# Load-Flow Solvability under Security Constraints in DC Distribution Networks

Zhe Chen, , Cong Wang Zhe Chen and Cong Wang share equal contribution. Zhe
Chen is with Department of Electrical Engineering, Technical University of
Denmark, 4000 Roskilde, Denmark (e-mail: zhech@elektro.dtu.dk). Cong Wang is
with College of Communication Engineering, Jilin University, 130022 Changchun,
China (e-mail: wangcong2020@jlu.edu.cn).

###### Abstract

We present sufficient conditions for the load-flow solvability under security
constraints in DC distribution networks. In addition, we show that a load-flow
solution that fulfills security constraints can be obtained via a convex
optimization.

###### Index Terms:

load flow, security constraints, nonsingularity, polynomial optimization,
feasibility, DC distribution networks.

##  I Introduction

Recently, DC distribution networks have received increasing attention. In
these networks, many control procedures (i) rely on the load-flow solvability,
and (ii) require that the load-flow solution (if exists) should satisfy the
security constraints. However, due to non-linearity of the load-flow
equations, this is hard to study. In this paper, we present sufficient
conditions for the load-flow solvability under security constraints in DC
distribution networks. Moreover, we show that a load-flow solution that
fulfills security constraints can be obtained by solving a convex
optimization.

##  II Problem Formulation

We consider a unipolar DC distribution network that has one slack bus, $N$
$PQ$ buses, and a generic topology (i.e., radial or meshed). The slack bus is
indexed by $0$, and the $PQ$ buses are indexed by $1,...,N$. For simplicity of
expression, we define

  * •

$\mathcal{N}=\\{0,...,N\\}$ as the set of all buses;

  * •

$\mathcal{N}^{PQ}=\mathcal{N}\setminus\\{0\\}$ as the set of $PQ$ buses;

  * •

$\mathcal{K}(n)\subseteq\mathcal{N}$ as the set of neighbouring buses for bus
$n\in\mathcal{N}$;

  * •

$I_{\mathcal{K}(n)}(k)$ as the indicator function that has value $1$ when
$k\in\mathcal{K}(n)$ and $0$ otherwise.

Now, let $v_{n}$, $p_{n}$ be the real-valued nodal voltage and nodal power
injection at bus $n\in\mathcal{N}$. In addition, let positive real constant
$g_{nk}$ be the line conductance between buses $n\in\mathcal{N}$ and
$k\in\mathcal{K}(n)$. Then, the load-flow equations at $PQ$ buses are
formulated as follows:

$\Big{(}\sum_{k\in\mathcal{K}(n)}g_{nk}\Big{)}v_{n}^{2}-\Big{(}\sum_{k\in\mathcal{K}(n)}g_{nk}v_{k}\Big{)}v_{n}=p_{n},\quad
n\in\mathcal{N}^{PQ}~{}.$ (1)

During power system operation, the nodal voltages and branch currents should
satisfy security constraints. In the paper, we express the security
constraints as follows:

$v^{\mathrm{min}}<v_{n}<v^{\mathrm{max}},\quad n\in\mathcal{N}^{PQ},$ (2)
$|g_{nk}(v_{n}-v_{k})|<i_{nk}^{\mathrm{max}},\quad
n\in\mathcal{N}^{PQ},~{}k\in\mathcal{K}(n),$ (3)

where $v^{\mathrm{min}}$, $v^{\mathrm{max}}$, and $i_{nk}^{\mathrm{max}}$ are
some pre-defined positive real constants. Note that, for the development of
our theoretical results, we need the security constraints to be strict
inequalities.

With the above grid model and notations, we study the following problem: Given
$v_{0}$ and $(p_{1},...,p_{N})$, is there a unique solution
$(v_{1},...,v_{N})$ that satisfies the load-flow equations (1) and the
security constraints (2)–(3) ?

##  III Main Results

###  III-A Existence

In Theorem 1, we provide sufficient conditions for the existence of a solution
to (1)–(3).

######  Theorem 1.

Assume that

  1. 1.

$2v^{\mathrm{min}}>v^{\mathrm{max}}>v_{0}>v^{\mathrm{min}}$;

  2. 2.

The following convex optimization $\mathrm{P1}$ is feasible.

$\displaystyle[\mathbf{P1}]\quad\max\limits_{\alpha_{n},\beta_{nk},v_{n}}\quad$
$\displaystyle\sum_{n=1}^{N}v_{n}$ $\displaystyle\mathrm{subject~{}to:}\quad$
$\displaystyle\mathrm{inequalities}~{}\eqref{eq:security1}-\eqref{eq:security2},$
$\displaystyle\quad\Big{(}\sum_{k\in\mathcal{K}(n)}g_{nk}\Big{)}\alpha_{n}-\sum_{k\in\mathcal{K}(n)\setminus\\{0\\}}g_{nk}\beta_{nk}$
$\displaystyle=p_{n}+\Big{(}g_{n0}v_{0}I_{\mathcal{K}(n)}(0)\Big{)}v_{n},\quad
n\in\mathcal{N}^{PQ},$ (4) $\displaystyle v_{n}^{2}\leq\alpha_{n},\quad
n\in\mathcal{N}^{PQ},$ (5) $\displaystyle\beta_{nk}\geq 0,\quad
n\in\mathcal{N}^{PQ},~{}k\in\mathcal{K}(n)$ (6)
$\displaystyle\beta_{nk}^{2}\leq\alpha_{n}\alpha_{k},\quad
n\in\mathcal{N}^{PQ},~{}k\in\mathcal{K}(n)~{}.$ (7)

Then, for any optimizer $\alpha_{n}^{\star}$, $\beta_{nk}^{\star}$,
$v_{n}^{\star}$ of $\mathrm{P1}$, we have that
$(v_{1}^{\star},...,v_{N}^{\star})$ is a solution to (1)–(3).

In practice, the condition
$2v^{\mathrm{min}}>v^{\mathrm{max}}>v_{0}>v^{\mathrm{min}}$ in Theorem 1 is
normally satisfied. Consequently, for any $(p_{1},...,p_{N})$ such that
$\mathrm{P1}$ is feasible, there exists at least one solution to (1)–(3).

It should be noticed that, Theorem 1 not only gives sufficient conditions on
the existence of a solution to (1)–(3), but also shows that a solution can be
obtained by solving the convex optimization $\mathrm{P1}$.

######  Remark 1.

Consider that the numerical solvers might not be able to handle strict
inequalities, we can solve the following convex optimization and verify
whether the optimal nodal voltages fulfill the strict inequality (2)–(3).

$\displaystyle\max\limits_{\alpha_{n},\beta_{nk},v_{n}}\quad$
$\displaystyle\sum_{n=1}^{N}v_{n}$ $\displaystyle\mathrm{subject~{}to:}\quad$
$\displaystyle\eqref{eq:alpha1}-\eqref{eq:beta2},$ $\displaystyle
v^{\mathrm{min}}\leq v_{n}\leq v^{\mathrm{max}},\quad n\in\mathcal{N}^{PQ},$
(8) $\displaystyle|g_{nk}(v_{n}-v_{k})|\leq i_{nk}^{\mathrm{max}},\quad
n\in\mathcal{N}^{PQ},~{}k\in\mathcal{K}(n)~{}.$ (9)

###  III-B Uniqueness

In Theorem 2, we provide sufficient conditions for the uniqueness of the
solution to (1)–(3).

######  Theorem 2.

If the following polynomial optimization $\mathrm{P2}$ is infeasible, then
there exists at most one solution to (1)–(3).

$\displaystyle[\mathbf{P2}]\quad\min\limits_{\gamma_{n},v_{n}}\quad$
$\displaystyle\sum_{n=1}^{N}v_{n}$ $\displaystyle\mathrm{subject~{}to:}\quad$
$\displaystyle\eqref{eq:security3}-\eqref{eq:security4},$
$\displaystyle\Bigg{(}2\Big{(}\sum_{k\in\mathcal{K}(n)}g_{nk}\Big{)}v_{n}-\sum_{k\in\mathcal{K}(n)}g_{nk}v_{k}\Bigg{)}\gamma_{n}$
$\displaystyle=\sum_{k\in\mathcal{K}(n)\setminus\\{0\\}}\Big{(}g_{nk}v_{n}\Big{)}\gamma_{k},\quad
n\in\mathcal{N}^{PQ},$ (10) $\displaystyle\sum_{n=1}^{N}\gamma_{n}^{2}=1~{}.$
(11)

As can be seen, the infeasibility of polynomial optimization $\mathrm{P2}$ is
determined by only the security bounds $v^{\mathrm{min}}$, $v^{\mathrm{max}}$,
$i_{nk}^{\mathrm{max}}$ and the slack-bus voltage $v_{0}$.

