# Renewable Energy Targets and Unintended Storage Cycling:
Implications for Energy Modeling

Martin Kittel<EMAIL_ADDRESS>Wolf-Peter Schill<EMAIL_ADDRESS>DIW Berlin,
Department of Energy, Transportation, Environment, Mohrenstraße 58, 10117
Berlin, Germany

###### Abstract

To decarbonize the economy, many governments have set targets for the use of
renewable energy sources. These are often formulated as relative shares of
electricity demand or supply. Implementing respective constraints in energy
models is a surprisingly delicate issue. They may cause a modeling artifact of
excessive electricity storage use. We introduce this phenomenon as “unintended
storage cycling”, which can be detected in case of simultaneous storage
charging and discharging. In this paper, we provide an analytical
representation of different approaches for implementing minimum renewable
share constraints in energy models, and show how these may lead to unintended
storage cycling. Using a parsimonious optimization model, we quantify related
distortions of optimal dispatch and investment decisions as well as market
prices, and identify important drivers of the phenomenon. Finally, we provide
recommendations on how to avoid the distorting effects of unintended storage
cycling in energy modeling.

APC

    Across-Period-Cycling
BEV

    battery-electric vehicles
DIETER

    Dispatch and Investment Evaluation Tool with Endogenous Renewables
KKT

    Karush-Kuhn-Tucker
LCOE

    Levelized Cost of Electricity
LCOS

    Levelized Cost of Storage
MV

    Market Value
NSL

    Normalized Storage Conversion Losses
PV

    photovoltaics
RLDC

    residual load duration curve
SLCR

    Storage Loss Coverage by Renewables
SPC

    Same-Period-Cycling
$STO^{LOSS}$

    storage conversion losses
VRE

    Variable renewable energy

###### keywords:

Minimum Renewable Shares , Renewable Energy Targets , Energy storage , Energy
Modeling , Open Source Modeling

††journal: arXiv

2.5cm2.4cm1.5cm2cm

##  1 Introduction

Replacing conventional electricity generation with renewable energy sources is
a prime option for mitigating greenhouse gas emissions in the power sector.
Technological progress, economies of scale, and support measures have led to
substantially growing shares of renewables in power sectors around the world.
Firm renewable capacity potentials, i.e., for hydroelectric and biomass power,
are limited in many regions. Variable renewable energy (VRE), such as wind and
solar photovoltaics (PV), will thus play a major role in the further
transition. These technologies have variable generation characteristics, as
their temporal availability depends on weather conditions, such as wind speeds
and cloud cover [1]. Investment costs for VRE have tremendously decreased in
recent years [2]. Due to an insufficient internalization of greenhouse gas
emissions [3] and low marginal generation costs of existing thermal power
plants, policy support continues to be relevant for low-cost renewable
deployment above penetration levels that would emerge without policy
intervention [4].

Many governments on state, national, and even intergovernmental levels have
adopted a minimum renewable energy share by a target date to promote the
deployment of renewable energy [5]. Essentially, this share describes the
ratio of energy from renewable sources to the total energy supplied or
consumed in a specific economic sector, or in the entire energy system. Such
targets and related policy measures can facilitate investment in VRE
technologies, and usually rise over time.

For instance, Germany has a long tradition of setting renewable energy targets
for its power sector, in particular via the Renewable Energy Sources Act
(EEG). The 2021 version of the EEG includes a renewable share of 65% in gross
electricity consumption by 2030 [6].111Gross electricity consumption includes
end energy consumption, grid and storage losses, internal power plant
consumption required for operation, and electricity exports net of imports.
France and Spain aim for a share of 40% and 74% of total generation by 2030,
respectively, while Sweden envisages a fully renewable power sector by 2040
[7]. In the United States, renewable portfolio standards require a certain
share of a utility’s total generation to come from renewable energy sources.
In 2020, 30 states had adopted mandatory standards, while in eight states
voluntary standards are in place [8]. For instance, California aims at a 60%
(100%) renewable share by 2030 (2045) [9]. China targets a renewable share of
35% in its total electricity generation by 2030 [10]. Japan aims at 22-24% for
the same target year. Overall, 137 countries had renewable energy targets in
place for the power sector in 2020 [7]. While some of these countries have set
only absolute renewable power capacity targets (e.g., Switzerland or New
Zealand), most make use of some sort of minimum renewable energy target.

Capacity expansion models of the power sector or the entire energy system are
frequently used to explore future scenarios with high shares of renewables and
determine least-cost solutions [11]. To comply with climate policy, such
models are often constrained by renewable energy targets. In this case, an
unexpected modeling artifact related to VRE generation may occur. Figure 1
illustrates the general mechanism. At high shares of VRE, situations arise in
which the VRE generation potential temporarily exceeds the demand for
electricity. In these situations, curtailing the renewable surplus would be an
obvious option (dashed green arrow). However, certain minimal renewable share
constraint formulations may lead to a situation where renewable curtailment is
replaced by storage conversion losses (red dashed error), facilitated by
additional, and often simultaneous, storage charging and discharging. Such
unintended storage cycling drives up generation from VRE compared to the
first-mentioned option of curtailing renewable surplus generation, and may
thus facilitate achieving renewable energy constraints at lower costs. Note
that this mechanism may arise in energy models, but does not relate to real-
world storage applications.

Figure 1: Stylized illustration of unintended storage cycling: Instead of
curtailing (dashed green) VRE surplus, the optimization may lead to unintended
storage cycling, i.e., to additional storage conversion losses (dashed red),
which remove excess renewable electricity from the system.

The phenomenon of unintended storage cycling has, to the best of our
knowledge, not been described or discussed in the energy system literature
[12]. There is reason to believe that this artifact is undetected or ignored,
yet prevailing in the power sector and energy system modeling domain. Our
paper aims at closing this gap. To this end, we first provide a systematic
representation of different approaches for implementing minimum renewable
share constraints in energy models. Using a parsimonious optimization model of
a power sector, we analytically derive some intuition as to how these
constraints may lead to unintended storage cycling and examine their impact on
equilibrium conditions of storage operators. For a stylized model
parameterization, we numerically analyze how this phenomenon may distort model
outcomes. We quantify the effects on dispatch and investment decisions of both
storage and other technologies, and on market prices. We also explore
important drivers of unintended storage cycling. Finally, we discuss different
options for avoiding the distorting effects of unintended storage cycling in
energy modeling and provide recommendations for energy modelers.

In energy modeling applications, constraints requiring a certain renewable
share may be manifold. A review by Gillich et al. [13] shows that some
applications use constraints for installed renewable generation capacity
(e.g., in GW). Other model applications use constraints on the share of
renewable energy (e.g., in TWh). Table 1 shows a selection of well-established
European models and model applications that employ renewable capacity and
energy targets. Additionally, two commonly implemented climate policy
instruments are listed, which also influence the share of renewable energy
sources in model outcomes: an annual green house gas emission budget and an
explicit CO2 price. The information is based on a review of research articles,
model documentations as well as other references such as dissertations, and,
wherever possible, on personal validation by model developers or affiliated
modelers.

Renewable energy constraints in power sector and energy system models may come
in various forms. Energy provision from renewable (conventional) energy
sources has to achieve (must not exceed) a certain share in total demand or
generation within a particular sector, or in the entire energy system.
Additionally, storage losses may be covered to a different extent in such
constraints. Based on this survey, we systematize possible ways of
implementing renewable energy targets, and examine their impact on modeling
results.

The artifact of unintended storage cycling potentially can occur in all cost-
optimizing power sector and energy system models that include binding
renewable targets. Personal communication further confirms that the phenomenon
has already been observed in various European energy models, which are marked
with an asterisk (*) in Table 1. For entries without an asterisk, it was not
possible to clarify whether the artifact has been observed so far.

Not all of the listed models consider storage losses in the formulation of the
renewable target, and only a minority does so to the full extent. We discuss
how this impacts the results of linear optimization models. As we show in this
paper, an incomplete consideration of storage losses in the renewable energy
constraint causes unintended storage cycling. To the best of our knowledge,
this has not been discussed in the energy modeling literature before.

The remainder of the paper is structured as follows: Section 2 introduces a
parsimonious power sector model and a numerical model implementation. Section
3 systematizes possible renewable energy target constraint formulations,
provides intuition for the cause of unintended storage cycling, and discusses
the artifact’s implications for storage in the long-term equilibrium. Section
4 illustrates different types of unintended storage cycling. Section 5
contains a numerical analysis, and determines the artifact’s impact on
dispatch and investment decisions, storage system effects, prices, and its
drivers. Section 6 discusses other instances of and potential remedies for
unintended storage cycling. Section 7 summarizes and concludes. Additional
material is available in the supplementary information (SI).

Table 1: Model features that affect the shares of renewable energy sources in selected European energy models and model applications. | renewable | conventional | renewable | conventional | renewable |  CO2 |  CO2 | docu-
---|---|---|---|---|---|---|---|---
Model | share in demand | share in generation | capacity target | budget | price | mentation
AnyMOD |  |  |  |  |  |  xa |  xa | [14]
Calliope* |  x[15]b |  xa |  xa |  xa |  x[16, 17] |  x[18] |  x[18] | [19]
DIETER* |  |  x[20, 21, 22] |  x[23, 24] |  |  |  xb |  xb | [20, 25]
DIMENSION* |  xc |  xc |  xc |  xc |  x[26] |  x[27] |  xa | [28, 27]
dynELMOD* |  |  |  |  |  x[29] |  x[30] |  | [31]
EMMA* |  xb |  |  |  |  x[32, 33, 34, 35] |  xa |  x[32, 33, 34, 35] | [32]
ELTRAMOD |  |  |  |  |  x[36] |  x[37, 38] |  x[39] | [40]
EU-REGEN* |  |  |  xa |  |  |  x[41, 42, 43] |  x[44, 45] | [46]
E2M2 |  x[47] |  |  x[48] |  |  x[49] |  x[13, 47, 50, 48] |  x[48, 49] | [47]
ENTIGRIS* |  x[51] |  x[51] |  |  |  |  |  x[51] |  n/ad
GENESYS |  |  |  |  |  |  xc |  xc | [52, 53]
GENeSYS-MOD* |  |  |  x[54] |  |  |  x[55] |  x[56] | [56]
ISAaR |  |  |  |  |  |  xa |  xc | [57]
LIMES-EU* |  x[58] |  |  xa |  |  |  x[58] |  x[59] | [60, 61]
LUT Model* |  xb |  x[62] |  x[63] |  x[63] |  x[64] |  xa |  x[65] | [62]
oemof |  xc |  xc |  |  |  x[66] |  xc |  xc | [66]
PyPSA* |  xc |  |  xc |  |  xc |  x[68] |  x[69] | [70]
REMix* |  x[71] |  |  x[72, 73] |  |  |  x[74] |  x[75] | [72, 73]
|  Notes: * Personal communication confirms that the artifact has been
observed in this model. a Possible, but not used or documented in model
applications so far. b Used in model applications, but not published yet. c
Applicable according to personal communication, but insufficient information
for linking specific model applications to this particular constraint. d More
information available at
www.ise.fraunhofer.de/de/geschaeftsfelder/leistungselektronik-netze-und-
intelligente-systeme/energiesystemanalyse/energiesystemmodelle-am-fraunhofer-
ise/entigris.html.
---

##  2 Methods

###  2.1 Formal definition of a parsimonious dispatch and investment model

We use a parsimonious power sector optimization model to analytically show how
different minimum renewable constraint formulations relate to costs and market
values of storage. Adopting a long-run equilibrium perspective, the model
minimizes total system costs of a full year. These include annualized
investment and annual variable costs of generation and storage technologies.
The model covers a single region and abstracts from a spatial resolution and
grid congestion. We assume linear cost functions and price-inelastic demand,
perfect foresight, and a perfect energy-only market. This set of assumptions
renders a long-term market equilibrium with zero profits for market
participants. The solution coincides with the optimal allocation determined by
a benevolent social planer. The model set-up and derivations are similar to
Brown & Reichenberg [67] and Biggar & Hasemzadeh [76, Ch.4]. Throughout the
exposition, capital letters denote endogenous decision variables. Small
letters refer to sets and exogenous parameters.

We denote $\sum_{t}d_{t}$ as the total demand for electricity over all hours
$t$ of the year. The set of generators consists of firm and variable renewable
generators $s\in\mathcal{R}$ and firm conventional generators
$s\in\mathcal{C}$, while an unspecified $s$ applies to the set of all
generators. $G_{s\in\mathcal{R},t}$ represents hourly renewable energy
generation net of curtailment of renewable surplus $CU_{s,t}$. Equivalently,
$G_{s\in\mathcal{C},t}$ is the hourly dispatch of conventional generators.
Total annual electricity generation is
$\sum_{s,t}G_{s,t}=\sum_{s\in\mathcal{R},t}G_{s,t}+\sum_{s\in\mathcal{C},t}G_{s,t}$.
The parameter $\bar{g}_{s,t}\in[0,1]$ renders a time-variant availability of
generator $s$. We denote a storage unit $r$, its charging from the grid
$G^{in}_{r,t}$, and its discharging back to the grid $G^{out}_{r,t}$. Total
storage loss is $\sum_{r,t}(G^{in}_{r,t}-G^{out}_{r,t})$, which has to be
covered by either renewable or conventional generators. We denote $i_{s}$ and
$i_{r}$ as technology-specific investment costs, $o_{s}$ and $o_{r}$ as
operation and maintenance costs accruing for hourly dispatch. $C_{s}$,
$C_{r}^{in}$ and $C_{r}^{out}$ denote installed capacity (power rating), and
$C_{r}^{l}$ storage energy capacity. The hourly storage energy level is
$G_{r,t}^{l}$.

