# Insurance Contract for High Renewable Energy Integration

Dongwei Zhao, Hao Wang, Jianwei Huang, Xiaojun Lin This work is supported in
part by the Shenzhen Science and Technology Program (Project
JCYJ20210324120011032), Guangdong Basic and Applied Basic Research Foundation
(Project 2021B1515120008), the Shenzhen Institute of Artificial Intelligence
and Robotics for Society, NSF through grant ECCS-2129631, and the Australian
Research Council (ARC) Discovery Early Career Researcher Award (DECRA) under
Grant DE230100046. Dongwei Zhao is with the MIT Energy Initiative,
Massachusetts Institute of Technology. Hao Wang is with the Department of Data
Science and Artificial Intelligence, Monash University. Jianwei Huang is with
the School of Science and Engineering, The Chinese University of Hong Kong,
Shenzhen and Shenzhen Institute of Artificial Intelligence and Robotics for
Society (corresponding author). Xiaojun Lin is with the Elmore Family School
of Electrical and Computer Engineering, Purdue University.

###### Abstract

The increasing penetration of renewable energy poses significant challenges to
power grid reliability. There have been increasing interests in utilizing
financial tools, such as insurance, to help end-users hedge the potential risk
of lost load due to renewable energy variability. With insurance, a user pays
a premium fee to the utility, so that he will get compensated in case his
demand is not fully satisfied. A proper insurance design needs to resolve the
following two challenges: (i) users’ reliability preference is private
information; and (ii) the insurance design is tightly coupled with the
renewable energy investment decision. To address these challenges, we adopt
the contract theory to elicit users’ private reliability preferences, and we
study how the utility can jointly optimize the insurance contract and the
planning of renewable energy. A key analytical challenge is that the joint
optimization of the insurance design and the planning of renewables is non-
convex. We resolve this difficulty by revealing important structural
properties of the optimal solution, using the help of two benchmark problems:
the no-insurance benchmark and the social-optimum benchmark. Compared with the
no-insurance benchmark, we prove that the social cost and users’ total energy
cost are always no larger under the optimal contract. Simulation results show
that the largest benefit of the insurance contract is achieved at a medium
electricity-bill price together with a low type heterogeneity and a high
renewable uncertainty.

††publicationid: pubid: 978-1-6654-3254-2/22/$31.00 ©2022 IEEE

##  I Introduction

###  I-A Background and motivation

The transition from fossil energy sources to renewable energy sources is
accelerating at the global scale as the threat of climate change continues to
grow. From 2014 to 2020, the globally installed capacity of solar PV has
increased from about 40 GW to 140 GW, while the wind energy capacity has seen
a growth from about 50 GW to 90 GWh [1]. Many countries have set
sustainability goals to increase the penetration of renewable energy in
electricity supplies. For example, the California electricity market in the US
sets a target of generating over two-thirds of its electricity from non-fossil
and non-nuclear resources by 2030 [2].

The increasing penetration of renewable energy generation, however, poses
challenges to system reliability. When a significant amount of renewable
energy supply fluctuates based on weather conditions, users may experience
lost loads (i.e., unsatisfied demands) more often [3]. In principle, lost load
could be alleviated by investing in other costly low-carbon energy resources.
For example, energy storage could provide flexibility, which is however
costly. Alternatively, the utility can simply invest in more renewable
capacity to reduce the probability of low supply, which can also be costly.
Therefore, it is crucial to provide the right financial arrangement so that
not only can end-users hedge the potential risk due to lost load, but the
utility is also incentivized to invest in the right amount of additional
resources to improve grid reliability.

The traditional financial arrangement between the utility and the end-users is
usually through the electricity bill at either static or dynamic rates.
Unfortunately, such an electricity bill does not provide a way for the end-
users to get compensation for lost load, nor does it enable the utility to
directly recover the cost of reliability-improving investment. In contrast, in
this paper we are interested in more advanced financial tools, in particular
insurance, to address the above open question. With insurance, a user pays a
premium fee to the utility in advance, so that he will get compensated if his
demand is not fully satisfied [4]. In return, the utility can use the premium
fees to invest in reliability-improving resources. Note that by choosing among
different insurance contracts, the end-users can express their heterogeneous
valuation of lost load. Therefore, we expect that the insurance contract will
align the utility’s cost with end-users’ value of lost load more accurately,
so that the utility is incentivized to invest in the right amount of
additional resources. As a first step towards this direction, in this paper,
we focus on a very-high renewable penetration setting, where the sole
investment option that the utility considers is to invest in more renewable
generation. We leave as future work the study of the investment of other
resources, such as energy storage.

To design a proper insurance scheme, however, the utility needs to resolve two
challenges. First, different users have distinct preferences of reliability
[5], which will influence the type of insurance that they want to purchase.
However, such user preferences are often private information unavailable to
the utility. Second, since the utility’s renewable energy investment can
affect the system reliability, it will also impact users’ insurance choices.
Hence, it is crucial for the utility to jointly consider the renewable energy
planning and insurance design together. These challenges motivate us to answer
the following key question in the scenario of very-high renewable energy
penetration:

  * •

How to jointly optimize an insurance contract and the renewable energy
planning, which helps users hedge the risks of lost load and helps the utility
improve social welfare?

###  I-B Related works

Several recent work has started to explore the benefits and design of
reliability insurance for electricity users. Billimoria et al. in [6] and
Fuentes et al. in [4] introduced the concepts of reliability insurance for end
users. However, these studies did not provide quantitative results in the
insurance design.

Regarding insurance design using quantitative methods, Fuentes et al. in [7]
conducted numerical studies to demonstrate the benefits of insurance for
users, assuming that users have homogeneous preferences towards reliability.
Abedi et al. in [8], Dong et al. in [9], Niromandfam et al. in [10], and
Fumagalli et al. in [11] considered users’ heterogeneity in reliability
preference and risk aversion in the insurance design. However, these studies
did not consider the planning of generation resources. As we mentioned
earlier, since the electricity supply can affect the system reliability and
users’ energy costs, it is important to consider the coupling between
insurance design and renewable energy investment.

The most relevant work to our study is [12], where Billimoria et al. studied
users’ heterogeneity in reliability preferences and optimized investment
decisions of electricity suppliers. However, the authors in [12] assumed that
users would voluntarily express to the insurance provider their reliability
preferences and compensation values, which is a rather unrealistic assumption
given users’ private information.

None of the above studies considered the joint optimization of insurance
design and renewable energy planning under users’ private reliability
preference. Thus, our work represents an important step towards filling this
gap, by adopting contract theory to study such a joint optimization.

###  I-C Main results

Focusing on an extreme future scenario of 100% renewable energy generation, we
summarize the key results and contribution of this paper as follows:

  * •

Insurance contract for reliability: To the best of our knowledge, our work is
the first to jointly design the insurance contract for end-users’ lost load
and optimize the planning of renewable energy. Such an insurance scheme can
play an important role in the future scenario of high renewable energy
integration.

  * •

Contract design and customized solution: We adopt contract theory to solve the
challenge of eliciting users’ private preferences of reliability. Due to the
use of contract theory, however, the joint optimization of contract design and
renewable energy planning involves solving a challenging non-convex
optimization problem. To resolve this difficulty, we characterize important
structural properties of the optimal solution, using the help from two
benchmark problems: the no-insurance benchmark and the social-optimum
benchmark.

  * •

Benefits of insurance contract: Compared with the no-insurance benchmark, we
prove that the optimal insurance contract will never reduce social welfare or
increase the users’ total energy cost. Simulation results show that the
optimal insurance contract reduces the social cost and users’ costs by up to
20% and 19%, respectively.

  * •

Impact of electricity-bill price, type heterogeneity, and renewable
uncertainty: The specific level of insurance benefits in reducing user cost
and improving social welfare is affected by a number of factors. Simulation
results show that the highest benefit is achieved at a medium electricity-bill
price together with a low heterogeneity of types and a high uncertainty of
renewables.

##  II System Model

We consider one electricity utility serving a group of
$\mathcal{I}=\\{1,2\ldots I\\}$ users. Besides charging users electricity
bills, the utility provides a set of insurance contract items for users. Users
can purchase insurance to hedge the risk of lost load.

Figure 1 illustrates the decision-making of the utility and users in two
different timescales. At the beginning of an investment horizon of $D$ days
(e.g., $D$ corresponds to a few years depending on the lifespan of
renewables), the utility decides the invested renewable energy capacity and
announces the insurance contract to users. Users decide whether to purchase
the insurance and which type of contract items to purchase. The investment
horizon is divided into operational horizons. Each operational horizon
corresponds to one day, which is divided into a set of $T$ periods (e.g., 24
hours) $\mathcal{T}=\\{1,2,3,...,T\\}$. On each day, the utility supplies
users with renewable energy. Users without insurance just pay the electricity
bill, but will not get reimbursed if his demand is not fully satisfied. In
contrast, users who have purchased insurance will get reimbursed for the
unserved demand (i.e., lost load).

Figure 1: Two timescales of decision-making.

###  II-A Users

For simplicity, we assume that each user $i$’s demand $D_{i}^{t}$ in time
period $t\in\mathcal{T}$ is the same across different days.111We can also
model the stochastic nature of the demand. This will make the system model
more realistic, although it does not change the main analysis and insights of
this paper. The reason is that what matters is the difference between the
renewable energy generation and the user’s total demand in each time period,
which is already stochastic in our current model. We will leave such an
extended study in the journal version. If a user has demand unserved by the
utility, the lost load will incur a cost to users due to service interruption.
We denote user $i$’s private value of lost load (VOLL) at time $t$ as
$V_{i}^{t}$. If user $i$’s unserved demand is $x_{i}^{t}$, his cost of lost
load at time $t$ is $V_{i}^{t}x_{i}^{t}$.

