is short=IS, long=In-sample, oos short=OOS, long=Out-of-sample, tso short=TSO,
long=Transmission System Operator, tsos short=TSOs, long=Transmission System
Operators, jcc short=JCC, long=Joint Chance-Constraint, jccp short=JCCP,
long=Joint Chance-Constrained Program, fcrd short=FCR-D, long= Frequency
Containment Reserve - Disturbance, fcrn short=FCR-N, long=Frequency
Containment Reserve - Normal operation mfrr short=mFRR, long=Manual Frequency
Restoration Reserve ler short=LER, long=Limited Energy Reservoir ev short=EV,
long=Electric Vehicle cvar short=CVaR, long=Conditional Value at Risk var
short=VaR, long=Value at Risk soc short=SoC, long=State of Charge lp short=LP,
long=Linear Program milp short=MILP, long=Mixed-Integer Linear Program

# Aggregator of Electric Vehicles Bidding in Nordic FCR-D Markets: A Chance-
Constrained Program

Gustav A. Lunde‡, Emil V. Damm‡, Peter A.V. Gade*†, Jalal Kazempour*
*Department of Wind and Energy Systems, Technical University of Denmark, Kgs. Lyngby, Denmark
†IBM Client Innovation Center, Copenhagen, Denmark
‡ Emil and Gustav are equal contributors to this work and designated as co-
first authors  Email addresses<EMAIL_ADDRESS>(E.V. Damm),
<EMAIL_ADDRESS>(G.A. Lunde<EMAIL_ADDRESS>(P.A.V. Gade<EMAIL_ADDRESS>(J. Kazempour).

###### Abstract

Recently, two new innovative regulations in the Nordic ancillary service
markets, the P90 rule and LER classification, were introduced to make the
market more attractive for flexible stochastic resources. The regulations
respectively relax market requirements related to the security and volume of
flexible capacity from such resources. However, this incentivizes aggregators
to exploit the rules when bidding flexible capacity. Considering the Nordic
ancillary service fcrd, we consider an aggregator with a portfolio of ev using
real-life data and present an optimization model that, new to the literature,
uses jcc for bidding its flexible capacity while adhering to the new market
regulations. Using different bundle sizes within the portfolio and the
approximation methods of the jcc, ALSO-X and cvar, we show that a significant
synergy effect emerges when aggregating a portfolio of ev, especially when
applying ALSO-X which exploits the rules more than cvar. We show that ev
owners can earn a significant profit when participating in the aggregator
portfolio.

###### Index Terms:

Synergy effect, demand-side flexibility, EV-chargers, joint chance
constraints, ancillary services

##  I Introduction

###  I-A Motivation

A new regulation in the Nordic area has emerged, called the P90 rule, which
specifically addresses how fluctuating units such as ev can provide ancillary
services by allowing some level of uncertainty in the capacity bid. The P90
rule has the potential to increase liquidity in these markets by lowering
barriers to entry for stochastic flexible resources. For example, fcrd has
seen soaring prices these past few years, even allowing for a pay-back time of
almost one year for a one MW/MWh battery. If fluctuating demand or production,
such as grid-connected ev, can participate with their flexibility, the tso can
lower their costs for fcrd procurement while ev owners can benefit monetarily.
To enable even higher fcrd capacity bids, aggregators can look to exploit the
P90 rule and the emerging synergy of pooling ev. In this paper, the synergy
effect thus refers to the increase in bidding when aggregating ev.
Furthermore, additional requirements have been approved for ler units (such as
ev), thus significantly impacting the available flexible capacity for
aggregators when bidding for such a portfolio of ev.

The importance of flexibility has increasingly been emphasized in recent years
as the tso look into a future with huge growth in both power demand and
renewable production with varying and unpredictable characteristics, likely
leading to more frequent and severe frequency disturbances in the power grid
[1, 2]. Accessing the flexibility of already existing infrastructure, such as
flexible demands, can help undertake the green transition and make it more
efficient [3]. Demand-side resources constitute a significant amount of
unrealized flexibility [4]. However, flexible demand has proved challenging to
integrate into ancillary services markets, both due to technical challenges
related to their baseline uncertainty [5], but also market and business
barriers [6].

In an effort to accommodate and incentivize flexible demand and production to
deliver ancillary services, the Nordic tso, including the Danish tso
Energinet, are revising their prequalification regulations. In Denmark,
Energinet has invented the aforementioned P90 rule for all ancillary services,
but also the ler classification scheme for specific frequency services (e.g.,
fcrd) [7]. They respectively allow for uncertainty related to the reserved
capacity bids and lessen the duration requirements for activation which
historically have proven to be two significant barriers for market entry of
fluctuating units such as ev [8].

Consequently, providing flexibility has rapidly become a hot topic in the
electricity-consuming industry. In particular, the fcrd service is
experiencing significant traction, with the prequalified capacity more than
doubling through 2023 summed for both up and down-regulation [2, 9]. Prices in
fcrd up and down-regulation have skyrocketed and since only negligible energy
delivery is expected, it serves as an extremely attractive market right now in
the Nordics (as opposed to tertiary reserves like mfrr). Notably, companies
are rapidly entering this market, but mainly with batteries.

However, there is a big potential for an aggregator to exploit the P90 rule
and ler classification to bid grid-connected ev into fcrd which is amplified
as a synergy effect upon aggregation. The focus of this paper is indeed to
quantify this synergy effect for ev in fcrd by incorporating these new market
rules in an optimization framework. To this end, a case study using real-life
data of grid-connected ev is used to illustrate their synergy effect for fcrd
bidding.

###  I-B Research questions and our contributions

In this paper, we will specifically be investigating how ev, categorized as
ler units, can provide fcrd bids in DK2111Denmark has two synchronous areas,
DK1 for the western part connected to continental Europe and DK2 for the
eastern part connected to the Nordic area. by directly incorporating and
exploiting the P90 rule from Energinet. The focal point of the paper will be
to answer the two following research questions: Given the current market
regulation for ancillary services in the Nordic area, (i) how can an
aggregator with a portfolio of ev participate in the fcrd market? And, (ii)
using real data from ev, how can a potential synergy effect of aggregating ev
be exploited for fcrd bidding?

