# Broad Ranges of Investment Configurations for Renewable Power Systems,
Robust to Cost Uncertainty and Near-Optimality

Fabian Neumann<EMAIL_ADDRESS>Tom Brown  Department of Digital
Transformation in Energy Systems, Institute of Energy Technology, Technische
Universität Berlin (TUB), Einsteinufer 25 (TA 8), 10587, Berlin, Germany
Institute for Automation and Applied Informatics (IAI), Karlsruhe Institute of
Technology (KIT), Hermann-von-Helmholtz-Platz 1, 76344, Eggenstein-
Leopoldshafen, Germany

###### Abstract

To achieve ambitious greenhouse gas emission reduction targets in time, the
planning of future energy systems needs to accommodate societal preferences,
e.g. low levels of acceptance for transmission expansion or onshore wind
turbines, and must also acknowledge the inherent uncertainties of technology
cost projections. To date, however, many capacity expansion models lean
heavily towards only minimising system cost and only studying few cost
projections. Here, we address both criticisms in unison. While taking account
of technology cost uncertainties, we apply methods from multi-objective
optimisation to explore trade-offs in a fully renewable European electricity
system between increasing system cost and extremising the use of individual
technologies for generating, storing and transmitting electricity to build
robust insights about what actions are viable within given cost ranges. We
identify boundary conditions that must be met for cost-efficiency regardless
of how cost developments will unfold; for instance, that some grid
reinforcement and long-term storage alongside a significant amount of wind
capacity appear essential. But, foremost, we reveal that near the cost-optimum
a broad spectrum of regionally and technologically diverse options exists in
any case, which allows policymakers to navigate around public acceptance
issues. The analysis requires to manage many computationally demanding
scenario runs efficiently, for which we leverage multi-fidelity surrogate
modelling techniques using sparse polynomial chaos expansions and low-
discrepancy sampling.

††journal: undisclosed

###### Contents

  1. 1 Introduction
  2. 2 Methods
    1. 2.1 Least-Cost Investment Planning
    2. 2.2 Near-Optimal Alternatives
    3. 2.3 Model Inputs
    4. 2.4 Technology Cost Uncertainty
    5. 2.5 Surrogate Modelling with Polynomial Chaos Expansion
    6. 2.6 Multifidelity Approach
    7. 2.7 Experimental Design
    8. 2.8 Model Validation
  3. 3 Results and Discussion
    1. 3.1 Cost and Capacity Distribution of Least-Cost Solutions
    2. 3.2 Parameter Sweeps and Global Sensitivity Indices
    3. 3.3 Fuzzy Near-Optimal Corridors with Increasing Cost Slack
    4. 3.4 Probabilistic Near-Optimal Space in Two Technology Dimensions
    5. 3.5 Capacity Distributions at Minimal Onshore Wind and Transmission Grid
    6. 3.6 Critical Appraisal
  4. 4 Conclusion
  5. Data Availability

##  1 Introduction

Energy system models have become a pivotal instrument for policy-making to
find cost-efficient system layouts that satisfy ambitious climate change
mitigation targets. But even though they have proliferated in spatial,
temporal, technological and sectoral detail and scope in recent years, least-
cost optimisation models can easily give a false sense of exactness [1, 2].
Frequently, they present just a single least-cost solution for a single set of
cost assumptions, which not only neglects uncertainties inherent to technology
cost projections which capacity expansion models are susceptible to [3, 4, 5],
but also hides a wide array of alternative solutions that are equally feasible
and only marginally more expensive [6, 7, 8].

Trade-offs revealed by deviating from least-cost solutions are extremely
attractive for policymakers, because they allow them to make decisions based
on non-economic criteria without affecting the cost-effectiveness of the
system. Knowing that many similarly costly but technologically diverse
solutions exist, helps to accommodate political and social dimensions that are
otherwise hard to quantify; for instance, rising public opposition towards
reinforced transmission lines and onshore wind turbines or an uneven
distribution of new infrastructure [9, 8, 10].

Techniques like multi-objective optimisation and modelling-to-generate-
alternatives are designed to find such near-optimal solutions. Among others,
they have been applied to investment planning models of the European [6], the
Italian [7], and the United States power system [11], pathways to decarbonise
the power system of the United Kingdom [12], a single-node sector-coupling
model of Germany [13], global integrated assessment models [14], and were
combined with a quick hull algorithm to span a polytope of low-cost solutions
for a single set of cost parameters [15].

However, most of the studies above only use a central cost projection for each
considered technology. But recent decades have shown that many of these
projections contain a high level of uncertainty, particularly for fast-moving
technologies like solar, batteries and hydrogen storage. This uncertainty
propagates through the model to strongly affect the optimal and near-optimal
system compositions, thus undermining any analysis of the trade-offs. Hence,
it is crucial that apparent compromises are rigorously tested for robustness
to technology cost uncertainty to raise confidence in conclusions about
viable, cost-effective power system designs. To thoroughly sweep the
uncertainty space, we can fortunately avail of previous works on multi-
dimensional global sensitivity analysis techniques in the context of least-
cost optimisation [3, 16, 17, 18, 19]. We expand their application to
strengthening insights on the scope of near-optimal trade-offs, wherein the
novelty of this contribution lies.

Here, we systematically explore robust trade-offs near the cost-optimum of a
fully renewable European electricity system model, and investigate how they
are affected by uncertain technology cost projections. Thereby, we evaluate
both compromises between system cost and technology choices, as well as
between pairs of technologies. We do so by solving numerous spatially and
temporally explicit long-term investment planning problems that coordinate
generation, transmission and storage investments subject to multi-period
linear optimal power flow constraints, while employing methods from global
sensitivity analysis to account for a wide range of cost projections for wind,
solar, battery and hydrogen storage technologies.

To handle the immense computational burden incurred by searching for near-
optimal alternatives alongside evaluating many different cost parameter sets,
we employ multi-fidelity surrogate modelling techniques, based on sparse
polynomial chaos expansion that allow us to merge results from one simpler and
another more detailed model. This approach has been proven very effective in
Tröndle et al. [3]. Heavy parallelisation with high-performance computing
infrastructure allowed us to solve more than 50,000 resource-intensive
optimisation problems which, in combination with surrogate modelling, admit
spanning a probabilistic space of near-optimal solutions rather than putting
single scenarios into the foreground.

Thereby, we are able to present alternative solutions beyond least-cost that
have a high chance of involving a limited cost increase, just as we identify
regions that are unlikely to be cost-efficient. We derive both ranges of
options and technology-specific boundary conditions, that are not affected by
cost uncertainty and must be met to keep the total system cost within a
specified range. Our results show that indeed many such similarly costly but
technologically diverse solutions exist regardless of how technology cost
developments will unfold within the considered ranges.

##  2 Methods

In this section, we first outline how we obtain least-cost and near-optimal
solutions for a given cost parameter set. We then describe the model of the
European power system and define the cost uncertainties. Finally, we explain
how we make use of multi-fidelity surrogate modelling techniques based on
polynomial chaos expansions and find an experimental design that efficiently
covers the parameter space.

###  2.1 Least-Cost Investment Planning

The objective of long-term power system planning is to minimise the total
annual system costs, comprising annualised capital costs $c_{\star}$ for
investments at locations $i$ in generator capacity $G_{i,r}$ of technology
$r$, storage capacity $H_{i,s}$ of technology $s$, and transmission line
capacities $F_{\ell}$, as well as the variable operating costs $o_{\star}$ for
generator dispatch $g_{i,r,t}$:

$\displaystyle\min_{G,H,F,g}\quad\left\\{\sum_{i,r}c_{i,r}\cdot
G_{i,r}+\sum_{i,s}c_{i,s}\cdot H_{i,s}+\sum_{\ell}c_{\ell}\cdot
F_{\ell}+\sum_{i,r,t}w_{t}\cdot o_{i,r}\cdot g_{i,r,t}\right\\}$ (1)

where the snapshots $t$ are weighted by $w_{t}$ such that their total duration
adds up to one year. The objective is subject to a set of linear constraints
that define limits on (i) the capacities of infrastructure from geographical
and technical potentials, (ii) the availability of variable renewable energy
sources for each location and point in time, and (iii) linearised multi-period
optimal power flow (LOPF) constraints including storage consistency equations,
which we describe in more detail in the following.