We classify users into a set $\mathcal{K}=\\{1,2,\ldots,K\\}$ of types based
on the values of lost load, i.e., user $i$ belongs to type $k\in\mathcal{K}$
if his VOLL is $\bm{V}_{k}=(V_{k}^{t},t\in\mathcal{T})$. The utility does not
know each user’s VOLL. However, we assume that utility knows each user’s
demand and the aggregate demand of each type, i.e.,
$\bm{D}_{k}=(D_{k}^{t},t\in\mathcal{T})$ for any type $k\in\mathcal{K}$. Such
demand information can be obtained from smart meters, and the aggregate demand
of each type can be estimated in reliability surveys [13]. We assume that
${V}_{1}^{t}\leq{V}_{2}^{t}\leq\ldots\leq{V}_{K}^{t}$ for each hour $t$, i.e.,
Type-1 has the lowest value.

###  II-B Utility

We consider the challenging scenario where the utility invests in only
renewable energy to supply electricity to users due to the zero-carbon target.

The daily energy supply from the utility is uncertain due to the variability
of renewable energy. If the utility invests in a total capacity $r\geq 0$ of
renewable energy, the daily random generation at $t\in\mathcal{T}$ as
$r\Theta^{t}$, where $\Theta^{t}$ is a random variable with support $[0,1]$.
Thus, the total random electricity supplied to users at time $t$ is
$s^{t}(r,\Theta^{t})=\min(D_{a}^{t},r\Theta^{t})$ where
$D_{a}^{t}=\sum_{i\in\mathcal{I}}D_{i}^{t}$ is the total demand. When the
energy generation cannot satisfy users’ total demand, i.e.,
$D_{a}^{t}>r\Theta^{t}$, the utility will allocate the energy according to the
following rule:222This rule can be easily implemented in expectation, e.g.,
the utility can shed users’ load in a round-robin fashion. We leave as future
work a more comprehensive study of power allocation considering network
constraints.

######  Assumption 1 (Energy allocation rule).

Under the capacity of renewable energy $r$, in case of insufficient energy
generation, the utility allocates energy to users proportionally to the users’
demands, i.e., the energy $d_{i}^{t}$ allocated to user $i$ is given by

$\displaystyle d_{i}^{t}(r,\Theta^{t})=\frac{D_{i}^{t}}{D_{a}^{t}}\cdot
s^{t}(r,\Theta^{t}).$ (1)

Note that $s^{t}(r,\Theta^{t})$ and $d_{i}^{t}(r,\Theta^{t})$ are functions of
$(r,\Theta^{t})$. In the rest of paper, we sometimes omit the dependency of
$s^{t}$ and $d_{i}^{t}$ on $(r,\Theta^{t})$ for ease of exposition. However,
they should always be considered as functions of $(r,\Theta^{t})$.

We assume that the utility charges a flat electricity-bill price $p$ for the
actual energy consumption of each user. Since the focus of this study is
insurance contract design, we will fix the value of $p$ and assume that
$p\leq{V}_{1}^{t},\forall t\in\mathcal{T}$. Otherwise, some users with low
valuation may be incentivized to curtail the load instead of paying the
electricity bill. Next, we will introduce the insurance contract design in
more details.

##  III Insurance Contract Design

Since the utility does not know each user’s private information of VOLL, we
adopt contract theory to elicit users’ VOLL. Specifically, the utility designs
a set of contract items for all user types, such that each user will
voluntarily choose the item corresponding to his own type. We will first
introduce the premium and reimbursement fees in the contract. Then, we
formulate the user’s problem and the utility’s problem under the insurance
contract, respectively.

###  III-A Contract items

The utility designs a set of contract items for premium fees, one for each
type (of users) in $\mathcal{K}$. The premium fee for each type
$k\in\mathcal{K}$ is $\bm{\pi}_{k}=(\pi_{k}^{t},t\in\mathcal{T})$ per unit
demand paid by users. If a user chooses the type-$k$ contract and pays
$\bm{\pi}_{k}$, the utility is obligated to compensate the user with a unit
reimbursement fee for any lost demand. Note that for the demand satisfied, the
user needs to pay at the electricity-bill price $p$. For the demand lost, user
$i$ has a cost $V_{i}^{t}$ for the unit lost load. Since the user will not pay
the electricity bill for the lost load, we let user $i$ of type $k$ be
compensated at $\bm{V}_{k}-\bm{p}=(V_{k}^{t}-p,t\in\mathcal{T})$.

###  III-B Users’ problem in choosing insurance contract

A user can decide whether to purchase the insurance. If he does, he needs to
choose which type of contract to purchase.

If user $i$ chooses the contract item of type $o_{i}\in\mathcal{K}$, i.e.,
premium fee $\bm{\pi}_{o_{i}}$ and reimbursement $\bm{V}_{o_{i}}-\bm{p}$, his
net cost (scaled to one day) $CU_{i}({o}_{i})$ is

$\displaystyle\sum_{t\in\mathcal{T}}{\pi_{o_{i}}^{t}}\cdot
D_{i}^{t}+\sum_{t\in\mathcal{T}}p\cdot\mathbb{E}_{\Theta^{t}}[d_{i}^{t}]+\sum_{t\in\mathcal{T}}\mathbb{E}_{\Theta^{t}}\left[{V_{i}^{t}\cdot(D_{i}^{t}-d_{i}^{t})}\right]$
$\displaystyle-\sum_{t\in\mathcal{T}}\mathbb{E}_{\Theta^{t}}\left[{(V_{o_{i}}^{t}-p)}\cdot(D_{i}^{t}-d_{i}^{t})\right].$
(2)

There are four terms in the cost formulation (2). The first term is the cost
of premium fee for the type-$o_{i}$ contract; the second term is the
electricity bill under the allocated energy amount $d_{i}^{t}$; the third part
is the cost of lost load under its VOLL $V_{i}^{t}$; and the fourth term is
the reimbursement from the utility. Note that if user $i$ of type $k$ chooses
his own type of contract, his cost is simplified to

$\displaystyle CU_{i}(k):=$
$\displaystyle\sum_{t\in\mathcal{T}}{\pi_{k}^{t}}\cdot
D_{i}^{t}+\sum_{t\in\mathcal{T}}p\cdot D_{i}^{t},$ (3)

which only includes the premium fee and electricity bill under the original
demand. The cost of lost load is exactly reimbursed by the utility. However,
if a user does not choose his own type, he will get reimbursed with a higher
or lower compensation rate than his actual VOLL. Our goal is to design the
contract so that each user will voluntarily choose the contract item designed
for his type.

If user $i$ does not purchase insurance, his cost $CU_{i}(0)$ is

$\displaystyle CU_{i}(0):=$
$\displaystyle\sum_{t\in\mathcal{T}}p\cdot\mathbb{E}_{\Theta^{t}}[d_{i}^{t}]+\sum_{t\in\mathcal{T}}\mathbb{E}_{\Theta^{t}}\left[{V_{i}^{t}\cdot(D_{i}^{t}-d_{i}^{t})}\right],$
(4)

which only includes the costs of electricity bill and lost load.

###  III-C Utility’s problem of the optimal contract design

At the beginning of the investment horizon, the utility optimizes the invested
renewable energy capacity $r$ and premium fee $\bm{\pi}$ of the contract. We
first introduce the conditions to ensure that each user chooses the contract
item of his own type. Then we formulate the utility’s problem to optimize the
premium fee and the investment of renewable energy.

####  III-C1 Validation of the contract

The utility announces the set of contract items to users, and users can freely
choose which type. To ensure that users will participate in the insurance
contract and are reimbursed based on their actual VOLL, the insurance contract
needs to satisfy Individual Rationality (IR) and Incentive Compatibility (IC)
[14] defined as follow.

######  Definition 1 (Individual Rationality).

The insurance contract is individual rational if a user $i$ of type
$k\in\mathcal{K}$ prefers purchasing the insurance item for that type rather
than not purchasing any contract, i.e., $CU_{i}(k)\leq CU_{i}(0)$.

######  Definition 2 (Incentive Compatibility).

The insurance contract is incentive compatible if a user $i$ of type
$k\in\mathcal{K}$ minimizes his cost by choosing the contract intended for his
type i.e., for any $m\in\mathcal{K}$ and $m\neq k$, $CU_{i}(k)\leq CU_{i}(m)$.

A contract is valid if it can satisfy both IR and IC conditions. Theorem 1
characterizes sufficient conditions to ensure that our insurance contract is
valid.

######  Theorem 1.

The contract is valid if the following sufficient IC and IR conditions are
satisfied:

$\displaystyle\textbf{IC}:\pi_{k}^{t}-\pi_{m}^{t}=$
$\displaystyle{(V_{k}^{t}-V_{m}^{t})\cdot\mathbb{E}_{\bm{\Theta}}\left[1-\frac{s^{t}}{D_{a}^{t}}\right]},$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\forall
t\in\mathcal{T},\forall m,k\in\mathcal{K}.$ (5)
$\displaystyle\textbf{IR}:0\leq\pi_{k}^{t}\leq{(V_{k}^{t}-p)\cdot\mathbb{E}_{\bm{\Theta}}\left[1-\frac{s^{t}}{D_{a}^{t}}\right]},\forall
t\in\mathcal{T},\forall k\in\mathcal{K}.$ (6)

The contract is always valid.