To answer these questions, an optimization approach using jcc to represent the
P90 rule has been developed, thus taking risk into account when bidding. In
addition, the paper explores and quantifies the significance of the emerging
synergy effect when aggregating EVs in bigger portfolios. The synergy effect,
in this study, is defined as the added value achieved in terms of increased
fcrd capacity bids by bundling ev in bigger portfolios as opposed to
individual or smaller clusters while ensuring that the reliability, i.e.,
security of supply, is kept in accordance with the P90 rule.

###  I-C Literature review

A vast amount of literature explores the feasibility of demand-side
technologies to provide much-needed flexibility to the power grid through
engagements in the power balancing market [10, 11, 12, 13]. The references
[10, 11] discuss the general concept of demand response and outline market
requirements, identified barriers, and the role of aggregators yet, neither of
them quantifies the value of participation from an entity perspective. More
concretely, [12] looks into the optimal strategy of a centralized battery
system participating in multiple ancillary service markets and quantifies the
economic viability. The paper proposes an optimization approach but, the
battery system is sorely used for providing flexibility thus lacking the
uncertainty of another primary usage. Similarly, [13] has been made using ev
but here the potential synergy among demand-side assets has not been
investigated. Also, the benefits of utilizing demand-side flexibility have
been investigated extensively in various other aggregation contexts and market
settings such as for energy communities [14, 15], microgrids [16, 15], and
prosumer markets [17].

However, neither of the existing literature has been looking at the recent
market changes with the P90 which intends to improve the incentive for demand-
side flexibility (and production flexibility) participation today.
Furthermore, none has specifically tried to explore and exploit the P90 rule
within an optimization framework. Thus. although many have looked into the
flexibility of ev for ancillary services [13, 18, 19], none with this purpose
of maximizing capacity deliverance.

The same goes for fcrd which is a well-examined service to the literature. For
example, in [20] the economic potential of utilizing a battery system in,
among others, the fcrd market is being quantified. And in [19] which, like
this study, has developed a day-ahead bidding strategy via. a stochastic
optimization model to assess the value of ev participating in the Nordic FCR
markets (includes fcrd). In this study, public charges have been used.
However, like the P90 rule, the FCR exclusive ler classification is brand new
and has yet to be seen represented in the models representing the market
conditions in the literature, up until now.

Various publications have explicitly or indirectly showcased that aggregating
fluctuating assets can generate a synergy effect leading to e.g. higher
profits as aggregating can help increase the capacity quantity and
predictability of available flexibility [21, 11]. For example in [22] it is
proven in a bidding context that a coalition of identical flexible assets,
here operating on the manual Frequency Restoration Reserve market, always will
be incentivized to stay in the coalition as they can increase their respective
profits. However, what has not been covered in the literature yet is the
potential for utilizing this well-documented synergy effect to progressively
exploit the market regulation in an optimization approach, as the P90 rule
allows for deliberately taking risks.

The addition of the P90 rule also imposes the possibility of introducing jcc
in the mathematical model which is new to the literature. However, in another
power system context, they are commonly used when optimizing power flows [23,
24, 25]. Here they are used to incorporate the uncertain penetration coming
from renewables. Furthermore, due to the non-convexity induced by chance
constraints, the references utilize approximation methods to obtain a
tractable optimization problem. Here CVaR is used in [24, 25] whereas both
CVaR and ALSO-X can be found applied to a generic chance constraint problem in
[26].

With the identified gaps in the literature, this paper uses real ev data, to
investigate and quantify the possibility of an aggregator exploiting the
synergy effect imposed by aggregation to enable larger bids which have
recently been made possible by the introduction of the P90 and ler market
regulations.

###  I-D Paper organization

The rest of the paper is organized as follows. Section II gives an overview of
the fcrd market, incumbent regulations of P90 and ler, and the ev data used.
Section III describes the simulation setup for the case study, i.e.,
assumptions, sample generation, and an exact definition of the synergy effect
in this context. Section IV introduces the mathematical optimization model
incorporating the P90 rule and ler classification as a jcc. This section also
introduces two approximations of the jcc, i.e., cvar and ALSO-X. Finally,
Section V shows and discusses the results of the case study while Section VI
concludes.

##  II Overview

In this section, we start by introducing the fcrd market with corresponding
definitions of the P90 rule and ler units and subsequently provide an
appropriate objective function for optimizing fcrd earnings. Second, we
illustrate and define the flexibility of a portfolio of ev with respect to the
ler regulation using real data222Kindly provided by the ev charge box company
Spirii..

###  II-A fcrd

The fcrd market handles large frequency disturbances between 49.5 to 49.9Hz
and 50.1 to 50.5Hz and requires a rapid frequency response from fcrd providers
(as fast as 2.5 seconds) [7]. Such frequency deviations happen rarely and they
typically last for very short durations. Thus, total energy delivery upon fcrd
activation is nearly negligible. For example, a 1MW bid of both up and down-
regulation for all hours from March 24 (2022) until March 21 (2023) yields
3.25MWh ($\uparrow$) and 4.05MWh ($\downarrow$) activation energy,
respectively, equaling only 0,037% ($\uparrow$) and 0,046% ($\downarrow$) of
the total bid capacity. Hence, fcrd provision is attractive for an aggregator
of ev as activation will not significantly influence their operation schedule
or quality of service.333Frequencies used in this paper are from [27] and have
been down-sampled to a minute resolution based on the maximum and minimum grid
frequencies recorded on the millisecond level within that given minute.

fcrd is an asymmetrical hourly reservation market partitioned into two
distinct markets. There is a market for upward regulation and another for
downward regulation. The historical hourly reservation prices are shown in
Figure 1, showing a market with very volatile and high prices. There are long
periods with extreme prices, e.g., fcrd down for most of 2023.