The capacities of generation, storage and transmission infrastructure are
limited to their geographical potentials from above and existing
infrastructure from below:

$\displaystyle\underline{G}_{i,r}\leq G_{i,r}\leq\overline{G}_{i,r}$
$\displaystyle\qquad\forall i,r$ (2) $\displaystyle\underline{H}_{i,s}\leq
H_{i,s}\leq\overline{H}_{i,s}$ $\displaystyle\qquad\forall i,s$ (3)
$\displaystyle\underline{F}_{\ell}\leq F_{\ell}\leq\overline{F}_{\ell}$
$\displaystyle\qquad\forall\ell$ (4)

The dispatch of a renewable generator is constrained by its rated capacity and
the time- and location-dependent availability $\overline{g}_{i,r,t}$, given in
per-unit of the generator’s capacity:

$\displaystyle 0\leq g_{i,r,t}\leq\overline{g}_{i,r,t}G_{i,r}\qquad\forall
i,r,t$ (5)

The dispatch of storage units is described by a charge variable
$h_{i,s,t}^{+}$ and a discharge variable $h_{i,s,t}^{-}$, each limited by the
power rating $H_{i,s}$.

$\displaystyle 0\leq h_{i,s,t}^{+}\leq H_{i,s}$ $\displaystyle\qquad\forall
i,s,t$ (6) $\displaystyle 0\leq h_{i,s,t}^{-}\leq H_{i,s}$
$\displaystyle\qquad\forall i,s,t$ (7)

The energy levels $e_{i,s,t}$ of all storage units are linked to the dispatch
by

$\displaystyle e_{i,s,t}=\>$ $\displaystyle\eta_{i,s,0}^{w_{t}}\cdot
e_{i,s,t-1}+w_{t}\cdot h_{i,s,t}^{\text{inflow}}-w_{t}\cdot
h_{i,s,t}^{\text{spillage}}$ $\displaystyle\quad\forall i,s,t$
$\displaystyle+\eta_{i,s,+}\cdot w_{t}\cdot
h_{i,s,t}^{+}-\eta_{i,s,-}^{-1}\cdot w_{t}\cdot h_{i,s,t}^{-}.$ (8)

Storage units can have a standing loss $\eta_{i,s,0}$, a charging efficiency
$\eta_{i,s,+}$, a discharging efficiency $\eta_{i,s,-}$, natural inflow
$h_{i,s,t}^{\text{inflow}}$ and spillage $h_{i,s,t}^{\text{spillage}}$. The
storage energy levels are assumed to be cyclic and are constrained by their
energy capacity

$\displaystyle e_{i,s,0}=e_{i,s,T}$ $\displaystyle\qquad\forall i,s$ (9)
$\displaystyle 0\leq e_{i,s,t}\leq\overline{T}_{s}\cdot H_{i,s}$
$\displaystyle\qquad\forall i,s,t.$ (10)

To reduce the number of decisison variables, we link the energy capacity to
power ratings with a technology-specific parameter $\overline{T}_{s}$ that
describes the maximum duration a storage unit can discharge at full power
rating.

Kirchhoff’s Current Law (KCL) requires local generators and storage units as
well as incoming or outgoing flows $f_{\ell,t}$ of incident transmission lines
$\ell$ to balance the inelastic electricity demand $d_{i,t}$ at each location
$i$ and snapshot $t$

$\displaystyle\sum_{r}g_{i,r,t}+\sum_{s}h_{i,s,t}+\sum_{\ell}K_{i\ell}f_{\ell,t}=d_{i,t}\qquad\forall
i,t,$ (11)

where $K_{i\ell}$ is the incidence matrix of the network.

Kichhoff’s Voltage Law (KVL) imposes further constraints on the flow of AC
lines. Using linearised load flow assumptions, the voltage angle difference
around every closed cycle in the network must add up to zero. We formulate
this constraint using a cycle basis $C_{\ell c}$ of the network graph where
the independent cycles $c$ are expressed as directed linear combinations of
lines $\ell$ [20]. This leads to the constraints

$\displaystyle\sum_{\ell}C_{\ell c}\cdot x_{\ell}\cdot
f_{\ell,t}=0\qquad\forall c,t$ (12)

where $x_{\ell}$ is the series inductive reactance of line $\ell$.
Controllable HVDC links are not affected by this constraint.

Finally, all line flows $f_{\ell,t}$ must be operated within their nominal
capacities $F_{\ell}$

$\displaystyle\left|f_{\ell,t}\right|\leq\overline{f}_{\ell}F_{\ell}$
$\displaystyle\qquad\forall\ell,t,$ (13)

where $\overline{f}_{\ell}$ acts as a per-unit buffer capacity to protect
against the outage of single circuits.

This problem is implemented in the open-source tool PyPSA [21] and is solved
by Gurobi. Note, that it assumes perfect foresight for a single reference year
based on which capacities are optimised. It does not include pathway
optimisation, nor aspects of reserve power, or system stability. Changes of
line expansion to line impedance are ignored.

###  2.2 Near-Optimal Alternatives

Using the least-cost solution as an anchor, we use the $\epsilon$-constraint
method from multi-objective optimisation to find near-optimal feasible
solutions [6, 22]. For notational brevity, let $c^{\top}x$ denote the linear
objective function Equation 1 and $Ax\leq b$ the set of linear constraints
Equations 2 to 13 in a space of continuous variables, such that the minimised
system cost can be represented by

$C=\min_{x}\left\\{c^{\top}x\mid Ax\leq b\right\\}.$ (14)

We then encode the original objective as a constraint such that the cost
increase is limited to a given $\epsilon$. In other words, the feasible space
is cut to solutions that are at most $\epsilon$ more expensive than the least-
cost solution. Given this slack, we can formulate alternative search
directions in the objective. For instance, we can seek to minimise the sum of
solar installations $x_{s}\subseteq x$ with

$\overline{x_{s}}=\min_{x_{s}}\left\\{\>1^{\top}x_{s}\mid Ax\leq b,\quad
c^{\top}x\leq(1+\epsilon)\cdot C\>\right\\}.$ (15)

To draw a full picture of the boundaries of the near-optimal feasible space,
we systematically explore the extremes of various technologies: we both
minimise and maximise the system-wide investments in solar, onshore wind,
offshore wind, any wind, hydrogen storage, and battery storage capacities, as
well as the total volume of transmission network expansion. Evaluating each of
these technology groups for different cost deviations
$\epsilon\in\\{1\%,2\%,4\%,6\%,8\%\\}$ allows us to observe how the degree of
freedom regarding investment decisions rises as the optimality tolerance is
increased, both at lower and upper ends. The boundaries delineate Pareto
frontiers on which no criterion, neither reducing system cost nor extremising
the capacity of a technology, can be improved without depressing the other. By
arguments of convexity, these extremes even define limits within which all
near-optimal solutions are contained. Moreover, although this scheme primarily
studies aggregated capacities, the solutions are spatially explicit, and we
can inspect for each case how the capacities of each technology are
distributed within the network.

The near-optimal analysis above only explores the extremes of one technology
at a time, i.e. one direction in the feasible space. But actually the space of
attainable solutions within $\epsilon$ of the cost-optimum is multi-
dimensional. To further investigate trade-offs between multiple technologies,
in addition to the $\epsilon$-constraint and the objective to extremise
capacities of a particular technology, we formulate a constraint that fixes
the capacity of another technology. For instance, we search for the minimum
amount of wind capacity $x_{w}\subseteq x$ given that a certain amount of
solar is built

$\overline{x_{w}}=\min_{x_{w}}\left\\{\>1^{\top}x_{w}\mid Ax\leq b,\quad
c^{\top}x\leq(1+\epsilon)\cdot C,\quad
1^{\top}x_{s}=\underline{x_{s}}+\alpha\cdot(\overline{x_{s}}-\underline{x_{s}})\>\right\\}.$
(16)

The $\alpha$ denotes the relative position within the near-optimal range of
solar capacities at given $\epsilon$. For example, at $\alpha=0\%$ we look for
the least wind capacity given that minimal solar capacities are built. An
alternative but more complex approach to spanning the space of near-optimal
solutions in multiple dimensions at a time using a quick hull algorithm was
presented by Pedersen et al. [15].