In Theorem 1, the IC constraint (5) specifies the conditions of the premium
fees between any two types, which ensures that each user will not choose other
types’ contract items. Note that the difference between premium fees depends
on the VOLL and renewables supply. The IR constraint (6) ensures that the
premium fee for each contract item is not too high, so that users will always
choose the contract. In the later analysis, we will focus on the utility’s
optimization problem subject to the conditions in Theorem 1.333We present the
necessary and sufficient conditions in the appendix [15]. The sufficient
conditions in (5) and (6) make the constraints independent of individual
user’s private information and make the contract design tractable.

####  III-C2 Formulation of utility’s problem

We first denote the reimbursement payment to all users as
$C_{s}^{t}(r,\Theta^{t})$:

$\displaystyle
C_{s}^{t}(r,\Theta^{t}):=\sum_{k\in\mathcal{K}}(V_{k}^{t}-p)\cdot(D_{k}^{t}-d_{k}^{t}).$
(7)

Then, based on the sufficient conditions of IC and IR in Theorem 1, we present
the profit $f^{\text{Ins}}(r,\bm{\pi})$ of the utility (scaled to one day) as
follows.

$\displaystyle f^{\text{Ins}}(r,\bm{\pi})$
$\displaystyle:=\sum_{k\in\mathcal{K}}\sum_{t\in\mathcal{T}}\pi_{k}^{t}D_{k}^{t}+\sum_{t\in\mathcal{T}}p\cdot\mathbb{E}_{\Theta^{t}}[s^{t}]-c_{r}\cdot
r$
$\displaystyle~{}~{}-\sum_{t\in\mathcal{T}}\mathbb{E}_{{\Theta}^{t}}[C_{s}^{t}(r,{\Theta^{t}})].$
(8)

There are four terms in the profit formulation (8). The first term is the
revenue from all the users’ premium fees; the second term is the total
electricity bill under the supply $s^{t}$ to users; the third term is the
investment cost of renewables under the unit capacity cost $c_{r}$ (scaled
into one day); and the fourth term is the reimbursement payment to all users.

In Problem-Ins, we focus on an insurance-providing utility (Ins-utility) whose
profit is regulated to protect users [16]. The regulated utility is required
to design the insurance to minimize all users’ collective energy costs (recall
single user’s energy cost in (3)), constrained on a minimum regulated profit
of $\xi\geq 0$.444We also consider a profit-seeking utility that maximizes her
profit in the appendix [15]. However, we show that the insurance contract has
no benefit compared with the no-insurance case if the utility is profit-
seeking. As shown in (3), when a user chooses his own type of contract, his
cost only includes the premium fee and the electricity bill under the original
demand. Since the electricity-bill price and the original demand are fixed, we
can simply minimize the total premium fees as in the objective (9) to minimize
users’ collective energy costs.

Problem-Ins: Regulated utility under insurance contract

$\displaystyle\min~{}$
$\displaystyle\sum_{k\in\mathcal{K}}\sum_{t\in\mathcal{T}}\pi_{k}^{t}D_{k}^{t}$
(9) s.t. $\displaystyle f^{\text{Ins}}(r,\bm{\pi})\geq\xi,$ (10)
$\displaystyle\eqref{eq:IC}-\eqref{eq:IR},$ $\displaystyle\text{var}:$
$\displaystyle r\geq 0,\bm{\pi},$

The constraint (10) regulates the utility’s profit to be no greater than a
lower bound $\xi\geq 0$. The constraints (5) and (6) are the IC and IR
conditions, respectively. We denote the optimal solution to Problem-Ins as
$(r^{*},\bm{\pi}^{*})$. Problem-Ins is challenging to solve due to the non-
convexity brought by (5). We will present the solution method in Section V.
Before the solution method, we present two performance benchmarks in the next
section.

##  IV Benchmarks

We consider two benchmarks to compare the results under the insurance
contract. One is the social-optimum case, and the other is the no-insurance
case.

###  IV-A Social-optimum benchmark (SO)

For the SO benchmark, a social planner invests in renewable energy to minimize
the system cost. The system cost, denoted as $C^{so}(r)$, consists of the
investment cost of renewable energy $c_{r}\cdot r$ and the expected cost of
lost load
$\sum_{t\in\mathcal{T}}\mathbb{E}_{\Theta^{t}}[C_{s}(r,{\Theta^{t}})]$. The
social-cost minimization problem is formulated as

Problem-SO: Social-cost minimization

$\displaystyle\min~{}C^{\text{SO}}(r):=$ $\displaystyle c_{r}\cdot
r+\sum_{t\in\mathcal{T}}\mathbb{E}_{\Theta^{t}}[C_{s}(r,{\Theta^{t}})]$ (11)
s.t. $\displaystyle r\geq 0,$ (12) $\displaystyle\text{var}:$ $\displaystyle
r.$

The objective function $C^{\text{SO}}(r)$ is convex in $r$. Problem-SO can be
easily solved by standard context optimization techniques. We let
$r^{\dagger}$ be the optimal solution to Problem-SO.

###  IV-B No-insurance benchmark (NoIns)

For the NoIns benchmark, we consider a regulated utility that does not provide
insurance (NoIns-utility). To reduce users’ lost load and thus minimize users’
total energy cost, the utility maximizes the invested capacity of renewable
energy under a regulated profit lower bound. The no-insurance optimization
problem is formulated as

Problem-NoIns: Regulated utility without insurance

$\displaystyle\max~{}$ $\displaystyle r$ s.t. $\displaystyle
f^{\text{No}}(r)\geq\xi,$ (13) var: $\displaystyle r\geq 0,$

where the profit of the utility is denoted as a function $f^{No}(r)$ of the
invested capacity $r$, given by

$\displaystyle f^{\text{No}}(r)=$
$\displaystyle\sum_{t\in\mathcal{T}}p\cdot\mathbb{E}_{\Theta^{t}}[s^{t}]-c_{r}\cdot
r.$ (14)

We let $r^{\ddagger}$ be the optimal solution to Problem-NoIns. As
$f^{\text{No}}(r)$ is concave, Problem-NoIns is a concave maximization
problem. More specifically, we have $f^{\text{No}}(0)=0$ and
$f^{\text{No}}(+\infty)<0$. Thus, if there exists $r^{\prime}$, such that
$f^{\text{No}}(r^{\prime})\geq\xi$, the optimal solution $r^{\ddagger}$ to
Problem-NoIns is achieved at $f^{\text{No}}(r^{\ddagger})=\xi$ because
$f^{\text{No}}(r)$ is either decreasing or first increasing and then
decreasing in $r$. The equation $f^{\text{No}}(r)=\xi$ has either one root if
$\max_{r}f^{\text{No}}(r)\leq 0$, or two roots if
$\max_{r}f^{\text{No}}(r)>0$. If there are two roots, we select the larger one
to maximize the capacity. Such a root can be found through standard numerical
methods like bisection search [17].

##  V Analysis of insurance contract

Problem-Ins is challenging to solve due to non-convexity. With the help of the
two benchmarks, we will first show how to solve Problem-Ins for the Ins-
utility, and then present analytical results regarding the performance of the
insurance contract. We include all the proofs in the online appendix [15] due
to the page limit.

###  V-A Solution method for Ins-utility

Problem-Ins is non-convex due to the equality constraint (5). To solve it, we
first solve the optimal capacity $r^{*}$ by constructing a convex problem from
the original problem. Then, under $r^{*}$, we solve the optimal premium fee
$\bm{\pi}^{*}$ based on the binding condition of the IR constraint (6).

Specifically, the convex problem to derive $r^{*}$ and the binding condition
of the IR constraint (6) are affected by the electricity-bill price $p$. We
present the results in Proposition 1.

######  Proposition 1 (Solution structure in Problem-Ins).

Assume that the profit lower-bound (10) is feasible. Recall that $r^{\dagger}$
is the optimal solution to Problem-SO. We let

$\displaystyle\underline{L}=\frac{c_{r}\cdot
r^{\dagger}+\xi}{\sum_{t}\mathbb{E}[s^{t\dagger}]},$ (15)
$\displaystyle\overline{L}=\frac{\xi+c_{r}\cdot
r^{\dagger}+\sum_{t}V_{1}^{t}(D_{a}^{t}-\mathbb{E}[s^{t\dagger}])}{\sum_{t}D_{a}^{t}}.$
(16)

  * •

If $\underline{L}<p<\overline{L}$, we have $r^{*}=r^{\dagger}$.

  * •

if $p\leq\underline{L}$, the IR constraint (6) reaches the upper bound for all
types and hours. We have $r^{*}=r^{\ddagger}$, which is the optimal solution
to Problem-NoIns.

  * •

if $p\geq\overline{L}$, IR constraint (6) of Type-1 reaches the lower bound
for all hours. The optimal capacity $r^{*}$ is equal to the optimal solution
of the following concave problem.

$\displaystyle\max~{}$ $\displaystyle r$ s.t. $\displaystyle
f^{\text{No}}(r)-\sum_{t\in\mathcal{T}}{(V_{1}^{t}-p)\cdot\mathbb{E}_{\bm{\Theta}}\left[D_{a}^{t}-{s^{t}}\right]}\geq\xi,$
(17) var: $\displaystyle r\geq 0$

Furthermore, the utility’s profit always achieves the lower bound $\xi$ in all
the above three cases, i.e., $f^{\text{Ins}}(r^{*},\bm{\pi}^{*})=\xi$.