By Theorem 2, under the infeasibility of $\mathrm{P2}$, if power injection
$(p_{1},...,p_{N})$ has a solution to (1)–(3), then this solution must be
unique.

######  Remark 2.

Due to the non-linear equality constraints, $\mathrm{P2}$ is not convex. This
means that the numerical solvers might not be able to directly check the
infeasibility of $\mathrm{P2}$. However, we know that $\mathrm{P2}$ is
infeasible if any convex relaxation of $\mathrm{P2}$ is infeasible. For
efficiency and practicality, we suggest to take the sparsity-exploiting SDP
relaxation in [1], which has already been successfully applied to the optimal
power flow problem in [2, 3, 4].

##  IV Proofs

###  IV-A Proof of Theorem 1

###### Proof.

Since $(v_{1}^{\star},...,v_{N}^{\star})$ satisfies (2)–(3), we only need to
prove that $(v_{1}^{\star},...,v_{N}^{\star})$ satisfies (1).

First, we show $\Big{(}v_{n}^{\star}\Big{)}^{2}=\alpha_{n}^{\star},~{}\forall
n\in\mathcal{N}^{PQ}$. Suppose that
$\Big{(}v_{\tilde{n}}^{\star}\Big{)}^{2}<\alpha_{\tilde{n}}^{\star}$ for some
$\tilde{n}\in\mathcal{N}^{PQ}$. As a result, we have

$\displaystyle\Big{(}\sum_{k\in\mathcal{K}(\tilde{n})}g_{\tilde{n}k}\Big{)}\Big{(}v_{\tilde{n}}^{\star}\Big{)}^{2}-\Big{(}g_{\tilde{n}0}v_{0}I_{\mathcal{K}(\tilde{n})}(0)\Big{)}v_{\tilde{n}}^{\star}$
$\displaystyle<$
$\displaystyle\Big{(}\sum_{k\in\mathcal{K}(\tilde{n})}g_{\tilde{n}k}\Big{)}\alpha_{\tilde{n}}^{\star}-\Big{(}g_{\tilde{n}0}v_{0}I_{\mathcal{K}(\tilde{n})}(0)\Big{)}v_{\tilde{n}}^{\star}$
$\displaystyle=$
$\displaystyle\sum_{k\in\mathcal{K}(\tilde{n})\setminus\\{0\\}}g_{\tilde{n}k}\beta_{\tilde{n}k}^{\star}+p_{\tilde{n}}~{}.$

Observe that the quadratic form
$\Big{(}\sum_{k\in\mathcal{K}(\tilde{n})}g_{\tilde{n}k}\Big{)}\Big{(}v_{\tilde{n}}^{\star}\Big{)}^{2}-\Big{(}g_{\tilde{n}0}v_{0}I_{\mathcal{K}(\tilde{n})}(0)\Big{)}v_{\tilde{n}}^{\star}$
is convex in $v_{\tilde{n}}^{\star}$. Moreover, this quadratic is
monotonically increasing with respect to $v_{\tilde{n}}^{\star}$, as

$\frac{g_{\tilde{n}0}v_{0}I_{\mathcal{K}(\tilde{n})}(0)}{2\sum_{k\in\mathcal{K}(\tilde{n})}g_{\tilde{n}k}}\leq\frac{v_{0}}{2}\leq
v^{\mathrm{min}}<v_{\tilde{n}}^{\star}~{}.$

Therefore, $v_{\tilde{n}}^{\star}$ can be further increased, which contradicts
the optimality of $v_{\tilde{n}}^{\star}$. Hence, we must have
$\Big{(}v_{n}^{\star}\Big{)}^{2}=\alpha_{n}^{\star},~{}\forall
n\in\mathcal{N}^{PQ}$.

Next, we show
$\Big{(}\beta_{nk}^{\star}\Big{)}^{2}=\alpha_{n}^{\star}\alpha_{k}^{\star},~{}\forall
n\in\mathcal{N}^{PQ},~{}k\in\mathcal{K}(n)$. Since $\beta_{nk}\geq
0,~{}\forall n\in\mathcal{N}^{PQ},~{}k\in\mathcal{K}(n)$, we could show
$\beta_{nk}^{\star}=v_{n}^{\star}v_{k}^{\star},~{}\forall
n\in\mathcal{N}^{PQ},~{}k\in\mathcal{K}(n)$. Suppose that
$\beta_{\tilde{n}\tilde{k}}^{\star}<v_{\tilde{n}}^{\star}v_{\tilde{k}}^{\star}$
for some $\tilde{n}\in\mathcal{N}^{PQ},~{}\tilde{k}\in\mathcal{K}(\tilde{n})$.
Then, we have

$\Big{(}\sum_{k\in\mathcal{K}(\tilde{n})}g_{\tilde{n}k}\Big{)}\Big{(}v_{\tilde{n}}^{\star}\Big{)}^{2}-\Big{(}g_{\tilde{n}0}v_{0}I_{\mathcal{K}(\tilde{n})}(0)+\sum_{k\in\mathcal{K}(\tilde{n})\setminus\\{0\\}}g_{\tilde{n}k}v_{k}^{\star}\Big{)}v_{\tilde{n}}^{\star}<p_{\tilde{n}}~{}.$

Observe that the quadratic form
$\Big{(}\sum_{k\in\mathcal{K}(\tilde{n})}g_{\tilde{n}k}\Big{)}\Big{(}v_{\tilde{n}}^{\star}\Big{)}^{2}-\Big{(}g_{\tilde{n}0}v_{0}I_{\mathcal{K}(\tilde{n})}(0)+\sum_{k\in\mathcal{K}(\tilde{n})\setminus\\{0\\}}g_{\tilde{n}k}v_{k}^{\star}\Big{)}v_{\tilde{n}}^{\star}$
is convex in $v_{\tilde{n}}^{\star}$. Moreover, this quadratic is
monotonically increasing with respect to $v_{\tilde{n}}^{\star}$, as

$\frac{g_{\tilde{n}0}v_{0}I_{\mathcal{K}(\tilde{n})}(0)+\sum_{k\in\mathcal{K}(\tilde{n})\setminus\\{0\\}}g_{\tilde{n}k}v_{k}^{\star}}{2\sum_{k\in\mathcal{K}(\tilde{n})}g_{\tilde{n}k}}<\frac{v^{\mathrm{max}}}{2}<v^{\mathrm{min}}<v_{\tilde{n}}^{\star}~{}.$

Therefore, $v_{\tilde{n}}^{\star}$ can be further increased, which contradicts
the optimality of $v_{\tilde{n}}^{\star}$. Hence, we must have
$\beta_{nk}^{\star}=v_{n}^{\star}v_{k}^{\star},~{}\forall
n\in\mathcal{N}^{PQ},~{}k\in\mathcal{K}(n)$. ∎

###  IV-B Proof of Theorem 2

###### Proof.

First, for all $n\in\mathcal{N}^{PQ}$, we have

$\displaystyle\Bigg{(}2\Big{(}\sum_{k\in\mathcal{K}(n)}g_{nk}\Big{)}v_{n}-\sum_{k\in\mathcal{K}(n)}g_{nk}v_{k}\Bigg{)}\gamma_{n}-\sum_{k\in\mathcal{K}(n)\setminus\\{0\\}}\Big{(}g_{nk}v_{n}\Big{)}\gamma_{k}$
$\displaystyle=$
$\displaystyle\frac{\partial\Bigg{(}\Big{(}\sum_{k\in\mathcal{K}(n)}g_{nk}\Big{)}v_{n}^{2}-\Big{(}\sum_{k\in\mathcal{K}(n)}g_{nk}v_{k}\Big{)}v_{n}\Bigg{)}}{\partial\Big{(}v_{1},...,v_{N}\Big{)}}\begin{pmatrix}\gamma_{1}\\\
\vdots\\\ \gamma_{N}\end{pmatrix}$

where $\displaystyle\frac{\partial}{\partial(v_{1},...,v_{N})}$ is the
$1\times N$ Jacobian.

Let the $N\times N$ Jacobian matrix of equation system (1) be

$\mathbf{J}(v_{1},...,v_{N})=\begin{pmatrix}\frac{\partial\Bigg{(}\Big{(}\sum_{k\in\mathcal{K}(1)}g_{1k}\Big{)}v_{1}^{2}-\Big{(}\sum_{k\in\mathcal{K}(1)}g_{1k}v_{k}\Big{)}v_{1}\Bigg{)}}{\partial\Big{(}v_{1},...,v_{N}\Big{)}}\\\
\vdots\\\
\frac{\partial\Bigg{(}\Big{(}\sum_{k\in\mathcal{K}(N)}g_{Nk}\Big{)}v_{N}^{2}-\Big{(}\sum_{k\in\mathcal{K}(N)}g_{Nk}v_{k}\Big{)}v_{N}\Bigg{)}}{\partial\Big{(}v_{1},...,v_{N}\Big{)}}\end{pmatrix}.$

Then, it can be seen that $\mathbf{J}(v_{1},...,v_{N})$ is non-singular in
$\\{v_{n},n\in\mathcal{N}^{PQ}:\mathrm{inequalities~{}\eqref{eq:security1}-\eqref{eq:security2}}\\}$
whenever the optimization $\mathrm{P2}$ is infeasible.