Due to computational constraints, we focus on technologies which are assumed
to lend themselves to substitution and limit the corresponding analysis to a
single cost increase level of $\epsilon=6\%$. We consider the three pairs, (i)
wind and solar, (ii) offshore and onshore wind, (iii) hydrogen and battery
storage, by minimising and maximising the former while fixing the latter at
positions $\alpha\in\\{0\%,25\%,50\%,75\%,100\%\\}$ within the respective
near-optimal range.

###  2.3 Model Inputs

(a) low-fidelity: | (b) high-fidelity:
---|---
37 nodes and 4-hourly | 128 nodes and 2-hourly
|
|
Figure 1: Spatial and temporal resolution of the low and high fidelity model.
Green lines represent controllable HVDC lines. Red lines represent HVAC lines.
The exemplary capacity factors for wind and solar are shown for four days in
March at the northernmost node in Germany, alongside the normalised load
profile.

The instances of the coordinated capacity expansion problem (Section 2.1) are
based on PyPSA-Eur, which is an open model of the European power transmission
system that combines high spatial and temporal resolution [23]. Because it
only uses open data and every processing step is defined in a workflow [24],
we achieve a high level of transparency and reproducibility. In the following,
we outline the main features and configurations, and refer to the
supplementary material and Hörsch et al. [23] for more details.

#### Scenario

We target a fully renewable electricity system based on variable resources
such as solar photovoltaics, onshore wind and offshore wind, that has not
carbon emissions. We pursue a greenfield approach subject to a few notable
exceptions. The existing hydro-electric infrastructure (run-of-river, hydro
dams, pumped-storage) is included but not considered to be extendable due to
assumed geographical constraints. Furthermore, the existing transmission
infrastructure can only be reinforced continuously but may not be removed. In
addition to balancing renewables in space with transmission networks, the
model includes storage options at each node to balance renewables in time. We
consider two extendable storage technologies: battery storage representing
short-term storage suited to balancing daily fluctuations and hydrogen storage
which exemplifies long-term synoptic and seasonal storage.

#### Spatial and Temporal Resolution

Since the spatial and temporal resolution strongly affects the size of the
optimisation problem, running the model at full resolution is computationally
infeasible. Throughout the paper, we will therefore make use of two levels of
aggregation, reflecting a compromise between the computational burden incurred
by high-resolution models and the growing inaccuracies regarding transmission
bottlenecks and resource distribution in low-resolution models. We consider a
low-fidelity model with 37 nodes at a 4-hourly resolution for a full year that
models power flow via a transport model (i.e. excluding KVL of Equation 12)
and a high-fidelity model with 128 nodes at a 2-hourly resolution that is
subject to linearised load flow constraints (Figure 1).

#### Transmission Grid and Hydro-Electricity

The topology of the European transmission network is retrieved from the
ENTSO-E transparency map and includes all lines at and above 220 kV.
Capacities and electrical characteristics of transmission lines and
substations are inferred from standard types for each voltage level, before
they are transformed to a uniform voltage level. For each line, $N-1$ security
is approximated by limiting the line loading to 70% of its nominal rating. The
dataset further includes existing high-voltage direct current (HVDC) links and
planned projects from the Ten Year Network Development Plan (TYNDP). Existing
run-of-river, hydro-electric dams, pumped-hydro storage plants are retrieved
from powerplantmatching, a merged dataset of conventional power plants.

#### Renewable Energy Potentials

Eligible areas for developing renewable infrastructure are calculated per
technology and the grid nodes’ Voronoi cells, assuming wind and solar
installations always connect to the closest substation. How much wind and
solar capacity may be built at a location is constrained by eligible codes of
the CORINE land use database and is further restricted by distance criteria,
allowed deployment density, and the natural protection areas specified in the
NATURA 2000 dataset. Moreover, offshore wind farms may not be developed at sea
depths exceeding 50 metres, as indicated by the GEBCO bathymetry dataset.

#### Renewables and Demand Time Series

The location-dependent renewables availability time series are generated based
on two historical weather datasets from the year 2013. We retrieve wind
speeds, run-off and surface roughness from the ERA5 reanalysis dataset and use
the satellite-aided SARAH-2 dataset for the direct and diffuse surface solar
irradiance. Models for wind turbines, solar panels, and the inflow into the
basins of hydro-electric dams convert the weather data to hourly capacity
factors and aggregate these to each grid node. Historical country-level load
time series are taken from ENTSO-E statistics and are heuristically
distributed to each grid node to 40% by population density and to 60% by gross
domestic product.

###  2.4 Technology Cost Uncertainty

Technology | Lower Annuity | Upper Annuity | Unit
---|---|---|---
Onshore Wind | 73 | 109 | EUR/kW/a
Offshore Wind | 178 | 245 | EUR/kW/a
Solar | 36 | 53 | EUR/kW/a
Battery | 30 | 125 | EUR/kW/a
Hydrogen | 111 | 259 | EUR/kW/a
Table 1: Technology cost uncertainty using optimistic and pessimistic
assumptions from the Danish Energy Agency [25].

Uncertainty of technology cost projections is driven by two main factors:
unknown learning rates (i.e. how quickly costs fall as more capacity is built)
and unclear deployment rates (i.e. how much capacity will be built in the
future) [26, 27]. As modelling technological learning endogeneously is
computationally challenging due to the nonconvexity it entails [28, 29],
technology cost uncertainty is typically defined exogenously by an interval
within which costs may vary and a distribution that specifies which segments
are more probable.

Ranges of cost projections are best chosen as wide as possible to avoid
excluding any plausible scenarios [30, 9]. When uncertainty has been
considered in the literature, cost assumptions have commonly been modelled to
vary between $\pm 20\%$ and $\pm 65\%$ depending on the technology’s maturity
[30, 31, 17, 32, 3]. In this study, we consider uncertainty regarding the
annuities of onshore wind, offshore wind, solar PV, battery and hydrogen
storage systems. The latter comprises the cost of electrolysis, cavern
storage, and fuel cells. For solar PV we assume an even split between utility-
scale PV and residential rooftop PV. Evaluating uncertainties based on
annuities has a distinct advantage. They can be seen to simultaneously
incorporate uncertainties about the overnight investments, fixed operation and
maintenance costs, their lifetime, and the discount rate, since multiple
combinations lead to the same annuity. We built the uncertainty ranges
presented in Table 1 from the optimistic and pessimistic technology cost and
lifetime projections for the year 2050 from the Danish Energy Agency, which
correspond to 90% confidence intervals [25]. In cases where no uncertainty
ranges were provided for the year 2050, such as for rooftop PV, projections
for the year 2030 define the upper end of the uncertainty interval.

Distributions of cost projections have been assumed to follow normal [16] or
triangular distributions [32]. But independent uniform distributions are the
most prevalent assumption [30, 33, 31, 3, 34, 12, 35, 36]. This approach is
backed by the maximum entropy approach [3], which states that given the
persistent lack of knowledge about the distribution the independent uniform
distribution, that makes fewest assumptions, is most appropriate. Although the
assumed independence may neglect synergies between technologies, for example,
between offshore and onshore wind turbine development, we follow the
literature by assuming that the cost are independent and uniformly distributed
within the ranges specified in Table 1.

###  2.5 Surrogate Modelling with Polynomial Chaos Expansion

Searching for least-cost solutions (Section 2.1) and many associated near-
optimal alternatives (Section 2.2) of a highly resolved power system model
(Section 2.3) on its own is already labour-intensive from a computational
perspective. Repeating this search for a large variety of cost assumptions
(Section 2.4), to be able to make statements about the robustness of
investment flexibility near the optimum under uncertainty, adds another layer
to the computational burden.