As Proposition 1 shows, the binding condition of IR (6) is affected by the
electricity-bill price. If the price is too low, the premium fees need to
ensure enough revenues for the utility, so that the IR constraints reach the
upper bound. If the price $p$ is too high, the premium fees should be set low
to provide benefits to users, and the IR constraints reach the lower bound.

In Figure 2, we illustrate the results in Proposition 1, as the electricity-
bill price varies. We show the optimal invested capacity under the benchmark
SO (black curve), NoIns-utility (blue curve), and Ins-utility (red curve),
with respect to the electricity-bill price. The red curve (Ins-utility)
consists of three parts. Within the lowest-price range (a-b), the investment
level overlaps with that of the NoIns-utility. Within the medium price range
(b-c), the investment level reaches that of the social optimum (SO). Within
the highest-price range (c-d), the investment level coincides with neither
NoIns-utility nor the social optimum (SO).

Based on Proposition 1, it is straightforward to first solve $r^{*}$ by the
equivalent convex problems. Then, under the solved $r^{*}$, we can compute the
optimum premium fee $\bm{\pi}^{*}$ based on the IC constraint (5) and the IR
constraint (6), which we provide a sketch in the following.

  * •

If $p\leq\overline{L}$, we compute $\bm{\pi}^{*}$ directly using the upper
bound of (6) for all types and hours.

  * •

If $\underline{L}<p<\overline{L}$, we derive $\bm{\pi}^{*}$ using the solution
to the set of linear equations $f^{\text{Ins}}(r^{*},\bm{\pi})=\xi$ and (5).

  * •

If $p\geq\overline{L}$, we compute $\bm{\pi}^{*}$ using (5) and the lower
bound of (6) constraint for all types and hours.

Figure 2: Illustration of solution structure for Ins-utility.

###  V-B Benefits of insurance contract

Compared with Problem-NoIns, Problem-Ins will consider the VOLL of users in
the insurance design and renewable planning, which improves social welfare and
benefits the users. Next, we present two propositions to demonstrate the
benefits of the Ins-utility compared with the NoIns-utility.

######  Proposition 2.

(Social cost reduction) The optimal social cost under Ins-utility is no larger
than that under NoIns-utility.

######  Proposition 3.

(Reduction in users’ energy cost) All users’ total cost under InsR-utility is
no larger than that under NoIns-utility.

Proposition 2 shows that the insurance contract will not increase the social
cost compared with the case that the utility does not provide insurance.
Proposition 3 shows that users’ collective cost is always no larger under Ins-
utility than that under NoIns-utility.

Although individual users may not always have lower costs under Ins-utility
than under NoIns-utility, they can still see a significant cost reduction if
the electricity-bill price is medium and the heterogeneity of types is at a
low level. We will demonstrate this in simulations of Section VI.

Finally, since the IR constraint is satisfied in the insurance, under InsR-
utility, a user’s energy cost when purchasing insurance is always no larger
compared to not purchasing.

##  VI Numerical studies

We conduct extensive numerical studies to show how the insurance contract can
improve social welfare and reduce users’ costs. In particular, we look into
the impact of several system parameters, including the electricity-bill price,
heterogeneity of types, and the uncertainty of renewable energy.

###  VI-A Setup

As an illustrative example, we consider four users555The utility makes
decisions based on the aggregate demand of each type. Thus, the simulation
results in this section can still hold if we fix the aggregate demand and
consider multiple users. of four types with distinct values of lost load
$V_{1}^{t}\leq V_{2}^{t}\leq V_{3}^{t}\leq V_{4}^{t}$ to capture the key
feature of type heterogeneity. The VOLL for each user is the same for all the
time slots $t\in\mathcal{T}$. We consider the identical demand $D_{k}^{t}=D$
for each type $k\in\mathcal{K}$ at each time slot $t\in\mathcal{T}$. We
simulate both uniform distribution and truncated normal distribution within
$[0,1]$ for the output factor $\Theta^{t}$. We let the factor $\Theta^{t}$ be
independent and identically distributed for all the time slots
$t\in\mathcal{T}$. To examine the cost reduction under insurance, what matters
are relative values between the capital cost $c_{r}$, electricity-bill price
$p$, and VOLLs. Thus, we set the capital cost at $c_{r}=1$, and scale the
parameters $p$ and VOLLs accordingly.666The range of practical electricity-
bill prices and capital costs of renewables can be quite wide. The US
electricity-bill price is about 0.1-0.3 $/kWh[18]. The capacity cost of PV
solar and onshore wind in 2020 can be about 800-2000$/kWh for 20 years [19],
i.e., about 0.1-0.3$/kWh per day. Thus, if we scale the renewable-energy
capacity cost to $c_{r}=1$, the electricity-bill price $p$ can be in the range
about 0.33-3.

Figure 3: (a) Social-cost reduction under Ins-utility compared with NoIns-
utility; (b) Users’ cost reduction under Ins-utility compared with NoIns-
utility; (c) Under Ins-utility, users’ cost reduction by purchasing insurance
compared with not purchasing.

Figure 4: (a) Social cost reduction under Ins-utility comapred with NoIns-
utility; (b) Type-1 user’s cost reduction under Ins-utility comapred with
NoIns-utility; (c) Under Ins-utility, Type-1 user’scost reduction by
purchasing insurance compared with not purchasing.

In Figures 3 and Figures 4, the horizontal axis denotes the electricity-bill
price $p$. In Figures 3, we fix the uniform distribution of renewable energy
and compare different VOLLs of users. We let $V_{1}+V_{4}=40$,
$V_{4}-V_{1}=\Delta$. Each curve corresponds to a different value of $\Delta$.
In Figure 4, we fix the VOLLs of users at $V_{1}=V_{2}=15$ and
$V_{3}=V_{4}=25$, and compare different levels of uncertainty of the truncated
normal distribution. Each curve corresponds to a different value of the
standard deviation (std).

###  VI-B Benefits of insurance contract

####  VI-B1 Social cost reduction

Comparing the Ins-utility with NoIns-utility, adopting the optimal insurance
can significantly reduce the social cost when the electricity bill price is
moderate. As shown in Figure 3(a), the social cost reduction can be nearly 20%
at $\Delta=0$ and about 14% at $\Delta=10$ when the electricity price is
moderate (around $1<p<1.5$).

####  VI-B2 Users’ cost reduction

Comparing the Ins-utility with NoIns-utility, adopting the optimal insurance
can significantly reduce the users’ cost. As shown in Figure 3(b), Type-1
user’s cost reduction can be up to 17%$\sim$19% when the electricity price is
moderate (around $1<p<1.5$). Type-4 user’s cost can be reduced by over 10% if
the difference $\Delta$ is not too high (e.g., $\Delta=0$ and $10$).

Furthermore, under Ins-utility, users’ costs never increase if they buy
insurance compared with not buying insurance. As shown in Figure 3(c), by
purchasing insurance, Type-1 user’s cost reduction can be up to 23%$\sim$32%
when the electricity price is moderate. Type-4 user’s cost reduction can be
about 15%$\sim$32% (around $1<p<1.5$).

###  VI-C Impact of the electricity-bill price, the type heterogeneity, and
the renewable uncertainty

As we elaborate below, our simulation results show that the largest benefit of
the optimal insurance contract (in terms of increasing social welfare and
reducing users’ costs) is achieved at a medium electricity-bill price together
with a low level of type heterogeneity and a high level of renewable
uncertainty.

####  VI-C1 Electricity-bill price

The maximum cost reduction is achieved at a medium electricity-bill price
(e.g., $1<p<1.5$ in Figures 3). When the price is too high (e.g., $p>2.5$ in
Figures 3) or too low (e.g., $p<0.6$ in Figure 3), the cost reduction becomes
close to zero. The reason is that too high (or too low) a price will lead to
large (or zero) investment of renewables, reducing the benefit of insurance.

####  VI-C2 Type heterogeneity

In Figure 3(a)-(c), the blue curves correspond to the identical VOLL of two
users with $\Delta=0$, the red curves correspond to $\Delta=10$, and the green
curves correspond to $\Delta=20$. A higher $\Delta$ implies a higher
heterogeneity. In all three subfigures, as the difference $\Delta$ increases,
the benefit of the insurance, including the social cost reduction and users’
cost reduction, will go down. The reason is that the utility needs to satisfy
the IC constraint (5) in the insurance design. A higher heterogeneity will
incur an additional cost due to this IC constraint. In contrast, if all the
types are identical, the IC constraint is trivially satisfied.

####  VI-C3 Renewable uncertainty

In Figures 4, a higher variance of the truncated normal distribution will
increase the maximum cost reduction. For example, when the variance of
renewable energy is close to zero (e.g., $\text{std}=0.1$), the maximum
social-cost reduction in Figure 4(a) is close to 0 (below 2%). The higher
uncertainty makes the insurance more important.

##  VII Conclusion

This work proposes the optimal insurance contract design between the utility
and end-users, which can hedge end-users’ risk and incorporate their
heterogeneous reliability preferences in the utility’s planning of renewable
energy. The joint optimization of contract design and renewable energy
planning involves solving a challenging non-convex optimization problem. We
resolve this difficulty by revealing important structural properties of the
optimal solution, using the help from two benchmark problems: the no-insurance
benchmark and the social-optimum benchmark. Compared with the no-insurance
benchmark, we demonstrate that the optimal insurance contract can improve
social welfare and reduce users’ cost. However, the specific performance gain
is affected by a number of factors, such as the electricity-bill price, the
heterogeneity of types, and the uncertainty of renewables. Our future work
will model the random demand and incorporate energy storage.