Figure 1: Hourly prices for the fcrd down and up markets from January 1, 2022,
to February 29, 2024 [28]. Only the grey area is used in this paper for the
case study.

The fcrd auction is composed of an early and late market, both cleared the day
before operation, D$-1$, using uniform pricing444Before February 2024, a pay-
as-bid scheme was used. . As shown in Figure 2, we only consider the early
market, where capacity bids, $(c_{h}^{\uparrow},c_{h}^{\downarrow})$, are
submitted each for hour $h$ of the following day. These bids are based on
available is information, $\omega\in\Omega$, of the ev portfolios flexibility,
$F_{m,\omega}$ for every minute, $m$. On the operation day $D$, the activation
energy, $A_{m}$, must be delivered according to the realized frequency
deviations and the realized flexibility of the ev portfolio, $F_{m}$. Finally,
participants are remunerated at fcrd capacity price, $\pi_{h}$, and penalized
according to missed activations and excess bids,
$(p_{h}^{\uparrow},p_{h}^{\downarrow})$, at price $\lambda_{h}$.

FCR-D (early) reservation bids$c_{h}^{\uparrow},c_{h}^{\downarrow}$
$(F_{m,\omega})$D$-$1(0:30am)FCR-D activation($A_{m},F_{m}$) D(12pm – 12am)Ex-
post Settlement$p_{h}^{\uparrow},p_{h}^{\downarrow}$
$(\pi_{h},\lambda_{h})$D+1(12pm)

Figure 2: Timeline of fcrd markets in DK2, Denmark. Notice that only the early
market is considered here. Optimization variables are denoted with lowercase
letters, while uppercase letters are used for parameters. D-1: capacity bids
$(c_{h}^{\uparrow},c_{h}^{\downarrow})$ with is flexibility $F_{m,\omega}$. D:
Activation energy $A_{m}$ with realized oos flexibility $F_{m}$. D+1: ex-post
calculation of penalties $(p_{h}^{\uparrow},p_{h}^{\downarrow})$ at penalty
prices $\pi_{h},\lambda_{h}$.

###  II-B P90 and ler definitions

For conventional units bidding into fcrd, full activation should be possible
continually for two hours. For ler units such as ev, the activation duration
is relaxed to twenty minutes. However, for ler units, 20% of the submitted bid
must be reserved in the opposite direction as follows:

######  Definition 1 (LER [29])

[…]E.g., If you wish to prequalify 1 MW for fcrd upwards, you must reserve 0.2
MW in the downwards direction for NEM as well as 20 minutes of full fcrd
upwards delivery, or 0.33 MWh of energy.

Consequently, ev are considered as ler units for downward regulation, but not
for upward regulation purposes (as they can not discharge to the grid). A
successful bid thus constitutes an energy reservoir allowing for the downwards
activation for 20 minutes continuously, with upwards flexibility being larger
than the upwards bid and 20% of the downwards bids, while the downwards
flexibility should also be larger than the downwards bid.

Furthermore, Energinet invented the P90 rule specifically for fluctuating
resources with stochastic demand or production (such as ev, wind and solar,
etc.):

######  Definition 2 (P90 rule [29])

[…]This means, that the participant’s prognosis, which must be approved by
Energinet, evaluates that the probability is 10% that the sold capacity is not
available. This entails that there is a 90% chance that the sold capacity or
more is available. This is when the prognosis is assumed to be correct. The
probability is then also 10%, that the entire sold capacity is not available.
If this were to happen, it does not entail that the sold capacity is not
available at all, however just that a part of the total capacity is not
available. The available part will with a high probability be close to the
sold capacity.

Hence, Definition 2 allows aggregators to bid their 10% quantile of forecasted
flexibility. Additionally, Energinet has issued a non-pursuable buffer of 5%,
meaning aggregators are firstly excluded from the market when there generally
is less than an 85% probability that the sold capacity is not available.
Definition 2 is evaluated as the average over at least 3 months of data.

###  II-C ev flexibility

This study has been provided with real-life data of 1,400 ev charge boxes
located at residential homes in Denmark. The data is recorded from March 24,
2022, to March 21, 2023. It is assumed that each charge box is only utilized
by the same ev. On average, measurements were registered every 2.84 minutes,
from which minute resolution consumption profiles have been interpolated for
each ev. The ev consumption serves as the baseline for capacity estimates. In
Figure 3 the baselines of two different portfolio sizes of ev are illustrated
with their maximum charging power for a given hour. Based on that, the
available upward $F^{\uparrow}$ and downward $F^{\downarrow}$ flexibility can
be derived for each minute. The figure showcases that downward regulation is
provided by increasing power consumption while decreasing power consumption
delivers upward regulation. Moreover, subject to the baseline consumption, the
maximum additional energy that can be consumed by the ev for the upcoming 20
minutes, without saturating the soc or violating the power rate, is found,
i.e., its energy flexibility $F^{\text{E}}$. Here, we note that the dataset
only contains information regarding the SoC of the specific charging sessions,
meaning the aggregator of ev does not know the SoC of the connected vehicle.
We assume that every charging session concludes with their being 10% of the
total session’s charged energy left on the EV’s battery, [30], which is
accessible for deliverance downward flexibility. We simple refer to
flexibility as the three-tuple $F=(F^{\uparrow},F^{\downarrow},F^{\text{E}})$.

Figure 3(a) illustrates how a single ev cannot provide any flexibility when it
is not connected. Figures 3(b) and 3(c) show the flexibility contained in a
portfolio of 1400 ev with and without ler constraints as per Definition 1. In
Figure 3(c) the upward flexibility is prioritized to facilitate the downward
capacity, as it can enable five times as much capacity by doing so compared to
bidding it into the upward market. Therefore, most of it can be seen to be
dissolved when compared to Figure 3(c). Moreover, a substantial amount of the
downward flexibility is unavailable at the beginning of the hour, as the
discrepancy between upward and downward flexibility is too big.