Second,
$\\{v_{n},n\in\mathcal{N}^{PQ}:\mathrm{inequalities~{}\eqref{eq:security1}-\eqref{eq:security2}}\\}$
is convex, since (2)–(3) are linear constraints on $v_{1},...,v_{N}$.

By Theorem 2 in [5], we know that there exists at most one solution to (1)–(3)
if $\mathrm{P2}$ is infeasible. ∎

## References

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  * [3] B. Ghaddar, J. Marecek, and M. Mevissen, “Optimal power flow as a polynomial optimization problem,” _IEEE Transactions on Power Systems_ , vol. 31, no. 1, pp. 539–546, Jan. 2016.
  * [4] D. K. Molzahn and I. A. Hiskens, “Sparsity-exploiting moment-based relaxations of the optimal power flow problem,” _IEEE Transactions on Power Systems_ , vol. 30, no. 6, pp. 3168–3180, Nov. 2015.
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Through energy droughts: Hydropower’s ability to sustain a high output

Hanna Ek Fälth 1,3 ✉, Fredrik Hedenus 1, Lina Reichenberg 1, Niclas Mattsson 2
1 Department of Space, Earth and Environment, Division of Physical Resource
Theory, Chalmers University of Technology, Gothenburg, Sweden
2 Department of Space, Earth and Environment, Division of Energy Technology,
Chalmers University of Technology, Gothenburg, Sweden
✉<EMAIL_ADDRESS>
### Summary

Previous research has raised concerns about energy droughts in renewables-
based energy systems. This study explores the ability of reservoir hydropower
to sustain a high output and, thereby, mitigate such energy droughts. Using
detailed modelling, we estimate that Swedish hydropower can sustain 67–92% of
its installed capacity for 3 weeks, with higher values possible in springtime.
The variation of the sustained output, equivalent to the capacity of 3–4
Swedish nuclear reactors, under-scores the importance of understanding the
potential output levels when devising strategies to counteract energy
droughts. Moreover, we find that regulations imposed on the flows in river
bottlenecks hinder higher sustained output levels. With the upcoming renewal
of environmental permits for hydropower plants in Sweden, these findings
provide valuable insights for policymakers. Furthermore, the sustained output
capabilities demonstrated in this study challenge the prevalent simplified
representations of hydropower in energy models, suggesting a need for more-
sophisticated modelling approaches.

### Keywords

Hydropower, Modelling, Energy systems, Energy droughts, Dunkelflaute event,
Resilience, Sustained output, Renewables

##  1 Introduction

Variation management strategies are essential in renewables-based energy
systems to ensure that the demand can be met at all times despite variations
in wind and solar power production. Energy storage, expansion of transmission
grids, demand-side management, and dispatchable generation technologies are
the key strategies discussed in the literature [1, 2, 3, 4, 5, 6, 7, 8, 9].
Furthermore, reservoir hydropower has been argued to provide flexibility in
renewables-based systems [10, 11, 12, 13, 14, 15, 16]. In this study, we
analyse the ability of reservoir hydropower to provide capacity during a
sustained period of low availability of supply relative to demand, a so-called
_energy drought_.

In contrast to many other variation management strategies, hydropower has been
used in power systems for decades, and has long been recognised for its
operational flexibility. In Sweden, hydropower has contributed significantly
to system flexibility by providing diurnal production to follow load, seasonal
storage capabilities, and grid stability. However, the shift toward renewable
energy systems could alter the way in which hydropower is optimally utilised.
A recent study by [17] has revealed a notable shift in hydropower’s
operational dynamics, in that the conventional daily production cycle is
becoming less pronounced, influenced by the increasing integration of wind
power into the electricity system. This trend suggests a re-evaluation of
hydropower’s role, which may be transitioning from its traditional focus on
meeting intra-day demand fluctuations to accommodating the variability of
renewable energy sources on different time-scales.

Energy systems that have a large share of renewables face numerous challenges
related to variability, ranging from short-term issues such as maintaining
frequency control and adapting to hourly load changes to persistent supply
shortages due to factors such as low-wind periods or technology failures.
Researchers have explored hydropower’s role in addressing these challenges
across different time-scales. [15] have developed a framework to assess
hydropower’s potential for enhancing short-term grid resilience after a
disturbance, highlighting reservoir hydropower’s critical role in managing
disruptions. [18] have quantified the quality of short-term regulation of
hydropower and the burden placed on generation equipment, and they have also
evaluated burden relief strategies under different future variable renewable
energy (VRE) scenarios. Extending the research to intra-day variations, [16]
have examined the capacity of a cascaded reservoir hydropower system to meet
daily demand peaks under various operational constraints, and they have
identified some key factors influencing intra-day flexibility. However,
hydropower’s potential to mitigate week-long energy droughts is poorly
explored. Such droughts can result from, for instance, weather phenomena [19,
20, 21, 22, 23, 24], technical failures in transmission lines [25, 26], and
emergency shutdowns of nuclear power plants [27, 28]. This study aims to fill
this knowledge gap by evaluating hydropower’s ability to sustain a high output
over extended periods, thereby assessing its capacity to counteract energy
droughts.

Historical hydropower production data do not show instances of hydropower
operating at near-maximum capacity for week-long periods. This could be due to
either inherent operational limitations of the hydropower systems or the lack
of economic incentives in the past. Therefore, rather than relying solely on
historical data, employing a model becomes crucial to understanding
hydropower’s ability to sustain high output levels over longer periods of time
with strong incentives. However, the energy systems models used to date often
simplify the representation of hydropower, neglecting key elements such as
river network effects, water delay times, and head-dependent production,
thereby over-estimating the flexibility of hydropower [29]. Both the
International Energy Agency (IEA) and the National Renewable Energy Laboratory
(NREL) advocate for more-detailed modelling that can accurately assess
hydropower’s flexibility [30, 31]. Overly simplified hydropower
representations imply that hydropower can operate continuously at maximum
output for as long as water is available in the reservoirs. However, is that
the case in reality? This study aims to answer this question by employing a
detailed model to assess hydropower’s sustained output capabilities and to
offer insights into its potential to mitigate the impacts of energy droughts.

The importance of such an analysis extends globally, especially to regions
that are heavily dependent upon reservoir hydropower, like South America,
Canada, China, Central Africa, and parts of the USA and Europe. As emphasised
above, a detailed representation of the hydropower system is essential for
accurate evaluation of its flexibility. Thus, this paper focuses on Sweden as
a case study, leveraging a detailed model of its hydropower infrastructure to
examine hydropower’s ability to sustain a high level of output for 1–3 weeks.
The installed hydropower capacity in Sweden is about 16 GW, and in 2022 about
70 TWh of the Swedish electricity generation originated from hydropower,
corresponding to 41% of the total electricity production [32]. Our model
covers 92% of the hydropower capacity in Sweden, encompassing nine major
rivers and incorporating around 240 reservoirs and power plants, each of which
is equipped with 1–10 turbines. Further details of the model and coverage are
provided in the Experimental procedure section.

The methodology that we use to represent hydropower in a detailed manner is
derived from the approach outlined previously by [29]. Briefly, it is an
optimisation model that maximises profit for a set of hydropower plants
subjected to deterministic spot prices over a full year. This model represents
each component of the river systems, including reservoirs, flow paths with
their respective flow times, and individual turbines with their respective
performance curves. The head dependency of hydropower production is modelled,
and environmental constraints, such as minimum flows and seasonally dependent
limits on water levels, are also represented. Notably, the spillage
representation is improved for this study to depict more accurately the
bottlenecks in the rivers, which are shown to heavily influence hydropower’s
ability to sustain output.

To assess the ability of hydropower to sustain a high output during energy
droughts, we simulate high-demand conditions by introducing constant, high
market prices for periods of 1–3 weeks into the historical price profiles.
This method allows us to mimic the economic signals that would trigger
hydropower plants to operate at increased output levels, mimicking real-world
energy droughts. By analysing the hydropower output during these high-price
periods, we can enhance our understanding of hydropower’s potential role in
addressing energy system challenges, such as prolonged energy shortages.
Initially, we investigate the potential to sustain a high output and how it is
affected by the upper limits imposed on flows in the river. Subsequently, we
map the seasonal patterns to explore the relationship between sustained output
variations and energy drought occurrences.