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## Appendix I. Proof of Theorem 1

We characterize the sufficient and necessary conditions for incentive
compatibility and individual rationality, respectively. Then, we provide the
sufficient conditions.

###  -A Incentive compatibility

Considering any user $i$ of type $k$, the condition of IC requires
$CU_{i}(o_{i}^{k})\leq CU_{i}(o_{i}^{m}),\forall m\in\mathcal{K}$, which leads
to

$\displaystyle\sum_{t}{{\pi_{k}^{t}}}\cdot
D_{i}^{t}\leq\sum_{t}{{\pi_{m}^{t}}}\cdot D_{i}^{t}$
$\displaystyle+\sum_{t}\mathbb{E}\left[{V_{k}^{t}\cdot(D_{i}^{t}-d_{i}^{t})}-V_{m}^{t}\cdot(D_{i}^{t}-d_{i}^{t})\right],\forall
m\in\mathcal{K}.$ (18)

The inequality set above is the sufficient and necessary condition of IC for
user $i$ of type $k$. Considering all the users, these inequality sets make up
the sufficient and necessary conditions of IC for the contract design.

One sufficient condition that makes (18) feasible for all the users is

$\displaystyle\pi_{k}^{t}-\pi_{m}^{t}=\mathbb{E}\left[{V_{k}^{t}\cdot(1-\frac{s^{t}}{D_{a}^{t}})}-{V_{m}^{t}}\cdot(1-\frac{s^{t}}{D_{a}^{t}})\right],$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\forall
t\in\mathcal{T},\forall m,k\in\mathcal{K},$ (19)

which makes (18) always achieve at the equality. This sufficient condition
makes the constraint of IC independent of individual user’s information, which
only depends on the type information. Under this condition, the costs of any
user choosing any types will be the same. Furthermore, (19) is feasible.

###  -B Individual rationality

Considering any user $i$ of type $k$, we have $CU_{i}(o_{i}^{k})\leq
CU_{i}(o_{i}^{o})$ for individual rationality, which leads to

$\displaystyle\sum_{t}{{\pi_{k}^{t}}}D_{i}^{t}\leq\sum_{t}{\mathbb{E}\left[(V_{k}^{t}-p)\cdot(1-\frac{s^{t}}{D_{a}^{t}})D_{i}^{t}\right]}.$
(20)

The inequality set above is the sufficient and necessary condition of IR for
user $i$ of type $k$. Considering all the users, these inequality sets make up
the sufficient and necessary conditions of IR for the contract design.

Considering non-negative premium fees, we have one sufficient condition is

$\displaystyle
0\leq\pi_{k}^{t}\leq{(V_{k}^{t}-p)\cdot\mathbb{E}[(1-\frac{s^{t}}{D_{a}^{t}})]},\forall
t\in\mathcal{T},\forall k\in\mathcal{K},$ (21)

which also makes the constraint of IR independent of individual user’s
information.

To sum up, we have Theorem 1 proved. ∎

## Appendix II. Proof of Proposition 1

We will first prove that the optimal solution is achieved at the lower bound
of (10) and then we prove the solution structures under the three electricity-
bill price conditions respectively.

To begin with, we can rewrite the IC constraint (5) and IR constraint (6) of
Problem-Ins based on the Type 1’s variable $\bm{\pi}_{1}$ into (22) and (23)
as follows.

$\displaystyle\pi_{k}^{t}=\pi_{1}^{t}+{(V_{k}^{t}-V_{1}^{t})\cdot\mathbb{E}_{\bm{\Theta}}\left[1-\frac{s^{t}}{D_{a}^{t}}\right]},\forall
t\in\mathcal{T},\forall k\in\mathcal{K}.$ (22) $\displaystyle
0\leq\pi_{1}^{t}\leq{(V_{1}^{t}-p)\cdot\mathbb{E}_{\bm{\Theta}}\left[1-\frac{s^{t}}{D_{a}^{t}}\right]},\forall
t\in\mathcal{T},\forall k\in\mathcal{K}.$ (23)

Based on (22), we have

$\displaystyle\sum_{t}\sum_{k}\pi_{k}^{t}D_{k}^{t}=\sum_{t}\left(\pi_{1}^{t}D_{a}^{t}+\sum_{k}D_{k}^{t}{(V_{k}^{t}-V_{1}^{t})\cdot\mathbb{E}\left[1-\frac{s^{t}}{D_{a}^{t}}\right]}\right).$
(24)

Based on (23) and (24), we have

$\displaystyle
f^{l}(r)\triangleq\sum_{t}\left(\sum_{k}D_{k}^{t}{(V_{k}^{t}-V_{1}^{t})\cdot\mathbb{E}\left[1-\frac{s^{t}}{D_{a}^{t}}\right]}\right)$
$\displaystyle\leq\sum_{t}\sum_{k}\pi_{k}^{t}D_{k}^{t}$
$\displaystyle\leq\sum_{t}\left(\sum_{k}D_{k}^{t}{(V_{k}^{t}-p)\cdot\mathbb{E}\left[1-\frac{s^{t}}{D_{a}^{t}}\right]}\right)\triangleq
f^{u}(r).$ (25)

### A. The optimal solution is achieved at the lower bound of (10)

We prove that the optimal solution is achieved at the lower bound of (10) by
contradiction. Suppose that the optimal solution is only achieved with
$f^{\text{Ins}}({r}^{*},\bm{\pi}^{*})>\xi$.

  * •

If (23) is not lower-bounded for some hour $t$, we can always reduce
$\pi_{1}^{t*}$ a bit to $\pi_{1}^{t^{\prime}}$ such that
$f^{\text{Ins}}({r}^{*},\bm{\pi}^{\prime})>\xi$. This reduces the objective,
which contradicts the optimal solution.

  * •

If (23) is lower-bounded for all the hours, we can always find
$r^{\prime}>{r}^{*}$ such that $f^{\text{Ins}}({r}^{\prime},\bm{\pi}^{*})=\xi$
because $f^{\text{Ins}}({r},\bm{\pi})\rightarrow-\infty$ as
$r\rightarrow+\infty$. Based on (22), since $s$ is non-decreasing in $r$, the
total premium fee, i.e., the objective will be not be increased, which
contradicts that the optimal solution is only achieved with
$f^{\text{Ins}}({r}^{*},\bm{\pi}^{*})>0$.

Therefore, with the equal constraint of (10), the premium fee
$\sum_{k}\sum_{t}\pi_{k}^{t}D_{k}^{t}$ is equal to

$\displaystyle
g(r)\triangleq-\sum_{t\in\mathcal{T}}p\cdot\mathbb{E}_{\Theta^{t}}[s^{t}]+c_{r}\cdot
r+\sum_{t\in\mathcal{T}}\mathbb{E}_{{\Theta}^{t}}[C_{s}^{t}(r,{\Theta^{t}})]+\xi.$
(26)

### B. Condition $\underline{L}<p<\overline{L}$

We will first solve the optimal solution to Problem-Ins by relaxing the
constraints (22) and (23), and then verify whether the solution satisfies (22)
and (23).

We first relax the constraints (22) and (23) in Problem-Ins. With the equal
constraint of (10), we equivalently minimize $g(r)$ to obtain the optimal
solution. We can easily prove that minimizing $g(r)$ is equivalent to
minimizing the social cost in Problem-SO. We solve the optimal solution
$r^{*}=r^{\dagger}$, which is the optimal capacity solution in Problem-SO, and
the optimal $\sum_{k}\sum_{t}\pi_{k}^{t*}D_{k}^{t*}$ is given by
$g(r^{\dagger})$.

Recall that we have relaxed the constraints (22) and (23). If the optimal
$\sum_{k}\sum_{t}\pi_{k}^{t*}D_{k}^{t*}$ satisfies (22) and (23), i.e., (25),
we have the optimal solution at $r^{*}=r^{\dagger}$. Thus, we check whether
$g^{l}(r^{\dagger})\leq g(r^{\dagger})\leq g^{u}(r^{\dagger})$ is satisfied as
shown in (25), which is equivalent to $\underline{L}\leq p\leq\overline{L}$.

### C. Condition $p\leq\underline{L}$

When $p\leq\underline{L}$, we have $g(r^{\dagger})>f^{u}(r^{\dagger})$. The IC
and IR constraints cannot be satisfied under the capacity solution
$r^{\dagger}$ in SO. In other words, for Problem-Ins, the constraint (23) at
the optimum is not slack for all the hours, which must reach the lower bound
or upper bound for some hours. The key to solving this problem is to discuss
the binding conditions (23). We prove by contradiction that when
$p\leq\underline{L}$, the constraint (23) is upper bounded at the optimum for
all the hours. Note that the condition that (23) is upper bounded for all the
hours is equivalent to that (6) is upper bounded for all the hours and all the
types.

First, we write the total premium fee, i.e., the objective, as a function of
$\bm{\pi}_{1}$ in the following

$\displaystyle\sum_{t}\sum_{k}\pi_{k}^{t}D_{k}^{t}$
$\displaystyle=\sum_{t}\pi_{1}^{t}D_{a}^{t}+f^{l}(r)$ (27) $\displaystyle\leq
f^{u}(r).$ (28)

Second, suppose that at the optimal solution, $\pi_{1}^{t}$ is not upper
bounded for all the hours. We denote the objective
$\sum_{t}\pi_{1}^{t}D_{a}^{t}+f^{l}(r)$ by $f^{m}(r)$ based on the given
optimal binding scenarios. We have $f^{l}(r)\leq f^{m}(r)<f^{u}(r)$. With the
equality constraint (10), we have $f^{m}(r^{*})-g(r^{*})=0$ and $g(r^{*})$ is
the minimal objective value.