(a) One ev without LER req.

(b) 1,400 ev without LER req.

(c) 1,400 ev with 20% LER req.

Figure 3: The consumption of different portfolio sizes of ev either with or
without the 20% LER requirements applied for a given sample hour.

Figure 4 shows how the flexibility is distributed throughout the day for a
portfolio containing all 1400 ev. Each data point represents a given minute’s
flexibility. The figure illustrates the bimodal nature of residential home
charging with ev primarily charging in the evening and night.

The conditional cumulative distribution function describes the associated
certainty of an aggregator having a flexibility higher than a given level.
Generally, there is a larger spread in the distribution for the upwards
flexibility, as shown in Figure 4(a), compared to the two other distributions,
as illustrated in Figures 4(b) and 4(c). For example, at midday for upward
regulation, there is a large right-tailed skewness in the data. Further, there
is a large discrepancy between the available upward and downward flexibility,
meaning the primary flexibility of ev consists of their ability to increase
consumption. To compare the energy flexibility with the momentarily up and
downwards, the energy can be converted into power by dividing it (kWh) by the
duration of 20 minutes ($\frac{1}{3}$h). By doing so, two elements become
apparent: (i) the energy flexibility will always be lower than the momentarily
downward flexibility, hence making the constraint on $F^{\downarrow}$
redundant, and (ii), the energy flexibility is generally more than five times
larger than the upward flexibility, meaning the binding ler requirement will
presumably be the scarcity of upward flexibility. For notation ease,
$F_{m}^{E}(\mathcal{\omega})$ is represented by its power property in the
following mathematical models, while the constraint $F^{\downarrow}$ is for
kept for consistency with Energinet’s documentation.

(a) Upwards flexibility $F^{\uparrow}$

(b) Downwards flexibility $F^{\downarrow}$

(c) Energy flexibility $F^{\mathrm{E}}$

Figure 4: Conditional cumulative distribution of available flexibility for a
portfolio of 1,400 ev from March 24, 2022, to March 21, 2023.

##  III Simulation setup

This section describes the simulation setup used to investigate how an
aggregator of ev can bid into fcrd in this paper. First, we revisit the time
sequence for decision-making in our model, along with a cross-validation
scheme to evaluate the model. Further, sampling of ev flexibility used for
bidding is presented. Lastly, we define the synergy effect in terms of two
metrics from which the aggregator’s bidding strategy is evaluated.

The code is publicly available at [31], although the ev data is proprietary.

###  III-A Evaluation strategy

We simulate the fcrd market in the three stages as previously illustrated in
Figure 2. In the first stage, D$-1$, an aggregator makes upward and downward
capacity bids $(c_{h}^{\mathrm{\uparrow}}$,$c_{h}^{\mathrm{\downarrow}})$, for
each hour of the upcoming day, $D$. The bids are generated using samples
$\omega\in\Omega$ of the uncertain available flexibility
$F_{m,\omega}=(F^{\uparrow}_{m,\omega},F^{\downarrow}_{m,\omega},F^{\text{E}}_{m,\omega})$.
A unique distribution of the available flexibility to each minute $m$ is thus
made prior to the first stage from which $|\Omega|$ samples are drawn. The
samples are drawn randomly without replacement for the first minute, while the
samples for all other minutes depend on the day sampled in the previous
minutes thus maintaining the correlation between minutes. The number of
samples $|\Omega|$ is set to 216 according to [32] elaborated in Appendix A.
Accordingly, all samples are assumed to be equiprobable.

As mentioned, stages two and three are merged into one stage. Here the set of
uncertain parameters for day $D$ is realized, i.e., required activation power
and realized available flexibility $F_{m}$. For all minutes, the bids are
evaluated by assessing whether activation could be met, and if the full bid
was possible to deliver. Thus, we define the maximum activation not met for
both up and down-regulation for each hour as the penalty quantities
$(p_{h}^{\uparrow},p_{h}^{\downarrow})$. Energinet defines the penalty cost as
the substitution cost of replacing the missed activation. However, as no
historical register of the replacement cost exists, the penalty costs
$(\pi_{h}^{\uparrow},\pi_{h}^{\downarrow})$ are assumed to be five times the
reservation prices $(\lambda_{h}^{\uparrow},\lambda_{h}^{\downarrow}$) which
are historical volume-weighted average prices [28]. For simplicity, the
profits/costs related to the energy delivery are considered negligible due to
the low activation rate of fcrd. Accordingly, the objective of the aggregator
is:

$\displaystyle\underbrace{c^{\uparrow}\pi^{\uparrow}+c^{\downarrow}\pi^{\downarrow}}_{\text{Reservation
payment}}-\underbrace{\lambda^{\uparrow}p^{\uparrow}-\lambda^{\downarrow}p^{\downarrow}}_{\text{Penalty
cost}}$ (1)

A three-fold cross-validation scheme is used to obtain oos results, where 363
days are randomly divided into three folds. The training fold consists of is
flexibility, $F_{m,\omega}$, from which bids
$(c^{\uparrow}_{h},c^{\downarrow}_{h})$ are derived using an optimization
model (as explained in Section IV) while the test fold is used to evaluate the
bids on realized portfolio flexibility, $F_{m}$, activation power, and fcrd
prices. Hence, for each iteration in the cross-validation scheme, only one
unique bid for up and down-regulation for each hour is made and evaluated on
121 days of the testing fold.

We note that this choice of cross-validation scheme is used to determine the
historical performance of an aggregator bidding into fcrd for a portfolio of
ev, which should be held up against a lookback strategy using, e.g., only most
recent days in a rolling horizon manner. However, we emphasize that this paper
is intended to provide an optimization model for how an aggregator can exploit
the P90 rule for ler units, and not to provide the best possible model (which
we leave for future work). To this end, a regular cross-validation scheme is
suitable.