For the sake of readability, we define $\circ=\\{l,in,out\\}$ and
$*=\\{in,out\\}$, and use the following aggregate cost terms for storage
investment and usage in the objective function.

$\sum_{r,\circ}i_{r}^{\circ}C_{r}^{\circ}=\sum_{r}i_{r}^{l}C_{r}^{l}+\sum_{r}i_{r}^{in}C_{r}^{in}+\sum_{r}i_{r}^{out}C_{r}^{out}$
(1)

$\sum_{r,t,*}o_{r}^{*}G_{r,t}^{*}=\sum_{r,t}o_{r}^{in}G_{r,t}^{in}+\sum_{r,t}o_{r}^{out}G_{r,t}^{out}$
(2)

Investment costs (1) comprise annualized infrastructure cost of the storage
(dis)charge unit, and storage energy costs. Storage usage cost (2) are
variable and accrue whenever a storage unit (dis)charges. We formulate the
optimization problem of the stylized power sector as follows:

—l— Cs, Gs,t,Cr∘, Gr,t∘∑_s i_s C_s + ∑_s,t o_s G_s,t + ∑_r,∘ i_r^∘C_r^∘+
∑_r,t,* o_r^* G_r,t^* -d_t + ∑_s G_s,t + ∑_r G_r,t^out - ∑_r G_r,t^in=0 ⟂λ_t
∀t G_s,t ≥0 ⟂¯μ_s,t ∀s, t ¯g_st C_s - G_s,t - CU_s,t= 0 ⟂λ^CU_s,t ∀s ∈R, t C_s
- G_s,t≥0 ⟂¯μ_s,t ∀s ∈C, t G^∘_r,t≥0 ⟂¯μ^∘_r,t ∀r, t C^∘_r - G^∘_r,t≥0
⟂¯μ^∘_r,t ∀s, t G^l_r,t - G^l_r,t-1 - η^in_r G^in_r,t + (η^out_r)^-1
G^out_r,t= 0 ⟂λ^l_r,t ∀r, t ∑_s ∈R,t G_s,t - Θ≥0 ⟂μ_θ Ω- ∑_s ∈C,t G_s,t≥0 ⟂μ_ω

The objective function (2.1) minimizes total fixed and variable costs of all
technologies. The market clearing condition (2.1) balances supply and demand
for electricity at all times. Constraints (2.1) and (2.1) impose non-
negativity. Variable renewable generation is limited by VRE availability, and
curtailment of VRE surplus (2.1), which comes at no costs. Generation of
conventional generators may not exceed installed capacity (2.1). Storage
(dis)charge and levels may be no larger than installed capacities (2.1).
Constraint (2.1) ensures inter-temporal consistency of storage filling and
withdrawal. As storage incurs conversion losses, storage (dis)charge effects
on the storage level are corrected by the efficiency rates $\eta^{in}_{r}<1$
and $\eta^{out}_{r}<1$ (2.1). To avoid free lunch, we further demand that the
storage level in the first and last hour of the year need to be equal
$G^{l}_{r,0}=G^{l}_{r,T}$. Constraint (2.1) imposes a minimum renewable share,
requiring total renewable generation aggregated overall $s\in\mathcal{R}$ to
achieve at least $\Theta$. In contrast, constraint (2.1) imposes a maximum
share $\Omega$ on total conventional generation aggregated over all
$s\in\mathcal{C}$. Constraints (2.1) and (2.1) are mutually exclusive
substitutes. Only one is considered by the optimization problem. Table SI.1 in
the supplementary information provides unequivocal constraint formulations for
our 12 model specifications aligned with the notation above.

A constraint’s dual variable indicates how much the objective value changes if
the constraint relaxed, also called shadow price. $\lambda_{t}$ represents the
system’s marginal cost of meeting an additional unit of demand for
electricity. $\lambda^{CU}_{s,t}$ represents the change in total cost when
marginally changing $CU_{s,t}$. In the optimum, it takes the value of zero.
The dual variables $\underaccent{\bar}{\mu}_{s,t}\geq 0$, $\bar{\mu}_{s,t}\geq
0$ and $\underaccent{\bar}{\mu}^{\circ}_{r,t}\geq 0$,
$\bar{\mu}^{\circ}_{r,t}\geq 0$, $\lambda^{l}_{r,t}$ are the shadow prices of
the capacity and generation constraints of generators and storages,
respectively. $\mu_{\theta/\omega}\geq 0$ indicate how much total system cost
would increase, if the boundary of the binding constraint was relaxed. That
is, if one more (less) unit of electricity from renewables (fossil fuels) had
to be supplied. Loosely speaking, it is the marginal cost of tightening the
VRE policy.

###  2.2 Numerical implementation

In section 5, we use a numerical implementation of our power sector model to
derive optimal dispatch and capacity expansion decisions. It is a stylized
version of the larger Dispatch and Investment Evaluation Tool with Endogenous
Renewables (DIETER) capacity expansion model [20]. A similar model set-up has
been previously used for analyzing electricity storage needs for renewable
energy integration [77], for reflections on the changing role of electricity
storage in the renewable energy transition [78], and for an illustration of
the economics of renewables and electricity storage [79]. The implementation
used here is integrated in a Python-GAMS interface, which enables Python-based
data pre- and post-processing, scenario analysis, and visualization [25]. For
transparency and reproducibility, we provide the model code, all input data,
and a manual in public repositories under permissive
licenses.222https://gitlab.com/diw-evu/dieter_development/dieterpy_reduced

We apply the model to a stylized setting loosely parameterized to the German
power sector. Annual electricity demand is $520$ TWh, which can be supplied by
a mix of two renewable technologies (solar PV and wind power) and two
conventional generation technologies (hard coal and open-cycle gas turbines).
Assumptions on fixed and variable costs can be found in Table SI.2 in the
supplementary information. We further include a generic electricity storage
technology that is parameterized to resemble pumped hydro storage with a
round-trip efficiency of 80%. The energy- and power-related costs of this
technology make its application most plausible for medium-duration storage,
which suits the cases studied here well. All individual generation units of a
technology are modeled as one technology aggregate. This parsimonious model
setup allows to focus on the effects of unintended storage cycling for
different levels of renewable penetration. Unless stated otherwise, we use a
minimum renewable penetration level of 80%.

##  3 An analytical approach to unintended storage cycling

###  3.1 Alternative renewable energy constraint formulations

Renewable energy constraints in energy models may be generalized as follows:

$base\;element\gtreqless share^{ref}\times reference\;element\pm
share^{loss}\times losses$ (3)

Varying the base and reference element, four basic constraint formulations are
possible. For the sake of argument, we denote them constraint families: (1) a
minimum renewable share in total demand, (2) a minimum renewable share in
total generation; and, conversely, (3) a maximum conventional share in total
demand, and (4) a maximum conventional share in total generation. Note that
total generation includes storage losses, while total demand does not.
Generation from renewables in constraint families (1) and (2) may include both
firm and variable renewable energy sources. Formulations (3) and (4)
indirectly enforce minimum renewable shares, as the limited generation from
conventional technologies has to be complemented by renewables.

Reflecting different practices in the modeling literature, we investigate
three variations for each constraint family as to what degree renewable
generators cover storage losses. We denote this characteristic  Storage Loss
Coverage by Renewables (SLCR). Let $\phi$ be the targeted renewable share, the
SLCR then may take the following values: (a) storage losses are completely
supplied from conventional energy sources (zero SLCR); (b) storage losses are
shared between renewable and conventional energy sources according to the
targeted renewable share $\phi$ and $(1-\phi)$, respectively (proportionate
SLCR); or (c) storage losses are completely covered by renewables (complete
SLCR).

Given current economics, power sector costs generally increase with VRE
penetration, as the need for providing temporal flexibility rises. Note that
we are interested in settings with high VRE penetration where the renewable
energy constraint is binding, i.e., with a penetration beyond the optimal
share of an unconstrained model without an explicit renewable energy
constraint. Model specifications with zero SLCR are the least restrictive
ones, as these require the lowest use of VRE. In contrast, models with
complete SLCR are most restrictive, as these require storage losses to be
completely covered by VRE. Model specifications with proportionate SLCR may
appear most intuitive and appropriately restrictive. Yet, we show that
incomplete SLCR levels lead to unintended storage cycling.

In total, the four constraint families and three SLCR levels yield twelve
different model specifications (Table 2).

Table 2: Investigated constraint formulations based on Equation (3), differentiated by constraint family and SLCR level. The right-hand sides of the inequalities define $\Theta$ or $\Omega$ of the optimization problem introduced in Section 2.1. Model | base |  |  shareref & |  |  shareloss | associated
---|---|---|---|---|---|---
specification | element | sign | reference element | $\pm$ | storage losses | constraint
1a | $\sum\limits_{s\in\mathcal{R},t}G_{s,t}$ | $\geq$ | $\phi\sum\limits_{t}d_{t}$ |  |  |  (2.1)
1b | $\sum\limits_{s\in\mathcal{R},t}G_{s,t}$ | $\geq$ | $\phi\sum\limits_{t}d_{t}$ | $+$ | $\phi\sum\limits_{r,t}(G^{in}_{r,t}-G^{out}_{r,t})$ |  (2.1)
1c | $\sum\limits_{s\in\mathcal{R},t}G_{s,t}$ | $\geq$ | $\phi\sum\limits_{t}d_{t}$ | $+$ | $\sum\limits_{r,t}(G^{in}_{r,t}-G^{out}_{r,t})$ |  (2.1)
2a | $\sum\limits_{s\in\mathcal{R},t}G_{s,t}$ | $\geq$ | $\phi\sum\limits_{s,t}G_{s,t}$ | $-$ | $\phi\sum\limits_{r,t}(G^{in}_{r,t}-G^{out}_{r,t})$ |  (2.1)
2b | $\sum\limits_{s\in\mathcal{R},t}G_{s,t}$ | $\geq$ | $\phi\sum\limits_{s,t}G_{s,t}$ |  |  |  (2.1)
2c | $\sum\limits_{s\in\mathcal{R},t}G_{s,t}$ | $\geq$ | $\phi\sum\limits_{s,t}G_{s,t}$ | $+$ | $(1-\phi)\sum\limits_{r,t}(G^{in}_{r,t}-G^{out}_{r,t})$ |  (2.1)
3a | $\sum\limits_{s\in\mathcal{C},t}G_{s,t}$ | $\leq$ | $(1-\phi)\sum\limits_{t}d_{t}$ | $+$ | $\sum\limits_{r,t}(G^{in}_{r,t}-G^{out}_{r,t})$ |  (2.1)
3b | $\sum\limits_{s\in\mathcal{C},t}G_{s,t}$ | $\leq$ | $(1-\phi)\sum\limits_{t}d_{t}$ | $+$ | $(1-\phi)\sum\limits_{r,t}(G^{in}_{r,t}-G^{out}_{r,t})$ |  (2.1)
3c | $\sum\limits_{s\in\mathcal{C},t}G_{s,t}$ | $\leq$ | $(1-\phi)\sum\limits_{t}d_{t}$ |  |  |  (2.1)
4a | $\sum\limits_{s\in\mathcal{C},t}G_{s,t}$ | $\leq$ | $(1-\phi)\sum\limits_{s,t}G_{s,t}$ | $+$ | $\phi\sum\limits_{r,t}(G^{in}_{r,t}-G^{out}_{r,t})$ |  (2.1)
4b | $\sum\limits_{s\in\mathcal{C},t}G_{s,t}$ | $\leq$ | $(1-\phi)\sum\limits_{s,t}G_{s,t}$ |  |  |  (2.1)
4c | $\sum\limits_{s\in\mathcal{C},t}G_{s,t}$ | $\leq$ | $(1-\phi)\sum\limits_{s,t}G_{s,t}$ | $-$ | $(1-\phi)\sum\limits_{r,t}(G^{in}_{r,t}-G^{out}_{r,t})$ |  (2.1)

###  3.2 Intuition for unintended storage cycling

We first provide some intuition for model specification (1a). Suppose the
optimal renewable share of an unconstrained model is below
$\phi\sum_{t}d_{t}$. Introducing constraint (2.1) forces the model to increase
renewable generation. Further suppose there are hours with renewable
curtailment. There are two options to increase
$\sum_{s\in\mathcal{R},t}G_{s,t}$. Either increase renewable generation
capacity, which incurs additional investment costs, or transform renewable
curtailment into storage losses, which only incurs variable storage costs. The
latter is done by increasing $G_{s\in\mathcal{R},t}$ in curtailment hours, and
by subsequently charging and discharging storage in these hours. This leads to
additional storage losses from unintended storage cycling, which are not
accounted for in the renewable constraint formulation of model specification
(1a). Suppose demand is 100 MWh in one of these curtailment hours, and storage
charging and discharging capacity is 10 MW with a round-trip efficiency of
80%. Without unintended storage cycling, $G_{s\in\mathcal{R},t}$ in this hour
is 100 MWh. Making use of unintended storage cycling, $G_{s\in\mathcal{R},t}$
can increase to 102 MWh, consisting of 10 MWh charged into storage and 92 MWh
to satisfy demand. In the same hour, 8 MWh are discharged from storage to
serve the load of 100 MWh. In this example, renewable generation increases by
2 MWh compared to a setting without unintended storage cycling. This is done
without additional renewable investments, but by transforming 2 MWh of
renewable curtailment into storage losses, which allows to meet constraint
(2.1), i.e., $\sum_{s\in\mathcal{R},t}G_{s,t}\geq\phi\sum_{t}d_{t}$, at lower
costs. Note that the additional 2 MWh of renewable generation contribute to
meeting the renewable energy target (Equation (2.1)), but do not serve
electricity demand (Equation (2.1)), as the energy is lost in the storage
conversion process.