##  2 Measuring sustained output

To address energy droughts effectively, we require technologies that can meet
the demand throughout these critical periods. This study introduces two
metrics to evaluate the ability to sustain a high output: Sustained capacity
and Sustained production.

  * •

Sustained capacity is defined as the consistent power output that hydropower
plants can guarantee throughout a specified period. This metric is measured as
a percentage of the maximum capacity of the plants.

  * •

Sustained production is defined as the total electrical energy generated
during a specific period. This metric is expressed as the percentage of the
maximum possible production, i.e., the total production from hydropower plants
running at maximum capacity throughout this period.

Figure 1 shows the hydropower production profile over 20 days with a simulated
14-day energy drought in the middle, and shows the metrics of Sustained
capacity and Sustained production applied to this example.

Figure 1: Illustration clarifying the definitions of the metrics of Sustained
capacity and Sustained production. Shown are the levels of hydropower
production over a 20-day period, and the metrics during a simulated 14-day
energy drought.

##  3 Results & Discussion

###  3.1 Hydropower capacity can be sustained at high levels over several
weeks

Our findings reveal that hydropower can sustain high production levels over
several weeks. Under current infrastructure and regulatory conditions, the
Sustained capacity of Swedish hydropower ranges from 67% to 96%, depending on
the time of year and the durations of high-price periods. On average, across
all months spanning the period of 2016–2019, the Sustained capacity for a
1-week period was 84%, decreasing to 78% for a 3-week period. Regardless of
the season or year, Swedish hydropower consistently maintained at least 67% of
its total capacity for up to 3 weeks. These results are illustrated in Figure
2 for the Present regime scenario, which showcases hydropower’s Sustained
capacity over one to three consecutive weeks. The results obtained for
Sustained production are very similar to those for Sustained capacity and are,
therefore, not shown here, although they are provided in Figure A.1 in the
Supplementary material. Furthermore, the results for the sustained output
divided according to Nord Pool price areas in Sweden are presented in Figures
A.2 and A.3 in the Supplementary material.

Figure 2: An illustration of the Sustained capacity of hydropower over a
period of one to three consecutive weeks of high electricity prices for three
different operational scenarios. Each violin shows the results for each month
over four different years. The dots represent the mean values, while the lines
extending from the dots indicate the 25th to 75th percentile range. The color-
coding corresponds to the following operational scenarios: Present regime with
current regulations and infrastructure (green); Reduced bottlenecks with a
higher upper limit on spillage in bottlenecks (blue); and Unrestricted, with
unrestricted spillage (red). The different scenarios highlight the influences
of operational constraints on the ability of hydropower to sustain a high
output during energy droughts. The results shown are based on approximately
6,500 model runs. Refer to the Experimental procedure section for details of
the test set-up.

Further examination of Figure 2 indicates that the variation in hydropower’s
ability to sustain output over 1 week versus 3 weeks is less significant than
the variation observed across different months and years, as highlighted by
the magnitudes of the variability within each violin. This suggests that the
timing of energy droughts plays a more critical role than drought duration in
determining hydropower’s capacity to maintain output during these periods.

We have modelled and analysed 92% of Sweden’s installed hydropower capacity,
focusing on the largest and most flexible river systems. Since the remaining
capacity primarily consists of plants with lower levels of operational
flexibility, the results for sustained output should not be extrapolated to
this remaining capacity. It is also worth noting that we assumed that all
power plants would be available for production at all times. In reality, owing
to planned and unplanned shutdowns for maintenance and failures, the actual
sustained production may occasionally be lower than is indicated by our
results.

Furthermore, the unpredictable natures of the inflow and prices pose
challenges for production planning, limiting the flexibility of hydropower
compared to the deterministic model’s idealised approach used in this study.
For instance, deterministic models may completely empty or fill reservoirs
with confidence because they know precisely when new water will arrive and
when a high-price period will appear. This cannot be accomplished in practice,
as it may lead to exceeding the upper and lower bounds of reservoir content
due to incorrect inflow forecasts. We introduced the high-price period only 1
week after the initial modelled hour, to reflect more accurately the actual
conditions and to reduce the over-estimation of flexibility from the
deterministic model. This adjustment restricts the model’s ability to optimise
the water levels before encountering the high-price period. Nevertheless, even
with this modelling approach, using a deterministic model still somewhat over-
estimates the possibilities for hydropower to sustain output.

###  3.2 Adjusting the allowed water flows at bottlenecks enhances hydropower
output during energy droughts

The current regulatory regime for hydropower production in Sweden includes
limitations on allowed water flow rates in different sections along the river.
The regulations generally permit operators to run water through or bypass
power plants up to the maximum flow capacity of the installed turbines, except
during high-inflow periods when more-significant water flow rates are allowed
or forced. Bottlenecks arise when the maximum flow capacity of the turbines in
a power plant is lower than that of plants upstream or downstream, thereby
impeding water transfer. For more details on bottlenecks, see the Experimental
procedure section.

To explore the impacts of bottlenecks on sustained output, we examined two
alternative scenarios. First, we analysed an Unrestricted scenario, in which
the upper limits on water flow rates were removed at each hydropower plant,
while maintaining maximum turbine flows, thereby effectively increasing the
spillage limits. Allowing unrestricted spillage throughout the river could
raise the sustained output to ¿95% on average for periods of up to 3 weeks
(illustrated by the red violins in Figure 2), as compared to 77% under current
regulatory regimes (green violins in Figure A.2). However, unrestricted
spillage entails severe risks, such as flooding and erosion. Although
unrealistic, this scenario indicates the impact of current water flow rate
limits on sustained output and serves as an upper bound on the possibilities
for sustained output.

Second, we evaluated a more targeted approach that allowed water flows to
exceed the turbine flow capacities at identified bottleneck sites. In this
scenario, which we refer to as the Reduced bottlenecks scenario, we raised the
flow limits in approximately 30% of the power plants to match the maximum
turbine flow rates of the upstream turbines, effectively increasing their
water flow rate limits by an average of 20%. The adjustments to the allowed
water flows were made without increasing the maximum turbine flow capacities,
thus only increasing the spillage limits. When we compared these revised flow
rate limits to the maximum recorded inflow for each plant (including local and
upstream contributions) during the period of 2016–2020, the limits increased
from an average of 18% to 22% of the maximum inflow. Similarly, when compared
to the mean inflow during the same period, the spillage limits increased from
an average of 163% to 194% of the mean inflow. (See the Experimental procedure
section for more details on how and why flow limits were expanded at
bottlenecks). In the Experimental procedure section, adjustments to the flow
rate limits, as shares of both the maximum and average inflows, are detailed
in Figure 7.

In this Reduced bottlenecks scenario, Sustained capacity averages 90% for 1
week and 85% for 3 weeks (blue violins in Figure 2). This marked improvement
over the Present regime scenario demonstrates that adjusting spillage
regulations at bottlenecks notably enhances hydropower’s performance during
energy droughts. Nevertheless, while the benefits of increasing flow limits
are clear, they must be carefully balanced against potential environmental and
social impacts, which could have negative effects on the river ecosystems and
the communities that depend on them.

Our analysis did not capture all the complexities related to spillage, since
we were constrained by data limitations. Each power plant operates under
individual permits that regulate releases of water. We generalised these
regulations by setting the upper limits on water flows through the turbines or
in spillways to be equal to the maximum installed turbine flow. This
assumption is based on consultations with experts from power production
companies and water regulation authorities. Furthermore, water flows are
restricted during wintertime due to ice formation in rivers and the risk of
mechanical issues, likely resulting in a reduced ability to sustain output
during the colder months. Given the generalisation of permits and the omission
of the ice-related problems, our results should be viewed as indicative of the
potential for sustained high output in Swedish hydropower rather than
definitive calculations.

Furthermore, it is important to note that achieving sustained high-level
production in rivers fundamentally depends on maintaining high flow rates
throughout the entire river system. Our approach focused on increasing the
spillage limits to improve flow at bottlenecks. Another effective solution
could be to expand the turbine flow capacity by installing additional turbines
at these critical points. Such an expansion would enhance the river system’s
ability to sustain high output and increase capacity while avoiding the
increased annual energy losses associated with higher spillage.