###  III-B Synergy effect

In this study, we hypothesize that the synergy effect of aggregating ev
increases the quantities of flexibility bids for fcrd. To validate this
hypothesis, two metrics are used. First, the aggregator profit as stated in
(1), and second, the percentage of total flexibility bid, defined as

$\displaystyle\text{Utilized bidding
capacity}=\frac{c^{\downarrow}+c^{\uparrow}}{F^{\downarrow}+F^{\uparrow}}$ (2)

An increase in both metrics thus indicates a synergy effect. Furthermore, a
third metric is used to measure the compliance with the P90 rule, i.e., how
often bids are higher than the realized flexibility (also called frequency of
overbidding):

$\displaystyle\text{Frequency of overbidding}=\frac{1}{H\times
N}\sum_{m,h}y_{m,h}$ (3)

Here, $y_{m,h}$ takes a value of one whenever an overbid occurs and zero if
not, while $H$ denotes the number of hours and $N$ the number of minutes being
tested for in the oos fold.

##  IV Mathematical modeling

In this section, we present the optimization problem that provides an optimal
bidding strategy for an ev aggregator participating in fcrd adhering to the
P90 rule using a jccp. First, a compact formulation of the problem is
presented, followed by two reformulations: (i) ALSO-X [33] and (ii) CVaR [34].
The optimization problem is a one-stage stochastic program with uncertain
power consumption of the ev portfolio, and therefore uncertain flexibility.

###  IV-A General mathematical formulation

As mentioned, only capacity bids are considered in this study since activation
energy is negligible. We assume an ev portfolio can deliver the required
frequency response as per the regulations [29]. The problem can thus simply be
represented in a one-stage stochastic optimization model with a jcc
representing the P90 rule with respect to available flexibility and ler
requirements:

$\displaystyle\underset{(c^{\downarrow},c^{\uparrow})\geq 0}{\text{Max}}\quad
c^{\downarrow}+c^{\uparrow}$ (4a) s.t.
$\displaystyle\mathbb{P}\left(\begin{array}[]{@{}l@{}}\frac{1}{5}c^{\downarrow}+c^{\uparrow}\leq
F_{m}^{\uparrow}(\mathcal{\omega}),\quad\forall m\\\ c^{\downarrow}\leq
F_{m}^{\downarrow}(\mathcal{\omega}),\quad\forall m\\\ c^{\downarrow}\leq
F_{m}^{\mathrm{E}}(\mathcal{\omega}),\quad\forall m\end{array}\right)\geq
1-\epsilon$ (4e)

Problem (4) looks to utilize as much of the ev’ flexible capacity as possible,
and maximizes upward and downward flexibility. For simplicity, the Problem is
decomposed per hour as there are no coupling constraints. The P90 rule applies
across all ler requirements as seen in the jcc in (4e). It expresses that
there should be at least $1-\epsilon$ probability that fcrd bids,
$(c^{\mathrm{\uparrow}},c^{\mathrm{\downarrow}})$, adhere to all ler
requirements as defined in Definition 1, for all minutes $m$ within a given
hour. The first ler constraint states that the available upward flexibility,
$F_{m}^{\uparrow}(\mathcal{\omega})$ in a given sample must be greater than
the upward bid plus 20% of the downward bid for all minutes. Likewise, the
second and third constraints state that the available downward flexibility,
$F_{m}^{\downarrow}(\mathcal{\omega})$, and energy flexibility,
$F_{m}^{E}(\mathcal{\omega})$, in a given sample, respectively, have to be
greater than the downward bid for all minutes. The safety level $\epsilon$ is
set to 0.1 as per Definition 2, hence allowing for a 10% risk of violating any
of the three ler requirements. Thereby, (4e) fully adheres to the regulations
for bidding uncertain demand into fcrd. To our knowledge, this is the first
attempt at explicitly modeling the P90 rule and ler requirements using a jcc
for fcrd.

###  IV-B Approximating jcc

jcc are generally computationally intractable [35]. In some cases, they can be
reformulated analytically [36]. However, several approximations of jcc exist
which provide a trade-off between optimality and computational speed. In this
study, we opt for two different convex approximation approaches commonly used
in the literature: (i) ALSO-X and (ii) cvar.

Both approximation methods are sample-based approaches, that reformulate
Problem (4) to a lp using a finite set of samples $\omega\in\Omega$ drawn from
historical data.

####  IV-B1 ALSO-X

The ALSO-X method [33] consists of two steps: a) Transforming the jccp to a
milp and b) an iterative solution strategy. The corresponding milp for Problem
(5) directly constrains the number of overbids by introducing binary variables
$y_{m,\omega}$ and the Big-M method:

$\underset{(c^{\downarrow},c^{\uparrow})\geq
0,y_{m,\omega}\in\\{0,1\\}}{\operatorname{Max}}c^{\downarrow}+c^{\uparrow}$
(5a) st.
$\displaystyle\frac{1}{5}c^{\downarrow}+c^{\uparrow}-F^{\uparrow}_{m,\omega}$
$\displaystyle\leq y_{m,\omega}M^{\uparrow}$ $\displaystyle\forall m,\omega$
(5b) $\displaystyle c^{\downarrow}-F^{\downarrow}_{m,\omega}$
$\displaystyle\leq y_{m,\omega}M^{\downarrow}$ $\displaystyle\forall m,\omega$
(5c) $\displaystyle c^{\downarrow}-F^{\mathrm{E}}_{m,\omega}$
$\displaystyle\leq y_{m,\omega}M^{E}$ $\displaystyle\forall m,\omega$ (5d)
$\displaystyle\sum_{m}\sum_{\omega}y_{m,\omega}$ $\displaystyle\leq q$ (5e)

Constraints (5b)-(5d) represent the ler requirements, implying that
$y_{m,\omega}=1$ if at least one of them is violated for a given minute.
Constraint (5e) controls the frequency of overbid by counting the number of
times that an overbid occurs using $y_{m,\omega}$. The count is then bound by
the sum of $y_{m,\omega}$ for all minutes and samples. The upper bound is
specified by parameter $q$, which is set according to the desired allowance
rate, i.e., the P90 rule in Definition 2. Thus, $q$ represents the 10% overbid
allowance, which equals $\lfloor\epsilon\cdot|\Omega|\cdot N\rfloor$, where
$N$ is the number of minutes. The three Big M’s are set equal to the sample
sets’ largest available flexibility.

