2. Geneva School of Economics and Management (GSEM), University of Geneva, Uni Mail, Bd du Pontd'Arve 40, CH1211 Genève 4, Switzerland
A Future highly renewable power system is foreseen to integrate a significant contribution from wind turbines and solar photovoltaic (PV) electricity [1] although both energy sources exhibit strong variability related to their weatherdriven nature. Integration of intermittent generation into the electric network is a challenging task as supply must always match demand. Balancing requirements are usually supported by adding reserve capacity in the form of quickly adjustable backup power plants (BPP), operating for example on gas. Because of variable renewable energy sources (VREs^{1}) overall low reliability and lack of flexibility, a highly renewable power system is likely to necessitate huge amounts of such backup units whereas conventional baseload generation will vanish [2]. Adding energy storage capacity is of course another way to maintain the systems' equilibrium and has the advantage to allow time domain transfer of excess VREs generation.
^{1}In the context of this paper VREs will stand for solar PV and wind energy.
Another possible integration strategy is to spatially extend the power system [3][6]. Present power systems are mainly designed within mesoscale areas, e.g., related to territorial boundaries, around dispatchable and centralized supply units in proximity to load centers. Another important feature introduced by VREs is distributed generation (DG) which entails deep structural changes in network dynamics. Facing a fast growing share of VREs generation the European transmission grid is already showing weaknesses as large scale interconnections are more and more solicited by widely spread fluctuating infeeds. Following the authors of [7] a strong panEuropean transmission backbone appears to be a necessary condition for further VREs deployment.
As far as quantity is concerned some locations are best suited than others for VREs units siting. Although, past the tipping point where renewable power exceeds the load it becomes unclear if it is not more beneficial to favor quality over quantity, i.e., to favor a set of dispersed locations that minimize the supplydemand mismatch. This question is also related to BPP and storage optimal sizing and is at the heart of the present study. More precisely this paper introduces a simple yet effective methodological framework which explicitly investigates the interplay between storage, balancing needs and optimal geographical dispersion of VREs generators.
As an introductory example, let us consider the optimal allocation problem of deploying efficiently a given VREs power capacity with respect to a timevarying consumers' load. At large enough grid scales there should be spatial and technological distributions of VREs units that are better than others to match demand. Taking a social planner point of view there is in fact a fundamental ambiguity on defining what is optimal. A first option would be to seek a spatial distribution of units that favor smoothness and thus minimize BPP power rating. This makes sense in practical terms as it is economically undesirable to secure every gigawatt (GW) of renewable power by an equivalent amount of BPP capacity. Another option would be to maximize renewable penetration, i.e., minimize BPP output, which is preferable when it comes to power system sustainability and efficient VREs load factors. In essence once renewable capacity is in peaking range the question of quantity versus quality becomes relevant, where quality must be understood here as the ability to match demand. This said it is likely that any optimal distribution will strongly depend on defining parameters such as the size of accessible space for unit deployment, the power gap between peak load and aggregated rated power of renewable units as well as the existing storage capacity within the system. Our analysis will show that there are several threshold effects that have to be accounted for.
The main goal of this study is to investigate how will evolve grid and storage integration options within growing VREs capacities. In order to do this we use an optimal distribution search algorithm in combination with a parametric physical simulation of hourly power mismatches between supply and demand. As a case study the Swiss electricity consumption is related to varying extents of transmission grid, thus specifically exploring how overall performance is affected by the definition of transmission network boundaries. Switzerland is geographically well centralized in Europe and already a crossing point of growing interregional power flows. It also happens to be on the way of nuclear phase out, which represents about a third of actual electricity production. How to replace this power capacity is still under examination and notably raises questions about the possible building up of electricity importations [8]. At the local level hydropower is the main renewable source at disposition, but it has also already attained its deployment limit. Accumulation dams still offer electricity storage prospects such as illustrated by the important "Nant de Drance" pumpedhydro project which will be able to deliver a power rating of 900 MW [9]. Further renewable power deployment in Switzerland is anticipated to rely mainly on distributed solar PV. Although we look into a specific case, narrowing underlaying hypothesis, this work has a more general scope in that it is addressing the local versus global debate surrounding the ongoing power system transformation of all European countries.
The paper proceeds as follows: in Section Ⅱ we review related work. The model is described in detail in Section Ⅲ. Results and discussions are developed in Section Ⅳ before concluding in Section V.