###  3.3 Higher sustained hydropower capacity can be obtained during periods
of high natural inflow and high reservoir levels

Our findings reveal notable variability in the sustained output across the
Present regime and Reduced bottlenecks scenarios, as illustrated by the green
and blue violins, respectively, in Figure 2. Figure 3 shows the 1-week
Sustained capacity by month for three operational scenarios: Present regime,
Reduced bottlenecks, and Unrestricted. The Unrestricted scenario consistently
demonstrates sustained capacities ¿98%, except from March to May. This
decrease can be explained by current operational practices, whereby reservoir
levels are typically at their lowest before the spring run-off due to
intentional winter draw-downs to meet the high demand and to accommodate
anticipated spring recharge. Consequently, diminished reservoir volumes limit
the capacity for full output over 1 week, and reduced head heights in plants
with significant reservoir level variability decrease the energy potential,
thereby affecting the power output. This also explains the downward trend in
Sustained capacity observed from November to March in both the Present regime
and Reduced bottlenecks scenarios (Figure 3).

In addition, Figure 3 charts the total natural inflow to the modelled rivers
over the year as a percentage of the maximum observed, illustrating the
correlation between inflow and Sustained capacity. A fluctuating inflow
throughout the year affects the permitted spillage levels, as high-level
spillage is only allowed during periods of naturally high inflow, such as
during snowmelt or consistent rainfall, as detailed in the Experimental
procedure section. Consequently, the potential for high sustained output is
greater during these high-inflow periods, as is evident in both the Present
regime and the Reduced bottlenecks scenarios. This pattern arises from the
design of the spillage constraints; natural inflow typically exceeds turbine
capacity mainly in the springtime and occasionally in autumn, dictating
spillage limits only during these periods. Allowing higher flows leads to a
higher Sustained capacity because it enables water to bypass bottlenecks, and
downstream plants can maintain high production levels using water from
upstream reservoirs. For the same reasons, months with higher variability in
terms of inflow also exhibit greater variability of Sustained capacity.
Notably, the sustained output in April appears low despite there being a high
inflow. This discrepancy is due to two factors: 1) the peak inflows typically
occur at the end of April, whereas we introduced the high price period in the
middle of the month; and 2) as discussed above, reservoir levels are generally
lower before the spring run-off due to the high demand during wintertime and
the anticipated recharge.

Figure 3: A comparison of the monthly Sustained capacity of Swedish hydropower
under three operational scenarios and the natural inflow throughout the year,
illustrating how today’s hydrological cycle influences hydropower’s ability to
sustain a high output. The Sustained capacity values under the Present regime
(green), Reduced bottlenecks (blue), and Unrestricted (red) scenarios are
shown in box plots, which display the range and mean values for each month.
The line graph represents the total inflow as a percentage of the maximum
recorded. The shaded area behind the line graph indicates the variability of
the inflow for the modelled period of 2016–2020.

Thus, two primary factors determine the ability of hydropower to sustain a
high hydropower output: 1) the water levels in the reservoirs; and 2) more
critically, the permitted flow rates. These factors contribute to significant
variability in hydropower’s ability to sustain a high output throughout the
year and across different years, as illustrated in Figure 3. However, it is
important to consider that future changes in inflow patterns driven by climate
change, as well as adjustments made to reservoir management in response to
these inflows and evolving energy demands, will alter the yearly differences
in hydropower’s potential to sustain output.

###  3.4 High Sustained capacity creates annual energy losses

Achieving a high sustained hydropower capacity inevitably reduces the annual
energy output, primarily due to increased spillage at bottlenecks and, to a
lesser extent, reduced turbine efficiencies at maximum flows. Figure 4
illustrates the annual production losses that are entailed by a high Sustained
capacity, contrasting the outputs from scenarios involving elevated market
prices with scenarios that have historical price levels. In the Unrestricted
scenario (Figure 2), sustaining an average capacity of 96%–98% results in a
loss of about 0.8% of the annual production for each week of sustained high
output, as depicted by the red violins in Figure 4. Meanwhile, with the
current regulations and infrastructure, an average Sustained capacity of
78%–84% is achievable (as shown in the Present regime scenario in Figure 2),
with a corresponding lower loss of about 0.2% of yearly production per week.

Figure 4: Losses in yearly production associated with sustaining high output
levels, comparing the yearly production obtained with elevated market prices
with the yearly production obtained with historical price levels. Each violin
represents the results for each month over 5 different years. The dots
represent the mean values, while the lines extending from the dots indicate
the 25th to 75th percentile range. The color-coding corresponds to the
following operational scenarios: Present regime, with the current regulations
and infrastructure (green); Reduced bottlenecks, with a higher upper limit on
spillage in bottlenecks (blue); and Unrestricted, with unrestricted spillage
(red).

In summary, sustaining a high output from hydropower, so as to counteract
energy droughts, leads to some energy losses, amounting to 0.2%–0.8% of annual
production for each week of sustained output. This should be considered within
the broader context of energy system needs. While losing renewable energy
might appear to be negative in terms of the pursuit of a transition to a
carbon-neutral energy system, the flexibility provided by hydropower is
critical for renewables-based energy systems. It is generally more challenging
to find renewable sources that offer this level of flexibility than to
generate renewable energy in bulk.

###  3.5 What does hydropower’s Sustained capacity, as measured in this study,
imply for the management of energy droughts?

Our analysis demonstrates that Swedish hydropower can deliver between 67% and
96% of its installed capacity during critical periods of 1–3 weeks. This
capability under-scores the significant potential of hydropower to ensure the
energy supply during multi-week energy droughts. However, evaluating the
adequacy of this sustained output requires the consideration of several key
factors.

First, the relevance of sustained hydropower output depends on its share of
the energy mix. In Sweden, the lowest observed sustained output of 67%,
approximately 10 GW, can fulfil about 40% of the peak electricity demand and
68% of the average electricity demand. At the higher end, a Sustained capacity
of 96%, equivalent to 14.5 GW, could meet almost 60% of the peak demand and
almost 100% of the average electricity demand. These values highlight the
significant role that hydropower plays in enhancing the resilience of a
renewable energy system, although they also point to the need for additional
energy sources during severe energy droughts. The broad range of observed
sustained outputs, with a 4.5 GW difference between the highest and lowest
outputs – equivalent to the capacity of three to four Swedish nuclear reactors
– underlines the need to evaluate carefully hydropower’s sustained output
capabilities when developing strategies to counteract energy droughts.

Moreover, Sweden’s energy system’s interconnection with the rest of Europe
adds complexity to the assessment. While this interconnectivity often acts as
a buffer against local shortages by enabling trade in energy, it also poses
risks when energy droughts occur simultaneously across the continent, thereby
amplifying the demand for reliable back-up solutions.

Second, the timing and seasonality of the hydropower output relative to energy
droughts are crucial. Energy droughts are more likely to occur during the
wintertime in today’s European electricity system [22, 20, 23], and this trend
may intensify with climate change [24]. Moreover, historical data indicate
that the winter months (December to February) are the periods of peak demand
in Sweden [33], due to the use of electric heating. Our findings reveal that
the ability of Swedish hydropower to sustain a high output is weakest during
the winter months (November to March). This observation aligns with periods
identified in recent literature as most susceptible to energy droughts.
Unfortunately, this means that the seasons with the highest risk of energy
shortages coincide with the periods during which there is a reduced capacity
for hydropower to deliver high outputs.

###  3.6 Understanding hydropower’s role in future energy systems

While our study does not directly evaluate the role of hydropower in future
energy systems by integrating a detailed representation within an energy
system model, it offers valuable insights that can facilitate such assessments
in future research. Our findings can be used to establish more realistic
assumptions regarding hydropower’s ability to sustain a high output, compared
to those used in the simplified representations that are commonly found in
existing energy system models.

Energy system model studies have shown that reservoir hydropower can provide
important flexibility in renewables-based systems [10, 11]. However, these
models often assume that hydropower’s Sustained capacity is 100% as long as
there is water in the reservoirs. Our research indicates that the actual
capacity to sustain high output can sometimes be as low as 67%. Models that
over-simplify hydropower’s capabilities may thus overestimate the resilience
provided by hydropower and underestimate the need for additional assets that
can provide backup during energy droughts.

In contrast, some models that explore renewable energy systems use historical
hydropower data to parameterise their representations. Such data have not
historically shown the high sustained outputs that our study suggests are
possible, due to the lack of economic incentives for such performance in
previous energy systems. Therefore, relying on historical data to predict
hydropower’s operation in future energy systems will likely under-estimate its
flexibility.

The influences of economic incentives on sustained output are significant, as
sustaining a high output involves a trade-off between maximising output during
high-price periods and saving water for hours with lower prices. In addition,
as we have shown, sustaining a high output entails a reduction of annual
production, adding to this trade-off. We conducted a sensitivity analysis to
determine whether the price level used in our study (5,000 SEK/MWh,
approximately 430 €/MWh) effectively encourages maximising the sustained
output during high-price periods. This analysis demonstrated that even at a
substantially higher price of 50,000 SEK/MWh, the sustained output did not
increase, thereby affirming that our chosen price point efficiently motivates
Swedish hydropower plants to reach maximum output during these critical
periods, given that prices during the remainder of the year align with the
historical levels recorded from 2016 to 2020. Details of this sensitivity
analysis are depicted in Figure A.6 in the Supplementary material.