Step b) in the ALSO-X algorithm obtains feasible solutions for Problem (5)
that are computationally tractable as described in Algorithm 1.

Algorithm 1 ALSO-X

1:Input: Stopping tolerance parameter $\delta$

2:Require: $\text{Relax the integrality of }y$

3:$\underline{q}\leftarrow
0,\quad\bar{q}\leftarrow\lfloor\epsilon\times\Omega\times M\rfloor$

4:while $\bar{q}-\underline{q}\geq\delta$ do

5: $\text{Set }q=\frac{(\underline{q}+\bar{q})}{2}$

6: $\text{Retrieve }\Theta^{*}\text{ as an optimal solution to relaxed of}$

7: Problem (5).

8: $\text{Set }\underline{q}=q\text{ if }\mathbb{P}(y_{m,\omega}*=0)\geq
1-\epsilon\text{; otherwise, }\bar{q}=q$

9:end while

10:Output: A feasible solution to Problem (5)

The first step of the algorithm is to relax the integrality of $y$, i.e.,
$0\leq y\leq 1$. The relaxation converts the MILP to an LP thus providing
convexity to the problem and making it faster to solve. To obtain a feasible
solution to the original problem, $q$ is iteratively tightened as seen through
steps 2 to 6 in Algorithm 1. The model terminates with a feasible solution
once the lower and upper bound difference is less than $\delta$ which
determines the threshold for convergence. In this study, setting $\delta$ to
be $10^{-5}$ was found to provide an optimal trade-off between computational
time and feasibility.

####  IV-B2 cvar

The cvar method [34] approximates the jcc by controlling magnitude of overbids
using a reformulated lp. Thus, cvar minimizes the expected overbid violation
for the worst $(1-\alpha)$ samples which is the var. Here $\alpha=0.1$ as per
the P90 rule in Definition 2. The cvar approximation problem is shown in (6):

$\underset{(c^{\downarrow},c^{\uparrow})\geq 0,\beta\leq
0,\zeta_{m,\omega}}{\operatorname{Maximize}}c^{\downarrow}+c^{\uparrow}$ (6a)
st.
$\displaystyle\frac{1}{5}c^{\downarrow}+c^{\uparrow}-F^{\uparrow}_{m,\omega}$
$\displaystyle\leq\zeta_{m,\omega}$ $\displaystyle\forall m,\omega$ (6b)
$\displaystyle c^{\downarrow}-F^{\downarrow}_{m,\omega}$
$\displaystyle\leq\zeta_{m,\omega}$ $\displaystyle\forall m,\omega$ (6c)
$\displaystyle c^{\downarrow}-F^{\mathrm{E}}_{m,\omega}$
$\displaystyle\leq\zeta_{m,\omega}$ $\displaystyle\forall m,\omega$ (6d)
$\displaystyle\sum_{m}\sum_{\omega}(\zeta_{m,\omega})\phi-(1-\alpha)\beta$
$\displaystyle\leq 0$ (6e) $\displaystyle\beta$
$\displaystyle\leq\zeta_{m,\omega}$ $\displaystyle\forall m,\omega$ (6f)

The three ler constraints are shown in (6b)-(6d). $\zeta_{m,\omega}$ is an
auxiliary variable that captures the potential overbid to each sample and
links the risk across all three ler constraints. Further constraints (6e)-(6f)
ensures that the model punishes making excessive overbids and conservative
bids, by respectively stating that the average $\zeta_{m,\omega}$ has to be
below var, and the implantation of $\beta$ as well as forcing it to take the
lowest $\zeta_{m,\omega}$.

In summary, the cvar approximation in (6) looks to optimize the aggregator’s
bids based on the worst samples determined by the var risk level. Given the
structure of the problem, it often leads the method to find the optimal cvar
threshold to be around the average missing subset below var, ultimately
leading cvar to be a conservative approximation method [37].

##  V Results and discussion

In this section, we show the synergy effect of aggregating ev for bidding into
fcrd while adhering to the P90 rule and the ler requirements using the
reformulations presented in the previous section and simulation setup
described in Section III. Furthermore, an Oracle model with perfect foresight
of the available ev flexibility will serve as an upper bound for the monetary
potential.

We start by assessing the synergy effect by presenting by solving (4) for
different portfolio sizes. Here, the metrics for synergy defined in (1), (2),
and (3) are used. Second, the sensitivity of the ler regulation is evaluated.
Lastly, we investigate how the optimization model exploits the P90 rule with
respect to the frequency of overbids.

###  V-A Synergy effect

Figure 5 shows the average hourly profit for all three models subject to eight
different portfolio configurations with bundle sizes of 70, 28, 14, 10, 7, 4,
2, and 1, meaning that for a specific configuration, the 1,400 ev will be
divided into one of the bundles. When assessing a specific portfolio
configuration, the total bid capacity and profit are determined by adding the
capacities and profits across all bundles. Thus, the comparison of ALSO-X and
cvar results are ensured to be comparable.

Overall, Figure 5 reveals a significant synergy effect, as increasing the
portfolio sizes within each bundle consistently increases earnings. The
synergy effect occurs immediately, whereafter the gradient decreases during
the last steps, indicating a diminishing return rate (although more ev would
be needed to conclude this). Increasing the portfolio size thus leverages the
P90 rule as uncertainty in realized ev consumption, and thereby flexibility,
is reduced. This interim finding is supported by comparing it to the Oracle
model, where the relative difference in earnings decreases for larger
portfolio sizes. However, even when bundling all 1,400 ev into a single
portfolio, the Oracle method outperforms the stochastic methods by a factor of
two to three. This indicates that a significant unfilled potential lies in
improving the representation of the probabilistic expectation of flexibility
for the ev used in this study.