Ⅱ. EXISTING LITERATUREScenarios with close to 100% renewable production are investigated in numerous studies within a wide range of methodological frameworks and geographical boundaries. Some carry out advanced operational and economic modeling [1], [10][13], while others concentrate on fundamental systemic aspects of VREs integration in order to gain a better understanding of existing options, opportunities and threats [5], [6], [14][16].
The role of storage in prospective power systems has naturally drawn a lot of attention in the energy research community as flexibility and curtailment losses issues are building up [2], [6], [15], [16]. Among others Rasmussen et al. [15] explore the interplay between storage sizing, solarwind optimal mix and balancing needs in an idealized panEuropean power system. Existing literature on energy storage is reviewed in depth by Zucker et al. [17].
A number of contributions investigate the statistical correlations of distributed wind and/or solar energy supply. According to Heide et al. [18] there is an interesting seasonal complementarity between wind and solar resources over Europe. Optimal mix considerations should thus be accounted for when looking at systemic VREs integration. Considering the Italian territory the authors of [19] investigate the statistical correlation between solar and wind energy supply, while the authors of [20] look at complementarity options for the Ontario (Canada) region. The authors of [21] introduce a probabilistic methodology to integrate stochastic wind power inflows into transmission load analysis. The work presented in [22] explores the smoothing effect of an Europeanwide grid on wind power output.
Benefits of global electricity grid deployment for VREs integration has been examined in [23]. More recently the influence of European transmission grid reinforcements has been deeply analyzed from a technicoeconomic perspective in various studies [3][6], [13]. The authors of [3] introduce an investment and dispatch optimization model to explore grid extension strategies considering leastcost objectives. It is shown that large scale grid interconnection is essential in order to achieve a costefficient highly renewable power system. Schaber et al. [4] use a regionally resolved technicoeconomic model to analyze how grid extensions interact with electricity market in Europe within projected VREs penetration for 2020. Future transmission network design is foreseen to have substantial impact on price dynamics as well as conventional generation power plant operation. In [5] a similar methodological set up has been used but to investigate grid extension as a function of wind and solar penetration and optimal mix. The interplay between backup energy demand and storage capacity is evaluated within transmission constraints in [6] while the authors of [13] presents a largescale spatial model of the European electricity market intended to analyze the effects of load flow congestion regarding investments strategies and market design, considering a social welfare maximization objective.
The main original contribution of this paper is to explicitly derive highly resolved optimal spatial distribution of VREs units using a parametrized search algorithm approach, spanning varying prospective power system designs. The goal is to investigate the intertwined scaling effects of both storage deployment and grid extension on the transitional path towards a highly renewable power system. We believe that our explicit approach has never been presented in scientific literature and has practical interest for a transparent evaluation of VREs integration options at a system level. In contrast with the technoeconomical approach developed in e.g., in [3][6], [13] we stay focused on the physically driven dynamics. Our geographically centralized case study approach also differs from more general assessments, as for instance in [2], [15]. By taking this narrower point of view we intend to reduce underlying hypothesis and complexity to get robust order of magnitude from a welldefined perspective. We think this will contribute to shed light on important dynamical patterns, helping to achieve an efficient integration of weather driven power supply. Our contribution also provides an effective methodological approach to the system wide optimal spatial distribution problem that can be implemented in more advanced modeling.
Ⅲ. METHODOLOGYThe backbone of our methodology relies on a parameterized weatherdriven simulation model of hourly power mismatches between generation and load. This basic modeling tool is designed to evaluate the overall performance of a given spatial distribution of renewable units. It is associated with a heuristic optimization process, namely a genetic algorithm, in order to find the nearoptimal distribution of VREs units over a specific territory. Later, we will present how these elements exactly interact but first start with the fundamental building blocks of our methodological design.
A. Grid and LoadIn this study the Swiss aggregated electricity consumption will serve as our demand side reference. 15 minutes end user load data were obtained from the national transmission system operator Swissgrid (www.swissgrid.ch) and from which is computed the hourly load time series
Our first objective is to simulate the weatherdriven electric generation delivered by a given set of spatially distributed solar and wind generators. Power system simulations are conducted on a 3 years period (20032005) and based on hourly timeseries of wind and PV electricity output calculated for each cell of a
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Fig. 1 Illustration of the transmission network and point based spatial representation of potential VREs sittings. 
In order to take into account electricity transmission limitations our spatial representation integrates the European high voltage transmission grid provided by ENTSOEE (www.entsoe.eu[7]). Furthermore we assume shortest path connection between each cell center and the nearest high voltage line (Fig. 1). Swiss electricity consumption is assumed to be geographically concentrated on a single node represented by the central dot in Fig. 2 where all distributed VREs generation is routed. Grid extension is then defined by selecting all cells within a distance
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Fig. 2 Illustration of the expanding grid domains considered in this work. The smallest grid extent is limited to the Swiss territory. The distance then extends by 300 km steps along transmission lines. 