Flow rate limitations in the rivers are crucial when considering periods of
elevated market prices. Interestingly, limits on flow rates had a marginal
impact when only historical prices were considered, as in the past there were
limited incentives for hydropower plants to sacrifice water to sustain high
output. This represents a critical take-away message with respect to the use
of historical data for model validation: factors that may seem to be
irrelevant under the conditions of current energy systems and price structures
could take on importance as new incentives emerge in future energy systems.

###  3.7 Policy implications and future research

Sweden is initiating a comprehensive process of reissuing environmental
permits for approximately 2000 hydropower plants across the country, starting
in Year 2024 [34, 35]. Our study highlights the critical role of bottlenecks
in influencing hydropower’s ability to sustain high output and, consequently,
the hydropower sector’s capacity to mitigate energy shortages during periods
of low wind or other critical phases for the energy system. While discussions
surrounding the new environmental permits often revolve around minimum flows,
the present study underlines the importance of considering upper flow limits.
We highlight the potential to enhance hydropower’s ability to sustain high
output through updates to environmental permits, particularly by allowing
increased spillage or maximum turbine flows. However, these considerations
must be carefully balanced against the risks for significant environmental
consequences.

Given that hydropower constitutes a significant share of the energy system in
many parts of the world, it is important to understand whether hydropower can
sustain output at, for example, 50% or close to 100%, so as to evaluate the
need for other types of back-up assets to ensure the supply during energy
droughts. However, generalising the results of this study to other hydro-rich
nations requires careful consideration, as each region presents unique
technical and environmental challenges that affect the ability to sustain high
output levels. Therefore, further studies of other regions are warranted to
generate more robust conclusions about the broader applicability of our
findings.

##  4 Experimental procedure

###  4.1 Resource availability

####  4.1.1 Lead contact

For inquiries regarding the paper’s content, data, and model, please contact
the lead author: Hanna Ek Fälth (hanna.ek.falth@chalmers.se).

####  4.1.2 Materials availability

No materials were used in this study.

####  4.1.3 Data and code availability

The model, written in Julia, utilised for the analysis in this study is
available on GitHub [36]. Some of the data used as model input for this study
could not be shared due to the confidentiality of the data. However,
alternative data that can be used to run the model are provided. Licensing
information can be found within the repository.

###  4.2 Model overview

The goal of this work was to determine the maximum possible capacity that
hydropower plants within a river and across multiple rivers can sustain
simultaneously for 1–3 weeks in the presence of strong economic incentives. To
this end, we conducted stress tests using a detailed hydropower optimisation
model that maximises profits [36]. These stress tests involved modifying the
historical price profiles to include constant, high market prices over one,
two or three consecutive weeks, simulating the economic signals that would
prompt hydropower plants to operate at high output levels, akin to real-world
scenarios of energy droughts.

Our modelling approach is based on the detailed hydropower model for rivers
with multiple hydropower plants that has previously been comprehensively
described as model B:L [29]. This linear optimisation model maximises profit
for a set of hydropower plants throughout the year, with an hourly resolution,
given deterministic inflows and spot prices for each river system. The model
represents all the reservoirs, waterways, flow times, and individual turbines
at each power plant. Turbine efficiencies are modelled using piece-wise linear
approximations of actual non-linear efficiency curves, while the head-
dependent production is linearised via a Taylor approximation. Furthermore,
the model incorporates environmental constraints, such as minimum flow
requirements and seasonally dependent minimum and maximum water levels. To
ensure the accuracy of the linearised model for evaluations of sustained
output, we conducted a sensitivity analysis using a full non-linear model
(described as model A in [29]). This sensitivity analysis confirmed that the
linearised model used in the present study accurately replicates the results
regarding the ability to sustain high output levels, as compared to using a
full non-linear model.

In addition to the technical features in the original model described in [29],
we implemented more realistic constraints on the allowed water flow at each
hydropower plant. These constraints are dictated by technical capacities and
ecological and social considerations, thus enhancing the model’s realism.
Moreover, as demonstrated in this study, the constraints on water flow rates
significantly impact the potential for sustained hydropower output,
underscoring their importance in accurately addressing the main research
question. Further details of these water flow limitations are presented in the
section on Water flow limitations and scenarios.

We have modelled hydropower in nine major rivers in Sweden111Luleälven,
Skellefteälven, Umeälven, Indalsälven, Ångermanälven, Ljungan, Ljusnan,
Dalälven and Göta älv (Klarälven + the stretch below Vänern), accounting for
about 15 GW or 92% of the nation’s hydropower capacity, as reported in the
Year 2022 official statistics [32]. Table 1 presents the geographic coverage
of the modelled hydropower, segmented by Nord Pool price area.

| Total | SE1 | SE2 | SE3 | SE4
---|---|---|---|---|---
Share [%] | 92 | 98 | 99 | 72 | 0
Modelled capacity [GW] | 15.13 | 5.24 | 7.99 | 1.90 | 0
Installed capacity [GW] [32] | 16.40 | 5.36 | 8.08 | 2.64 | 0.32
Table 1: Modelled shares of Swedish hydropower capacity, detailing the
percentages of total installed capacity in Sweden, both overall and according
to Nord Pool price area.

###  4.3 Data overview

For the parameterisation of our model across the nine rivers, we employed an
extensive data-set spanning from Year 2016 to Year 2020. This data-set
includes the hourly production levels, water levels at both the intake and
outlet, turbine discharge rates, and spillage across the 240 reservoirs and
power plants included in our analysis. All the companies that own these
facilities provided the data under confidentiality agreements.

The input data for the parameterised model include the spot prices for each
bidding area from Year 2016 to Year 2020, sourced from Nord Pool, and the
hourly site-specific water inflow data for all rivers, as provided by the
Swedish Meteorological and Hydrological Institute (SMHI) and the Water
Regulation companies (Vattenregleringsföretagen). In addition, the hourly
intake water levels were used to determine the start and end reservoir levels
for each model run.

###  4.4 Test design

####  4.4.1 Representing energy droughts

To evaluate the ability of hydropower to sustain high outputs during periods
of energy droughts, we simulated conditions of high demand by incorporating
constant high market prices into historical price profiles. This technique
enables us to emulate the economic signals that would prompt hydropower plants
to operate at increased output levels, mirroring real-world scenarios of
energy droughts.

We supplied the detailed optimisation model, previously described, with
modified historical price profiles from 2016 to 2020. These profiles were
adjusted to include one, two or three consecutive weeks with a constant high
price of 5,000 SEK/MWh (approximately 430€/MWh). This pricing level was
selected as a substantial incentive for high production levels; for reference,
the peak price on the Swedish spot market in Year 2023 was 3,760 SEK/MWh.
Figure 5 demonstrates an example of a 1-week, high-price period introduced in
January 2020 for the SE3 area.

We conducted a sensitivity analysis to confirm that the price level of 5,000
SEK/MWh effectively motivates maximum sustained output during these high-price
intervals. The findings, detailed in Figure A.6 in the Supplementary material,
reveal that even a significantly higher price of 50,000 SEK/MWh did not
increase the sustained output. This result confirms that our chosen price
level effectively encourages Swedish hydropower plants to maximise their
outputs during critical high-demand periods, assuming that prices during the
remainder of the year align with the historical levels recorded between 2016
and 2020.

Figure 5: Example of a price profile with an introduced high-price period of 1
week in region SE3 in January 2020.

We utilised a model that has perfect foresight, incorporating deterministic
water inflows and spot prices. To reduce the risk of over-estimating the
sustained outputs – a consequence of using a model that is capable of
precisely planning water releases and levels with complete foresight – we
strategically introduced the high-price period just 1 week after the initial
modelled hour. This approach restricts the model’s ability to optimise water
levels in advance, thus enhancing the realism of our results for sustained
output.

####  4.4.2 Model runs

To investigate how timing influences the sustained output capabilities of
Swedish hydropower, we carried out an extensive series of model runs. Across
each of the nine rivers, we executed 36 full-year model runs for each year
between 2016 and 2019. These 36 runs (3 x 12) consisted of: one run with one
high-price week; one run with two high-price weeks; and one run with three
high-price weeks, repeated for each month. Thus, we modelled the period of
January 2016 to January 2017 with a high-price period in January of one, two,
and three consecutive weeks separately, and replicated this for all months and
years up to December 2019 to December 2020. This methodological approach
enabled us to assess how the timing of energy drought periods affects
sustained production levels, as influenced by variations in inflow and initial
reservoir levels.