Surrogate models111Surrogate names are also known by names such as
approximation models, response surface methods, metamodels and emulators.
offer a solution for such cases, where the outcome of the original model
cannot be obtained easily. In contrast to the full model, they only imitate
the input/output behaviour for a selection of aggregated outputs, but take
much less time to compute [37]. Like other machine learning techniques, they
generalise from a training dataset that comprises only a limited number of
samples. As surrogate models interpolate gaps in the parameter space that are
not contained in the sample set, which would otherwise be computationally
expensive to fill, they are well suited to use cases such as parameter space
exploration and sensitivity analysis.

Consequently, in this paper we will make use of surrogate models that map the
cost of onshore wind, offshore wind, solar, hydrogen, and battery storage
(Table 1) onto a selection of eight system-level outputs. These are the total
system cost and the installed onshore wind, offshore wind, solar, hydrogen,
battery, and transmission network capacities. We construct surrogate models
for least-cost and near-optimal solutions separately for each system cost
slack, search direction, fixed total capacity, and output variable. This
results in a collection of 808 individual surrogate models based on 101 solved
optimisation problems per set of cost assumptions. The method we choose from
an abundance of alternatives is based on polynomial chaos expansion (PCE) [38,
39, 40]. We select this approach because the resulting approximations allow
efficient analytical statistical evaluation [38] and can conveniently combine
training data from variously detailed models [37].

The general idea of surrogate models based on PCE is to represent uncertain
model outputs as a linear combination of orthogonal basis functions of the
random input variables weighted by deterministic coefficients [41]. It is a
Hilbert space technique that works in principle analogously to decomposing a
periodic signal into its Fourier components [41]. Building the surrogate model
consists of the following steps: (i) sampling a set of cost projections from
the parameter space, (ii) solving the least-cost or near-optimal investment
planning problem for each sample, (iii) selecting an expansion of orthogonal
polynomials within the parameter space, (iv) performing a regression to
calculate the polynomial coefficients, and ultimately (v) using the model
approximation for statistical analysis. In the following, we will formalise
this approach mathematically, which we implemented using the chaospy toolbox
[42], and elaborate on individual aspects in more detail.

We start by defining the vector of random input variables as

$\bm{\mathsf{x}}=\\{\mathsf{x}_{1},\dots,\mathsf{x}_{m}\\}$ (17)

that represents the $m$ uncertain cost projections. Further, we let

$\mathsf{y}=f(\bm{\mathsf{x}})$ (18)

describe how the uncertainty of inputs $\bm{\mathsf{x}}$ propagates through
the computationally intensive model $f$ (i.e. the solving a large optimisation
problem) to the outputs $\mathsf{y}\in\mathbb{R}$.

We can represent the computational model $f$ with its polynomial chaos
expansion

$\mathsf{y}=f(\bm{\mathsf{x}})=\sum_{{\bm{\alpha}}\,\in\,\mathbb{N}^{m}}r_{{\bm{\alpha}}}\psi_{{\bm{\alpha}}}(\bm{\mathsf{x}}),$
(19)

where $\psi_{\bm{\alpha}}$ denotes multivariate orthogonal polynomials that
form a Hilbertian basis and $r_{\bm{\alpha}}\in\mathbb{R}$ are the
corresponding polynomial coefficients [38]. The multiindex
${\bm{\alpha}}=\\{\alpha_{1},\dots,\alpha_{m}\\}$ denotes the degree of the
polynomial $\psi_{\bm{\alpha}}$ in each of the $m$ random input variables
$\mathsf{x}_{i}$. As Equation 19 features an infinite number of unknown
coefficients, it is common practice to approximate by truncating the expansion
to get a finite number of coefficients

$f(\bm{\mathsf{x}})\approx
f^{\prime}(\bm{\mathsf{x}})=\sum_{{\bm{\alpha}}\,\in\,\mathcal{A}^{m,p}}r_{\bm{\alpha}}\psi_{\bm{\alpha}}(\bm{\mathsf{x}}).$
(20)

In the standard truncation scheme [40, 38], all polynomials in $m$ input
variables where the total degree is less than $p$ are selected. We can write
this as a set of indices

$\mathcal{A}^{m,p}=\left\\{{\bm{\alpha}}\in\mathbb{N}^{m}\,:\,\left|{\bm{\alpha}}\right|\leq
p\right\\},$ (21)

where $\left|{\bm{\alpha}}\right|=\sum_{i=1}^{m}\alpha_{i}$. Given the joint
distribution of $\bm{\mathsf{x}}$ and a maximum degree, a suitable collection
of orthogonal polynomials can be constructed using a three terms recurrence
algorithm [42]. The cardinality of the truncated PCE,

$q=\text{card}\,\mathcal{A}^{m,p}=\left(\begin{matrix}m+p\\\
p\end{matrix}\right)=\frac{(m+p)!}{m!p!},$ (22)

indicates the number of unknown polynomial coefficients.

We determine these coefficicients by a regression based on a set of cost
parameter samples and the corresponding outputs,

$\mathcal{X}=\left\\{\bm{x}^{(1)},\dots,\bm{x}^{(n)}\right\\}\quad\text{and}\quad\mathcal{Y}=\left\\{f\left(\bm{x}^{(1)}\right),\dots,f\left(\bm{x}^{(n)}\right)\right\\}.$
(23)

Using this training dataset, we minimise the least-square residual of the
polynomial approximation across all observations. We add an extra $L_{1}$
regularisation term, that induces a preference for fewer non-zero
coefficients, and solve

$\hat{\bm{r}}=\operatorname*{\arg\\!\min}_{\bm{r}\,\in\,\mathbb{R}^{q}}\left[\frac{1}{n}\sum_{i=1}^{n}\left(f\left(\bm{x}^{(i)}\right)-\sum_{{\bm{\alpha}}\,\in\,\mathcal{A}^{m,p}}r_{\bm{\alpha}}\psi_{\bm{\alpha}}\left(\bm{x}^{(i)}\right)\right)^{2}+\lambda\left\lVert\bm{r}\right\rVert_{1}\right],$
(24)

where we set the regularisation penalty to $\lambda=0.005$. This results in a
sparse PCE that has proven to improve approximations in high-dimensional
uncertainty spaces and to reduce the required number of samples for comparable
approximation errors [40]. Knowing the optimised regression coefficients, we
can now assemble the complete surrogate model

$\mathsf{y}=f(\bm{\mathsf{x}})\approx
f^{\prime}(\bm{\mathsf{x}})=\sum_{{\bm{\alpha}}\,\in\,\mathcal{A}^{m,p}}\hat{r}_{\bm{\alpha}}\psi_{\bm{\alpha}}(\bm{\mathsf{x}}).$
(25)

###  2.6 Multifidelity Approach

To construct a sufficiently precise PCE-based surrogate model, it is desirable
to base it on many samples from a high-fidelity model. However, this is likely
prohibitively time-consuming. On the other hand, relying only on samples from
a low-fidelity model may be too inaccurate [43]. For example, an investment
model that features only a single node per country will underestimate
transmission bottlenecks and regionally uneven resource or demand
distribution. In Section 2.3 we already alluded to using two models with
varying spatial and temporal resolution in this paper. We integrate both in a
multi-fidelity approach [43, 37], and demonstrate how we can simultaneously
avail of high coverage of the parameter space by sampling the simpler model
many times, and the high spatio-temporal detail yielded by fewer more complex
model runs.