Transmission losses are also accounted for by reducing delivered energy at a rate of
The next step is to derive, for both wind and solar resources, a tempospatial VREs hourly power output dataset. Formally we derive for each cell on the spatial footprint a resource specific hourly electricity generation timeseries based on highly resolved reanalysis data.
PV energy output depends primarily on the amount of incident solar radiation received which in turn is a function of the geographical location and plane orientation of the module as well as atmospheric conditions. For the present study we assume generic PV cells characterized by a global conversion efficiency of 13%. We also assume PV modules to be fixed, ideally oriented south and tilted at latitude angle in order to maximize inplane irradiation over the year.
Solar electricity output is calculated using hourly mean of horizontalplane surface incoming shortwave (SIS) radiation available from the Climate Monitoring Satellite Application Facility (CMSAF, www.cmsaf.eu [24], [25]). The SIS data set is given at
Cell specific hourly wind power generation is evaluated using the European Center for Mediumrange Weather Forecast (ECMWF, www.ecmwf.int [27]) ERAinterim reanalysis database [28]. Longitudinal and latitudinal wind magnitudes timeseries are available from the ECMWF at
The power system sizing is defined according to the four following parameters:
1)
2)
3)
4)
We assume that the total renewable power
Given a set of distributed VREs generators
$ \begin{align} \Delta_t := P_{\rm ref} \left[\sum_{x\in I_w}W_{(x, t)}+\sum_{x\in I_s}PV_{(x, t)} \right]  L(t) \end{align} $  (1) 
where
Let us point out that the hourly power mismatch can be either positive (
Two different balancing mechanisms will be considered. First in the merit order, the system will relay on a generic and centralized roundtrip storage technology. This storage is sized by definition of both its maximal stored energy capacity
As the
$ \begin{align} \tilde{\Delta}_t= \max\big(\min (\Delta_t, \zeta), 0\big)+\min\big(\max (\Delta_t, \zeta), 0\big). \end{align} $  (2) 
Generally speaking optimal storage dispatch strategy is a complex task [15][30]. We assume here a straightforward policy where any excess VREs generation is stored unless storage is full and deficits are first covered by stored energy unless storage is empty. All transactions are of course limited by the storage ramping constraints, capacity limits and conversion losses. Thus the storage filling level time series
$ \begin{align} S_t= \left\{\begin{array}{lll} \min (S_{t1}+\eta _{\rm in}\tilde{\Delta}_{t}, C_s), &{\rm if}& \tilde{\Delta}_t\geq 0 \\ \max (S_{t1}+\frac{1}{\eta _{\rm out}}\tilde{\Delta}_t, 0), &{\rm if}& \tilde{\Delta}_t<0. \end{array}\right.\nonumber \end{align} $  (3) 
To avoid boundary value problems a one year spin up period has been used to settle initial storage level
The time dependent storage power flow
$ \begin{align} B_t=\left\{\begin{array}{lll} 0, &{\rm if}&\;\Delta_t\geq 0\\ \Delta _t\eta _{\rm out}F_t, &{\rm if}&\; \Delta_t<0 \end{array}\right.\nonumber \end{align} $  (4) 
where we have chosen to express backup energy in negative values for better clarity.
To summarize the simulation set up we start with a parameterized power system
One key metric that we are interested in is the total backup energy use
$ \begin{align} \Psi_{_{(T)}} := \sum_{t=1}^{T} \left B_t \right  \end{align} $  (5) 
from which we define the renewable fraction
$ \begin{align} \Omega _{_{(T)}}:=1\dfrac{\Psi _{_{(T)}}}{\sum\limits_{t=1}^{T}L(t)}. \end{align} $  (6) 
The renewable fraction is a measure of overall adequacy between load and VREs generation. As such an optimal spatial distribution of VREs generators should tend to maximize this quantity. Another metric of interest is the minimal power sizing of BPP required to cover demand, that is the maximal hourly power call of the backup time series. This last quantity is divided by the peak load as to make it a normalized quantity taking values between 0 and 1:
$ \begin{align} \beta_{_{(T)}}:=\frac{\max\limits_{t \in T} \left ( B_t \right )}{\max\limits_{t \in T} ( L_t )}. \end{align} $  (7) 
Backup power capacity is definitely a critical aspect of system design when considering high share of solar and wind energy. On the transitional path toward a highly renewable power system the value of reliability needs to be evaluated carefully as competing strategies could easily develop between backup power minimization and renewable fraction maximization.