To evaluate the potential losses in annual hydropower production due to a
sustained high output, we conducted additional model runs using historical
spot prices as baseline cases. These base cases consisted of 12 full-year
model runs for each year between 2016 and 2019 (one for each month) to match
the high-price week runs. These model runs allowed us to compare the annual
production levels from the base case runs with those from the high-price
periods, thereby quantifying the loss in annual production resulting from a
sustained high output.

Maximum turbine capacities are not a direct input parameter in our model; they
are inferred from the maximum turbine flows, varying head levels derived from
the intake and outlet levels, and the configuration of the turbine efficiency
curves. To determine the maximum capacity for each river, we ran the model
with a high-price period that lasted just 1 hour and permitted any optimal
initial reservoir levels. Establishing the maximum capacity enabled a
meaningful comparison between the Sustained capacity during high-price weeks
and the maximum capacity.

Furthermore, to examine how regulations imposed on water flow limitations
impact the ability to sustain a high output, we analysed three scenarios
concerning allowed water flows, as further explained in the Water flow
limitations and scenarios section.

In total, we conducted close to 6,500 model runs, considering five different
price profiles (1-, 2-, and 3-week high-price runs, base case runs, and
maximum capacity runs) for each year, for 4 years, across 12 months and nine
rivers for three different spillage scenarios.

###  4.5 Water flow limitations and scenarios

In Sweden, each hydropower plant operates under a permit that specifies the
allowed water flows. These water flow limitations can create bottlenecks at
specific points along the river, significantly affecting the abilities of the
hydropower plants in that river to sustain high outputs, as demonstrated by
the findings of this study. This section provides an overview of how we define
river bottlenecks and how they impact the ability to sustain high output
levels. It further explains how water flow limitations were integrated into
our model and details the three scenarios analysed in this study.

####  4.5.1 Bottlenecks affecting the ability of hydropower to sustain a high
output

River flow rates typically increase as one moves downstream, due to the
convergence of tributaries as the river progresses towards the sea. However,
the larger reservoirs in Swedish river systems, which serve as substantial
seasonal water storage facilities, are predominantly located upstream. In
contrast, some downstream reservoirs can only hold enough water for a few days
of full-capacity generation. Therefor, sustaining high production levels at
downstream plants for prolonged periods requires the release of water from
these larger upstream reservoirs, so as to compensate for the limited
capacities of smaller downstream reservoirs.

Bottlenecks arise where the allowed water flow rate of one plant is lower than
others upstream or downstream, impeding water transfer. Figure 6 conceptually
illustrates such a bottleneck. Consider a river system with a large upstream
reservoir connected to a turbine and spillway with high-flow capability.
Further downstream lies a smaller reservoir with a turbine and spillway that
is capable of lower flows, followed by another small reservoir that is
equipped with a high-flow turbine because it receives additional water from a
tributary. The most-downstream turbine cannot sustain maximum capacity without
rapidly depleting its reservoir, because even if ample water remains upstream,
it cannot bypass the bottleneck (the middle turbine) quickly enough. Moreover,
the most-upstream turbine cannot run at full capacity indefinitely without
risking an overflow at the intermediary reservoir.

Figure 6: Conceptual representation of a river system, illustrating the
concept of bottlenecks that can exist at various points along a river. The
diagram depicts a series of reservoirs and turbines, with a large upstream
reservoir followed by smaller ones downstream. It highlights how varying
turbine capacities can create bottlenecks, particularly when a lower-capacity
turbine is situated between higher-capacity turbines. Given the present
regulatory frameworks and infrastructure, whereby the allowed flow through a
plant (either through the turbine or in the spillway) generally is limited to
the maximum flow capacity of the plant’s turbines, these bottlenecks diminish
the ability to sustain high levels of hydropower production.

####  4.5.2 Flow limits in the model

The permit under which hydropower plants operate can limit the amount of water
that they can release through turbines or in spillways, as well as the rate at
which they can change water flows. For instance, some power plants can operate
with so-called short-term regulation, i.e., frequently changing the water
flows withdrawn from their closest reservoir, while others do not.

Since some plants have the right to conduct short-term regulation, whereas
some can only change their water flows at a limited rate of change, we
introduced two variables to represent the flow constraints in our model: long-
term spillage, $L_{t,p,p2}$; and short-term spillage, $S_{t,p,p2}$. The long-
term spillage has the upper bound $l_{t,p,p2}$, as in Equation (1). The right
to regulate water in the short term includes both spillage and the water
released in the turbines. Thus, we set the sum of the short-term spillage,
$S_{t,p,p2}$, and the turbine discharge, $D_{t,p,p2}$, to be below the upper
bound for short-term regulation, $s_{t,p,p2}$, as in Equation (2). Index $t$
represents time, and the indices $p$ and $p2$ represent the passage from plant
$p$ to downstream plant $p2$.

$L_{t,p,p2}<=l_{t,p,p2}$ (1) $S_{t,p,p2}+D_{t,p,p2}<=s_{t,p,p2}$ (2)

To restrict the rate of change in long-term spillage, we introduced
constraints that limit the increase and decrease of long-term spillage from
hour to hour. The parameter $i$ represents the allowed increase each hour, and
$d$ represents the allowed decrease each hour.

$L_{t,p,p2}<=L_{t-1,p,p2}+i_{t,p,p2}$ (3)
$L_{t,p,p2}>=L_{t-1,p,p2}-d_{t,p,p2}$ (4)

In our scenarios, we applied either short-term spillage or long-term spillage
to each plant. In reality, it could be that one is allowed to release water
above the limit for short-term regulation if that water release is carried out
with slow changes to the flow. However, allowing for both long-term and short-
term spillage in our model requires some model development since the current
implementation would lead to the possibility of having short-term spillage on
top of the long-term spillage, thereby changing the flow rates faster than is
allowed above the upper limit for short-term regulation.

####  4.5.3 Scenarios

To explore the impacts of bottlenecks on sustained output, we examined the
following three scenarios with different upper bounds on spillage: Present
regime, Reduced bottlenecks, and Unrestricted. The scenarios are briefly
explained below, and the entire code for all scenarios is available in the
model at [36].

Present regime scenario
Sweden’s current regulatory framework for hydropower production involves the
issuing of specific permits for each plant. These permits typically impose
restrictions on permissible water flow rates at specific points along the
river. Some power plants can operate with short-term regulation, while others
do not. For instance, reservoirs that are not directly linked to power plants
frequently lack such permissions. Based on discussions with specialists from
power production companies and water regulation authorities in Sweden, we
conclude that power plants that are authorised to engage in short-term
regulation are generally permitted to adjust flows rapidly between any minimum
flow requirement and the maximum capacity of their installed turbines. In this
study, for the ’Present regime scenario’, we have assumed that all reservoirs
that are connected directly to power plants are entitled to regulate their
water flows on a short-term basis. We have further assumed that flows that
exceed the maximum capacity of the installed turbines, plus any mandatory
minimum spillage requirements, are not permitted except during periods when
the inflow exceeds the turbine’s maximum flow capabilities. Consequently, for
these reservoirs, the upper limit on short-term regulation [$s$ in Equation
(2)] is set each hour by the greater of the maximum flow capacity of installed
turbines or the present inflow to that reservoir. Furthermore, the upper limit
on long-term spillage [$l$ in Equation (1)] is set at zero.

For reservoirs that are not directly connected to a power plant, we set the
short-term regulation [$s$ in Equation (2)] to zero and permit long-term
spillage [$l$ in Equation (2)] every hour, equal to either the maximum inflow
recorded that calendar month during the period of 2016–2020 or the mean inflow
for the same period, whichever is greater during that hour.

Reduced bottlenecks scenario
In this scenario, we identified all power plants in which the maximum flow
capacity of the installed turbines was lower than that of any of the upstream
plants. Following this approach, approximately 30% of all power plants were
classified as bottlenecks. For these identified bottlenecks, we increased the
allowed short-term regulation [$s$ in Equation (2)] to match the maximum flow
capacity of the turbines in the upstream plant. This adjustment resulted in an
average increase of 20% in the maximum water flows at these plants. When
comparing these revised flow rate limits to the maximum recorded inflow for
each plant (including the inflow to all upstream plants) over the period of
2016–2020, we found that the limits increased from an average of 18% to 22% of
the maximum inflow. Similarly, when compared to the average inflow during the
same period, the spillage limits rose from an average of 163% to 194% of the
mean inflow. Details of these adjustments to the flow rate limits, in relation
to both the maximum and average inflows, are illustrated in Figure 7.