The idea of the multi-fidelity approach is to build a corrective surrogate
model $\Delta^{\prime}(\bm{\mathsf{x}})$ for the error of the low-fidelity
model $f_{\ell}$ compared to the high-fidelity model $f_{h}$

$\Delta(\bm{\mathsf{x}})=f_{h}(\bm{\mathsf{x}})-f_{\ell}(\bm{\mathsf{x}}),$
(26)

and add it to a surrogate model of the low-fidelity model to approximate the
behaviour of the high-fidelity model

$f_{h}^{\prime}(\bm{\mathsf{x}})=f_{\ell}^{\prime}(\bm{\mathsf{x}})+\Delta^{\prime}(\bm{\mathsf{x}}).$
(27)

Typically, the corrective PCE rectifies only the lower order effects of the
low-fidelity surrogate model [37]. The advantage is that this way the
correction function can be determined based on fewer samples analogous to
Section 2.5. To sample the errors, it is only required that the high-fidelity
samples are a subset of the low-fidelity samples, e.g.

$\mathcal{X}_{h}=\left\\{\bm{x}^{(1)},\dots,\bm{x}^{(n_{h})}\right\\}\quad\text{and}\quad\mathcal{X}_{\ell}=\left\\{\bm{x}^{(1)},\dots,\bm{x}^{(n_{h})},\dots,\bm{x}^{(n_{\ell})}\right\\},$
(28)

which we can easily guarantee by using deterministic low-discrepancy series in
the experimental design (Section 2.7). With $p_{c}<p_{\ell}$ and consequently
$\mathcal{A}_{c}\subset\mathcal{A}_{\ell}$, the multi-fidelity surrogate model
can be written as a combination of low-fidelity and corrective polynomial
coefficients

$f_{h}^{\prime}(\bm{\mathsf{x}})=\sum_{{\bm{\alpha}}\,\in\,\mathcal{A}_{\ell}^{m,p_{\ell}}\,\cap\,\mathcal{A}_{c}^{m,p_{c}}}\left(r_{\ell,{\bm{\alpha}}}+r_{c,{\bm{\alpha}}}\right)\psi_{\bm{\alpha}}(\bm{\mathsf{x}})+\sum_{{\bm{\alpha}}\,\in\,\mathcal{A}_{\ell}^{m,p_{\ell}}\,\setminus\,\mathcal{A}_{c}^{m,p_{c}}}r_{\ell,{\bm{\alpha}}}\psi_{\bm{\alpha}}(\bm{\mathsf{x}}).$
(29)

In this work, we apply a multi-fidelity surrogate model that considers effects
up to order three observed in the low-fidelity model. These are then corrected
with linear terms derived from insights from the high-fidelity model. We
justify this choice by experimentation in Section 2.8, by testing against
other typical choices between orders one to five [40]. Given the polynomial
expansion order, the remaining question is how many samples are necessary to
attain an acceptable approximation.

###  2.7 Experimental Design

The experimental design covers strategies to find sufficiently high coverage
of the parameter space at low computational cost [39, 19]. It deals with how
many samples are drawn and what sampling method is used.

Traditional Monte-Carlo sampling with pseudo-random numbers is known to
possess slow convergence properties, especially in high-dimensional parameter
spaces. So-called low-discrepancy series can greatly improve on random
sampling. Because they are designed to avoid forming large gaps and clusters,
these deterministic sequences efficiently sample from the parameter space
[39]. Thus, we choose to draw our samples from a low-discrepancy Halton
sequence.

For the question about how many samples should be drawn, we resort to the
oversampling ratio (OSR) as a guideline. The OSR is defined as the ratio
between the number of samples and the number of unknown coefficients [37]. The
literature recommends values between two and three [44, 37, 39, 40]. In other
words, for a sufficiently accurate approximation, there should be
significantly more samples than unknown coefficients. If the OSR is lower, the
regression is prone to the risk of overfitting. On the other hand, a high OSR
may lead to a very coarse approximation [37].

According to Equation 22, targeting an OSR of two and considering the five
uncertain technology cost parameters (Table 1), approximating linear effects
would require at least 12 samples, whereas cubic relations would already need
112 samples. Even 504 samples would be necessary to model the dynamics of
order 5. To investigate the quality of different PCE orders and retain a
validation dataset, we draw 500 samples for the low-fidelity model. Due to the
computational burden carried by the high-fidelity models, we settle on a
linear correction in advance, such that 15 samples for the high-fidelity model
are acceptable. In combination with 101 least-cost and near-optimal
optimisation runs for each sample, this setup results in a total number of
50,500 runs of the low-fidelity model and 1,515 runs of the high-fidelity
model. On average a single high-fidelity model run took 20 GB of memory and 5
hours to solve. Each low-fidelity model run on average consumed 3 GB of memory
and completed within 5 minutes. This setup profits tremendously from
parallelisation as it involves numerous independent optimisation runs.
Moreover, it would have been infeasible to carry out without high-performance
computing.

###  2.8 Model Validation

(a) number of samples

(b) polynomial order

Figure 2: Cross-validation errors by output for varying sample sizes and
polynomial orders of least-cost low-fidelity surrogate models.

We justify the use of surrogate modelling by cross-validation. Out of the 500
low-fidelity samples, 100 samples are not used in the regression. This
validation dataset is unknown to the surrogate model and is consulted to
assess the approximation’s quality. Because the high-fidelity sample size is
limited and approximating near-optimal solutions is not assumed to
fundamentally differ, we base the validation on low-fidelity least-cost
solutions only. We experimentally evaluate the approximation errors between
predicted and observed data for different combinations of polynomial order and
sample size to decide on a suitable parameterisation. We present the
coefficient of determination (R2) for the variance captured, the mean absolute
(percentage) errors (MAE/MAPE) for absolute and relative deviations, and the
root mean squared error (RMSE).

Regarding the number of samples required, Figure 2(a) foremost illustrates
that, given enough samples, we achieve average relative errors of less than 4%
for most output variables. This is comparable to the cross-validation errors
from Tröndle et al. [3] at rates below 5%. Only for offshore wind and battery
storage, we observe larger errors. However, this can be explained by a
distortion of the relative measure when these technologies are hardly built
for some cost projections. On the contrary, the prediction of total system
costs is remarkably accurate. Figure 2(a) also demonstrates that for a
polynomial order of 3, we gain no significant improvement with more than 200
samples. In fact, thanks to the regularisation term used in the regression, we
already attain acceptable levels of accuracy with as few as 50 samples.
Moreover, the high R2 values underline that the surrogate model can explain
most of the output variance.

Regarding the polynomial order, Figure 2(b) shows that an order of 2 and below
may be too simple to capture the interaction between different parameters. On
the other hand, an order of 4 and above yields no improvement and, were it not
for the moderating regularisation term, would even result in a loss of
generalisation properties due to overfitting. As higher-order approximations
require significantly more samples, an order of 3 appears to be a suitable
compromise to limit the computational burden.

##  3 Results and Discussion

In this section, we approach the uncertainty analysis to near-optimal
solutions by reviewing the propagation of input uncertainties into least-cost
solutions first and expanding gradually from there. This includes inspecting
cost and capacity distributions induced by unknown future technology cost and
conducting a global sensitivity analysis that identifies the most influential
cost parameters for least-cost solutions. We then expand the uncertainty
analysis to the space of nearly cost-optimal solutions, which yields us
insights about the consistency of near-optimal alternatives across a variety
of cost parameters.

###  3.1 Cost and Capacity Distribution of Least-Cost Solutions

Figure 3:  Distribution of total system cost, generation, storage, and
transmission capacities for least-cost solutions.

Based on the uncertainty of cost inputs, the total annual system costs vary
between 160 and 220 billion Euro per year, as displayed in Figure 3. This
means the most pessimistic cost projections entail about 40% higher cost than
the most optimistic projections. All least-cost solutions build at least 350
GW solar and 600 GW wind, but no more than 1100 GW. While wind capacities tend
towards higher values, solar capacities tend towards lower values. We observe
that least-cost solutions clearly prefer onshore over offshore wind, yet
onshore wind features the highest uncertainty range alongside battery storage.
The cost optimum gravitates towards hydrogen storage rather than battery
storage unless battery storage becomes very cheap. There are no least-cost
solutions without hydrogen, only some without battery storage. Transmission
network expansion is least affected by cost uncertainty and consistently
doubled compared to today’s capacities. The question arises, what we can
conclude from these insights. The interpretation of the observed ranges may be
limited because they are not robust when we look beyond the least-cost
solutions and acknowledge structural modelling uncertainties, such as social
constraints. Moreover, the pure distribution of outputs does not yet convey
information about how sensitive results are to particular cost assumptions.
But knowing the technologies for which lowering overnight costs has a
significant impact is important to promote technological learning in that
direction.