Figure 5: Mean hourly earnings for all 1,400 ev for different bundle
configurations. The hourly mean is determined oos for all 363 days using
three-fold cross-validation as described in Section III.

Table I underpins these findings as it shows how the Oracle model bids a
significantly higher proportion of the available flexibility. Looking at the
approximations, ALSO-X manages to bid more flexibility than cvar and thus
outperforms cvar from a monetary point of view. The same is seen in Figure 5,
where ALSO-X’s higher bids explicitly correspond to higher profits.

An average ev included in the portfolio of 1,400 ev would earn an annual
profit of $857$ DKK using ALSO-X and $638$ DKK with cvar, respectively bidding
24.4% and 18.5% of the total flexibility. Hence, from an aggregator’s
perspective, there is a significant monetary benefit of pooling all available
ev and by using ALSO-X as opposed to cvar. However, cvar might prove better
for new, unseen data in non-stationary environments. For example, as new ev
join the portfolio or when grid tariffs change due to political decisions. In
these cases, ALSO-X is fairly aggressive and is not guaranteed to adhere to
the P90 rule while cvar is much more conservative.

| CVaR | ALSO-X | Oracle
---|---|---|---
Upwards flexibility | 2.0% | 4.7% | 40.0%
Downwards flexibility | 10.8% | 16.0% | 40.0%
Overall | 9.5% | 14.0% | 40.0%
TABLE I: Average percentage of utilized bidding capacity bid across the 1,400
ev for all bundle sizes.

###  V-B Relaxation of ler regulation

Interestingly, all three models are far from bidding 100% of their total
flexibility. In particular, it is noteworthy that the Oracle model only bid
$40$% of its total available flexibility. This is due to two factors: (i) Bids
are required to be uniform within hours, meaning some flexibility is simply
lost and (ii) as illustrated in Figure 3, subsets of both upward and downward
flexibility are inaccessible due to the ler requirements of reserving 20% of
the downward bid in upward flexibility. Little upward flexibility is directly
bid into the market, as it is reserved to facilitate the downward bid. The
downwards bids are restrained by the discrepancy between downward and upward
flexibility often being greater than a factor of five, as illustrated in
Figures 4(a) and 4(b), which ultimately causes a lot of downwards capacities
to be unavailable.

The impact of relaxing the ler constraints, is presented in Table II. Here,
the portfolio of 1,400 ev is tested on two relaxed sets of ler requirements
using the ALSO-X bidding scheme. In the testing case referred to as energy
relaxation, the rule of reserving 20% of the downwards capacity bid in upwards
flexibility is removed, while the 20 duration requirements are removed in the
testing case referred to as energy relaxation. In the upwards relaxation case
upwards flexibility is not required reserved for downwards bids, meaning a
massive increase in upwards flexibility bid, while it also enables extra
downwards capacity bid. For the energy relaxation case, the downward bids also
increase, concluding the energy reservoir indeed is limiting the bids in the
base case. However, the upward flexibility bid decreases, as a larger quantity
of it is reserved to facilitate the extra enabled downward capacity. The
general results signify the immense influence of the specific constraint. In
particular, the upwards relaxation is binding, as the total amount of
flexibility bid increases from 24.4% to 35.6%, with an annual revenue increase
of $61$%. Consequently, the ler requirements are particularly constraining for
an ev aggregator and perhaps of concern to the tso as less flexibility is
being procured.

Figure 6 illustrates the case where the average downward bids are placed
relative to the distribution of available flexibility for the three cases. It
reveals that bids in the relaxed cases are generally placed closer to the 10%
risk level allowed by the P90 rule. Further, it illustrates how each of the
two LER requirements is the binding constraint for the base ler case at
different timesteps. The base ler’s downward bids are bounded by its energy
reservoir level between 22:00-03:00 and 10:00-14:00, as the energy relaxation
produces larger bids, while the base case follows the bids in the upwards
relaxation case in these hours. As illustrated in Figures 4(a) and 4(b) these
are hours where the average discrepancy between upwards and downwards
flexibility is lower than a factor of five, meaning there is enough upwards
flexibility to facilitate the full downwards flexibility being bid. In the
remaining hours, the discrepancy is larger than a factor of five, thus
indicating that it is a shortage of upward flexibility that restrains the
downward capacity bid. Accordingly, both requirements established to handle
duration issues are restraining the bids, with the constraint on the upward
flexibility especially being the most restraining.

From an aggregator’s perspective, the challenges of the ler regulations could
be mitigated by increasing the heterogeneity of its portfolio. Here, it would
be beneficial to include technologies with more upward flexibility, and a more
diverse consumption pattern, allowing for a more consistent soc of its energy
reservoir during the day. From a system perspective, the tso might be
interested in carefully assessing the ler requirements as Figure 6 and Table
II clearly shows how liquidity increases when relaxing them. Ultimately, it is
a trade-off between the security of supply, liquidity, and costs, as more
liquidity drives down tso expenditure per MW of fcrd, but decreases the
security of supply. Further research is needed to determine the exact added
security of supply of the ler requirements. However, they are in place to
ensure the duration of the bid capacity, which might be insignificant to the
security of supply, as longer activation periods hardly ever occur. Therefore,
it would be of interest to analyze alternative approaches for preserving the
security of supply.

|  | Annual
---
profit
|  $\uparrow$ utilized
---
bidding capacity
|  $\downarrow$ utilized
---
bidding capacity
Base LER | 1,200 kDKK | 8.6% | 27.3%
Upwards relaxation | 1,934 kDKK | 36.4% | 35.4%
Energy relaxation | 1,344 kDKK | 3.1% | 32.8%
TABLE II: Synergy metrics evaluated oos for a 1,400 ev portfolio using the
ALSO-X approximation over all 363 days with three-folded cross-validation as
described in Section III. Figure 6: The average amount of downwards
flexibility bid using ALSO-X and a portfolio size of 1,400 ev with and without
the respective ler requirements of 20 minutes of continuous down-regulation
(Energy) and 20% capacity reservation in the opposite direction (Upwards).