Under model specification (1b), a similar reasoning applies. The only
difference is that storage losses are partially accounted for in the renewable
constraint with $\phi\sum_{r,t}(G^{in}_{r,t}-G^{out}_{r,t})$. Only the
remaining fraction of storage losses,
$(1-\phi)\sum_{r,t}(G^{in}_{r,t}-G^{out}_{r,t})$, can be used to transform
curtailment into storage losses via unintended storage cycling. Suppose the
targeted renewable share is $\phi=80\%$ in the setting described above.
Unintended storage cycling still leads to a renewable generation increase of 2
MWh in the exemplary hour, which is transformed into storage losses. Yet, 80%
of these losses are accounted for in the renewable constraint. Thus, only 20%
of the increase in renewable generation, i.e., 0.4 MWh, help to relax the
renewable constraint. To achieve a similar effect as under model (1a), more
unintended storage cycling would be required.

Under model specification (1c), storage losses are completely covered by the
renewable energy constraint. Unintended storage cycling would only increase
total variable storage costs, without providing any benefit in terms of
meeting the renewable constraint. Accordingly, renewable generation can only
be increased via additional renewable capacity investments.

A similar reasoning applies to the other renewable share constraints listed in
Table 2. For models using constraint family (2), the only difference is that
the fraction of storage losses covered by renewables is slightly less
intuitive than in the example described above, as storage losses are already
included in overall generation $\sum_{s,t}G_{s,t}$. The effects in constraint
families (3) and (4) are similar to the ones in (1) and (2), respectively.

###  3.3 Long-term equilibrium conditions for storage

We first consider an unconstrained long-term equilibrium of the model defined
in Section 3.3 without binding renewable energy targets, i.e., discarding
constraints (2.1) and (2.1). Joskow in [1] stresses that both cost and value
of a technology matter in such a setting. The Levelized Cost of Storage (LCOS)
of a storage unit $r$ is the average cost of each unit energy discharged [80].
It includes investment and variable costs as well as the cost of the
electricity used for charging. As it relates to the storage output, the LCOS
also consider the value of conversion losses.

$LCOS_{r}=\frac{\sum_{\circ}i_{r}^{\circ}C_{r}^{\circ}+\sum_{t,*}o_{r}^{*}G_{r,t}^{*}+\sum_{t}\lambda_{t}G_{r,t}^{in}}{\sum_{t}G^{out}_{r,t}}$
(4)

When interpreting the LCOS metrics, the temporal heterogeneity of electricity
as a good has to be considered. The value of electricity is time-variant due
to a number of reasons [1, 32, 81]. Both the electricity demand and the output
of many generation technologies are fluctuating in nature. Especially the
value of VRE depends on the time when they generate, i.e., their variable
availability profile [79], and the location of generation in the geographical
context. Consequently, the electricity price fluctuates over time and space,
too. Transmission and distribution grid constraints might further affect the
locational value of VRE.

The Market Value (MV) accommodates these variations in electricity value [1].
The MV of a storage technology $r$ refers to the average value of each unit of
energy supplied to the system. Storage operators generate revenue by
dispatching stored energy at market prices $\lambda_{t}$. They exploit
arbitrage opportunities, ideally dispatching during periods with high prices,
while charging during low-price periods [67].

$MV_{r}=\frac{\sum_{t}\lambda_{t}G_{r,t}^{out}}{\sum_{t}G^{out}_{r,t}}$ (5)

Storage achieves economic efficiency if LCOS equals its marginal economic
value, and competitiveness if LCOS is less or equal to its MV. For perfect and
complete markets, the marginal economic value and the MV coincide [81]. In the
long-term equilibrium of an unconstrained optimum, it follows that LCOS equal
storage MV, i.e., storage operators generate zero profit as costs perfectly
match revenue (zero-profit condition) [67]. Using the Lagrangian function and
Karush-Kuhn-Tucker (KKT) conditions (Section SI.3), Section SI.4 proves this
optimality condition.

$LCOS_{r}=MV_{r}$ (6)

Imposing a binding renewable energy target, i.e., including constraints (2.1)
and (2.1), may alter the zero-profit condition of storage in a long-term
equilibrium. For clarity, we define a new metric, the Normalized Storage
Conversion Losses (NSL) of a storage unit $r$. It indicates storage conversion
losses related to each unit of storage output.

$NSL_{r}=\frac{\sum_{t}\big{(}G_{r,t}^{in}-G_{r,t}^{out}\big{)}}{\sum_{t}G^{out}_{r,t}}$
(7)

Table 3 shows that a binding renewable energy target may affect the
composition of storage costs and revenue streams of all investigated model
specifications. Exemplary proofs of those conditions can be found in Section
SI.5.

Within each constraint family, the LCOS increase by a certain margin, the more
storage losses are covered by renewables. This margin is a multiple of the
factor $\mu_{\theta/\omega}NSL_{r}$, which represents the storage loss per
unit storage output, valued at the marginal system costs of an additional unit
of renewable generation. For a binding renewable target, renewable energy
becomes a scarce good with additional inherent value. When renewable energy is
lost in storage conversion processes, its inherent value for achieving the
renewable constraint is annihilated. Restoring the lost energy incurs
additional costs. We thus interpret this factor as storage integration costs
that arise in a system with a binding renewable energy constraint. If the
constraint is not binding, the dual variable of the minimum renewable
constraint $\mu_{\theta/\omega}$ is zero, and the zero-profit condition of
storage remains unchanged compared to the unconstrained optimum without
renewable policy targets.

Table 3: Storage operators’ zero-profit conditions. model | costs |  | revenue
---|---|---|---
1a | $LCOS_{r}$ | $=$ | $MV_{r}$
1b | $LCOS_{r}+\phi\mu_{\theta}NSL_{r}$ | $=$ | $MV_{r}$
1c | $LCOS_{r}+\mu_{\theta}NSL_{r}$ | $=$ | $MV_{r}$
2a, 4a | $LCOS_{r}-\phi\mu_{\theta}NSL_{r}$ | $=$ | $MV_{r}$
2b, 4b | $LCOS_{r}$ | $=$ | $MV_{r}$
2c, 4c | $LCOS_{r}+(1-\phi)\mu_{\theta}NSL_{r}$ | $=$ | $MV_{r}$
3a | $LCOS_{r}-\mu_{\omega}NSL_{r}$ | $=$ | $MV_{r}$
3b | $LCOS_{r}-(1-\phi)\mu_{\omega}NSL_{r}$ | $=$ | $MV_{r}$
3c | $LCOS_{r}$ | $=$ | $MV_{r}$

Models with complete SLCR include all storage losses arising in the system in
the formulation of the renewable energy constraint. The zero-profit condition
of storage operators properly accounts for all storage loss costs related to
the binding renewable target. The economic trade-off between storage use and
renewable curtailment is properly specified such that unintended storage
cycling is prevented.

In contrast, models with zero SLCR neglect storage losses in the minimum
renewable constraint. They can be completely covered by conventional
generators. Consequently, the zero-profit condition of storage operators does
not include any storage loss costs related to the binding renewable target.
Within in the same constraint family, the zero-profit condition is reduced by
the factor $\mu_{\theta/\omega}NSL_{r}$ compared to models with complete SLCR.
Similarly, models with proportionate SLCR only partially require renewables to
cover storage losses. The system costs of storage losses are thus only
partially imposed on storage operators in the long-term equilibrium. Compared
to models with complete SLCR within in the same constraint family, the zero-
profit condition is reduced by the factor
$(1-\phi)\mu_{\theta/\omega}NSL_{r}$, which corresponds to the share of
storage loss costs that is not accounted for in the renewable constraint. In
other words, models based on zero and proportionate SLCR do not fully account
for the underlying LCOS related to the binding renewable target. This causes
excessive storage use and too little VRE surplus curtailment in the long-term
equilibrium.

##  4 An illustration of different types of unintended storage cycling

Unintended storage cycling can be observed in case of simultaneous charging
and discharging of storage. This involves energy that is cycled through the
storage within the same period, which we refer to as Same-Period-Cycling
(SPC). Further, there is also an inter-temporal instance of the artifact:
Across-Period-Cycling (APC). It refers to a situation in which unintendedly
discharged energy has been charged prior to the period of unintended
discharge. Both SPC and APC are excessive and constitute unintended storage
use.

There are four conceivable types of unintended storage cycling. They differ in
terms of the ratio of storage charging to discharging within the same period.
Further, they vary in terms of the temporality of unintendedly cycled energy,
which may be cycled either within the period (SPC only), or within the same
and across periods (SPC and APC). Figure 2 illustrates these types with
stylized numbers. For the sake of illustration, they are based on an assumed
low efficiency for charging and discharging of 80%, respectively.

(a) Type 1: Storage charging equals discharging.

(b) Type 2: Storage discharging exceeds charging.

(c) Type 3: Storage charging exceeds discharging; discharging exceeds same-
period unintendedly cycled energy.

(d) Type 4: Storage charging exceeds discharging; but not the full cycling
potential is exploited.

(e)

Figure 2: Four types of unintended storage cycling.

The first type occurs in a situation in which storage charging and discharging
within the same period are equal (Figure 2a). Due to conversion losses, out of
10 MWh charged energy (left bar) only 6.4 MWh can be discharged (dark area,
right bar). This energy amount represents the unintendedly cycled energy
within the same period (orange area, middle bar). Corresponding losses when
charging and discharging amount to 2.0 and 1.6 MWh (light and dark red areas,
middle bar), respectively. We define the total energy required for cycling
these 10 MWh within the same hour as SPC. It comprises the unintendedly cycled
energy (6.4 MWh) plus conversion losses (3.6 MWh in total), totaling 10 MWh.
To achieve a total discharge of 10 MWh in this period, it takes another 3.6
MWh (lighter area, right bar) of energy that already had to be stored during a
previous period. This additional unintendedly cycled energy of 3.6 MWh (light
green, middle bar) is cycled across periods. Again, it entails storage
conversion losses when charging (1.125 MWh, medium green, middle bar) and
discharging (0.9 MWh, dark green, middle bar). We define the energy for
unintended cycling across periods (3.6 MWh plus 2.025 MWh losses), totaling
5.625 MWh, as APC. The total energy required for equal storage charging and
discharging of 10 MWh within the same period is 25.625 MWh. It contains the
total charging, which is the sum of SPC and APC (15.625 MWh); plus the total
unintended discharging (10 MWh). The net effect of such storage use on the
system’s energy balance in the respective hour is zero. Thus, the entire
25.625 MWh are considered as unintended storage use. We then define unintended
storage losses as the energy removed from the system through SPC and APC
losses. Essentially, unintended storage cycling transforms VRE surplus into
unintended storage losses and circumvents renewable curtailment. In our
running example, the unintended storage losses is 5.625 MWh.

The second type of unintended storage cycling occurs when storage discharge
exceeds the same-period charging, i.e., effective discharging (Figure 2b).
Both SPC and APC occur in this case, similar to the first type. However, the
difference between the same-period charging and discharging (dark area, right
bar) exerts a net effect on the system’s energy balance in this period. This
is intended storage use that helps to serve demand, and needs to be
distinguished from storage use solely arising to generate additional losses to
remove excess electricity from the system.

The third and fourth types of unintended storage cycling occur when more
energy is charged than discharged within the same period. In the third type,
total discharged energy consists of both SPC and APC, with less use of APC
compared to the first type (Figure 2c). In the fourth type, effective charging
occurs. There is no APC, and the unintended cycling potential is not
necessarily fully exhausted (Figure 2d). In the depicted case, the SPC is
smaller than the charged energy, leaving room for some level of intended
charging.