(a) onshore wind

(b) offshore wind

(c) solar

(d) battery storage

(e) hydrogen storage

(f) transmission

Figure 4:  Sensitivity of capacities towards their own technology cost. The
median (Q50) alongside the 5%, 25%, 75%, and 95% quantiles (Q5–Q95) display
the sensitivity subject to the uncertainty induced by other cost parameters.

(a) first-order Sobol indices [%]

(b) total Sobol indices [%]

Figure 5:  Sobol indices. These sensitivity indices attribute output variance
to random input variables and reveal which inputs the outputs are most
sensitive to. The first-order Sobol indices quantify the share of output
variance due to variations in one input parameter alone. The total Sobol
indices further include interactions with other input variables. Total Sobol
indices can be greater than 100% if the contributions are not purely additive.

###  3.2 Parameter Sweeps and Global Sensitivity Indices

Figure 4 addresses a selection of local self-sensitivities, i.e. how the cost
of a technology influences its deployment while displaying the remaining
uncertainty induced by other cost parameters. The overall tendency is easily
explained: the cheaper a technology becomes, the more it is built. However,
changes of slope and effects on the uncertainty range as one cost parameter is
swept are insightful nonetheless. For instance, Figure 4 reveals that battery
storage becomes significantly more attractive economically once its annuity
falls below 75 EUR/kW/a (including 6h energy capacity at full power output)
hydrogen storage features a steady slope. A low cost of onshore wind makes
building much onshore wind capacity attractive with low uncertainty, whereas
if onshore wind costs are high how much is built greatly depends on other cost
parameters. The opposite behaviour is observed for offshore wind and solar.
The cost of hydrogen storage mostly causes the limited uncertainty about cost-
optimal levels of grid expansion. As the cost of hydrogen storage falls, less
grid reinforcement is chosen. But since the presented self-sensitivities only
exhibit a fraction of all sensitivities, in the next step we formalise how
input uncertainties affect each outcome systematically by applying variance-
based global sensitivity analysis techniques, which have been applied in the
context of energy systems e.g. in [3, 16].

Sensitivity indices, or Sobol indices, attribute the observed output variance
to each input and can be computed analytically from the polynomial chaos
expansion [38]. For our application, the Sobol indices can, for instance, tell
us which technology cost contributes the most to total system cost or how much
of a specific technology will be built. The first-order Sobol indices describe
the share of output variance due to variations in one input alone averaged
over variations in the other inputs. Total Sobol indices also consider higher-
order interactions, which are greater than 100% if the relations are not
purely additive.

The first-order and total Sobol indices for least-cost solutions in Figure 5
show that the total system cost is largely determined by how expensive it is
to build onshore wind capacity, followed by the cost of hydrogen storage. The
amount of wind in the system is almost exclusively governed by the cost of
onshore and offshore wind parks. Other carriers yield a more varied picture.
The cost-optimal solar capacities additionally depend on onshore wind and
battery costs. The amount of hydrogen storage is influenced by battery and
hydrogen storage cost alike. Although there are noticeable higher-order
effects, which are most extensive for transmission, the first-order effects
dominate. Strikingly, the volume of transmission network expansion strongly
depends on the cost of hydrogen storage, which can be explained by the
synoptic spatio-temporal scale of wind power variability across the European
continent which both hydrogen storage and transmission networks seek to
balance from different angles. While hydrogen storage typically balances
multi-week variations in time, continent-spanning transmission networks
exploit the circumstance that as weather systems traverse the continent, it is
likely always to be windy somewhere in Europe.

###  3.3 Fuzzy Near-Optimal Corridors with Increasing Cost Slack

So far, we quantified the output uncertainty and analysed the sensitivity
towards inputs at least-cost solutions only. Yet, it has been previously shown
that even for a single cost parameter set a wide array of technologically
diverse but similarly costly solutions exists [6]. In the next step, we
examine how technology cost uncertainty affects the shape of the space of
near-optimal solutions.

By identifying feasible alternatives common to all, few or no cost samples, we
outline low-cost solutions common to most parameter sets (e.g. above 90%
contour) as well as system layouts that do not meet low-cost criteria in any
circumstances for varying $\epsilon$ in Figure 6. The wider the displayed
contour lines are apart, the more uncertainty exists about the boundaries. The
closer contour lines are together, the more specific the limits are. The
height of the quantiles quantifies flexibility for a given level of certainty
and slack; the angle presents information about the sensitivity towards cost
slack.

Figure 6:  Space of near-optimal solutions by technology under cost
uncertainty. For each technology and cost sample, the minimum and maximum
capacities obtained for increasing cost penalties $\epsilon$ form a cone,
starting from a common least-cost solution. By arguments of convexity, the
capacity ranges contained by the cone can be near-optimal and feasible, given
a degree of freedom in the other technologies. From optimisation theory, we
know that the cones widen up for increased slacks. As we consider technology
cost uncertainty, the cone will look slightly different for each sample. The
contour lines represent the frequency a solution is inside the near-optimal
cone over the whole parameter space. This is calculated from the overlap of
many cones, each representing a set of cost assumptions. Due to discrete
sampling points in the $\epsilon$-dimension, the plots further apply quadratic
interpolation and a Gaussian filter for smoothing.

From the fuzzy upper and lower Pareto fronts in Figure 6 we can see that it is
highly likely that building 900 GW of wind capacity is possible within 3% of
the optimum, and that conversely building less than 600 GW has a low chance of
being near the cost optimum. Only a few solutions can forego onshore wind
entirely and remain within 8% of the cost-optimum, whereas it is very likely
possible to build a system without offshore wind at a cost penalty of 4% at
most. On the other hand, more offshore wind generation is equally possible.
Unlike for onshore wind, where it is more uncertain how little can be built,
uncertainty regarding offshore wind deployment exists about how much can be
built so that costs remain within a pre-specified range. For solar, the range
of options within 8% of the cost optimum at 90% certainty is very wide.
Anything between 100 GW and 1000 GW appears feasible. In comparison to onshore
wind, the uncertainty about minimal solar requirements is smaller.

The level of required transmission expansion is least affected by the cost
uncertainty. To remain within $\epsilon=8\%$ it is just as likely possible to
plan for moderate grid reinforcement by 30% as is initiating extensive
remodelling of the grid by tripling the transmission volume compared to what
is currently in operation. These results indicate that in any case some
transmission reinforcement to balance renewable variations across the
continent appears to be essential. Hydrogen storage, symbolising long-term
storage, also gives the impression of a vital technology in many cases.
Building 100 GW of hydrogen storage capacity is likely viable within 2% of the
cost optimum and, even at $\epsilon=8\%$, only 25% of cost samples require no
long-term storage; when battery costs are exceptionally low. Overall, 90% of
cases appear to function without any short-term battery storage while the
system cost rises by 4% at most. However, especially battery storage exhibits
a large degree of freedom to build more.

###  3.4 Probabilistic Near-Optimal Space in Two Technology Dimensions

(a) wind and solar

(b) offshore and onshore wind

(c) hydrogen and battery storage

Figure 7:  Space of near-optimal solutions by selected pairs of technologies
under cost uncertainty. Just like in Figure 6, the contour lines depict the
overlap of the space of near-optimal alternatives across the parameter space.
It can be thought of as the cross-section of the probabilistic near-optimal
feasible space for a given $\epsilon$ in two technology dimensions and
highlights that the extremes of two technologies from Figure 6 cannot be
achieved simultaneously.

The fuzzy cones from Figure 6 look at trade-offs between system cost and
single techologies, assuming that the other technologies can be heavily
optimised. But as there are dependencies between the technologies, in Figure 7
we furthermore evaluate trade-offs between technologies for three selected
pairs at fixed system cost increase of $\epsilon=6\%$, addressing which
combinations of wind and solar capacity, offshore and onshore turbines, and
hydrogen and battery storage are likely to be cost-efficient.