###  V-C Frequency of overbid

We now compare ALSO-X and cvar with respect to the compliance of the P90 rule.
Figure 7 shows cvar and ALSO-X daily frequency of overbidding for the
portfolio of 1,400 ev. ALSO-X hits the desired allowance rate of $10$% almost
perfectly, as the frequency of overbid across all simulated days is $10.15$%.
cvar is well below the threshold as only $3.89$% of all minutes are overbids.
cvar has 190 days with no overbids altogether, and almost no days exceeding
the exclusion rate of $15$%. The spread amongst the days is broader for
ALSO-X, with a larger subset of days above the exclusion threshold. The
individual days with very large overbids are not a problem according to
Definition 2 of the P90 rule, as it refers to the average violation rate over
the whole inspection period, which is required to be a minimum of three
months. Hence, both bidding schemes adhere to the P90 rule.

Nonetheless, if the sample period was shorter, ALSO-X could potentially
violate the P90 rule as there exist subsets that go above the $15$% excluding
threshold, whereas cvar supplies more security in that regard. Hence, the
methods present a trade-off between potential profits and security of supply,
meaning even though ALSO-X seems preferable from a profit viewpoint, cvar
could be preferable, as it is inherently conservative and offers a margin for
error with respect to the P90 rule. From a system perspective, the tso gets
less liquidity in the market if aggregators and other service providers use
robust approaches like cvar, but more security of supply, and vice versa if
methods like ALSO-X is used which fully exploit the rule set.

Having an adequate representation of the uncertain available flexibility
$F=(F^{\uparrow}$, $F^{\downarrow}$, $F^{E})$ is thus especially important if
applying ALSO-X, while cvar would be more likely to produce feasible results
oos with an inaccurate representation.

(a) ALSO-X

(b) cvar

Figure 7: Frequency of overbids tested for a portfolio of 500 ev for cvar and
ALSO-X for 363 days.

##  VI Conclusion and future work

The stability of the future power grid is challenged by increased renewable
and less stable fossil-fuel-based generation. Demand flexibility contains
significant, untapped potential in this regard. We investigated how a
portfolio of grid-connected ev can deliver fcrd in Denmark through an
aggregator, thus earning money while balancing large frequency disturbances in
the power grid. We showed how an aggregator can model the newly invented P90
from the Danish tso while adhering to ler requirements using a jccp with two
approximations: (i) ALSO-X and (ii) cvar.

A significant synergy was identified upon aggregation of ev, as the quantity
of flexibility enabled for fcrd bidding increased for larger portfolios,
yielding higher aggregator profits and less uncertainty of available
flexibility. cvar appeared as a conservative approximation method compared to
ALSO-X. ALSO-X fully explored the allowed frequency of overbids, i.e., the 10%
threshold, while cvar’s rate of overbids was less than half of ALSO-X’s, thus
not leveraging the full potential of the P90 rule.

In general, bids were especially limited by the ler constraint reserving 20%
of the downward bid in upward flexibility. Hence, future work should
investigate how these challenges could be mitigated, as it would be of
interest for both the aggregator and the tso to leverage more of the portfolio
flexibility into the market. From the aggregator perspective, we suggest
introducing a more diverse technology portfolio, which could further reduce
the uncertainty of available flexibility. From a tso’s perspective, it would
be of interest to further compare the added security of supply versus exposed
liquidity and procurement costs when modifying the ler constraints. Here the
requirement of reserving 20% flexibility in the opposite direction to control
the SoC was found significantly more constraining than the 20-minute full
response requirement.

Other avenues of future research should explore the influence of the
simplifications and assumptions inherent in this study. Moreover, our rather
simplistic approach to the probabilistic representation of the underlying
flexibility of the ev portfolio is an element with potential for improvement,
highlighted by the large discrepancy between the stochastic and Oracle models.
We recommend aggregators develop a more advanced forecasting method such as
generalized linear models. Furthermore, distributionally robust optimization
can help in cases where the empirical distribution of the underlying
flexibility is misspecified.

## Acknowledgement

The authors would like to acknowledge Spirii for data sharing and Edoardo
Simioni (Ørsted) for providing feedback. We also thank Thomas Dalgas
(Energinet) for his guidance on market regulations.

## Appendix A: Number of samples

Various approaches exist for selecting the number of samples for stochastic
optimization as seen in [36, 24, 25] where an empirical approach is used and
numerical results are tested subject to different sample sizes, while [38, 39]
uses analytical approaches. In this study, we have opted to use a theoretical
number of samples as seen in (7) [33]:

$N\geq\frac{2}{\epsilon}\log\left(\frac{1}{\delta}\right)+2n+\frac{2n}{\epsilon}\log\left(\frac{2}{\epsilon}\right)\rightarrow
N\geq 216$ (7)

Here, multiple parameters are selected according to the given problem. First,
$\epsilon\in(0,1)$ is the argument denoting the acceptable failure rate of the
jcc, which is set to 0.1 based on Energinet’s P90 requirement. Second, $n$
denotes the dimensionality of the model, which translates to the number of
decision variables (i.e., two). Third, $\delta\in(0,1)$ denotes the confidence
level of the probabilistic constraint holding. Accordingly, $\delta$ is
selected such that there is $1-\delta$ probability that the model’s decisions
($c^{\uparrow},c^{\downarrow}$) are a feasible solution in the
$\epsilon$-level problem formulation. We set $\delta$ to $0.01$, allowing a 1%
risk a given hour to violate the P90 rule. The relative high-risk rate is
chosen, as the P90 allows for 5% room for error, while the evaluation is on
the average violation frequency, allowing more risk on individual bids.

We note the (7) only holds when the underlying data-generating process is
stationary which might not be the case in reality. However, it works well
within a cross-validation framework to showcase the synergy effect of
aggregating ev, and to demonstrate the efficacy of the jccp representing the
P90 rule.

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