F. Optimal Spatial Distribution of VREs UnitsGiven a parameterized power system sizing (
GA is a search heuristics inspired by natural selection theory and as such is traditionally described using biological analogies. An individual is a candidate solution to the optimization problem and is defined by his genotype, i.e., the set
Within our setup, each of the
$ \begin{align} H_p=\frac{1}{\Psi _p\sum\limits_{i=1}^{n}\frac{1}{\Psi _i}}. \end{align} $  (8) 
A crossover operator is then applied to shuffle genes between the selected parents. To avoid local extremum convergence the next step is to introduce a random exploration of search space using a mutation operator which will replace genes of the offspring's with some given probability. Precisely, a renewable unit
For our experimental results we will concentrate on two classes of scenarios. The first class of scenario restricts to power systems without storage capacity, where integration schemes rely only on spatial distribution patterns. In the second class of scenarios we introduce a generic storage which allow time domain transfer of load. The motivation behind this is to shed light on possible tradeoffs between storage and grid strategies. To clarify scales we will express the installed renewable power
We first investigate the share of electricity demand that can be met without storage capacity as a function of the aggregated VREs power
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Fig. 3 Renewable fraction 
To better understand this behavior results of Fig. 3 should be put in perspective with the solarwind mix of solutions presented in Fig. 4 where
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Fig. 4 Solarwind mix of 120 optimal solutions as a function of spatial extent 
A general statement is that PV is better suited than wind energy for small grid integration but then the day night and seasonal cycles of solar resource also imposes important constraints on achievable penetration levels as well as needed reserve. Large grid VREs integration gives access to high potential wind sites which increases substantially the overall performances. Notably, for an installed renewable power equivalent to peak load, there is almost a
We concentrate in this section on a single class of storage scenarios where
^{2}Projected Swiss pumpedstorage needs are related to domestic VREs generation at penetration level of about
Results presented in Fig. 3 can be compared to their counterpart of Fig. 5 where values are again obtained from spatial distribution that maximizes renewable penetration but this time with the use of storage capacity.
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Fig. 5 Renewable fraction 
As expected the renewable fraction is increased proportionally to
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Fig. 6 Monotonic representation of power mismatch (continuous lines) and backup needs (dashed line) for selected scenarios with 
Looking at Figs. 6 (b) and 6 (d) we can see that the introduction of storage is accompanied by a redistribution of units which reinforces excess VREs generation but also reinforces negative mismatches. This is however not the case in Figs. 6 (a) and 6 (c) where both mismatch curves are nearly identical. While storage use is clearly growing with grid extent it appears that the interplay between spatial distribution and storage holds some subtleties. It seems logical that the possibility of time domain transfer will relax constraints on direct supplydemand equilibrium in presence of excess VREs generation and should thus favor spatial concentration, on the other hand growing renewable power and larger grids will widen spatial complementarity options. In order to clarify how this tradeoff takes place we will continue our investigation by further looking at unit distribution patterns.
C. Spatial Distribution of SolutionsAs a first insight, and to illustrate the pattern evolution of optimal solutions, Fig. 7 shows the geographical distribution of VREs units obtained for selected scenarios with
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Fig. 7 Distribution map for selected scenarios with R= 1500 km, k = 20 units and no storage. This Figure illustrates the evolution of optimal VREs units distribution from highly concentrated to widely spread out solutions. Notice however that when spreading out units still aggregate around several (coastal) distinct poles. 
$ \begin{align} \delta_R := \frac{\sum\limits_{i < j\leq k} d_h(x_i, x_j)}{\max\limits_{all sets}(\sum\limits_{p < q\leq k}d_h(x_p, x_q))} \end{align} $  (9) 
where
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Fig. 8 Dispersion δ_{R} as a function of P_{ren} for different grid extents. Without storage (a) and with storage (b). For all grid extents there is a minimum power threshold from which dispersion starts to monotonically increase. The introduction of storage does not significantly change dispersion values although there is globally a right side drift. 