First, Figure 7(a) addresses constraints between wind and solar. The upper
right boundary exists because building much of both wind and solar would be
too expensive. The absence of solutions in the bottom left corner means that
building too little of any wind or solar does not suffice to generate enough
electricity. From the shape and contours, we see a high chance that building
1000 GW of wind and 400 GW of solar is within 6% of the cost-optimum. On the
other hand, building less than 200 GW of solar and 600 GW of wind is unlikely
to yield a low-cost solution. In general, minimising the capacity of both
primal energy sources will shift capacity installations to high-yield
locations even if additional network expansion is necessary and boost the
preference for highly efficient storage technologies. Overall, we can take
away from this that, even considering combinations of wind and solar, a wide
space of low-cost options exists with moderate to high likelihood, although
the range of alternatives is shown to be more constrained.

The trade-off between onshore wind and offshore wind is illustrated in Figure
7(b). Here, the most certain area is characterised by building more than 600
GW onshore wind, and less than 250 GW offshore wind capacity. However, there
are some solutions with high substitutability between onshore and offshore
wind, shown in the upper left bulge of the contour plot. Compared to wind and
solar, the range of near-optimal solutions is even more constrained. The key
role of energy storage in a fully renewable system is underlined in Figure
7(c). Around 50 GW of each is at least needed in any case, while highest
likelihoods are attained when building 150 GW of each.

(a) minimal onshore wind with 8% system cost slack

(b) minimal transmission expansion with 8% system cost slack

Figure 8:  Distribution of total system cost, generation, storage, and
transmission capacities for two near-optimal search directions with
$\epsilon=8\%$ system cost slack.

###  3.5 Capacity Distributions at Minimal Onshore Wind and Transmission Grid

The aforementioned contour plots Figures 6 to 7 outline what is likely
possible within specified cost ranges and subject to technology cost
uncertainty, but do not expose the changes the overall system layout
experiences when reaching for the extremes in one technology. Therefore, we
show in Figure 8 how the system-wide capacity distributions vary compared to
the least-cost solutions (Figure 3) for two exemplary alternative objectives.
For that, we chose minimising onshore wind capacity and transmission expansion
because they are often linked to social acceptance issues.

Figure 8(a) illustrates that reducing onshore wind capacity is predominantly
compensated by increased offshore wind generation but also added solar
capacities. The increased focus on offshore wind also leads to a tendency
towards more hydrogen storage, while transmission expansion levels are
similarly distributed as for the least-cost solutions. From Figure 8(b) we can
further extract that avoiding transmission expansion entails more hydrogen
storage that compensates balancing in space with balancing in time, and more
generation capacity overall, where resources are distributed to locations with
high demand but weaker capacity factors and more heavily curtailed.

###  3.6 Critical Appraisal

The need to solve models for many cost projections and near-optimal search
directions in reasonable time means that compromises had to be made in other
modelling dimensions. For instance, the analysis would profit from a richer
set of technologies and further uncertain input parameters, including
efficiencies of fuel cells and electrolysis or the consideration of
concentrating solar power, geothermal energy, biomass, and nuclear to name
just a few. But as the number of considered technologies and parameters rises,
so does the computational burden. Given the already considerable computational
efforts involved in procuring our results, considering the full breadth of
technologies and uncertainties would not have been feasible with the
computational resources available. Moreover, limitations apply to the scope of
the analysis which is limited to the electricity sector does not consider
coupling to other energy sectors. However, accounting for interactions across
sectors at high resolution in similarly set future studies is desirable and in
development. Additionally, we assess no path dependencies via multi-period
investments and endogenous learning, but optimise for an emission reduction in
a particular target year based on annualised costs. We further disregard
interannual variations of weather data by basing the analysis just on a single
weather year for computational reasons. Lastly, aspects such as reserves,
system adequacy and inertia have not been considered.

##  4 Conclusion

In this work, we systematically explore a space of alternatives beyond least-
cost solutions for society and politics to work with. We show how narrowly
following cost-optimal results underplays an immense degree of freedom in
designing future renewable power systems. To back our finding that there is no
unique path to cost-efficiency, we account for the inherent uncertainties
regarding technology cost projections, and draw robust conclusions about the
range of options, boundary conditions and cost sensitivities:

#### Wide Range of Trade-Offs

We find that there is a substantial range of options within 8% of the least-
cost solution regardless of how cost developments will unfold. This holds
across all technologies individually and even when considering dependencies
between wind and solar, offshore and onshore wind, as well as hydrogen and
battery storage.

#### Must-Avoid Boundary Conditions

We also carve out a few boundary conditions which must be met to keep costs
low and are not affected by the prevailing cost uncertainty. For a fully
renewable power system, either offshore or onshore wind capacities in the
order of 600 GW along with some long-term storage technology and transmission
network reinforcement by more than 30% appears essential.

#### Technology Cost Sensitivities

We identify onshore wind cost as the apparent main determinant of system cost,
though it can often be substituted with offshore wind for a small additional
cost. Moreover, the deployment of batteries is the most sensitive to its cost,
whereas required levels of transmission expansion are least affected by cost
uncertainty.

The robust investment flexibility in shaping a fully renewable power system we
reveal opens the floor to discussions about social trade-offs and navigating
around issues, such as public opposition towards wind turbines or transmission
lines. Rather than modellers making normative choices about how the energy
system should be optimised, we offer methods that present a wide spectrum of
options and trade-offs that are feasible and within a reasonable cost range,
to help society decide how to shape the future of the energy system.

## Acknowledgement

F.N. and T.B. gratefully acknowledge funding from the Helmholtz Association
under grant no. VH-NG-1352. The responsibility for the contents lies with the
authors.

## CRediT Author Statement

Fabian Neumann: Conceptualization, Methodology, Investigation, Software,
Validation, Formal analysis, Visualization, Writing – Original Draft, Writing
– Review & Editing Tom Brown: Conceptualization, Writing – Review & Editing,
Supervision, Project administration, Funding acquisition

## Data Availability

The code to reproduce the experiments is available at github.com/fneum/broad-
ranges. We also refer to the documentation of PyPSA (pypsa.readthedocs.io) and
PyPSA-Eur (pypsa-eur.readthedocs.io).