##  5 Numerical results for unintended storage cycling

###  5.1 Occurrence and effects on total system costs

We test 12 different model specifications with alternative minimum renewable
share constraint formulations as detailed in Table 2, using the stylized
numerical model introduced in Section 2.2. We find that unintended storage
cycling is a frequently occurring modeling artifact. It arises when the SLCR
level of the imposed renewable share constraint is incomplete (i.e.,
constraints with suffix a and b in Table 2), regardless of the constraint
family (1, 2, 3, or 4). This is most pronounced for models with proportionate
SLCR. Only models with complete SLCR (constraints c) prevent the model
artifact (Figure 3a), i.e., requiring renewable generation to completely cover
storage losses averts unintended storage cycling. We further find the fourth
type of unintended storage cycling to be most common. Most model outcomes vary
across SLCR levels (i.e., constraints with suffix a, b, or c), but coincide
across constraint families. For the sake of illustration, we thus present our
findings differentiated by SLCR level, unless stated otherwise.

(a) Number of hours in which unintended storage cycling occurs.

(b) Total systems costs.

Figure 3: Number of hours in which unintended storage cycling occurs and total
system costs per SLCR level at a minimum renewable share of 80%. These graphs
are similar for all constraint families, with only marginal differences.

Confirming the intuition laid out in the previous sections, the artifact leads
to lower-cost solutions (Figure 3b). This essentially happens as the renewable
constraint is partly relaxed in the sense that storage losses also contribute
to meeting the renewable target. Compared to the setting with complete SLCR,
some renewable curtailment is avoided and transformed into additional VRE
generation, and subsequently, into storage losses. However, this energy is not
used to serve final demand - which is clearly an unintended effect of the
renewable constraint formulation.

###  5.2 Effects on optimal dispatch and investment decisions

Figure 4 illustrates generation, curtailment, and storage use for an exemplary
week for model specifications imposing a minimum renewable share in demand
with zero SLCR (1a), proportionate SLCR (1b), and complete SLCR (1c).
Respective illustrations for the other three constraint families (2, 3, 4) are
very similar with only marginal deviations. In the models with zero and
proportionate SLCR, unintended storage cycling takes place, e.g., between
hours 200 and 230, but not in the model with complete SLCR. The artifact’s
impact is not limited to the simultaneity of charging and discharging. The
hourly storage usage patterns also differ substantially.

(a) Model specification (1a) with zero SLCR.

(b) Model specification (1b) with proportionate SLCR.

(c) Model specification (1c) with complete SLCR.

Figure 4: Generation and price profiles for the second week of the target year
at a renewable target of 80% in total demand. The positive part of the
ordinate relates to storage discharging, its negative part to charging.

Unintended storage cycling also affects the generation profiles of renewable
and conventional generators, as well as residual load. The latter is demand
for electricity in a given time period, e.g., one hour, net of VRE generation
potential during this period. A positive residual load refers to situations in
which demand exceeds the VRE generation potential, and requires the use of
dispatchable generation technologies. In contrast, the residual load is
negative if the VRE generation potential exceeds demand. Residual load in the
exemplary week is lower during periods of renewable curtailment in models with
zero or proportionate SLCR compared to models with complete SLCR. This is a
consequence of lower curtailment of renewable generation, notably during the
first days of the week, which is transformed into additional storage losses
related to unintended storage cycling. During the last days of the week,
renewable energy is scarce as a result of prevailing weather conditions, and
residual load has to be met by dispatchable power plants. Generation from
conventional technologies in models with zero or proportionate SLCR slightly
exceed their complete SLCR equivalent. This is caused by lower VRE capacity in
the models with incomplete SLCR.

Unintended storage cycling also distorts optimal capacity and annual dispatch
decisions. The effects are largely similar in models using the same SLCR level
across all constraint families.333While investment decisions coincide across
all constraint families (Figure SI.1), constraint family (3) models use a bit
less wind power and more PV (Figure SI.2). In models with incomplete SLCR, VRE
generation capacity (Figure 5a) and storage energy capacity (Figure 5b)
decrease, while the dispatch of conventional generators as well as storage use
increase (Figure 5c). Renewable curtailment declines, and is partly converted
into unintended storage losses (Figure 5d).

(a) Installed generation capacity including storage charging and discharging.

(b) Installed storage energy capacity.

(c) Generation including storage charging (negative part of ordinate) and
discharging.

(d) Storage losses and curtailment.

Figure 5: Installed capacity, annual generation, curtailment, and storage
losses per technology at a renewable share of 80%.

If unintended storage cycling occurs, the increase in total storage charging
and discharging is related to unintended storage use (Figure 5c), generating
unintended storage losses (Figure 5d). This effect is most pronounced in
models with proportionate SLCR, as only the fraction of the additional storage
losses helps to achieve the renewable target that is not covered by the
renewable energy constraint (compare Section 3.2). About 96% of the unintended
storage use relates to SPC, while APC accounts for the remaining 4%.

Renewable curtailment decreases due to the conversion of renewable curtailment
into additional VRE generation, and subsequently, unintended storage losses.
This substitution is most effective in models with zero SLCR. Because of the
additional renewable generation that is not curtailed, but removed via
unintended storage cycling, much less VRE capacity is required in models with
incomplete SLCR to meet the renewable energy constraint (Figure 5a). This is
most pronounced for wind power, which is the most expensive renewable energy
source under the parameterization used here. PV capacity slightly increases,
while optimal wind power deployment disproportionately decreases compared to
the case with complete SLCR. The same holds true for generation from both
technologies (Figure 5c). Renewable curtailment decreases further as a
consequence of the capacity effect. Accordingly, renewable surplus energy
declines, such that the need for storing excess energy over a longer period of
time decreases, which yields a lower optimal storage energy capacity (Figure
5b).

Renewable energy that it is removed from the system in the form of unintended
storage losses contributes to the renewable energy constraint. However, it
does not provide useful energy for serving demand. To meet the system’s energy
balance, generation from coal plants thus increases during hours in which, due
to the VRE capacity effect, the VRE generation potential decreases compared to
a setting without unintended storage cycling. This is most pronounced for
models with zero SLCR. Because of the increase in generation from coal plants,
carbon dioxide emissions grow by 25% or 6% in models with zero or
proportionate SLCR, respectively.

Aside from effects directly related to unintended storage cycling, the
developments illustrated in Figure 5 are also driven by a lower ambition level
of the renewable energy constraints of models with zero or proportionate SLCR.
Since storage losses can, at least partly, be covered by conventional
generators here, these models require lower renewable generation than a
specification with complete SLCR by definition. Section SI.7 disentangles
these the impact of both factors. The factor separation raises complementary
insights, but also introduces new complexities. For practical model
applications, only the combined effects of varying renewable ambition levels
and unintended storage cycling are relevant, which are shown in Figure 5.

###  5.3 System effects of storage illustrated with residual load duration
curves

To illustrate system effects of electricity storage and VRE, a residual load
duration curve (RLDC) can be used [77, 78, 82]. A standard RLDC sorts all
hourly residual load values of a full year in descending order (solid lines in
Figure 7). With increasing VRE penetration, the RLDC shifts downwards
especially on the right-hand side [79]. Augmented versions of residual load
duration curves can be calculated that also consider optimal curtailment and
electricity storage use (Figures 6 and 7).

(a) Model specification (1c) with complete SLCR.

(b) Model specification (1b) with proportionate SLCR.

(c)

Figure 6: Different configurations of the RLDC at a targeted renewable share
of 80%.

In a case without unintended storage cycling, in the optimum storage takes up
a certain amount of renewable surplus generation (area A in Figure 6a, leading
to the plateau B) as well as some generation from dispatchable power plants
with low variable costs (area C). Storage shifts this energy to periods with
positive residual load and VRE shortage (area D), displacing conventional
generation. Optimal renewable curtailment yields a number of situations with
zero residual load in which storage is not being used (plateau E of the RLDC
after curtailment in Figure 6a).

Accordingly, RLDCs illustrate the trade-off between optimal storage sizing and
VRE curtailment. As storage investments incur costs, it is not optimal to
fully take up renewable surplus generation, but to also allow some VRE
curtailment. In general, this lowers the need for both storage charging and
energy capacity [77, 83].

Unintended storage cycling distorts these optimal curtailment and storage use
patterns (Figure 6b444The Figure shows the case with proportionate SLCR. The
RLDC of a model specification with zero SLCR is similarly affected and omitted
for the sake of clarity. The RLDCs of constraint family (3) models deviate
only marginally.). A larger part of VRE surplus generation is taken up by
storage and not curtailed (compare area in F in Figure 6b to area A in Figure
6a). Unintended storage cycling results in energy being charged (area G in
Figure 6b) and discharged again in the same period (area H). In contrast to
the case without unintended storage cycling, there are no hours left in which
VRE surplus is curtailed without storage being charged, which would result in
a plateau of the residual load at the zero line. Instead, there are hours in
which maximum possible unintended storage losses are generated (plateau I).
Here, both storage charging and discharging are at maximum capacity, while the
remaining VRE surplus is curtailed. Note that same-period storage cycling is
limited not only by the installed storage charging capacity, but also by the
discharging capacity. In our parameterization, optimal storage discharging
capacity is lower than the optimal charging capacity. In hours to the left and
right of those in plateau I, unintended storage cycling may also occur. Yet,
not the full cycling potential is exploited here. On the right-hand side of
plateau I, storage charging capacity is never at the maximum, while storage
discharging capacity is never at the maximum on the left-hand side of plateau
I.

Optimal dispatch and investment decisions of all technologies are affected by
unintended storage cycling, which is most pronounced in settings with zero
SLCR. Here, optimal storage charging capacity slightly increases to expand the
unintended storage cycling potential (vertical distance between dashed lines
in plateau J in Figure 7). Further, the VRE capacity is lower than in models
with complete SLCR, yielding an upward shift of the RLDC (area between solid
lines in Figure 7555While less pronounced, similar effects also occur in
settings with proportionate SLCR. Yet, for better illustration, we contrast
RLDCs of models with zero and complete SLCR.). While the shifted VRE surplus
energy from the negative part of the RLDC increases, the additionally charged
energy is not used to serve demand, but partly removed from the system via
unintended storage cycling. As a consequence, conventional generation
increases (area K between the dashed lines in Figure 7). This goes along with
additional capacity entry of coal in models with zero SLCR, which helps to
serve residual load during peak periods (vertical distance between dashed
lines in plateau L in Figure 7).

Figure 7: Different configurations of the RLDC for the model specification
(1a) with zero SLCR in black and the specification (1c) with complete SLCR in
gray at a targeted renewable share of 80%.

###  5.4 Effects on market clearing prices

The merit order ranks available generation technologies in ascending order of
variable generation costs. In our cost-minimizing setting, technologies with
lowest variable costs are dispatched first to meet demand. In the absence of
electricity storage, the technology that supplies the last unit of demand —
the marginal technology — sets the uniform price for the power sector at the
intersection of demand and supply. The dual variable $\lambda_{t}$ of the
energy balance, Equation (2.1), indicates the resulting price, which depends
on the optimality condition of the marginal technology. It can be derived from
the first-order condition of the underlying optimization model that applies to
the generation variable of the marginal technology [76, Ch.4]. The price
constitutes the sum of variable costs and the dual variables of constraints
that may be binding for the marginal technology.

If storage is deployed, the first-order condition of storage discharging
includes variable costs, the dual variable of capacity- and energy-related
constraints, and the derivative of the renewable energy constraint (Equations
(SI.15a) and (SI.15b)). While most of these summands coincide for all model
specifications, the derivative of the renewable constraint may vary over both
constraint families and SLCR levels (Equations (SI.17a) to (SI.17l)).

Similarly, the first-order condition of storage charging determines the price
at which storage charges (Equations (SI.16a) and (SI.16b)). This is the price
that results at the intersection of demand and supply. It may also change
across both constraint families and SLCR levels due to a variation in the
derivation of the renewable energy constraint with respect to storage charging
(Equations (SI.18a) to (SI.18l)).

Figure 8 illustrates the estimated distribution (kernel density estimation) of
prices,666We interpret the dual variables of the energy balance as hourly
wholesale prices, compare [67] for model specifications based on constraint
family (1). which vary across SLCR levels for two reasons. First, as the
capacity portfolio changes across SLCR levels, the shape of the merit order
and the intersection of the supply curve with demand may be affected. Second,
varying optimality conditions for storage charging and discharging drive the
differences in wholesale prices between different SLCR levels.

Figure 8 also shows that unintended storage cycling occurs only in low-price
periods, i.e., in hours with renewable curtailment. Note that prices may
become negative in models using constraint family (1), which can be
interpreted as the consequence of an energy-based renewable support mechanism
[67, 79].

Figure 8: Distribution of prices across different SLCR levels for a minimum
renewable share in demand. The left side of each violin in dark blue shows the
distribution of prices during hours without simultaneous storage charging and
discharging. The right side in teal refers to the prices of all hours in which
the unintended storage cycling occurs. The bubbles represent underlying
discrete prices to visualize accumulations and outliers. The numbers at the
top indicate average prices, while the numbers at the bottom refer to the
number of hourly prices for each of the violin’s sides.

Price formations of models based on constraint families (2) to (4) are
characterized by the same variation in the distribution across SLCR levels,
but a constant level shift may apply for each constraint family. This effect
is due to a variation of optimality conditions of generators across constraint
families. It may be explored in more detail in the future, but is beyond the
scope of this paper.

###  5.5 Drivers of unintended storage cycling

To disentangle the drivers of unintended storage cycling, we conduct
sensitivity analyses with respect to key input parameters: the renewable
energy target, storage efficiency, as well as the variable costs of storage
use, renewable generation, and renewable curtailment.