Third, we show the properties of some functions. All $f^{l}(r)$, $f^{m}(r)$,
and $f^{u}(r)$ are decreasing in $r$, and $g(r)$ is convex in $r$. Since
$r^{\dagger}$ minimizes the unconstrained $g(r)$, $g(r)$ decreases over
$(0,r^{\dagger})$ and increases $(r^{\dagger},+\infty)$. It is not difficult
to show that $f^{m}(r)-g(r)$, $f^{u}(r)-g(r)$, and $f^{l}(r)-g(r)$ are all
concave in $r$, and they decrease over $(r^{\dagger},+\infty)$.

Now, we construct the contradiction. When $p\leq\underline{L}$, we have
$f^{u}(r^{\dagger})-g(r^{\dagger})<0$, which leads to
$f^{m}(r^{\dagger})-g(r^{\dagger})<0$. Since $f^{m}(r^{*})-g(r^{*})=0$, and
$f^{m}(r)-g(r)$ is negative and decreasing over $(r^{\dagger},+\infty)$, it is
straightforward that $r^{*}<r^{\dagger}$. Since the continuous functions
$f^{u}(r^{*})-g(r^{*})>f^{m}(r^{*})-g(r^{*})=0$ and
$f^{u}(r^{\dagger})-g(r^{\dagger})<0$, we always have a new value
$r*<r^{\prime}<r^{\dagger}$ such that $f^{u}(r^{\prime})-g(r^{\prime})=0$.
Since $g(r)$ decreases over $(0,r^{\dagger})$, we have
$g(r^{\prime})<g(r^{*})$. This contradicts that $g(r^{*})$ is the minimal
objective value.

Finally, with the upper-bounded constraint (23) for all the hours, we easily
have $r^{*}=r^{\ddagger}$, which is the optimal solution to Problem-NoIns.

### D. Condition $p\geq\overline{L}$

This part proof is similar to Subsection.C. When $p\geq\overline{L}$, we have
$g(r^{\dagger})<f^{l}(r^{\dagger})$. The IC and IR constraints cannot be
satisfied under the capacity solution $r^{\dagger}$ in SO. In other words, for
Problem-Ins, the constraint (23) at the optimum is not slack for all the
hours, which must reach the lower bound or upper bound for some hours. We
prove by contradiction that when $p\geq\overline{L}$, the constraint (23) is
lower bounded at the optimum for all the hours.

First, still, suppose that at the optimal solution, $\pi_{1}^{t}$ is not lower
bounded for all the hours. We denote the objective
$\sum_{t}\pi_{1}^{t}D_{a}^{t}+f^{l}(r)$ by $f^{m}(r)$ based on the given
optimal binding scenarios. We have $f^{l}(r)<f^{m}(r)\leq f^{u}(r)$. With the
equality constraint (10), we have $f^{m}(r^{*})-g(r^{*})=0$ and $g(r^{*})$ is
the minimal objective value.

Third, we show the properties of some functions, just following the same
result in Subsection.C.

Now, we construct the contradiction. When $p\geq\overline{L}$, we have
$f^{l}(r^{\dagger})-g(r^{\dagger})>0$, which leads to
$f^{m}(r^{\dagger})-g(r^{\dagger})>0$. Since $f^{m}(r^{*})-g(r^{*})=0$, and
$f^{m}(r)-g(r)$ is positive at $r^{\dagger}$ and decreasing over
$(r^{\dagger},+\infty)$, it is possible that $r^{*}>r^{\dagger}$ or
$r^{*}<r^{\dagger}$.

  * •

Note that if $r^{*}>r^{\dagger}$, $f^{m}(r)-g(r)$ decreases between
$(r^{\dagger},r^{*})$.

  * •

If $r^{*}<r^{\dagger}$, $f^{m}(r)-g(r)$ increases between $(0,r^{*})$.

We have the continuous functions
$f^{l}(r^{*})-g(r^{*})<f^{m}(r^{*})-g(r^{*})=0$ and
$f^{l}(r^{\dagger})-g(r^{\dagger})>0$.

  * •

If $r^{*}>r^{\dagger}$, we always have a new value
$r^{\dagger}<r^{\prime}<r^{*}$ such that $f^{l}(r^{\prime})-g(r^{\prime})=0$.
Since $g(r)$ increases over $(r^{\dagger},r^{*})$, we have
$g(r^{\prime})<g(r^{*})$. This contradicts that $g(r^{*})$ is the minimum
objective value.

  * •

If $r^{*}<r^{\dagger}$, we always have a new value
$r^{*}<r^{\prime}<r^{\dagger}$ such that $f^{l}(r^{\prime})-g(r^{\prime})=0$.
Since $g(r)$ decreases over $(r^{*},r^{\dagger})$, we have
$g(r^{\prime})<g(r^{*})$. This contradicts that $g(r^{*})$ is the minimum
objective value.

Finally, with the lower-bounded constraint (23) for all the hours, it is not
difficult to show that the optimal capacity $r^{*}$ is equal to the optimal
solution of the following concave problem.

$\displaystyle\max~{}$ $\displaystyle r$ s.t. $\displaystyle
f^{\text{No}}(r)-\sum_{t\in\mathcal{T}}{(V_{1}^{t}-p)\cdot\mathbb{E}_{\bm{\Theta}}\left[D_{a}^{t}-{s^{t}}\right]}\geq\xi,$
var: $\displaystyle r\geq 0$

∎

## Appendix III. Proof of Proposition 2

We prove it by discussing the three electricity-bill price conditions
respectively.

If $\underline{L}<p<\overline{L}$, the social cost under Ins-utility is
equivalent to social optimum, which must be no greater than that under NoIns-
utility. If $p\leq\underline{L}$, the social cost under Ins-utility are just
equal to that under NoIns-utility.

If $p\geq\overline{L}$, we will first prove that the optimal capacity under SO
is no greater than the under Ins-utility, which is further no greater than the
under NoIns-utility. Then, we prove the social cost under under Ins-utility is
no higher than that under NoIns-utility.

  * •

(a) We prove $r^{*}\geq r^{\dagger}$. Based on the results in Appendix.II.D,
considering the optimal $r^{*}$ under Ins-utility, we have
$f^{l}(r^{*})-g(r^{*})=0$ and $f^{l}(r^{\dagger})-g(r^{\dagger})>0$. Since
$f^{l}(r)-g(r)$ is concave, there are actually two possible solutions, To
minimize the objective with the lower-bound at (23), we choose the maximum one
which satisfies $r^{*}\geq r^{\dagger}$.

  * •

(b) We prove $r^{*}\leq r^{\ddagger}$. Recall that $r^{\ddagger}$ is the
optimal solution to Problem-Noins. For $r^{*}$,
$f^{\text{No}}(r)-\sum_{t}{(V_{1}^{t}-p)\cdot\mathbb{E}_{\bm{\Theta}}\left[D_{a}^{t}-{s^{t}}\right]}$
is concave in $r$, which approaches $+\infty$ as $r\rightarrow+\infty$. Thus,
$r^{*}$ is achieved at the maximum solution to
$f^{\text{No}}(r)-\sum_{t}{(V_{1}^{t}-p)\cdot\mathbb{E}_{\bm{\Theta}}\left[D_{a}^{t}-{s^{t}}\right]}=\xi$.
Similarly, $r^{\ddagger}$ is achieved at the at the maximum solution to
$f^{\text{No}}(r)=\xi$. Since
$f^{\text{No}}(r)-\sum_{t}{(V_{1}^{t}-p)\cdot\mathbb{E}_{\bm{\Theta}}\left[D_{a}^{t}-{s^{t}}\right]}\leq
f^{\text{No}}(r)$, we have $r^{*}\leq r^{\ddagger}$.

Overall, we have $r^{\dagger}\leq r^{*}\leq r^{\ddagger}$. Note that social
cost increases over $(r^{\dagger},r^{\ddagger})$ following $g(r)$ as shown in
Appendix.II.C, so the social cost under under Ins-utility is no higher than
that under NoIns-utility.

Based on the discussion of all the three conditions of the electricity-bill
price, we complete the proof. ∎

## Appendix IV. Proof of Proposition 3

To prove that users’ total cost under InsR-utility are no larger than that
under NoIns-utility, we show that the NoIns-utility is a sub-case of Ins-
utility depending on the electricity bill price. Specifically, under Ins-
utility, if $p\leq\underline{L}$, the optimal invested capacity $r^{*}$ is
equivalent to that under NoIns-utility. Furthermore, each type’s IR constraint
is upper bounded for each hour, which means that the cost of each user is the
same as the case that he does not not purchase the insurance under the optimal
invested capacity $r^{*}$. Thus, in terms of users’ cost and invested
capacity, Ins-utility is equivalent to NoIns-utility if $p\leq\underline{L}$.
Since Ins-utility minimizes users’ total premium fee (equivalent to total
cost) and $p\leq\underline{L}$ is just a sub case of all conditions, users’
total cost under InsR-utility is always no larger than that under NoIns-
utility. ∎

## Appendix V. Profit-seeking utility

The profit-seeking utility aims to maximize her profit under the insurance
contract as follows.