For all grid extent spatial distribution (quasi) monotonically evolves from perfectly concentrated to spread out configurations. This is obviously in agreement with our initial intuition, less obvious is the fact that all curves exhibit similar path and all tend to accumulate around
Integration potential for a solarwind based power system has been investigated using an optimal distribution search algorithm, in combination with a parametric physical simulation of hourly power mismatches. A motivation for this research was to illuminate options and constraints on the transitional path towards a fully renewable power system, taking a central European perspective. In this respect our results show a strong incentive to expand the spatial distribution of renewable units as small scale deployment greatly limits what is achievable in terms of penetration level. This is however without considering decisive economic factors such as needed transmission investments or market design, which surely are aspects that have to be accounted for. Our findings that increasing spatial extension plays a strong role for VREs integration is also in good general concordance with previous studies, such as in [3], [5], [6], although direct comparison is made uneasy because of the difference in representation.
A main contribution of this study is to explicitly define best possible geographical distribution of VREs generators for growing power system sizing. As such it provides an upper bound on integration schemes useful for framing the ongoing energy transition debate. Implemented in a more detailed power system simulation this modeling approach could be part of a decisionmaking tool for electricity planners, allowing to identify advantageous system design.
Considering ambitious renewable targets our results reveal possible forthcoming systemic bottleneck. From the central European standpoint grid extension is already critical to achieve efficiently a
Introduction of storage also appears to be mostly beneficial when associated to large scale distribution as it enables a significant decrease in backup power sizing, on the contrary to small grid integration, although the overall smoothing effect of storage only occurs at a fairly high share of VREs power. Optimal distribution of units is also subject to variations when storage is added, allowing more quantity oriented configurations, but there is in any case a threshold point from which unit dispersion starts to improve performances. A key finding is that dispersion however does not increase steadily toward a fully atomized geographical dispersion but settles on multipole arrangements, independently of grid extent. One aspect that our study does not clarify is how narrow the distribution optimums are. This should be further investigated.
Swiss hydropower dynamics has not been directly integrated in our methodological setup in order to get a general sense of interacting strategies focused on VREs. Let's mention that the model responses should be weakly sensitive to the introduction of some baseload capacity, as far as peak load is redefined as the maximal power gap between load and baseload generation. However hydroelectricity entails its own variability, mainly seasonal, that will somehow affect optimality of solutions. This aspect along with a more advanced representation of storage scheduling is setting ground for further research prospects, notably in the context of forecast uncertainties.
Our case study findings may also not generalize to other load centers, notably for coastal locations that have near access to high wind profiles. Nevertheless for most central European locations we can expect to find similar characteristics. It would be interesting to carry out this same analysis on different regions and at higher spatial resolution to further enhance the general understanding of underlying dynamics.
Increasing transactions between European load centers indicates that the power system is already heading towards an extended network [7] although the ideal transmission grid design is still under question. Focusing on a wind and solar based power system we have identified some inherent physical limitations and opportunities, framing the range of possible tempospatial integration options from a localized perspective. Yet a number of other alternative options and synergies can be locally exploited to fully realize the ongoing energy transition. Among them demandresponse strategies are of particular interest in the context of highly renewable but lowly flexible power systems. Adding demand side flexibility would be a consistent next step to this study.
APPENDIX A Optimal Distribution Search AlgorithmGiven the setup introduced in Section Ⅲ our objective is to find the best geographical distribution of
$ \begin{align} S(\chi_R, k)=\frac{(2\chi_R+k1)!}{k!(2\chi_R1)!} \end{align} $  (10) 
Although values for
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Fig. 9 Relative fitness as a function of number of units k for selected scenarios. 
An alternative to the GA is to adopt a greedy approach on the problem. Given
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Fig. 10 Greedy solutions versus GA solutions as a function of P_{ren}. 
We now proceed to present in more details the GA we have designed for this research, the generic procedure is presented in Appendix C. Specific values used for GA parameters, such as generation size and operator rates, are presented in Table Ⅵ. We recall that the relative fitness of a candidate solution
Algorithm 1 Optimal Distribution Search Algorithm 
Given: DO: 1. Initial seeding procedure: 1.1 Calculate the greedy solution 1.2 Calculate an elite solution single individuals. 2. Populate initial generation for populate initial generation with greedy solution: for populate initial generation with elite solution: for populate initial generation with random individuals While (Current generation 3. Evaluate fitness criterion 4. Rank individuals according to fitness. 5. Reconduct most fit member in child generation : 6. Calculate the diversity factor of current generation: 7. Adapt the mutation rate: mr 8. Selection and Crossover: for Select two parents to fitness. if (rank ( Onepoint crossover between parents. else for With probability Assign resulting offspring's to child generation: and 9. Mutation: for for With probability 10. Add current generation to population. 11. Create new generation: End While 12. Return (Most fit member of population). End DO 
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