With increasing renewable penetration, the number of hours with unintended
storage cycling and the energy related to the artifact grows
disproportionately (Figure 9a and 10). Below a renewable share of 40%, no
storage is deployed due to very limited VRE surplus generation. Accordingly,
unintended storage cycling does not occur. As the required VRE penetration
increases, VRE surplus energy and optimal electricity storage capacities also
grow. Unintended storage cycling becomes available and is increasingly used.
In a fully renewable power sector, model specifications with proportionate and
complete SLCR converge, since the renewable share constraints coincide (see
Table 2 for $\phi=1$). Both of these SLCR levels then prevent unintended
storage cycling altogether (see top row in Figure 9a), while model
specifications with zero SLCR still suffer from the artifact (see also Figure
10b). By construction, the latter do not comply with the notion of a fully
renewable power sector even at a renewable share of 100%, as they do not
explicitly require complete storage losses to be covered by renewables.
Instead, some level of conventional generation always remains in the system
under zero SLCR constraint formulations.

(a) Different renewable penetration levels (no occurrence below 40%).

(b) Different levels of round-trip storage efficiency.

(c) Different levels of variable costs of storage use.

(d) Different levels of variable costs of renewable generation (no occurrence
above 9000 EUR/MWh for zero SLCR).

(e) Different levels of variable costs of renewable curtailment.

Figure 9: Sensitivity analysis of the annual energy of unintended storage
cycling for different SLCR levels. A renewable share of 80% applies to all
panels except panel 9a. Despite marginal discrepancies, results coincide
across all constraint families.

(a) Energy flows at a renewable share of 80%.

(b) Energy flows at a renewable share of 100%.

Figure 10: Energy flows for a power sector based on the model (1a) using a
renewable share constraint with zero SLCR. The height of each trace represents
the energy amount. Light green traces represent intended power flows withdrawn
from or fed back into the grid. Red traces indicate unintended power flows due
to unintended storage cycling. Unintended storage cycling and unintended
storage losses grow with the renewable share target.

The round-trip efficiency of the electricity storage technology determines
storage losses. Yet, unintended storage cycling does not increase with lower
round-trip efficiency (below a value of 1.0). Instead, LCOS decline as
efficiency increases. The system’s benefit of storage grows, and yields higher
optimal storage capacity. This enables higher levels of unintended storage
cycling (Figure 9b). A round-trip efficiency of 1.0, a strong (and
unrealistic) assumption for electricity storage, averts storage losses and
unintended storage cycling altogether.

Increasing variable costs of storage use, attributed to each unit of charging
and discharging, are another driver of unintended storage cycling (Figure 9c).
If these costs are zero, the economic trade-off between curtailment of VRE
surplus and unintended storage losses cannot be quantified. The optimization
becomes indifferent between the two options, and unintended storage cycling
may occur at undetermined levels. As variable cost for storage use rise, the
cost advantage of removing VRE surplus via unintended storage cycling
declines. Above a certain threshold, which is substantially higher than the
variable costs of current storage technologies [84], unintended storage
cycling becomes an unfavorable option in the cost minimization. In our
stylized setting, this threshold is much lower for models with proportionate
SLCR than for models with zero SLCR.

The same holds true for increasing variable costs of VRE generation, which
incentivize curtailment of VRE surplus rather than feeding it into the grid.
This makes unintended storage cycling less favorable, as converting renewable
surplus into additional storage losses increases the variable cost of
renewable generation. Above a certain threshold, the costs of unintended
storage cycling thus surpass its gains, and the artifact disappears (Figure
9d). Yet, this threshold is beyond of what appears plausible for real-world
VRE technologies [84].

All results presented so far were generated under the assumption that VRE
curtailment comes at no cost. Rising variable costs of curtailment leads to
decreasing curtailment levels. Instead, VRE generation at times of high VRE
availability increases, which is then removed from the system via additional
unintended storage cycling (Figure 9e) in settings with incomplete SLCR. At
very high curtailment costs this becomes effective even for model
specifications with complete SLCR.

##  6 Discussion

###  6.1 A remedy for unintended storage cycling using renewable energy
constraints

We have shown that unintended storage cycling can cause significant
disruptions of optimal dispatch and investment patterns in models that include
a constraint on annual renewable energy generation. To remedy this artifact,
the substitution of VRE curtailment with additional storage losses can be
prohibited by fully including storage losses to the renewable energy
constraint. This is done in all model specifications with complete SLCR,
regardless of the constraint family.

Real-world renewable energy targets are often defined as the ratio of annual
generation from renewable energy sources over total electricity generation (or
total demand) in the system. This implies that storage losses should also be
distributed among renewable and conventional generators according to the
targeted renewable share. Model specifications with proportionate SLCR conform
with this notion, thus appearing as most policy-relevant. In contrast, models
with complete SLCR are overly restrictive in the sense that they require
additional renewable generation to completely cover storage losses.

Notwithstanding, we recommend applying a renewable energy constraint with
complete SLCR, as this remedies unintended storage cycling. In post-
optimization, the achieved renewable share could still be reported according
to both complete and proportionate SLCR. The latter can then be interpreted
and communicated as the actual renewable share of the system, which is
slightly higher than the former. For example, in our model setting with
complete SLCR and a renewable target of 80%, the actual renewable share
reported according to the formulation with proportionate SLCR is 81.16%. We
consider such a slightly increased renewable share to be a reasonable price
for energy modelers to pay to avoid unintended storage cycling, especially as
renewable shares are set to increase anyway in light of ongoing ambitions to
achieve carbon-neutrality. Moreover, models with proportionate and complete
SLCR converge in a fully renewable setting, which, most likely, is required
for carbon-neutrality.

Alternatively, modelers may attempt to slightly decrease the desired renewable
target in models with complete SLCR to achieve a system configuration that
meets the desired target reported as per the proportionate SLCR model. For
example, we need to specify a renewable share of 79.3% in a model with
complete SLCR in order to achieve an actual renewable energy share of 80%
reported according to a formulation with proportionate SLCR. However, the
identification of the required share as per complete SLCR would require trial-
and-error model runs that need to be done for each model parameterization
individually. Depending on the scope of the model and its computational
burden, this may be infeasible because of significant modeling time and
effort, as well as a high number of trails.

Based on the results shown in Section 5.5, one option to avoid over-
restricting energy models with complete SLCR would be to increase the variable
costs of storage use and renewable generation. This makes unintended storage
cycling less favorable in the sense of the objective function and may allow
using a more intuitive constraint formulation with proportionate SLCR.
However, this strategy has to be taken with caution. In the real world,
variable storage costs are generally low, irrespective of the technology. A
manipulation of these costs distorts the relative costs of the available
technology portfolio, and thus also optimal dispatch and investment decisions.
The deployment of storage may decline, potentially causing far-reaching
system-wide effects, such as a hindered integration of VRE. Moreover,
identifying an appropriate increase in variable storage costs that does not
substantially affect optimal storage deployment while preventing unintended
storage cycling is not straightforward. Instead, it requires extensive
testing, and may even be infeasible, depending on the model parameterization.
We thus advise against the deliberate manipulation of storage cost assumptions
to remedy unintended storage cycling when using renewable energy targets.

###  6.2 Potential remedies for unintended storage cycling using alternative
renewable targets

There are two alternative options for avoiding unintended storage cycling
while using renewable targets with a basis other than annual energy. First,
instead of using the actual renewable energy generation (after curtailment),
the theoretical renewable generation potential (before curtailment) could
serve as the base element in the renewable energy constraint (see Equation
(3)) [72, 73, 85, 86]. Relaxing the renewable energy constraint via unintended
storage cycling is then impossible, since the substitution of curtailment with
additional storage losses cannot contribute useful energy to the renewable
target. However, this approach changes the nature of the renewable energy
target. Models constrained this way optimize the system such that there is
sufficient VRE capacity to achieve some renewable energy generation potential
relative to total demand or total generation. Yet, actual renewable generation
is not constrained and may be much lower. Modeling scenarios that adhere to
binding renewable energy policy targets in terms of actual renewable
generation is thus impossible. Furthermore, the composition of the optimal
technology portfolio may significantly deviate from models with complete SLCR,
as the role of conventional generators could be overestimated, while the need
for renewable generation capacity and appropriate balancing options may be
underestimated.

Second, an exogenous expansion target for renewable capacity could be used
[16, 32, 36]. This approach foresees a capacity mix with an exogenous
renewable part, while the remaining part is determined endogenously. Since the
actual generation of each technology is not optimized to realize a certain
share of renewable energy, the latter may be much lower than desired. Due to
this uncertainty of actual renewable generation, binding renewable energy
targets cannot be investigated. This could be resolved by a manual
determination of exogenous renewable capacity targets to achieve a desired
renewable energy target. However, identifying concrete renewable capacities a
priori is not straightforward and may require a large number of trials. This
becomes particularly relevant for comprehensive model settings that include
multiple sector coupling technologies, which puts the feasibility of this
approach into question.

###  6.3 Potential remedies for unintended storage cycling using emission
constraints

Instead of including renewable targets, modelers may constrain or penalize
carbon emissions to achieve high renewable shares and, at the same time, avoid
unintended storage cycling. Based on our literature review, we discuss two
options: a limitation of the allowable emission budget (carbon cap), and an
explicit carbon price.

Given $e_{s}$ being the specific emissions per generator $s$ and $CB$ the
available carbon budget, a constraint implementing a carbon cap could be
formulated as:

$\sum\limits_{s,t}e_{s}G_{s,t}\leq CB$ (8)

The dual variable of this constraint indicates how much total system costs
change if the carbon cap tightens by one marginal unit. We interpret this as
the system costs for one unit of emission reduction. In a long-term
equilibrium, this implicit carbon price can be interpreted as a carbon price
paid by emitting generators [67]. Note that setting policy-relevant carbon
budgets in electricity models may be challenging, as political targets for the
power sector often do not exist.

In contrast, setting an explicit carbon price on each unit of emissions is
more easy to implement, and it becomes directly effective as an additional
variable cost component in the objective function.

In both settings, unintended storage cycling cannot occur since renewable
energy generation is not constrained. The conversion of VRE curtailment into
additional storage losses thus never leads to lower-cost solutions.
Importantly, this only holds true if variable costs for storage use are
greater zero. Otherwise, the model may be indifferent between VRE curtailment
and storage losses.

In general, outcomes of models with similar renewable shares may substantially
differ, depending on what drives this share, a carbon constraint, or a
renewable constraint. The latter promotes additional renewable generation or
capacity beyond the unconstrained optimum. This is done by introducing an
energy- or capacity-oriented subsidy for renewable generators, which increases
their revenues. In contrast, carbon policy instruments impact the use of
conventional generators by increasing their costs. This forces a decline
especially of those generators with the highest carbon emissions, which may be
replaced by lower-emission technologies. The resulting capacity and energy mix
may significantly differ from the solution based on a binding renewable
target. A binding carbon cap favors lower-emission technologies (e.g., gas-
fired plants) over higher-emission technologies (e.g., lignite-fired plants).
As the former tend to have higher variable and lower capital costs, this shift
may decrease the optimal storage deployment and affect the overall capacity
mix.

Explicit carbon pricing affects the merit order of available generation
technologies, increasing the variable cost of emitting technologies. Neither
renewable energy nor carbon emissions are constrained. The resulting optimal
capacity and energy mix resembles the unconstrained optimum that reflects the
economics of the available technology portfolio. Depending on the costs and
availability of flexibility options in the model, substantial carbon prices
may be required to achieve very high shares of renewables. Moreover, the
determination of appropriate price levels a priori may be difficult since not
only economic but also political, environmental, and social aspects may impact
price formations on the electricity wholesale market.

While model formulations that constrain or penalize carbon emissions can
effectively avoid unintended storage cycling, they generally lead to different
outcomes than models with explicit renewable energy targets. Hence, they are
not an adequate substitute for cases where the effects of explicit renewable
targets are of interest, e.g., in many countries that have set such renewable
targets.

###  6.4 Unintended energy losses in settings with sector coupling

The use of renewable electricity for mobility, heating, and industrial
processes is considered a main strategy for decarbonizing the energy system
[87]. This strategy is often referred to as sector coupling. It also includes
indirect electrification via green hydrogen, which may be used where direct
electrification is not viable [88, 89].

The mechanism of unintended storage cycling described here for electricity
storage may also apply to various sector coupling technologies, as these also
come with inherent energy losses. Here, the model artifact may materialize in
the form of VRE surplus being converted into unintended energy losses.

There are three relevant types of such losses: First, losses inherent to
sector coupling options that allow for re-conversion to electricity, i.e.,
with an operation principle similar to electricity storage. These take up
energy from the power sector and convert it to some intermediate form. The
energy is then either consumed for end energy use in some other sector, or, if
beneficial from the system perspective, partly fed back into the grid at a
later point in time. Examples are battery-electric vehicles with a vehicle-to-
grid discharging option, or power plants fueled with green hydrogen.777The
same applies to load-shifting activities that incur energy losses. To remedy
unintended energy losses, we recommend to adjust the loss term of the
renewable energy constraint such that all relevant losses are covered by
renewable generators. Only losses related to processes enabling unintended
energy cycling are relevant in this context (e.g., losses related to
discharging electricity back from an electric vehicle battery to the grid).
Losses from energy consumption without re-conversion option (e.g., losses
related to actually driving an electric vehicle) are not relevant here, if the
respective demand of a sector coupling option is exogenous to the model (see
Section SI.8).