$\displaystyle\max~{}$ $\displaystyle f^{\text{Ins}}$ (29) s.t.
$\displaystyle\eqref{eq:IC}\eqref{eq:IR},$ (30) $\displaystyle\text{var}:$
$\displaystyle r\geq 0,\bm{\pi}$ (31)

To maximize the profit, it is straightforward that for the IR constraint (6),
each type’s premium fee at each hour will reach the upper bound. With the
upper bound (6), we can easily show that the optimal capacity under the
profit-seeking Ins-utility is equivalent to the following problem.

$\displaystyle\max~{}$ $\displaystyle f^{\text{No}}$ (32)
$\displaystyle\text{var}:$ $\displaystyle r\geq 0,$ (33)

which is exactly the case that NoIns-utility maximizes her profit.

Therefore, if the utility is profit-seeking, providing the insurance and not
providing lead to the same renewable investment and the same profit for the
utility. They also incur the same cost for users as all the IR constraints (6)
reach the upper bound under profit-seeking Ins-utility.


# A Method To Characterize Metalenses For Light Collection Applications

T. Contreras,11footnotetext: Corresponding author. A. Martins  C. Stanford  C.
O. Escobar  R. Guenette  M. Stancari  J. Martín-Albo  B. Lawrence-Sanderson
A. Para  A. Kish  F. Kellerer

###### Abstract

Metalenses and metasurfaces are promising emerging technologies that could
improve light collection in light collection detectors, concentrating light on
small area photodetectors such as silicon photomultipliers. Here we present a
detailed method to characterize metalenses to assess their efficiency at
concentrating monochromatic light coming from a wide range of incidence
angles, not taking into account their imaging quality.

##  1 Motivation

Light collection is essential for detectors in a wide range of applications.
Future particle physics detectors looking for dark matter, studying neutrino
interactions and oscillations, or searching for neutrinoless double beta
decay, would significantly benefit from increased light collection to reach,
and even enhance, their physics goals. Other applications for light collection
detectors include medical physics like positron emission tomography, material
science using photochemistry and mass spectroscopy, and photonics using non-
linear optics.

Metalenses, composed of dielectric nanostructures that focus light like a
lens, shown in Figure 1a, are a novel and versatile technology that could
greatly increase the efficiency of light collection in detectors while being
small, practical, durable, radiopure, and cost-effective [1, 2]. The
nanostructures in a metalens can be made out of a variety of materials and
shapes of subwavelength dimensions, allowing for a wide range of specific
optical properties. These devices are compatible with mass manufacturable
processes that are used in the chip industry, which enables their fabrication
in large scales, making metalenses cost-effective compared to traditional
lenses [3, 4, 5, 6].

With this versatility, metalenses could be paired with light detectors in many
ways to improve light collection. In previous work, we demonstrated that
metalenses can be an attractive solution to increase the light collection of
small photon detectors by a factor of 7, despite using a non-optimized
metalens design [7]. The use of metalenses in large-scale detectors could
provide a solution to increasing the light collection without increasing the
number of photon detecting channels. Metalenses could also be paired with
silicon photomulitpliers (SiPMs) or CMOS devices to improve the photon
detection efficiency [8, 9, 10].

(a)

(b)

Figure 1:  (a) A scanning electron microscope micrograph of a metalenses
designed for 632 nm used for the characterization. The arrangement of
nanopillars of different sizes (from 150 nm to 460 nm) can be seen. (b) A
metalens mounted to be characterized. This metalens was fabricated with only a
thin stripe across, for simplicity in characterization and fabrication, with
the understanding that the metalens would be circularly symmetric.

Metalens development has mainly focused on imaging purposes [1, 2, 11, 12, 13,
6], where the attention was put on the quality of the images produced from
fully illuminating the lenses rather than on the efficiencies of transmitting
and focusing the light for different parameters. When investigating a metalens
for high-efficiency light focusing, the characterization of the optical
properties need to be done for parameters such as incident angle, distance
from the lens, outgoing angle, and field of view. This is also essential to
allow detailed simulations of the impact of metalenses in a large-scale
detector.

In this paper, we present a method to characterize metalenses for light
collection, rather than image quality, for the development of metalenses for
light collection detectors. By measuring the efficiency for a range of
incident angles of light, position from the lens, and angles of outgoing
light, simulations of specific detector geometries and light profiles can
accurately estimate light collection improvements when incorporating
metalenses into light detector designs. We begin this development with an
optimized metalens designed for 632 nm light and compare it to the simulated
response. This wavelength is easier to fabricate and to test this method of
characterization, which can then be used with minimal changes for metalenses
of other wavelengths in future studies.

##  2 Description of the metalens used for the characterization

The metalenses are made of fused silica nanoposts combining two design
approaches, one for the low deflection angle region (phase map design
approach) and one for the higher deflection angle region (supercell design
approach), to optimize the efficiency in each region. The fabrication was
performed using lift-off based electron-beam lithography [6].

In the low deflection angle region (less than $\approx$20∘), we used the
conventional phase map design approach to modulate the transmitted field with
a hyperbolic phase profile [6]. We designed 1200 nm tall nanoposts on silica
with a unit cell size of 500 nm using the phase modulation design, as shown in
Figure 2 (a). Figure 2 (b) shows the post array phase shift and transmittance
as function of the post diameter used to encode the metalens phase profile.
However, the phase map design has a small efficiency when it is used to steer
light at higher deflection angles due to the excitation of higher order modes
when the array symmetry is broken [14]. In other words, for high deflection
angles (greater than $\approx$20∘) the local approximation assumed in this
design is no longer valid and the array phase shift of each post does not
correspond to the phase shift it can provide in the metasurface or metalens.
This is evident in the black dots shown in Figure 2 (d) that shows the
deflection efficiency of beam steering metasurfaces designed using the phase
control method of Figures 2 (a) and (b). Since the conventional nanopost
period is fixed at 500 nm, only supercells with unit cells that are multiples
of 500 nm were calculated. Note that the efficiency drops for deflection
angles higher than 20∘, approximately.

Figure 2:  (a) Phase map design based on 1200 nm tall silica nanoposts. The
unit cell is fixed at 500 nm and the structure is designed to operate at 632
nm. The insets in (a) and (c) show SEM micrographs of some of the
corresponding fabricated structures. The scale bar represents 1 $\mu$m. (b)
Transmission and relative phase shift of the nanoposts in (a) as function of
their diameter. In the design, we constrained the minimum diameter to 150 nm.
(c) Optimized silica-based metagrating structure, which uses a pair of coupled
nanoposts. The inset shows a top view of the structure with the optimized
parameters. The long period $L_{x}$ was fixed according to the diffraction
equation for each deflection angle, and the small period was fixed at 500 nm
in all cases. The operating wavelength is 632 nm. (d) Diffraction efficiency
of the supercell design metagratings (red dots) and of beam steering
metasurfaces designed with the phase map approach (black dots) shown in (a)
and (b).

Such reduction of efficiency at high deflection angles would limit the
numerical aperture (NA) and overall efficiency of our metalens. Thus, to avoid
this issue, we used the supercell design approach with metagratings to steer
light at high deflection angle regions [15]. The metagratings are optimized to
maximize the deflection efficiency at a certain angle using more than one post
per supercell or with a freeform shape [16]. Our design consists of two 1200
nm tall glass coupled nanoposts with rectangular cross-section, as shown in
Figure 2 (c). We fixed the orthogonal period at 500 nm, which is subwavelength
in air for a 632 nm excitation wavelength. The longer period, $L_{x}$ was
chosen using the grating equation to diffract at an angle $\theta$ as shown in
equation 2.1

$L_{x}=\frac{\lambda_{0}}{\sin{\theta}}$ (2.1)

We used a genetic algorithm to optimize the first order efficiency by varying
the remaining geometrical parameters, shown in Figure 2 (c). We constrained
the minimum gap between the posts at a minimum value of 50 nm and the minimum
post size at 200 nm. All calculations were performed using the rigorous
coupled wave analysis method (RCWA) [17, 18]. The red dots in Figure 2 (d)
show the resulting first order efficiency of the optimized structures using
the supercell design approach with metagratings as a function of the
deflection angle. The efficiency of the meta gratings remain higher than 65%
for diffraction angles as high as 32∘, which corresponds to
$\sin\theta\approx$ 0.5. On the contrary, the phase control design efficiency
remains higher than 70% only for deflection angles smaller than 20∘ and
quickly reduces for higher deflection angles. It should be noted that, the
design of metagratings is not practical for small deflection angles as their
periods are inversely proportional to the sine of the deflection angle,
according to equation 2.1. Therefore, in our metalens design we used the phase
control design for deflection angles $\theta$ up to 20∘ of deflection,
corresponding to $\sin\theta$ < 0.36, and the supercell design for deflection
angles larger than 20∘, corresponding to 0.36 < $\sin\theta$ < 0.75, resulting
in a metalens optimized for efficient first order deflection across a wide
range of deflection angles.

The data presented here were taken with a rectangular metalens of 1.2 mm width
and 13.7 mm length, with a focal length of 7 mm and a NA of 0.75, combining
the two designs described and placed on a glass substrate. The optimized
metalens was produced as a rectangle (shown in Figure 1b rather than a
circular metalens to increase speed of prototype production and testing, and
considering a symmetric response can be easily obtained in electron-beam and
photolithographic processes. The width of the rectangular metalens was
determined to completely enclose the width of the light beam, described in
section 3. We simulated a 300 $\mu$m Gaussian beam reaching a metalens on
different positions. Our model can accurately simulate a large area metalens
in a timely manner.