Figure 7: Comparing the flow rate limitations at hydropower bottleneck sites
in the Present regime and the Reduced bottlenecks scenarios. Highlighted are
the adjustments made to enhance performance during energy droughts. The left-
hand panel displays the flow rate limits as a percentage of maximum inflow to
the plant, while the right-hand panel shows these limits as a percentage of
the mean inflow to the plant.

Unrestricted scenario
In the unrestricted scenario, the upper limits on short-term regulation [$s$
in Equation (2)] were removed for all power plants and reservoirs. Thus, all
the plants, including reservoirs without directly connected power plants, were
allowed to regulate water without an upper limit. Note that we retained the
upper limits on turbine discharge. Thus, we effectively increased the spillage
limits.

###  4.6 Acknowledgements

We thank the Swedish water regulation authorities and power production
companies for valuable discussions and information regarding Sweden’s river
infrastructures and hydropower regulatory frameworks. Special thanks to Anna
Hedström Ringvall at Vattenregleringsföretagen and Emma Wikner at Statkraft
for their contributions. We also thank Vattenfall, Statkraft, Skelleftekraft,
Jämtkraft, Fortum and Energiföretagen for providing the data used in this
study.

###  4.7 Declaration of interest

The authors declare no competing interests.

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##  Appendix A Supplementary material

###  A.1 Sustained production

In this section, we present a figure on Sustained production to enhance
understanding of sustained output, complementing the Sustained capacity values
discussed in the main article.

(a) Sustained hydropower production levels in Sweden over a period of one to
three consecutive weeks with high electricity prices for three different
operational scenarios.

(b) This is the same figure as Figure 2 in the main article. It is reproduced
here to facilitate comparison of the Sustained production and Sustained
capacity.

Figure A.1: Illustrations of the sustained hydropower production and the
sustained hydropower capacity in Sweden over a period of one to three
consecutive weeks with high electricity prices for three different operational
scenarios. Each violin contains the results for individual full-year runs with
high-price periods for each month over four different years. The dots
represent the mean values, while the lines extending from the dots indicate
the 25th to 75th percentile range. The color-coding corresponds to the
following operational scenarios: Present regime, with current regulations and
infrastructure (green); Reduced bottlenecks, with an increased upper limit on
spillage in bottlenecks (blue); and Unrestricted, with unrestricted spillage
(red). The different scenarios highlight the influence of operational
constraints on the ability of hydropower to sustain a high output during
energy droughts. The results shown are based on approximately 6500 model runs.
Refer to the Experimental procedure for details of the test set-up.

###  A.2 Main results categorised by Nord Pool price areas

Here, we present the main results from the present study, now categorised
according to Nord Pool price areas, and arranged in the same sequence as in
the main text.

####  A.2.1 Sustained capacity

(a) The ability of hydropower in SE1 to sustain a high capacity. The results
are based on detailed modelling of 98 % of the installed capacity in SE1.

(b) The ability of hydropower in SE2 to sustain a high capacity. The results
are based on detailed modelling of 99 % of the installed capacity in SE2.

(c) The ability of hydropower in SE3 to sustain a high capacity. The results
are based on detailed modelling of 72 % of the installed capacity in SE3.

(d) The ability of hydropower in SE to sustain a high capacity. The results
are based on detailed modelling of 92 % of the installed capacity in Sweden.

Figure A.2: Illustrations of the sustained hydropower capacities per price
area over a period of one to three consecutive weeks with high electricity
prices for three different operational scenarios. Each violin contains the
results for individual full-year runs with high-price periods for each month
over four different years. The dots represent the mean values, while the lines
extending from the dots indicate the 25th to 75th percentile range. The color-
coding corresponds to the following operational scenarios: Present regime,
with current regulations and infrastructure (green); Reduced bottlenecks, with
an increased upper limit on spillage in bottlenecks (blue); and Unrestricted,
with unrestricted spillage (red). The different scenarios highlight the
influences of operational constraints on the ability of hydropower to sustain
a high output during energy droughts. The results shown are based on
approximately 6,500 model runs. Refer to the Experimental procedure section
for details of the test set-up.

####  A.2.2 Sustained production

(a) The ability of hydropower in SE1 to sustain high production levels. The
results are based on detailed modelling of 98 % of the installed capacity in
SE1.

(b) The ability of hydropower in SE2 to sustain high production levels. The
results are based on detailed modelling of 99 % of the installed capacity in
SE2.

(c) The ability of hydropower in SE3 to sustain high production levels. The
results are based on detailed modelling of 72 % of the installed capacity in
SE3.

(d) The ability of hydropower in SE to sustain high production levels. The
results are based on detailed modelling of 92 % of the installed capacity in
Sweden.

Figure A.3: Illustrations of the sustained hydropower production per price
area over a period of one to three consecutive weeks with high electricity
prices for three different operational scenarios. Each violin contains the
results for individual full-year runs with high-price periods for each month
over four different years. The dots represent the mean values, while the lines
extending from the dots indicate the 25th to 75th percentile range. The color-
coding corresponds to the following operational scenarios: Present regime,
with current regulations and infrastructure (green); Reduced bottlenecks, with
an increased upper limit on spillage in bottlenecks (blue); and Unrestricted,
with unrestricted spillage (red). The different scenarios highlight the
influence of operational constraints on the ability of hydropower to sustain a
high output during energy droughts. The results shown are based on
approximately 6500 model runs. Refer to the Experimental procedure section for
details of the test set-up.

####  A.2.3 Sustained capacity per month and its correlation with inflow

(a) Monthly Sustained capacity in relation to the natural inflow throughout
the year in SE1. The results are based on detailed modelling of 98 % of the
installed capacity in SE1.

(b) Monthly Sustained capacity in relation to the natural inflow throughout
the year in SE2. The results are based on detailed modelling of 99 % of the
installed capacity in SE2.

(c) Monthly Sustained capacity in relation to the natural inflow throughout
the year in SE3. The results are based on detailed modelling of 72 % of the
installed capacity in SE3.

(d) Monthly Sustained capacity in relation to the natural inflow throughout
the year in all of Sweden. The results are based on detailed modelling of 92 %
of the installed capacity in Sweden.

Figure A.4: A comparison of the monthly sustained capacities of Swedish
hydropower in each price area under three operational scenarios versus the
natural inflow throughout the year, illustrating how today’s hydrological
cycle influences hydropower’s ability to sustain output. Sustained production
levels under the Present regime (green), Reduced bottlenecks (blue), and
Unrestricted (red) scenarios are shown through box plots, which display the
range and mean values for each month. The line graph represents the total
inflow as a percentage of the maximum recorded. The shaded area under the line
graph indicates the variability of the inflow in the modelled period of
2016-2020.

####  A.2.4 Loss of total annual production

(a) Loss of yearly production induced by sustaining high output in SE1. The
results are based on detailed modelling of 98 % of the installed capacity in
SE1.

(b) Loss of in yearly production induced by sustaining high output in SE2. The
results are based on detailed modelling of 99 % of the installed capacity in
SE2.

(c) Loss of in yearly production induced by sustaining high output in SE3. The
results are based on detailed modelling of 72 % of the installed capacity in
SE3.

(d) Loss of in yearly production induced by sustaining high output in all of
Sweden. The results are based on detailed modelling of 92 % of the installed
capacity in Sweden.

Figure A.5: Visualisation of the losses in yearly production per price area
associated with sustaining high output levels, comparing the yearly production
levels obtained in the presence of high market prices with the yearly
production obtained with historical prices. Each violin contains the results
for each month over five different years. The dots represent the mean values,
while the lines extending from the dots indicate the 25th to 75th percentile
range. The color-coding corresponds to the following operational scenarios:
Present regime, with current regulations and infrastructure (green); Reduced
bottlenecks, with an increased upper limit on spillage in bottlenecks (blue);
and Unrestricted, with unrestricted spillage (red).

###  A.3 Sensitivity analysis

Figure A.6: Illustrations of the Sustained capacity in Sweden over three
consecutive weeks with high electricity prices. The different violins
correspond to the following sensitivity analyses: Present regime (left-most
violins), with the linearised model (B:L) and a price of 5,000 SEK/MWh (430
€/MWh) during the studied high-price period; Present regime non-linear (middle
violins), with the full non-linear model (A) and a price of 5,000 SEK/MWh (430
€/MWh) during the studied high-price period; and Present regime high-price
(right-most violins) with the linearised model (B:L) and a price of 50,000
SEK/MWh (4,300 €/MWh) during the studied high-price period. The Present regime
violin is the same as that presented as the main result in this study. Each
violin contains the results for individual full-year runs with high-price
periods for each month over 4 years. The dots represent the mean values, while
the lines extending from the dots indicate the 25th to 75th percentile range.