Second, there are standing losses through self-discharge. These may occur in
various types of energy storage, e.g., in different types of low- and high-
temperature heat storage, or in liquid hydrogen storage.888Standing losses may
also be relevant for some long-duration electricity storage technologies. The
model artifact may reoccur here in the form of distorted storage use patterns
that aim at increasing standing losses by stretching the storage period.
Again, properly accounting for standing losses in the loss term of the
renewable energy constraint would remedy the issue.

Third, there may be unintended losses arising from sub-optimal technology
choices. Suppose there are two different sector coupling options available,
one of them more energy-efficient than the other one. Optimal use and
investment decisions may then skew away from the alternative with lower losses
towards the one with higher losses. This would allow more VRE surplus from the
power sector to be converted into energy losses that relax the renewable
constraint. An example for such alternative technologies are different supply
chains of green hydrogen, such as liquid or gaseous hydrogen [24]. We leave it
to future research to find a remedy for distortions related to this type of
unintended energy losses.

###  6.5 Unintended energy losses in multi-regional settings with transmission
losses

Electricity trade facilitates spatial smoothing of VRE generation and electric
load in multi-regional settings, especially in the absence of other
flexibility options [90]. This causes losses in the transmission grid due to
the resistance of power lines. Unintended and excessive shifts of energy
across and between nodes or regions may arise to generate additional grid
losses that remove VRE surplus from the system instead of curtailing it at its
place of origin. The additional VRE generation would then help to relax the
renewable energy constraint, which lowers the need for VRE capacity and
renders a lower-cost solution possible. However, the exported energy does not
or only partly serve demand in the importing region, and generation from other
generators needs to serve demand. Unintended transmission losses may thus
affect optimal use and investment patterns of generation and transmission
infrastructure as well as their spatial allocation across regions.

A model specification with a system-wide renewable share referenced to either
total demand or generation accumulated across all regions is particularly
prone to unintended transmission losses. This is because the spatial
allocation of generation and transmission capacity is not constrained, but
freely relocatable across all regions. Optimal VRE generation capacity may
increase in regions that have low VRE Levelized Cost of Electricity (LCOE) due
to superior weather conditions. At least some of the additional renewable
surplus could be converted into additional renewable generation here, and,
subsequently, removed from the system in form of unintended transmission
losses. This decreases the need for VRE capacity installations in regions with
high LCOE related to inferior weather conditions. Instead, dispatchable
technologies with lower LCOE can serve demand here. This capacity shift across
regions may lead to lower-cost solution than settings without unintended
transmission losses. To prevent the artifact from occurring, the renewable
energy constraint would have to require renewable generators to completely
cover all transmission losses arising across all regions.

Unintended transmission losses may also arise in settings with disaggregated
renewable targets specific to each region. A capacity and portfolio effect
similar to the one caused by unintended storage cycling may arise here, only
that VRE surplus is exported and partly converted into unintended transmission
losses instead of unintended storage losses. Again, we recommend to add
transmission losses to the loss term of the renewable energy constraint. Note
that total domestic demand does not include the energy that is lost in the
transmission grid. Referenced to domestic demand, the loss term of the
renewable energy constraint should thus include total losses associated with
imports required to supply domestic demand. In contrast, total domestic
generation includes the energy lost in the transmission grid when exporting to
other countries. When referencing the domestic renewable energy constraint to
some fraction of domestic generation, only a fraction of the transmission
losses from exports to other regions are considered. The remainder should be
added to the loss term of the renewable energy constraint (cp. the treatment
of unintended storage losses in the loss term of constraint (2c) in Table 2).
Again, this merits further investigation in future research.

##  7 Summary and conclusion

In this paper, we describe and investigate the modeling artifact of unintended
storage cycling, which may arise in a wide range of cost-minimizing energy
models that make use of binding renewable energy constraints. In such models,
it can be beneficial not to curtail renewable surplus energy, but to convert
excess electricity into additional storage losses. This artifact can be
detected when there are periods with simultaneous storage charging and
discharging. The respective increase in renewable generation can be realized
without additional renewable capacity installations, thus helping to achieve
the renewable energy target at lower costs. This may distort optimal dispatch
and investment decisions of all technologies.

We specifically explore the impact of different constraint formulations for
renewable energy targets on model outcomes and unintended storage cycling.
Using a parsimonious power sector model, we analytically derive intuition for
the cause of unintended storage cycling from an energy modeler’s perspective.
We further investigate how including storage losses in the renewable energy
constraint affects optimality conditions for storage in the long-term
equilibrium. Only if storage losses have to be completely covered by renewable
generation, unintended storage cycling can be prevented. Loosely
parameterizing the model to the German power sector, we show that the artifact
can substantially distort optimal dispatch and investment decisions as well as
modeled market prices.

To prevent unintended storage cycling, we recommend to completely cover
storage losses in the renewable energy constraint, if such a constraint is
used in energy models. This comes at the expense of a slight over-restriction
of the model concerning the renewable target, but we argue that this is a
reasonable price to pay to avoid the artifact. Furthermore, this can be
addressed by adjusted reporting and target setting. Alternatively, other types
of renewable constraints not based on energy could be used to avoid the
problem, but these come with other drawbacks. Likewise, the artifact can be
resolved by implementing carbon emission constraints or penalties. Yet, as
these change overall power sector outcomes, they may not be suitable for some
types of policy-relevant analyses where binding renewable energy targets play
a role.

Unintended storage cycling may be considered a special case of a more general
phenomenon of unintended energy losses. It may also occur in models with
sector coupling technologies and transmission. Here, various types of energy
losses could be traded-off against renewable curtailment to relax the
renewable energy constraint in least-cost model solutions. In general,
including such losses in the renewable constraint may help in many cases. We
leave it to future research to explore the feasibility and the limits of this
approach for complex models with many sector coupling technologies and
transmission.

## Acknowledgments

We thank Mario Kendziorski and Alexander Zerrahn for valuable inputs and
discussions, as well as Alexander Roth and Mariza Montes de Oca León for
feedback on earlier drafts. We also thank the participants of various
conferences and seminars for helpful comments, including the DIW Berlin online
workshop “The role of energy storage in power sectors with fossil fuel phase
out” on 9 October 2020, the PhD seminar “Economics of Fossil Fuel Phase-out”
on 20 November 2020, the International Renewable Energy Storage (IRES)
Conference 2021, the International Conference on Energy Economics and
Technology (Enerday) 2021, the Young Energy Economists and Engineers Seminar
(YEEES) 2021, the International Association for Energy Economics (IAEE) Online
Conference 2021, and the Annual Conference of the European Association of
Environmental and Resource Economists (EAERE) 2021. We acknowledge funding by
the German Federal Ministry of Education and Research via the project “Future
of Fossil Fuels in the wake of greenhouse gas neutrality” (grant number FKZ
01LA1810B), and by the German Federal Ministry for Economic Affairs and Energy
via the project “MODEZEEN” (grant number FKZ 03EI1019D).

## Author Contributions

Martin Kittel: Conceptualization, methodology, software, formal analysis,
investigation, data curation, visualization, writing - original draft, writing
- review and editing, supervision, project administration. Wolf-Peter Schill:
Conceptualization, investigation, data curation, writing - review and editing,
project administration.

##

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##  SI Supplementary information

###  SI.1 Constraint formulations

Table SI.1: Minimum renewable and maximum conventional shares considered in the optimization problem discussed in Section 2.1. No. | SLCR | share | constraint
---|---|---|---
1a | zero | $\Omega=\phi\sum\limits_{t}d_{t}$ |  (2.1)
1b | proportionate | $\Omega=\phi\Big{(}\sum\limits_{t}d_{t}+\sum\limits_{r,t}(G^{in}_{r,t}-G^{out}_{r,t})\Big{)}$ |  (2.1)
1c | complete | $\Omega=\phi\sum\limits_{t}d_{t}+\sum\limits_{r,t}(G^{in}_{r,t}-G^{out}_{r,t})$ |  (2.1)
2a | zero | $\Omega=\phi\Big{(}\sum\limits_{s,t}G_{s,t}-\sum\limits_{r,t}(G^{in}_{r,t}-G^{out}_{r,t})\Big{)}$ |  (2.1)
2b | proportionate | $\Omega=\phi\sum\limits_{s,t}G_{s,t}$ |  (2.1)
2c | complete | $\Omega=\phi\sum\limits_{s,t}G_{s,t}+(1-\phi)\sum\limits_{r,t}(G^{in}_{r,t}-G^{out}_{r,t})$ |  (2.1)
3a | zero | $\Theta=(1-\phi)\sum\limits_{t}d_{t}+\sum\limits_{r,t}(G^{in}_{r,t}-G^{out}_{r,t})$ |  (2.1)
3b | proportionate | $\Theta=(1-\phi)\Big{(}\sum\limits_{t}d_{t}+\sum\limits_{r,t}(G^{in}_{r,t}-G^{out}_{r,t})\Big{)}$ |  (2.1)
3c | complete | $\Theta=(1-\phi)\sum\limits_{t}d_{t}$ |  (2.1)
4a | zero | $\Theta=(1-\phi)\sum\limits_{s,t}G_{s,t}+\phi\sum\limits_{r,t}(G^{in}_{r,t}-G^{out}_{r,t})$ |  (2.1)
4b | proportionate | $\Theta=(1-\phi)\sum\limits_{s,t}G_{s,t}$ |  (2.1)
4c | complete | $\Theta=(1-\phi)\Big{(}\sum\limits_{s,t}G_{s,t}-\sum\limits_{r,t}(G^{in}_{r,t}-G^{out}_{r,t})\Big{)}$ |  (2.1)

###  SI.2 Cost assumptions

Table SI.2: Overview of assumptions on investment, annual fixed and variable costs. | overnight investment costs | annual fixed costs | variable costs
---|---|---|---
technology | [EUR/kW(h)] | [EUR/kW] | [EUR/MWh]
coal | 1300 | 25 | 21.55
ocgt | 400 | 1.5 | 76.34
pv | 390 | 10.6 | -
wind | 1000 | 20 | -
storage charging | 1.1 | - | 0.5
storage discharging | 1.1 | - | 0.5
storage energy | 80 | - | -

###  SI.3 Mathematical background - Lagrangian function

Suppose that $f$, $g_{l}$, and $h_{k}$ are affine, continuously differentiable
functions. Our constrained optimization problem minimizes the objective
function $f$ over decision variables $x_{i}$, subject to equality constraints
$g_{l}$ and inequality constraints $h_{k}$:

—l— x_if(x_i) g_l(x_i)=0 ⟂λ_l ∀l h_k(x_i)≥0 ⟂μ_k ∀k

For optimality of a minimization problem with the inequality sign convention
as in Equation (SI.3), we require parallel objective and constraint gradients
at the tangential point [91, Ch. 18.5]:

$\nabla
f(x_{i})=\sum_{l}\lambda_{l}\nabla_{x}g_{l}(x_{i})+\sum_{k}\mu_{k}\nabla_{x}h_{k}(x_{i})$
(SI.1)

Based on the optimality condition and the problem set-up in Equation (SI.3),
we establish the Lagrangian function with non-negative multipliers of
inequality constraints [91, Ch. 18.5]:

$\mathcal{L}(x_{i},\lambda_{l},\mu_{k})=f(x_{i})-\sum_{l}\lambda_{l}g_{l}(x_{i})-\sum_{k}\mu_{k}h_{k}(x_{i})$
(SI.2)

With this formulation, $\mu_{k}$ indicates the change in the objective if
$h_{k}(x_{i})$ is relaxed. For instance, suppose $h_{k}(x_{i})$ includes a
constant $c$, such that $h_{k}(x_{i})\leq c$. If $c$ marginally increased,
that is $h_{k}(x_{i})\leq c+\epsilon$, $\mu_{k}$ represents the corresponding
rise in $f(x_{i})$. In the optimum, which is characterized by the vector
$(x_{i}^{*},\lambda_{l}^{*},\mu_{k}^{*})$, the following four KKT conditions
are satisfied. First, stationarity, i.e., there is no feasible direction to
improve the objective:

$\nabla_{x}\mathcal{L}(x_{i}^{*},\lambda_{l}^{*},\mu_{k}^{*})=\nabla_{x}f(x_{i}^{*})-\sum_{l}\lambda_{l}^{*}\nabla_{x}g_{l}(x_{i}^{*})-\sum_{k}\mu_{k}^{*}\nabla_{x}h_{k}(x_{i}^{*})=0$
(SI.3)

Second, primal feasibility ensures feasibility of the (in-)equality
constraints:

$\displaystyle g_{l}(x_{i}^{*})$ $\displaystyle=0$ (SI.4a) $\displaystyle
h_{k}(x_{i}^{*})$ $\displaystyle\leq 0$ (SI.4b)

Third, while the multipliers (also called dual variables) of the equality
constraints are free, dual feasibility establishes non-negative multipliers
for inequality constraints:

$\mu_{k}^{*}\geq 0$ (SI.5)

Finally, complementary slackness conditions require a positive multiplier
$\mu_{k}^{*}>0$ if an inequality constraint is binding, i.e.,
$h_{k}(x_{i}^{*})=0$, or $\mu_{k}^{*}=0$ if the constraint is not binding,
i.e., $h_{k}(x_{i}^{*})<0$:

$\mu_{k}^{*}h_{k}(x_{i}^{*})=0$ (SI.6)

###  SI.4 Derivation of optimality conditions in an unconstrained optimum

In the unconstrained optimum of the model introduced in Section 2.1, renewable
energy constraints (2.1) and (2.1) do not apply. In this case, KKT
stationarity yields the following first-order conditions for storage:

$\displaystyle\frac{\partial\mathcal{L}}{\partial C_{r}^{\circ}}$
$\displaystyle=0\Rightarrow i_{r}^{\circ}-\sum_{t}\bar{\mu}_{s,t}^{\circ}=0$
(SI.7) $\displaystyle\frac{\partial\mathcal{L}}{\partial G^{out}_{r,t}}$
$\displaystyle=0\Rightarrow
o^{out}_{r}-\lambda_{t}-\underaccent{\bar}{\mu}^{out}_{r,t}+\bar{\mu}^{out}_{r,t}-(\eta^{out}_{r})^{-1}\lambda^{l}_{r,t}=0$
(SI.8) $\displaystyle\frac{\partial\mathcal{L}}{\partial G^{in}_{r,t}}$
$\displaystyle=0\Rightarrow
o^{in}_{r}+\lambda_{t}-\underaccent{\bar}{\mu}^{in}_{r,t}+\bar{\mu}^{in}_{r,t}+\eta^{in}_{r}\lambda^{l}_{r,t}=0$
(SI.9) $\displaystyle\frac{\partial\mathcal{L}}{\partial G^{l}_{r,t}}$
$\displaystyle=0\Rightarrow-\underaccent{\bar}{\mu}^{l}_{r,t}+\bar{\mu}^{l}_{r,t}-\lambda^{l}_{r,t}+\lambda^{l}_{r,t+1}=0$
(SI.10)

KKT complementary slackness requires the following for the storage inequality
constraints:

$\displaystyle\underaccent{\bar}{\mu}^{\circ}_{r,t}G^{\circ}_{r,t}$
$\displaystyle=0$ (SI.11)
$\displaystyle\underaccent{\bar}{\mu}^{\circ}_{r,t}(C_{r}^{\circ}-G^{\circ}_{r,t})$
$\displaystyle=0$ (SI.12)

We establish the zero-profit rule for storage of the unconstrained optimum as
follows:

$\begin{split}\sum_{\circ}i_{r}^{\circ}C_{r}^{\circ}&+\sum_{t,*}o_{r}^{*}G_{r,t}^{*}+\sum_{t}\lambda_{t}G_{r,t}^{in}\\\
&=\sum_{t,\circ}\bar{\mu}^{\circ}_{r,t}C^{\circ}_{r}+\sum_{t}G_{r,t}^{out}\Big{(}\lambda_{t}+\underaccent{\bar}{\mu}^{out}_{r,t}-\bar{\mu}^{out}_{r,t}+\lambda^{l}_{r,t}(\eta^{out}_{r})^{-1}\Big{)}\\\
&\quad+\sum_{t}G_{r,t}^{in}\Big{(}-\lambda_{t}+\underaccent{\bar}{\mu}^{in}_{r,t}-\bar{\mu}^{in}_{r,t}-\lambda^{l}_{r,t}\eta^{in}_{r}\Big{)}+\sum_{t}\lambda_{t}G_{r,t}^{in}\\\
&=\sum_{t,\circ}\bar{\mu}^{\circ}_{r,t}C^{\circ}_{r}+\sum_{t}G_{r,t}^{out}\Big{(}\lambda_{t}+\underaccent{\bar}{\mu}^{out}_{r,t}-\bar{\mu}^{out}_{r,t}+\lambda^{l}_{r,t}(\eta^{out}_{r})^{-1}\Big{)}\\\
&\quad+\sum_{t}G_{r,t}^{in}\Big{(}-\lambda_{t}+\lambda_{t}+\underaccent{\bar}{\mu}^{in}_{r,t}-\bar{\mu}^{in}_{r,t}-\lambda^{l}_{r,t}\eta^{in}_{r}\Big{)}\\\
&=\sum_{t}\Big{(}\underaccent{\bar}{\mu}^{out}_{rt}G^{out}_{r,t}+\underaccent{\bar}{\mu}^{in}_{rt}G^{in}_{r,t}+\bar{\mu}^{out}_{rt}(C^{out}_{r}-G^{out}_{r,t})+\bar{\mu}^{in}_{rt}(C^{in}_{r}-G^{in}_{r,t})\\\
&\quad+G^{out}_{r,t}\big{(}\lambda_{t}+\lambda^{l}_{r,t}(\eta^{out}_{r})^{-1}\big{)}+G^{in}_{r,t}\big{(}-\lambda^{l}_{r,t}\eta^{in}_{r}\big{)}+G^{l}_{r,t}\big{(}\underaccent{\bar}{\mu}^{l}_{r,t}+\lambda^{l}_{r,t}-\lambda^{l}_{r,t+1}\big{)}\Big{)}\\\
&=\sum_{t}\Big{(}\lambda_{t}G_{r,t}^{out}+\underaccent{\bar}{\mu}^{l}_{r,t}G^{l}_{r,t}+\lambda^{l}_{r,t}\big{(}G^{l}_{r,t}-\eta_{r}^{l}G^{l}_{r,t-1}-\eta^{in}_{r}G^{in}_{r,t}+(\eta^{out}_{r})^{-1}G^{out}_{r,t}\big{)}\Big{)}\\\
&=\sum_{t}\lambda_{t}G_{r,t}^{out}\end{split}$ (SI.13)

The first step employs stationarity conditions (SI.7), (SI.8) and (SI.9). The
second step rearranges terms, uses complementarity from (SI.12), and
substitutes $\underaccent{\bar}{\mu}^{l}_{r,t}$ as in (SI.10). Third, the
complementarity conditions (SI.11) and (SI.12) are exploited to cancel out
terms, while the cycling sum over $G^{l}_{r,t}$ is shifted to relate the
timing to $\lambda^{l}_{r,t}$. Last, primal feasibility of (2.1) is exploited.

Dividing both sides of Equation (SI.13) by annual energy output of storage
$\sum_{t}G_{s,t}^{out}$ and rearranging terms renders the storage zero-profit
condition in the long-term equilibrium:

$LCOS_{r}=MV_{r}$ (SI.14)

###  SI.5 Derivation of optimality conditions with binding renewable targets

When imposing a binding renewable target, the renewable energy constraints
(2.1) and (2.1) become effective. The KKT stationarity conditions (SI.15a) and
(SI.15b) refer to models using constraints (2.1) or (2.1), respectively, and
replace the stationarity condition (SI.8) of the unconstrained model.
Similarly, (SI.16a) and (SI.16b) replace (SI.9):

$\displaystyle\frac{\partial\mathcal{L}}{\partial G^{out}_{r,t}}$
$\displaystyle=0\Rightarrow
o^{out}_{r}-\lambda_{t}-\underaccent{\bar}{\mu}^{out}_{r,t}+\bar{\mu}^{out}_{r,t}-(\eta^{out}_{r})^{-1}\lambda^{l}_{r,t}+\frac{\partial\Theta}{\partial
G^{out}_{r,t}}\mu_{\theta}=0$ (SI.15a)
$\displaystyle\frac{\partial\mathcal{L}}{\partial G^{out}_{r,t}}$
$\displaystyle=0\Rightarrow
o^{out}_{r}-\lambda_{t}-\underaccent{\bar}{\mu}^{out}_{r,t}+\bar{\mu}^{out}_{r,t}-(\eta^{out}_{r})^{-1}\lambda^{l}_{r,t}-\frac{\partial\Omega}{\partial
G^{out}_{r,t}}\mu_{\omega}=0$ (SI.15b)

$\displaystyle\frac{\partial\mathcal{L}}{\partial G^{in}_{r,t}}$
$\displaystyle=0\Rightarrow
o^{in}_{r}+\lambda_{t}-\underaccent{\bar}{\mu}^{in}_{r,t}+\bar{\mu}^{in}_{r,t}+\eta^{in}_{r}\lambda^{l}_{r,t}+\frac{\partial\Theta}{\partial
G^{in}_{r,t}}\mu_{\theta}=0$ (SI.16a)
$\displaystyle\frac{\partial\mathcal{L}}{\partial G^{in}_{r,t}}$
$\displaystyle=0\Rightarrow
o^{in}_{r}+\lambda_{t}-\underaccent{\bar}{\mu}^{in}_{r,t}+\bar{\mu}^{in}_{r,t}+\eta^{in}_{r}\lambda^{l}_{r,t}-\frac{\partial\Omega}{\partial
G^{in}_{r,t}}\mu_{\omega}=0$ (SI.16b)

Derivations (SI.15a) and (SI.15b) are alternatives, referring to model
configurations based on constraint family (1) and (2) as defined in equation
(2.1), or constraint family (3) and (4) as in equation (2.1), respectively.
The same holds true for (SI.16a) and (SI.16b). The optimality conditions for
storage discharging are:

$\displaystyle\Theta_{1a}$
$\displaystyle:\lambda_{t}=o^{out}_{r}-\underaccent{\bar}{\mu}^{out}_{r,t}+\bar{\mu}^{out}_{r,t}-(\eta^{out}_{r})^{-1}\lambda^{l}_{r,t}$
(SI.17a) $\displaystyle\Theta_{1b}$
$\displaystyle:\lambda_{t}=o^{out}_{r}-\underaccent{\bar}{\mu}^{out}_{r,t}+\bar{\mu}^{out}_{r,t}-(\eta^{out}_{r})^{-1}\lambda^{l}_{r,t}-\phi\mu_{\theta}$
(SI.17b) $\displaystyle\Theta_{1c}$
$\displaystyle:\lambda_{t}=o^{out}_{r}-\underaccent{\bar}{\mu}^{out}_{r,t}+\bar{\mu}^{out}_{r,t}-(\eta^{out}_{r})^{-1}\lambda^{l}_{r,t}-\mu_{\theta}$
(SI.17c) $\displaystyle\Theta_{2a}$
$\displaystyle:\lambda_{t}=o^{out}_{r}-\underaccent{\bar}{\mu}^{out}_{r,t}+\bar{\mu}^{out}_{r,t}-(\eta^{out}_{r})^{-1}\lambda^{l}_{r,t}+\phi\mu_{\theta}$
(SI.17d) $\displaystyle\Theta_{2b}$
$\displaystyle:\lambda_{t}=o^{out}_{r}-\underaccent{\bar}{\mu}^{out}_{r,t}+\bar{\mu}^{out}_{r,t}-(\eta^{out}_{r})^{-1}\lambda^{l}_{r,t}$
(SI.17e) $\displaystyle\Theta_{2c}$
$\displaystyle:\lambda_{t}=o^{out}_{r}-\underaccent{\bar}{\mu}^{out}_{r,t}+\bar{\mu}^{out}_{r,t}-(\eta^{out}_{r})^{-1}\lambda^{l}_{r,t}-(1-\phi)\mu_{\theta}$
(SI.17f) $\displaystyle\Omega_{3a}$
$\displaystyle:\lambda_{t}=o^{out}_{r}-\underaccent{\bar}{\mu}^{out}_{r,t}+\bar{\mu}^{out}_{r,t}-(\eta^{out}_{r})^{-1}\lambda^{l}_{r,t}+\mu_{\omega}$
(SI.17g) $\displaystyle\Omega_{3b}$
$\displaystyle:\lambda_{t}=o^{out}_{r}-\underaccent{\bar}{\mu}^{out}_{r,t}+\bar{\mu}^{out}_{r,t}-(\eta^{out}_{r})^{-1}\lambda^{l}_{r,t}+(1-\phi)\mu_{\omega}$
(SI.17h) $\displaystyle\Omega_{3c}$
$\displaystyle:\lambda_{t}=o^{out}_{r}-\underaccent{\bar}{\mu}^{out}_{r,t}+\bar{\mu}^{out}_{r,t}-(\eta^{out}_{r})^{-1}\lambda^{l}_{r,t}$
(SI.17i) $\displaystyle\Omega_{4a}$
$\displaystyle:\lambda_{t}=o^{out}_{r}-\underaccent{\bar}{\mu}^{out}_{r,t}+\bar{\mu}^{out}_{r,t}-(\eta^{out}_{r})^{-1}\lambda^{l}_{r,t}+\phi\mu_{\omega}$
(SI.17j) $\displaystyle\Omega_{4b}$
$\displaystyle:\lambda_{t}=o^{out}_{r}-\underaccent{\bar}{\mu}^{out}_{r,t}+\bar{\mu}^{out}_{r,t}-(\eta^{out}_{r})^{-1}\lambda^{l}_{r,t}$
(SI.17k) $\displaystyle\Omega_{4c}$
$\displaystyle:\lambda_{t}=o^{out}_{r}-\underaccent{\bar}{\mu}^{out}_{r,t}+\bar{\mu}^{out}_{r,t}-(\eta^{out}_{r})^{-1}\lambda^{l}_{r,t}-(1-\phi)\mu_{\omega}$
(SI.17l)

Equivalently, the optimality conditions for storage charging are:

$\displaystyle\Theta_{1a}$
$\displaystyle:\lambda_{t}=-o^{in}_{r}+\underaccent{\bar}{\mu}^{in}_{r,t}-\bar{\mu}^{in}_{r,t}-\eta^{in}_{r}\lambda^{l}_{r,t}$
(SI.18a) $\displaystyle\Theta_{1b}$
$\displaystyle:\lambda_{t}=-o^{in}_{r}+\underaccent{\bar}{\mu}^{in}_{r,t}-\bar{\mu}^{in}_{r,t}-\eta^{in}_{r}\lambda^{l}_{r,t}-\phi\mu_{\theta}$
(SI.18b) $\displaystyle\Theta_{1c}$
$\displaystyle:\lambda_{t}=-o^{in}_{r}+\underaccent{\bar}{\mu}^{in}_{r,t}-\bar{\mu}^{in}_{r,t}-\eta^{in}_{r}\lambda^{l}_{r,t}-\mu_{\theta}$
(SI.18c) $\displaystyle\Theta_{2a}$
$\displaystyle:\lambda_{t}=-o^{in}_{r}+\underaccent{\bar}{\mu}^{in}_{r,t}-\bar{\mu}^{in}_{r,t}-\eta^{in}_{r}\lambda^{l}_{r,t}+\phi\mu_{\theta}$
(SI.18d) $\displaystyle\Theta_{2b}$
$\displaystyle:\lambda_{t}=-o^{in}_{r}+\underaccent{\bar}{\mu}^{in}_{r,t}-\bar{\mu}^{in}_{r,t}-\eta^{in}_{r}\lambda^{l}_{r,t}$
(SI.18e) $\displaystyle\Theta_{2c}$
$\displaystyle:\lambda_{t}=-o^{in}_{r}+\underaccent{\bar}{\mu}^{in}_{r,t}-\bar{\mu}^{in}_{r,t}-\eta^{in}_{r}\lambda^{l}_{r,t}-(1-\phi)\mu_{\theta}$
(SI.18f) $\displaystyle\Omega_{3a}$
$\displaystyle:\lambda_{t}=-o^{in}_{r}+\underaccent{\bar}{\mu}^{in}_{r,t}-\bar{\mu}^{in}_{r,t}-\eta^{in}_{r}\lambda^{l}_{r,t}+\mu_{\omega}$
(SI.18g) $\displaystyle\Omega_{3b}$
$\displaystyle:\lambda_{t}=-o^{in}_{r}+\underaccent{\bar}{\mu}^{in}_{r,t}-\bar{\mu}^{in}_{r,t}-\eta^{in}_{r}\lambda^{l}_{r,t}+(1-\phi)\mu_{\omega}$
(SI.18h) $\displaystyle\Omega_{3c}$
$\displaystyle:\lambda_{t}=-o^{in}_{r}+\underaccent{\bar}{\mu}^{in}_{r,t}-\bar{\mu}^{in}_{r,t}-\eta^{in}_{r}\lambda^{l}_{r,t}$
(SI.18i) $\displaystyle\Omega_{4a}$
$\displaystyle:\lambda_{t}=-o^{in}_{r}+\underaccent{\bar}{\mu}^{in}_{r,t}-\bar{\mu}^{in}_{r,t}-\eta^{in}_{r}\lambda^{l}_{r,t}+\phi\mu_{\omega}$
(SI.18j) $\displaystyle\Omega_{4b}$
$\displaystyle:\lambda_{t}=-o^{in}_{r}+\underaccent{\bar}{\mu}^{in}_{r,t}-\bar{\mu}^{in}_{r,t}-\eta^{in}_{r}\lambda^{l}_{r,t}$
(SI.18k) $\displaystyle\Omega_{4c}$
$\displaystyle:\lambda_{t}=-o^{in}_{r}+\underaccent{\bar}{\mu}^{in}_{r,t}-\bar{\mu}^{in}_{r,t}-\eta^{in}_{r}\lambda^{l}_{r,t}-(1-\phi)\mu_{\omega}$
(SI.18l)

By way of example, the zero-profit rule for storage in the model specification
(1c) is as follows:

$\begin{split}\sum_{\circ}i_{r}^{\circ}C_{r}^{\circ}&+\sum_{t,*}o_{r}^{*}G_{r,t}^{*}+\sum_{t}\lambda_{t}G_{r,t}^{in}\\\
&=\sum_{t,\circ}\bar{\mu}^{\circ}_{r,t}C^{\circ}_{r}+\sum_{t}G_{r,t}^{out}\Big{(}\lambda_{t}+\underaccent{\bar}{\mu}^{out}_{r,t}-\bar{\mu}^{out}_{r,t}+\lambda^{l}_{r,t}(\eta^{out}_{r})^{-1}-\frac{\partial\Theta_{1c}}{\partial
G_{r,t}^{out}}\mu_{\theta}\Big{)}\\\
&\quad+\sum_{t}G_{r,t}^{in}\Big{(}-\lambda_{t}+\underaccent{\bar}{\mu}^{in}_{r,t}-\bar{\mu}^{in}_{r,t}-\lambda^{l}_{r,t}\eta^{in}_{r}-\frac{\partial\Theta_{1c}}{\partial
G_{r,t}^{in}}\mu_{\theta}\Big{)}+\sum_{t}\lambda_{t}G_{r,t}^{in}\\\
&=\sum_{t,\circ}\bar{\mu}^{\circ}_{r,t}C^{\circ}_{r}+\sum_{t}G_{r,t}^{out}\Big{(}\lambda_{t}+\underaccent{\bar}{\mu}^{out}_{r,t}-\bar{\mu}^{out}_{r,t}+\lambda^{l}_{r,t}(\eta^{out}_{r})^{-1}+\mu_{\theta}\Big{)}\\\
&\quad+\sum_{t}G_{r,t}^{in}\Big{(}-\lambda_{t}+\lambda_{t}+\underaccent{\bar}{\mu}^{in}_{r,t}-\bar{\mu}^{in}_{r,t}-\lambda^{l}_{r,t}\eta^{in}_{r}-\mu_{\theta}\Big{)}\\\
&=\sum_{t}\Big{(}\underaccent{\bar}{\mu}^{out}_{rt}G^{out}_{r,t}+\underaccent{\bar}{\mu}^{in}_{rt}G^{in}_{r,t}+\bar{\mu}^{out}_{rt}(C^{out}_{r}-G^{out}_{r,t})+\bar{\mu}^{in}_{rt}(C^{in}_{r}-G^{in}_{r,t})\\\
&\quad+G^{out}_{r,t}\big{(}\lambda_{t}+\lambda^{l}_{r,t}(\eta^{out}_{r})^{-1}+\mu_{\theta}\big{)}+G^{in}_{r,t}\big{(}-\lambda^{l}_{r,t}\eta^{in}_{r}-\mu_{\theta}\big{)}\\\
&\quad+G^{l}_{r,t}\big{(}\underaccent{\bar}{\mu}^{l}_{r,t}+\lambda^{l}_{r,t}-\lambda^{l}_{r,t+1}\big{)}\Big{)}\\\
&=\sum_{t}\Big{(}\lambda_{t}G_{r,t}^{out}-\mu_{\theta}\big{(}G_{r,t}^{in}-G_{r,t}^{out}\big{)}+\underaccent{\bar}{\mu}^{l}_{r,t}G^{l}_{r,t}\\\
&\quad+\lambda^{l}_{r,t}\big{(}G^{l}_{r,t}-\eta_{r}^{l}G^{l}_{r,t-1}-\eta^{in}_{r}G^{in}_{r,t}+(\eta^{out}_{r})^{-1}G^{out}_{r,t}\big{)}\Big{)}\\\
&=\sum_{t}\Big{(}\lambda_{t}G_{r,t}^{out}-\mu_{\theta}\big{(}G_{r,t}^{in}-G_{r,t}^{out}\big{)}\Big{)}\end{split}$
(SI.19)

The first step employs stationarity conditions (SI.7), (SI.15a) and (SI.16a).
Second, the remaining derivatives are computed. The third step rearranges
terms, uses complementarity from (SI.12), and substitutes
$\underaccent{\bar}{\mu}^{l}_{r,t}$ as in (SI.10). Fourth, the complementarity
conditions (SI.11) and (SI.12) are exploited to cancel out terms, while the
cycling sum over $G^{l}_{r,t}$ is shifted to relate the timing to
$\lambda^{l}_{r,t}$. Last, primal feasibility of (2.1) is exploited.

Dividing both sides of equation (SI.19) by the storage unit’s annual energy
output $\sum_{t}G_{s,t}^{out}$ and rearranging terms renders its long-term
equilibrium condition, as in Table 3:

$LCOS_{r}+\mu_{\theta}NSL_{r}=MV_{r}$ (SI.20)

The mathematical proofs zero-profit conditions of all other investigated model
specifications follow the same procedure and are available upon request.

###  SI.6 Specifics of numerical results of constraint family (3)

Dispatch from VRE is more similar between different SLCR levels within
constraint family (3) models compared to the other constraint families (Figure
SI.2). Furthermore, models based on constraint family (3) slightly favor PV
over wind compared to all other models using other constraint families. In
contrast, optimal investment decisions coincide across all constraint
families, and vary only with respect to the SLCR level (Figure SI.1).

Figure SI.1: Total capacity per technology at a renewable target of 80% in
absolute numbers for model specifications with complete SLCR (left panel), and
the deviation from the complete SLCR case for models with zero and
proportionate SLCR for constraint families (1), (2), and (4) (middle panel),
and for constraint family (3) (right panel). Figure SI.2: Total generation per
technology at a renewable target of 80% in absolute numbers for model
specifications with complete SLCR (left panel), and the deviation from the
complete SLCR case for models with zero and proportionate SLCR for constraint
families (1), (2), and (4) (middle panel), and for constraint family (3)
(right panel).

###  SI.7 Factor separation: unintended storage cycling and ambition levels of
the renewable constraint

The differences in optimal capacity and dispatch decisions presented in Figure
5 are caused by variations of the SLCR level in the renewable energy
constraint. Two overlapping factors drive results: (i) unintended storage
cycling, and (ii) different ambition levels regarding the required minimum
renewable penetration. The latter increases as more storage losses have to be
covered by renewable energy. As explained in Section 3.1, models with zero
SLCR require the lowest use of VRE, followed by models with proportionate
SLCR, and such with complete SLCR, which require the highest use of VRE. We
disentangle the impacts of both factors by carrying out additional model runs
with adjusted renewable energy targets.

First, the effects of unintended storage cycling can be isolated when
contrasting a model with incomplete SLCR to a setting with complete SLCR. To
rule out the impacts of different ambition levels regarding the renewable
target, the realized renewable share of both settings needs to coincide when
reported according to the renewable share formula of the model with incomplete
SLCR.

In our stylized setting, a complete SLCR model with a 77.0% renewable target
achieves the same renewable penetration as a zero SLCR model with an 80.0%
renewable target, when reporting both renewable penetration levels according
to the zero SLCR specification. Likewise, the renewable penetration of a model
with proportionate SLCR and an 80% target matches that of a model with a
complete SLCR and a 79.3% target, when reported according to the proportionate
SLCR formula. Note that determining such matching renewable targets for
complete SLCR settings, in which unintended storage cycling never occurs, may
require a time-consuming try-and-error approach, which needs to be carried out
for each individual model parameterization.

Second, the impact of different ambition levels of the required renewable
energy share can be determined by comparing complete SLCR specifications with
different renewable shares. In our stylized setting, contrasting the
additional specifications with complete SLCR and a 77.0% or 79.3% renewable
target to a complete SLCR model with 80% allows for separating the effects of
different ambition levels from unintended storage cycling for zero and
proportionate SLCR settings with an 80% renewable penetration, respectively.

Figure SI.3 illustrates the outcomes of the two additional model
specifications with adjusted renewable energy targets (77.0% and 79.3%), which
allow for disentangling the effects of both factors. Comparing the cases with
complete SLCR, i.e., models without unintended storage cycling, it can be seen
that total VRE capacity decreases with lower ambition levels (Figure SI.3a).
This decrease is the main driver for lower storage energy capacity needs
presented in Section 5.2 (Figure SI.3b).

(a) Installed generation capacity including storage charging and discharging.
(b) Installed storage energy capacity.

(c) Generation including storage charging (negative part of ordinate) and
discharging. (d) Storage losses and curtailment.

Figure SI.3: Installed capacity, annual generation, curtailment, and storage
losses per technology, including factor separation model runs. The numbers in
the labels refer to renewable targets specified in respective models.

Comparing the zero and proportionate SLCR models with their counterparts with
complete SLCR and adjusted renewable shares (i.e., comparing columns 1 and 2,