##  3 Characterization methodology and experimental setup

When studying the application of metalenses to particle physics detectors,
which have a wide range of geometries and photon detection arrangements, it is
important to have a detailed characterization of the optical components as a
function of different incoming angles and distances between the lens and the
photodetectors. This will provide the necessary optical parameters needed for
detailed particle physics detector simulations.

In order to fully characterize a metalens, we built an experimental setup
capable of automatically moving a sensor, a metalens, and a light source
linearly in addition to rotating the metalens to change the angle of incidence
of the light on the metalens. The setup can be seen in Figure 3 and is
illustrated in Figure 4.

Figure 3:  Experimental setup developed for the metalens characterization with
the laser diode on the far right, followed by the two pinholes, with the
metalens in the center and a power meter on the left. Arrows next to labels of
the devices in the experiment demonstrate the direction the motors can move
the given device.

A 3 mm $\times$ 3 mm S13370 Hamamatsu VUV SiPM paired with a red 632 nm LED
was used initially, as SiPMs can detect single photons and are often used in
noble element detectors. The VUV SiPM has sensitivity up to 900 nm, thus could
still be paired with the red
LED.222https://hamamatsu.su/files/uploads/pdf/3_mppc/s13370_vuv4-mppc_b_(1).pdf
To compare the results, we also used a Thorlabs S120SV Power Meter paired with
the 632 nm laser diode to provide sufficient power detectable by the power
meter. The power meter and laser diode setup allowed us to get a smaller spot
size on the metalens. We found the results comparable between the SiPM and
power meter setup, and thus continued using the power meter setup as we
optimized the metalens design due to the increases speed of data taking with
the power meter. As such, data presented here was taken only with the power
meter setup.

The light source, at one end of a dark box, is coupled to two pinholes, 200
$\mu$m and 400 $\mu$m, to create a beam of light with a spot size of
approximately 300 $\mu$m on the metalens. The metalens then redirects, or
focuses, light to the plane with the sensor paired with an adjustable slit
shown in Figure 4. The slit allows for alignment of the system between the
sensor center and the beam.

Figure 4:  Diagram of the experimental setup and power meter output for three
different positions of the light source relative to the metalens. The light
source (laser diode) is collimated by two pinholes forming a beam spot size of
300 $\mu$m on the metalens, which focuses the light onto a photon detection
device (power meter). An adjustable slit is in front of the photon detection
device, which is used for alignments and left completely open for data taking.
The light path is illustrated in red, where the 0th and 1st diffraction order
of the metalens are shown.

The experimental setup includes PRUSA stepper motors connected to Thorlabs
optics boards with 3D printed pieces for mounting.333https://www.prusa3d.com/
Three motors are assigned to linearly move the sensor, metalens, and light
source, and one motor to rotate the metalens. Each motor is connected to its
own easy driver, with all motors connected to one Arduino board. The stepper
motors allow for fine-grained linear movement down to 0.1 mm and angle
rotations down to 0.225 degrees. A Python-based software package was developed
to easily control each motor, by using the PyFirmata package to connect to the
Arduino board. The developed software package includes automatic alignments of
the system of sensor, metalens, and light source as well as the software for
data taking. With this software, we are able to move all motors and measure
the power from the power meter remotely. This allowed us to completely align
the system automatically, by scanning across the lens to find the center of
the metalens as well as the angle relative to the light beam. While aligning,
continuously updating plots give a live view of the varying power of the power
meter as it is moved perpendicular to the light beam.

The slit is only used for alignment, as the slit cannot be placed close enough
to the power meter such that it does not block the light reaching the power
meter at higher deflection angles. The slit is left open after aligning and
while taking data with the power meter. Thorlabs Optical Power Monitor
monitors the power meter, and was used to measure the average power from the
metalens-laser diode beam by averaging 15 power measurements found over 0.15
seconds.

While taking data, the light source stayed in place, focused by the pinholes
to approximately 300 $\mu$m on the metalens, creating a diffraction pattern on
a perpendicular plane to the lens. The power meter and metalens then move in
small steps of 0.2 mm and 1mm, respectively, across this light pattern. The
zeroth order of this pattern is the light going straight through the metalens.
The first order is the desired angle in which the metalens is optimized, as
shown in Figure 5 where the first order is the largest peak of each waveform,
and the deflection angle is determined from the power meter position combined
with the metalens offset relative to the light beam. A reference measurement,
used to normalize the metalens output, was taken with the light going straight
through the bare glass substrate.

(a)

(b)

Figure 5:  (a) Waveforms showing the light intensity seen by the power meter,
with zero representing the power meter centered with the light source. Each
waveform shows the response of the metalens when offset relative to the light
beam spot (metalens and light beam centered in blue, metalens offset 2 mm from
center in orange, 3 mm from center in green). The zeroth order, where the
light goes straight through the lens, is at the 0 position, while the first
order shifts based on the position of the light source on the metalens. (b) A
light map of the entire data set. The first order (m= -1 , focusing) is
clearly seen as the brightest band at an oblique angle, while the zeroth order
is the vertical line. The m=+1 order (diverging) can be seen as the lighter
tilted band. The overlay lines show the expected positions for the diffraction
orders according to the grating equations 2.1.

The orders were determined by their distance from the zeroth order, which is
always centered at zero, shown in Figure 5. The peak position of the m-th
diffraction order on the SiPM plane ($d_{m}$) can be found by

$d_{m}=d_{0}\tan{(\theta(r,m)+\theta_{0})}$ (3.1)

where $d_{0}=10cm$ is the distance between the power meter and the metalens,
$\theta_{0}$ is the rotation angle of the metalens, r is the metalens offset
and $\theta(r,m)$ is the diffraction angle, which for a focal length $f$ is
given by [19]

$\theta(r,m)=\arcsin{(\frac{mr}{\sqrt{r^{2}+f^{2}}}-\sin\theta_{0})}$ (3.2)

##  4 Results

The response of the metalens for various diffraction orders can be seen in
Figure 5a, with each line representing a different metalens offsets relative
to the light beam. With the light beam centered on the metalens, we get only
the zeroth order, shown in blue. As we move the metalens farther from the
light beam, we can clearly see the 1st order that the metalens is optimized
for as the largest peak, with the 0th and -1st order as small peaks. Figure 5b
shows a light map of the metalens, where each horizontal slice represents a
waveform like in Figure 5a. The black, blue, white, red and green lines mark
the position of the -2,-1,0,1,2 orders, respectively, according to equation
3.1.

The diffraction efficiency from a given order was found by taking the maximum
of that order’s peak, which represents the total power of that given order,
and dividing by the maximum of the reference waveforms peak, which is the
total incident power on the metalens. The reference waveform was taken using
only the substrate used by the metalens. By finding the efficiency of a given
order across the entire metalens, we can see the response of the metalens,
shown in Figure 6b. The measured and simulated efficiencies match quite well,
giving us confidence in the simulations to produce the metalens response. We
then repeated these measurements with the metalens at different incident
angles from the incoming light source, shown in Figure 7. The efficiency of
the metalens decreases with increasing angle of incidence.

Figure 6:  Light transmission efficiency measured (solid lines) and simulated
(dashed lines) for the 0th (blue) and 1st (black) orders, in function of the
metalens offset from the light source. The top axis relates this offset to the
resulting deflection angle to the first order. The region marked by horizontal
lines represents the area where the orders overlap, so no distinction can be
made between orders. Top: Metalens is perpendicular to the light source.
Bottom: Metalens is at a 20-degree angle relative to the light source. Figure
7: The light transmission efficiency of the 1st order as a function of the
incidence angle. The colored lines show the four metalens offsets relative to
the light source.

##  5 Discussion

The characterization described here provides a powerful method of measuring
the efficiency of metalenses across the surface of the lens and as a function
of light incidence angle. Combined with the focal length and NA, these provide
the full set of optical parameters needed to fully simulate the response of
these devices in a particle physics detector environment.

With the use of stepper motors, we were able to fully automate the system to
characterize the light collection as a function of angle of refraction, as
well as angle of incoming light. Typical measurements in the metalens
community fully illuminate the lens and measure the focusing efficiency,
lacking the response measurements necessary to use metalenses in particle
physics detectors. Our method provides a full, low cost, and local
characterization of the metalens by using the stepper motors to take automated
measurements of the angular and position response. We have demonstrated the
effectiveness of this characterization, as the ease of data taking allowed us
to quickly optimize a metalens design for 632 nm light, used in this setup.

With this method, we found the measured and simulated efficiency matches quite
well, giving us confidence in the simulations to produce the metalens
response. This is another major step in using metalenses in light collecting
detectors, which would require simulations to test the use of metalenses in
specific environments, light profiles, and detector geometries. Future work
includes measuring the efficiency with metalenses designed for UV, VUV and
visible light, where this method for characterization can be used after
replacing with the appropriate metalens and light source.

## Acknowledgments

Part of this research was funded by the Sloan Foundation under a 2021 Alfred
P. Sloan Research Fellowship. R. Guenette, T. Contreras are partly supported
by ERC-2020-SyG-951281. Fermilab participation was supported by a Laboratory
Directed Research and Development (LDRD) grant L2021.011. J. Martín-Albo is
supported by the Ramón y Cajal program (ref. RYC2021-033265-I) funded by the
Spanish MCIN/AEI/10.13039/501100011033 and by the EU (NextGenerationEU/PRTR).

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