## References

  * [1] E. Trutnevyte, Does cost optimization approximate the real-world energy transition?, Energy 106 (2016) 182–193.  doi:10/f8vrcz.
  * [2] S. Pye, O. Broad, C. Bataille, P. Brockway, H. E. Daly, R. Freeman, A. Gambhir, O. Geden, F. Rogan, S. Sanghvi, J. Tomei, I. Vorushylo, J. Watson, Modelling net-zero emissions energy systems requires a change in approach, Climate Policy (2020) 1–10doi:10/ghdkp9.
  * [3] T. Tröndle, J. Lilliestam, S. Marelli, S. Pfenninger, Trade-Offs between Geographic Scale, Cost, and Infrastructure Requirements for Fully Renewable Electricity in Europe, Joule (2020) S2542435120303366doi:10/gg8zk2.
  * [4] X. Yue, S. Pye, J. DeCarolis, F. G. Li, F. Rogan, B. O. Gallachòir, A review of approaches to uncertainty assessment in energy system optimization models, Energy Strategy Reviews 21 (2018) 204–217.  doi:10/gf75x4.
  * [5] S. Pye, F. G. Li, A. Petersen, O. Broad, W. McDowall, J. Price, W. Usher, Assessing qualitative and quantitative dimensions of uncertainty in energy modelling for policy support in the United Kingdom, Energy Research & Social Science 46 (2018) 332–344.  doi:10/gf8nvv.
  * [6] F. Neumann, T. Brown, The near-optimal feasible space of a renewable power system model, Electric Power Systems Research 190 (2021) 106690, arXiv:1910.01891.  doi:10/ghcpr2.
  * [7] F. Lombardi, B. Pickering, E. Colombo, S. Pfenninger, Policy Decision Support for Renewables Deployment through Spatially Explicit Practically Optimal Alternatives, Joule (2020) S2542435120303482doi:10/gg8z6v.
  * [8] J.-P. Sasse, E. Trutnevyte, Regional impacts of electricity system transition in Central Europe until 2035, Nature Communications 11 (1) (2020) 4972.  doi:10/ghdwjs.
  * [9] D. L. McCollum, A. Gambhir, J. Rogelj, C. Wilson, Energy modellers should explore extremes more systematically in scenarios, Nature Energy 5 (2) (2020) 104–107.  doi:10/ggk3hj.
  * [10] D. P. Schlachtberger, T. Brown, M. Schäfer, S. Schramm, M. Greiner, Cost optimal scenarios of a future highly renewable European electricity system: Exploring the influence of weather data, cost parameters and policy constraints, Energy 163 (2018) 100–114.  doi:10/gfk5cj.
  * [11] J. F. DeCarolis, S. Babaee, B. Li, S. Kanungo, Modelling to generate alternatives with an energy system optimization model, Environmental Modelling and Software 79 (2016) 300–310.  doi:10/f8n923.
  * [12] F. G. Li, E. Trutnevyte, Investment appraisal of cost-optimal and near-optimal pathways for the UK electricity sector transition to 2050, Applied Energy 189 (2017) 89–109.  doi:10/f9qtrf.
  * [13] L. Nacken, F. Krebs, T. Fischer, C. Hoffmann, Integrated renewable energy systems for Germany – A model-based exploration of the decision space, 2019, p. 8.
  * [14] J. Price, I. Keppo, Modelling to generate alternatives: A technique to explore uncertainty in energy-environment-economy models, Applied Energy 195 (2017) 356–369.  doi:10/ggfzqq.
  * [15] T. T. Pedersen, M. Victoria, M. G. Rasmussen, G. B. Andresen, Modeling all alternative solutions for highly renewable energy systems, arXiv:2010.00836 (Oct. 2020).
  * [16] G. Mavromatidis, K. Orehounig, J. Carmeliet, Uncertainty and global sensitivity analysis for the optimal design of distributed energy systems, Applied Energy 214 (2018) 219–238.  doi:10/gc6n48.
  * [17] A. Pizarro-Alonso, H. Ravn, M. Münster, Uncertainties towards a fossil-free system with high integration of wind energy in long-term planning, Applied Energy 253 (2019) 113528.  doi:10/ghfgtx.
  * [18] B. Fais, I. Keppo, M. Zeyringer, W. Usher, H. Daly, Impact of technology uncertainty on future low-carbon pathways in the UK, Energy Strategy Reviews 13-14 (2016) 154–168.  doi:10/f9gmpd.
  * [19] W. Usher, The Value of Global Sensitivity Analysis for Energy System Modelling (2015) 29.
  * [20] J. Hörsch, H. Ronellenfitsch, D. Witthaut, T. Brown, Linear optimal power flow using cycle flows, Electric Power Systems Research 158 (2018) 126–135, arXiv:1704.01881.  doi:10/gdb8kx.
  * [21] T. Brown, J. Hörsch, D. Schlachtberger, PyPSA: Python for Power System Analysis, Journal of Open Research Software 6 (2018) 4, arXiv:1707.09913.  doi:10/gfb7m9.
  * [22] G. Mavrotas, Effective implementation of the $\epsilon$-constraint method in Multi-Objective Mathematical Programming problems, Applied Mathematics and Computation 213 (2) (2009) 455–465.  doi:10/bwkzrt.
  * [23] J. Hörsch, F. Hofmann, D. Schlachtberger, T. Brown, PyPSA-Eur: An open optimisation model of the European transmission system, Energy Strategy Reviews 22 (2018) 207–215, arXiv:1806.01613.  doi:10/d294.
  * [24] J. Köster, S. Rahmann, Snakemake—a scalable bioinformatics workflow engine, Bioinformatics 28 (19) (2012) 2520–2522.  doi:10/gd2xzq.
  * [25] Danish Energy Agency, Technology data (2020).
URL https://ens.dk/en/our-services/projections-and-models/technology-data

  * [26] A. Gritsevskyi, N. Nakićenovi, Modeling uncertainty of induced technological change, Energy Policy 28 (13) (2000) 907–921.  doi:10/d4rkhb.
  * [27] S. Yeh, E. S. Rubin, A review of uncertainties in technology experience curves, Energy Economics 34 (3) (2012) 762–771.  doi:10/dzc2b4.
  * [28] C. F. Heuberger, E. S. Rubin, I. Staffell, N. Shah, N. Mac Dowell, Power capacity expansion planning considering endogenous technology cost learning, Applied Energy 204 (2017) 831–845.  doi:10/gcgs2v.
  * [29] N. Mattsson, Learning by modeling energy systems, Ph.D. thesis (2019).
  * [30] S. Moret, V. Codina Gironès, M. Bierlaire, F. Maréchal, Characterization of input uncertainties in strategic energy planning models, Applied Energy 202 (2017) 597–617.  doi:10/gbsxmz.
  * [31] B. Shirizadeh, Q. Perrier, P. Quirion, How sensitive are optimal fully renewable power systems to technology cost uncertainty? (2019) 45.
  * [32] P.-H. Li, S. Pye, I. Keppo, Using clustering algorithms to characterise uncertain long-term decarbonisation pathways, Applied Energy 268 (2020) 114947\.  doi:10/ghchgg.
  * [33] S. Moret, M. Bierlaire, F. Maréchal, Robust Optimization for Strategic Energy Planning, Informatica 27 (3) (2016) 625–648.  doi:10/f87mh8.
  * [34] S. Pilpola, P. D. Lund, Analyzing the effects of uncertainties on the modelling of low-carbon energy system pathways, Energy 201 (2020) 117652.  doi:10/ggtk2r.
  * [35] E. Trutnevyte, EXPANSE methodology for evaluating the economic potential of renewable energy from an energy mix perspective, Applied Energy 111 (2013) 593–601.  doi:10/ftdb.
  * [36] Lopion, Markewitz, Stolten, Robinius, Cost Uncertainties in Energy System Optimization Models: A Quadratic Programming Approach for Avoiding Penny Switching Effects, Energies 12 (20) (2019) 4006.  doi:10/ghchff.
  * [37] P. S. Palar, T. Tsuchiya, G. T. Parks, Multi-fidelity non-intrusive polynomial chaos based on regression, Computer Methods in Applied Mechanics and Engineering 305 (2016) 579–606.  doi:10/f8jsb6.
  * [38] B. Sudret, Global sensitivity analysis using polynomial chaos expansions, Reliability Engineering & System Safety 93 (7) (2008) 964–979.  doi:10/bjjp5r.
  * [39] N. Fajraoui, S. Marelli, B. Sudret, On optimal experimental designs for Sparse Polynomial Chaos Expansions, arXiv:1703.05312 (Mar. 2017).
  * [40] L. L. Gratiet, S. Marelli, B. Sudret, Metamodel-based sensitivity analysis: Polynomial chaos expansions and Gaussian processes, arXiv:1606.04273 (2015) 1–37doi:10/ggxtbp.
  * [41] T. Mühlpfordt, Uncertainty Quantification via Polynomial Chaos Expansion – Methods and Applications for Optimization of Power Systems (2020).  doi:10/ggxtbr.
  * [42] J. Feinberg, H. P. Langtangen, Chaospy: An open source tool for designing methods of uncertainty quantification, Journal of Computational Science 11 (2015) 46–57.  doi:10/gddwzk.
  * [43] L. W.-T. Ng, M. Eldred, Multifidelity Uncertainty Quantification Using Non-Intrusive Polynomial Chaos and Stochastic Collocation, in: 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, American Institute of Aeronautics and Astronautics, Honolulu, Hawaii, 2012.  doi:10/ghknms.
  * [44] S. Hosder, R. Walters, M. Balch, Efficient Sampling for Non-Intrusive Polynomial Chaos Applications with Multiple Uncertain Input Variables.  doi:10/ftdc.