ∈ 𝑁 𝑐 𝐺 𝑖,𝑠 𝑀𝑊 asset installable potential 𝐺 max 𝑐,𝑠 ∑ 𝑖 ∈ 𝑁 𝑐 𝐺 max 𝑖,𝑠 𝑀𝑊 Table 6
Aggregation rules for attributes of lines in seriesattribute aggregated attribute mapping values or unitslength (HVDC lines) 𝑙 𝑐,𝑑 min 𝓁 𝑖,𝑗 ∈ 𝑁 𝑐,𝑑 𝑙 𝑖,𝑗 kmpower capacity 𝐹 𝑐,𝑑 ∑ 𝓁 𝑖,𝑗 ∈ 𝑁 𝑐,𝑑 𝐹 𝑖,𝑗 MVAfraction of length underwater 𝑢 𝑐,𝑑 𝑙 𝑐,𝑑 ∑ 𝓁 𝑖,𝑗 ∈ 𝑁 𝑐,𝑑 𝑙 𝑖,𝑗 ⋅ 𝑢 𝑖,𝑗 per unit Table 7
Aggregation rules for attributes of lines in parallelattribute aggregated attribute mapping values or unitspower capacity 𝑠 nom 𝑐,𝑑 ∑ 𝓁 𝑖,𝑗 ∈ 𝑁 𝑐,𝑑 𝑠 nom 𝑖,𝑗 𝑀𝑉 𝐴 power capacity maximum 𝑠 min 𝑐,𝑑 ∑ 𝓁 𝑖,𝑗 ∈ 𝑁 𝑐,𝑑 𝑠 min 𝑖,𝑗 𝑀𝑉 𝐴 power capacity minimum 𝑠 max 𝑐,𝑑 ∑ 𝓁 𝑖,𝑗 ∈ 𝑁 𝑐,𝑑 𝑠 max 𝑖,𝑗 𝑀𝑉 𝐴 number of parallel lines 𝑛 parallel 𝑐,𝑑 ∑ 𝓁 𝑖,𝑗 ∈ 𝑁 𝑐,𝑑 𝑛 parallel 𝑖,𝑗 ℝ terrain factor for capital costs terr 𝑐,𝑑 | 𝑁 𝑐,𝑑 | ∑ 𝓁 𝑖,𝑗 ∈ 𝑁 𝑐,𝑑 terr 𝑖,𝑗 per unitMartha Maria et al.: Preprint submitted to Elsevier
Page 17 of 15he strong effect of network resolution on energy system models
10 0 10 20 303540455055606570 0.0000.0250.0500.0750.1000.1250.1500.1750.200 (a) Solar
10 0 10 20 303540455055606570 0.00.10.20.30.40.50.6 (b) Wind onshore
10 0 10 20 3040506070 0.00.10.20.30.40.50.6 (c) Wind offshore
Figure 9:
Wind and solar capacity factors in Europe for theweather year 2013 at full resolution.Martha Maria et al.:
Preprint submitted to Elsevier
Page 18 of 15he strong effect of network resolution on energy system models c o s t s [ y e a r ]
0% transmission expansion
25% transmission expansion t y p e = g e n .
50% transmission expansion
37 45 64 90 128 181 256 362 512 number of clusters
37 45 64 90 128 181 256 362 512 number of clusters
37 45 64 90 128 181 256 362 512 number of clusters t y p e = f l e x . Case 1 c o s t s [ y e a r ]
0% transmission expansion
25% transmission expansion t y p e = g e n .
50% transmission expansion
37 45 64 90 128 181 256 362 512 1024 number of clusters
37 45 64 90 128 181 256 362 512 1024 number of clusters
37 45 64 90 128 181 256 362 512 1024 number of clusters t y p e = f l e x . Case 2 c o s t s [ y e a r ]
0% transmission expansion
25% transmission expansion t y p e = g e n .
50% transmission expansion
37 45 64 90 128 181 256 362 512 number of clusters
37 45 64 90 128 181 256 362 512 number of clusters
37 45 64 90 128 181 256 362 512 number of clusters t y p e = f l e x . Case 3
Figure 10:
Technology breakdown of the annual system costs for generation (top) and flexibility options (bottom) as a functionof the number of clusters for Cases 1, 2 and 3. Cases correspond to the rows, while transmission expansion scenarios correspondto the columns.Martha Maria et al.:
Preprint submitted to Elsevier
Page 19 of 15he strong effect of network resolution on energy system models
Case 1 Case 2 c a rr i e r = o ff w i n d Case 3 c a rr i e r = o n w i n d
37 45 64 90 128 181 256 362 512 number of clusters
37 45 64 90 128 181 256 362 512 number of clusters
37 45 64 90 128 181 256 362 512 number of clusters c a rr i e r = s o l a r average capacityfactors in % Figure 11:
Average capacity factors for each technology for the no transmission expansion scenario in all three cases. c u r t a il m e n t [ T W h ] Case 1 Case 2 t y p e = b r e a k d o w n Case 3
37 45 64 90 128 181 256 362 512 number of clusters
37 45 64 90 128 181 256 362 512 number of clusters
37 45 64 90 128 181 256 362 512 number of clusters t y p e = t o t a l Figure 12:
Curtailment for the no transmission expansion scenario in all three cases.Martha Maria et al.:
Preprint submitted to Elsevier
Page 20 of 15he strong effect of network resolution on energy system models I n s t a ll e d c a p a c i t y [ G W ] Case 1
Renewable capacity distribution per country at a transmission expansion level of 0.0% I n s t a ll e d c a p a c i t y [ G W ] Case 2
AL AT BA BE BG CH CZ DE DK EE ES FI FR GB GR HR HU IE IT LT LU LV ME MK NL NO PL PT RO RS SE SI SK country I n s t a ll e d c a p a c i t y [ G W ] Case 3
Figure 13:
Capacities per country for the no transmission expansion scenario in all three cases.Martha Maria et al.:
Preprint submitted to Elsevier
Page 21 of 15he strong effect of network resolution on energy system models transmission expansion limit t r a n s [ M W k m a ] Underground cablesOverhead lines clusters
Figure 14:
Shadow (dual) price of the line volume constraint.Martha Maria et al.:
Preprint submitted to Elsevier
Page 22 of 15he strong effect of network resolution on energy system models R A M [ M B ] Case 1: Simultaneous Clusteringtransmission expansion [%]
Case 2: Clustering on generation sites Case 3: Clustering on transmission network
100 200 300 400 500 number of clusters s o l v i n g t i m e [ d a y s ]
100 200 300 400 500 number of clusters
100 200 300 400 500 number of clusters
Figure 15:
Memory consumption and solving time.Martha Maria et al.:
Preprint submitted to Elsevier
Page 23 of 15he strong effect of network resolution on energy system models s o l a r Capacity Factors per Technology for 1024 Sites w i n d o n s h o r e w i n d o ff s h o r e s o l a r Capacity Factors per Technology for 256 Sites w i n d o n s h o r e w i n d o ff s h o r e s o l a r Capacity Factors per Technology for 37 Sites w i n d o n s h o r e w i n d o ff s h o r e Figure 16:
Breakdown of capacity factors per technology for the weather cutout pixels inside each cluster region as a durationcurve (orange), with the median marked in red. The overall duration curve of pixel capacity factors for the whole of Europe isplotted in blue.Martha Maria et al.: