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1

Managing Risks Faced by Strategic Battery Storage

in Joint Energy-Reserve Markets

K. Pandˇ

zi´

c, K. Bruninx, Member,IEEE, H. Pandˇ

zi´

c, Senior Member,IEEE

Abstract—Securing proﬁts from energy, reserve capacity and

balancing markets is critical to ensure the proﬁtability of battery

energy systems (BES). However, the intimate connection between

offers on these trading ﬂoors combined with the limited energy

storage capacity of BES renders its scheduling very complex.

In this paper, we develop a bilevel optimization problem for

strategic participation of a BES in the day-ahead energy-reserve

and balancing markets, improving the state-of-the-art by (i)

considering the conditional-value-at-risk; (ii) ensuring the real-

time feasibility of the obtained day-ahead schedule; (iii) ad-

dressing the operational underperformance risk stemming from

inaccurate battery modeling. In a case study, we illustrate how

the proposed model allows risk-averse BES owners to hedge their

day-ahead position without jeopardizing their expected proﬁt,

while ensuring the feasibility of their day-ahead schedule.

Index Terms—Battery Energy Storage, MPEC, Joint Energy-

Reserve Market, Balancing Market, Conditional-Value-at-Risk

NOMENCLATURE

A. Sets and Indices

HSet of BES units, indexed by h.

ISet of generating units, indexed by i.

JSet of breakpoints of the linearized battery charging

curve, indexed by j.

LSet of lines, indexed by l.

PSet of wind scenarios, indexed by p.

SSet of buses, indexed by s, while s(h)is a bus where

BES his located.

TSet of time steps, indexed by t.

WSet of wind farms, indexed by w.

ΛSet of dual variables related to equalities.

Ξ[·]Decision variable set, where [·]stands for upper-level

(UL) or lower-level (LL). Ξ = ΞUL ∪ΞLL

B. Parameters

BlSusceptance of line l(S).

CD

sDemand bid at bus s(e/MWh).

CG

iGeneration cost of unit i(e/MWh).

CG↑

i,CG↓

iUpward (↑) or downward (↓) reserve capacity

offer of generating unit i(e/MW).

CBG↑

i,CBG↓

iUpward (↑) or downward (↓) balancing offer of

generating unit i(e/MWh).

K. Pandˇ

zi´

c is with the Croatian TSO (HOPS), K. Bruninx is with the

KU Leuven, and H. Pandˇ

zi´

c is with the University of Zagreb Faculty of

Electrical Engineering and Computing. The research leading to these results

has received funding from the European Union’s Horizon 2020 research and

innovation programme under grant agreement No 863876 in the context of the

FLEXGRID project. The contents of this document are the sole responsibility

of authors and can under no circumstances be regarded as reﬂecting the

position of the European Union.

CS↑

h,CS↓

hUpward (↑) or downward (↓) reserve capacity

offer of BES h(e/MW).

CBS↑

h,CBS↓

hUpward (↑) or downward (↓) balancing offer of

BES h(e/MWh).

Ds,t Demand at time step tat bus s(MW).

FlCapacity of line l(MW).

FjMaximum BES charging power over segment j

ranging from 0 to 1.

GiCapacity of generating unit i(MW).

Pw

p,w,t Available wind output of wind farm win scenario

p(MWh).

Pw

p,w,t Maximum wind output of wind farm wover all

scenarios (MWh).

Qdis

h,t Maximum discharging power of BES h(MW).

RjSize of BES state-of-energy segment j, ranging

from 0 to 1.

SOEhCapacity of BES h(MWh).

Vlol

sValue of lost load at bus s(e/MWh).

αWeighting factor between expected proﬁt and

CVAR.

βAuxiliary variable, value-at-risk (e).

Interval spanned by the CVAR.

ηch

hCharging efﬁciency of BES h.

ηdis

hDischarging efﬁciency of BES h.

ΠpProbability of scenario p.

C. Primal variables & select dual variables

1) Positive variables:

chh,t Charging offer of BES hat time step t(MWh).

dish,t Discharging bid of BES hat time step t(MWh).

ds,t Scheduled demand at bus sat time step t(MWh).

dshed

p,s,t Involuntarily load shed at bus sin scenario pat

time step t(MWh).

gi,t Scheduled output of generating unit iat time step

t(MWh).

pws

w,t Power sold by wind farm wat time step t(MWh).

pcurt

p,w,t Wind curtailment of wind farm wat time step t

in scenario p(MWh).

qch

h,t Charging energy of BES hat time step t(MWh).

qdis

h,t Discharging energy of BES hat time step t

(MWh).

rreq↑

t,rreq↓

tTotal upward (↑) or downward (↓) reserve require-

ment (MW).

rg↑

i,t,rg↓

i,t Scheduled upward (↑) or downward (↓) reserve

capacity of generating unit iat time step t(MW).

rbg↑

p,i,t,rbg↓

p,i,t Dispatched upward (↑) or downward (↓) reserve

of generating unit iat time step tin scenario p

(MWh).

2

rs↑

h,t,rs↓

h,t Upward (↑) or downward (↓) reserve capacity

offer of BES hat time step t(MW).

rs↑

h,t,rs↓

h,t Scheduled upward (↑) or downward (↓) reserve

capacity of BES hat time step t(MW).

rbs↑

p,h,t,rbs↓

p,h,t Dispatched upward (↑) or downward (↓) reserve

of BES htime step tin scenario p(MWh).

soeh,t State-of-energy of BES hat time step t(MWh).

soeh,t,j State-of-energy of segment jof BES hat time

step t(MWh).

∆soeh,t Maximum charging power of BES hat time step

t(MW).

2) Free variables:

λs,t Day-ahead LMP at bus sand time period t

(e/MWh).

λb

p,s,t Real-time balancing prices in scenario pat bus s

at time step t(e/MWh).

λR↑

t,λR↓

tUpward (↑) or downward (↓) reserve capacity

price (e/MW).

fl,t Power ﬂow of line lat time step t(MWh).

fb

p,l,t Real power ﬂow in scenario pon line lat time

step t(MWh).

fdev

p,l,t Deviation between scheduled and real power ﬂow

in scenario pon line lat time step t(MWh).

θVoltage angles of bus sat time step t(rad).

θb

p,s,t Voltage angles of bus sin scenario pat time step

t(rad).

I. INTRODUCTION

ENERGY storage systems (ESS) may enable cost-efﬁcient

and reliable operation of power systems with high

shares of electricity generated from renewable energy sources.

Amongst other factors, decreasing investment costs, local in-

centives and increasing opportunities in energy, reserve capac-

ity and balancing markets may trigger an accelerated uptake

of distributed and bulk ESS from a modest 9GW/17GWh

deployed as of 2018 to 1,095GW/2,850GWh by 2040 [1].

This massive ESS deployment will mostly be ordinated by

the cost-reduction of lithium-ion batteries, which are in the

focus in this paper.

In light of the storage evolution, researchers have exten-

sively studied the proﬁt-maximization problem faced by ESS

owners. Although a price-taking perspective offers valuable

insights, e.g. [2], [3], in recent years most researchers resort

to bilevel optimization to study the strategic participation of

large-scale ESS in the day-ahead wholesale electricity market.

For example, Wang et al. [4] analyze a network-constrained

market-clearing mechanism with ESS participation under per-

fect and imperfect competition. They reveal that a modest level

of local transmission congestion and imperfect competition

both increase the ESS proﬁts. Pandˇ

zi´

c et al. [5] quantify

the value of coordination of multiple ESS units scattered

throughout the network and emphasize the importance of

increasing the look-ahead horizon to two days in order to

precharge and/or preserve the stored energy from the previous

day, thus gaining higher overall proﬁt.

Introduction of uncertainty in strategic bilevel models is

deemed to reduce computational tractability. Already a price-

taking stochastic bidding model of an energy storage com-

bined with a wind power plant called for a heuristic solution

technique in [6], where a neural network was used to ﬁt the

uncertain functions, while a genetic algorithm was employed

to ﬁnd the optimal bidding solution. A stochastic bilevel

model, where a load-serving entity owns an energy storage

and acts in the day-ahead market, is modeled in [7]. The

uncertainty related to the net load, i.e. the actual load minus

the realization of wind generation, is modeled through a set

of credible scenarios. Participation of a battery energy storage

in European-style day-ahead energy and reserve markets is

presented in [8]. The presented model is bilevel, as the

battery storage acts strategically in the balancing market. The

uncertainty is presented through a set of reserve activation

scenarios. The battery storage is considered as a price taker in

the day-ahead energy market and the network constraints are

ignored, which is legitimate for European power exchanges.

In this paper, we develop a novel model to study the partici-

pation of risk-averse, merchant BES in joint day-ahead energy-

reserve capacity and balancing markets. We draw inspiration

from three streams of literature.

The ﬁrst stream of literature deals with the strategic par-

ticipation of energy storage in energy, reserve capacity and

balancing markets. This requires explicitly considering the

intrinsic link between the offers in the day-ahead energy

market, capacity offered in the day-ahead reserve capacity

market and the impact of the real-time dispatch of this capacity

in the balancing market. Indeed, if not properly accounted

for, the real-time activation of ESS-based reserves may lead

to violations of the ESS’ state-of-energy constraints. Whereas

Nasrolahpour et al. [9] account for the expected, average im-

pact of reserve activation on the state-of-energy, Schillemans

et al. [10] ensure that the worst-case real-time state-of-energy

respects the capacity limits of the ESS. In this paper, we will

follow the last approach. Note that one could alternatively

enforce state-of-energy constraints per balancing scenario.

However, capturing all possible real-time reserve activation

combinations to ensure the feasibility of reserve activation

would require a large scenario set and render the problem

intractable [9]. Therefore, we opt for conservative, robust

worst-case reserve activation constraints. The conservatism of

such constraints may be reduced by enforcing probabilistic

guarantees on the availability of the scheduled reserve capacity

via chance constrained programming [11], which is out-of-

scope of the current paper.

Second, although the risk-averse behavior of generating

companies has been shown to signiﬁcantly affect their operat-

ing and investment decisions [12], few authors have considered

risk-averse behavior of a BES participating in energy and/or

ancillary service markets. A price-taker model for the day-

ahead bidding strategy of an energy storage coupled with a

wind farm is presented in [13]. The uncertainty on the market

prices and the local wind power output is tackled using the

robust optimization. However, the only consideration of a

strategic risk-averse energy storage that captures the effect the

storage has on market prices was presented in [14], where the

authors solve an optimal energy storage management problem

under risk consideration that captures the market prices impact

through transactions costs. In our paper, we represent the

3

price-maker assumption of BES through a bilevel modeling

framework.

Lastly, as we are interested in BES, i.e. lithium-ion-based

energy storage, we challenge a common misconception of

modeling a BES using only a generic energy storage mixed-

integer linear model, e.g. [15]. More reﬁned battery models

have been developed for controlling a behind-the-meter battery

[16], a BES providing secondary reserve [17], [18]. However,

none of the strategic BES models considers any speciﬁc

battery characteristic. Dependency of the battery charging

power on its state-of-energy is a distinguished battery feature

that could have serious implications on the real-life feasibility

of the obtained BES operation schedule [19]. Thus, we incor-

porate the accurate battery charging model and analyze the

effects of using the generic constant-power battery charging

model instead.

In summary, this paper contributes to the body of literature

by developing a novel bilevel optimization problem that allows

deﬁning optimal bid strategies for a strategic, price-making

and risk-averse BES owner, considering the impact of its

bid strategies on the price formation in joint energy-reserve-

balancing markets. Contrary to [9], the proposed model (i)

explicitly represents the propagation in price formation from

the balancing market to the day-ahead energy market; (ii)

ensures that the activation of scheduled BES-based reserves

is feasible even if all reserves are consecutively activated in

the upward and/or downward direction by enforcing worst-

case reserve activation constraints, and (iii) does not require a

binary expansion approximation if one considers a risk-neutral

BES owner (see Section II-C). Compared to [10], we employ

a scenario-based representation of the balancing market. This

allows accounting for inter-temporal links between balancing

prices, introduced by the BES, and an accurate representation

of the state-of-energy and value of stored energy in each

balancing market scenario. Furthermore, we consider (i) trans-

mission constraints in our market clearing model, which may

inﬂuence the dispatch, reserve procurement and deployment,

hence, providing an opportunity for the BES owner to exercise

market power [4]; (ii) the CVaR, which allows balancing the

expected proﬁts and risk; and (iii) a detailed BES model suited

for lithium-ion batteries in order to accurately capture the

charging capabilities of these systems [19]. None of these

features are considered in [9], [10], but may have a signiﬁcant

impact on the BES’ proﬁtability.

The presented model may be directly integrated in the day-

to-day decision processes of BES owners or aggregators. In

addition, it sheds light on the relevance of reserve capacity

payments, which is crucial to market operators, policy makers

and regulators. Finally, it demonstrates the importance of using

accurate battery models in the BES scheduling process.

The remainder of this paper continues as follows. Section

II introduces the bilevel optimization problem that describes

the strategic participation of the BES owner in energy, reserve

capacity and balancing markets. Section III contains our case

study, which illustrates the effectiveness of the proposed

model. Computational complexity of the proposed solving

methodology is discussed in Section IV, while our conclusions

are articulated in Section V.

II. MATHEMATICAL MODEL

The bilevel structure of the proposed model is illustrated in

Figure 1. Before we present the formulation of the decision

problem faced by the BES owner (Section II-B) and the

associated solution procedure (Section II-C), we summarize

key assumptions made during model development below.

A. Assumptions

The decision problem of a strategic BES owner is for-

mulated as a bilevel optimization problem. The upper-level

problem determines the bid strategy of the BES owner in

the joint day-ahead energy and reserve capacity market in

the ﬁrst stage and the balancing market in the second stage,

whereas the lower-level problem describes the clearing and

price formation on those markets.

We assume that BES is the only strategic actor in the system,

as common in the price-maker market participation studies,

who attempts to maximize its proﬁt by bidding price-quantity

pairs in the hourly day-ahead energy-reserve capacity and

balancing markets. The BES owner competes with conven-

tional generation in all markets. In order to analyze strategic

market participation of other market participants, one would

need to derive a mathematical problem with equilibrium con-

straints (MPEC) for each market participant, thus forming an

equilibrium problem with equilibrium constraints (EPEC) and

solve this complex problem. An interested reader is advised

to examine the EPEC model proposed in [5] and the solution

technique for multiple BES owners acting strategically in the

day-ahead energy market presented therein.

The strategic BES knows the bidding strategies of non-

strategic players, i.e. all the generators, which is assumed in

all strategic offering models, e.g. [9]. In reality, these data are

available in some markets, e.g. historical data on participants

offers in the market of Alberta are available at [20]. Even if

they are not available, an inverse optimization procedure can

Lower-level problem (2.1)–(2.29)

Day-ahead and balancing market

clearing

ΞLL ={ds,t,dshed

p,s,t,gi,t ,rg↓

i,t,rg↑

i,t,rbg↑

p,i,t,

rbg↓

p,i,t,qch

h,t,qdis

h,t,rs↓

h,t,rs↑

h,t,rbs↑

p,h,t,rbs↓

p,h,t,

rreq↑

t,rreq↓

t,θs,t,θb

p,s,t,fl,t ,fb

p,l,t,

fdev

p,l,t,pws

w,t,pcurt

p,w,t }

Upper-level problem (1.1)–(1.15)

BES optimal bidding schedule

ΞUL ={chh,t ,dish,t ,rs↑

h,t,rs↓

h,t,soeh,t ,

soeh,t,j ,∆soeh,t}

qch

h,t,qdis

h,t,rs↓

h,t,rs↑

h,t,rbs↑

p,h,t,rbs↓

p,h,t,

λs,t,λR↑

t,λR↓

t,λb

p,s,t

chh,t,dish,t ,rs↑

h,t,rs↓

h,t

Fig. 1. An illustration of the proposed bilevel program and the interfaces

between the upper- and lower-level problems.

4

be employed to derive such data [21]. Naturally, the amount

of historical data and the number of scenarios affect the

quality of the solution. The BES owner endogenously forms a

deterministic anticipation of the day-ahead energy-reserve ca-

pacity market, whereas the uncertainty on the balancing stage

is represented through scenarios. The BES owner is solely

responsible for managing the state-of-energy of its system, i.e.,

to ensure a feasible dispatch in all reserve activation scenarios.

In the lower-level market clearing problem, we maximize

the expected social welfare in the day-ahead energy-reserve

capacity and balancing markets, given the uncertainty on the

available wind power output in real-time. The presented model

can be easily extended to accommodate other sources of

uncertainty, e.g. load uncertainty.

The market-clearing model performs simultaneous energy

and reserve capacity clearing and considers an anticipated

reserve activation. This type of setting is suitable for the

US-style markets, where the Independent System Operator

(ISO) is in charge of both the energy market and the ancillary

services market. Since the day-ahead market is cleared before

the realization of wind output uncertainty, the problem at hand

is structured as a two-stage stochastic problem, where the

ﬁrst stage represents the actual day-ahead market clearing,

providing the day-ahead energy prices and up and down

reserve capacity prices, while the second stage implements

presumable realizations of the wind power plants output,

providing the balancing energy prices. The market model is

built upon the setting proposed in [22], which is suitable

for power systems with high integration of non-controllable

renewable generation such as wind power. In such setting, the

ISO maximizes the overall social welfare, considering the day-

ahead energy, reserve capacity and reserve activation costs in

a single optimization problem.

The day-ahead energy and balancing markets are nodal,

meaning that the transmission constraints are enforced when

deploying reserves. Given a scenario-based description of the

uncertain wind output, the market model allows endogenously

determining a single day-ahead energy market clearing and the

required reserve capacity. To ensure feasibility, load shedding

is allowed in the balancing stage. The length of a time step is

one hour.

B. Formulation

The upper-level problem is formulated as follows:

Maximize

ΞUL α·X

p

Πp·φp(Ξ) + (1 −α)·CV aR(Ξ) (1.1)

subject to

φp(Ξ) = X

t∈T

X

h∈H

λs(h),t ·(qdis

h,t −qch

h,t) +X

t∈T

X

h∈H

λR↑

t·rs↑

h,t

+λR↓

t·rs↓

h,t+X

t∈T

X

h∈H

λb

p,s,t

Πp

·(rbs↑

p,h,t −rbs↓

p,h,t)(1.2)

CV aR(Ξ) = β−1

·X

p∈P

Πp·µp(1.3)

µp≥β−φp(Ξ),∀p(1.4)

µp≥0,∀p(1.5)

0≤chh,t ≤∆soeh,t

ηch

h

,∀h, t (1.6)

0≤dish,t ≤Qdis

h,t ·ηdis

h,∀h, t (1.7)

chh,t −dish,t +rs↓

h,t ≤∆soeh,t

ηch

h

∀h, t (1.8)

−chh,t +dish,t +rs↑

h,t ≤Qdis

h,t ·ηdis

h∀h, t (1.9)

soeh,t =soeh,t−1+qch

h,t ·ηch

h−qdis

h,t/ηdis

h,∀h, t (1.10)

0≤soeh,t −

t

X

k=1

rs↑

h,k/ηdis

h,∀h, t (1.11)

soeh,t +

t

X

k=1

rs↓

h,k ·ηch

h≤SOEh,∀h, t (1.12)

soeh,t =

J−1

X

j=1

soeh,t,j ,∀h, t (1.13)

0≤soeh,t,j ≤(Rj+1 −Rj)·SOEh,∀h, t, j (1.14)

∆soeh,t=F1·SOEh+

J−1

X

j=1

Fj+1 −Fj

Rj+1 −Rj

·soeh,t−1,j ,∀h, t

(1.15)

where ΞUL ={chh,t ,dish,t ,rs↑

h,t,rs↓

h,t,soeh,t ,soeh,t,j ,

∆soeh,t}.

The upper-level objective function (1.1) is a weighted aver-

age (0≤α≤1) between the expected proﬁt Pp∈P Πp·φp(Ξ)

and the conditional value at risk CV aR(Ξ). The proﬁt in each

scenario pconsists of (Eq. (1.2)): (i) the arbitrage proﬁt at the

scheduling stage (pool prices), (ii) the proﬁt from offering the

up and down reserve capacity and (iii) the proﬁt associated

with deploying reserves in real time (balancing prices). The

CVaR is calculated in Eqs. (1.3)–(1.5), with βthe lowest proﬁt

that is strictly exceeded with a probability of at most 1−.

Constraints (1.6)-(1.7) limit the BES charging and discharg-

ing quantities offered in the market to respective maximum

powers determined by the bidirectional power inverter and

battery technology limitations. Since maximum BES charging

power depends on the battery state-of-energy, it is calculated

based on the maximum energy that can be charged into the

battery, ∆soeh,t, determined via Eq. (1.15) based on the accu-

rate battery charging model presented in [19]. Equations (1.8)–

(1.9) limit the offered up and down reserve quantities with

respect to the day-ahead BES charging/discharging schedule

and maximum BES charging/discharging power. Constraint

(1.8) indicates that down reserve can be provided by stopping

the day-ahead scheduled discharging process and starting to

charge the battery. The opposite reasoning is valid for the up

reserve provision (Eq. (1.9)). Equation (1.10) calculates the

state-of-energy of each BES considering only the day-ahead

cleared quantities. The the state-of-energy deviations incurred

by the activation of BES up and down reserves are taken into

account in Eqs. (1.11)–(1.12). These deviations need to be

such that the minimum and maximum state-of-energy is not

violated. Constraints (1.13)–(1.15) determine the maximum

energy that can be charged into a battery within a single time

5

period considering the battery state-of-energy in the previous

time period (see [19] for details).

Variables chh,t ,dish,t,rs↑

h,t, and rs↓

h,t are used in the lower-

level problem as the quantities that BES hoffers in the day-

ahead market:

Maximize

ΞLL X

t∈T

X

s∈S

CD

s·ds,t +X

t∈T

X

h∈H

(Cch

h·qch

h,t

−Cdis

h·qdis

h,t −CS↑

h·rs↑

h,t −CS↓

h·rs↓

h,t)

−X

t∈T

X

i∈I

(CG

i·gi,t +CG↑

i·rg↑

i,t +CG↓

i·rg↓

i,t)

−X

p∈P

X

t∈T

X

i∈I

Πp·(CBG↑

i·rbg↑

p,i,t −CBG↓

i·rbg↓

p,i,t)

−X

p∈P

X

t∈T

X

h∈H

Πp·(CBS↑

h·rbs↑

p,h,t −CBS↓

h·rbs↓

p,h,t)

−X

p∈P

X

t∈T

X

s∈S

Πp·Vlol

s·dshed

p,s,t (2.1)

subject to

−X

i∈Is

gi,t +X

l∈Ls

fl,t −X

w∈Ws

pws

w,t +ds,t

+X

h∈Hs

(qch

h,t −qdis

h,t)=0 ∀s, t;λs,t (2.2)

X

i∈I

rg↑

i,t +X

h∈H

rs↑

h,t ≥rreq↑

t∀t;λR↑

t(2.3)

X

i∈I

rg↓

i,t +X

h∈H

rs↓

h,t ≥rreq↓

t∀t;λR↓

t(2.4)

−X

i∈Is

(rbg↑

p,i,t −rbg↓

p,i,t)+X

l∈Ls

fdev

p,l,t −X

h∈s

(rbs↑

p,h,t −rbs↓

p,h,t)−dshed

p,s,t

+X

w∈Ws

pws

w,t −(Pw

p,w,t−pcurt

p,w,t)= 0 ∀p, s, t;λb

p,s,t (2.5)

fl,t =Bl·X

l/s

θs,t ∀l, t;αl,t (2.6)

fb

p,l,t =Bl·X

l/s

θb

p,s,t ∀p, l, t;αb

p,l,t (2.7)

fdev

p,l,t =fb

p,l,t −fl,t ∀p, l, t;αdev

p,l,t (2.8)

θsref ,t = 0 ∀t;ωt(2.9)

θb

sref ,t = 0 ∀p, t;ωb

p,t (2.10)

−Fl≤fl,t ≤Fl∀l, t;γ+/−

l,t (2.11)

−Fl≤fb

p,l,t ≤Fl∀p, l, t;γb+/b−

p,l,t (2.12)

0≤ds,t ≤Ds,t ∀s, t;δ+/−

s,t (2.13)

gi,t −rg↓

i,t ≥0∀i, t;ψ−

i,t (2.14)

gi,t +rg↑

i,t ≤Gi∀i, t;ψ+

i,t (2.15)

0≤rbg↑

p,i,t ≤rg↑

i,t ∀p, i, t;ζbg↑+/bg↑−

p,i,t (2.16)

0≤rbg↓

p,i,t ≤rg↓

i,t ∀p, i, t;ζbg↓+/bg↓−

p,i,t (2.17)

0≤rbs↑

p,h,t ≤rs↑

h,t ∀p, h, t;ζbs↑+/bs↑−

p,h,t (2.18)

0≤rbs↓

p,h,t ≤rs↓

h,t ∀p, h, t;ζbs↓+/bs↓−

p,h,t (2.19)

0≤rs↑

h,t ≤rs↑

h,t ∀h, t;ρs↑+/s↑−

h,t (2.20)

0≤rs↓

h,t ≤rs↓

h,t ∀h, t;ρs↓+/s↓−

h,t (2.21)

0≤qch

h,t ≤chh,t ∀h, t;σc+/c−

h,t (2.22)

0≤qdis

h,t ≤dish,t ∀h, t;σd+/d−

h,t (2.23)

0≤dshed

p,s,t ≤ds,t ∀p, s, t;β+/−

p,s,t (2.24)

rreq↑

t≥X

w∈Ws

pws

w,t −(Pw

p,w,t −pcurt

p,w,t)∀p, t;τ↑

p,t (2.25)

rreq↓

t≥− X

w∈Ws

pws

w,t −(Pw

p,w,t −pcurt

p,w,t)∀p, t;τ↓

p,t (2.26)

0≤pcurt

p,w,t ≤Pw

p,w,t ∀p, w, t;φ+/−

p,w,t (2.27)

0≤pws

w,t ≤Pw

w,t ∀w, t;µws+/−(2.28)

gi,t, rg↓

i,t, r g↑

i,t, r req↑

t, rreq↓

t≥0 ; µg/g↓/g↑/req↑/req↓(2.29)

where ΞLL ={ds,t,dshed

p,s,t,gi,t ,rg↓

i,t,rg↑

i,t,rbg↑

p,i,t,rbg↓

p,i,t,qch

h,t,

qdis

h,t,rs↓

h,t,rs↑

h,t,rbs↑

p,h,t,rbs↓

p,h,t,rreq↑

t,rreq↓

t,θs,t,θb

p,s,t,fl,t ,

fb

p,l,t,fdev

p,l,t,pws

w,t,pcurt

p,w,t }.

The lower-level problem simulates the market clearing pro-

cedure in which the total surplus w.r.t. the cleared quantities of

the market participants is maximized (Eq. (2.1)). The ﬁrst term

is the summation of the demand-cleared quantities, followed

by the BES day-ahead cleared energy quantities and up and

down reserve capacity provision costs. The third row refers

to generator costs for cleared energy, as well as up and down

reserve capacity provision costs. The fourth and ﬁfth rows

include generator and BES up and down capacity activation at

the balancing stage for expected realizations of wind scenarios.

The last row penalizes load shedding per wind power scenario.

Equation (2.2) is the power balance equation at the day-

ahead clearing stage. The uncertain wind farm output is sched-

uled at pws

w,t. Constraints (2.3)–(2.4) ensure sufﬁcient up and

down reserve capacities. Both can be provided by conventional

generators and/or BES. Equation (2.5) calculates deviation

of the power balance constraint for realization of each wind

scenario. Equations (2.6)–(2.7) calculate the power ﬂows at the

day-ahead and balancing stages, while Eq. (2.8) calculates the

deviation in power ﬂows between the two stages used in Eq.

(2.5). Equations (2.9)–(2.10) set the reference voltage angle to

zero and Constraints (2.11)–(2.12) limit the line power ﬂows

at the day-ahead and balancing stages. Constraint (2.13) sets

limits on the demand at each bus, while Eqs. (2.14)–(2.15)

impose limits on generation output and reserves. Constraints

(2.16)–(2.19) limit the maximum activated reserve over all

scenarios to the scheduled up and down reserved capacity for

generators and BES. Constraints (2.20)–(2.23) limit the cleared

up and down reserve quantities, as well as the day-ahead

energy charging and discharging quantities of strategic BES

to the values determined in the upper-level problem. Since up

reserve can be activated by the means of load shedding, it is

limited by the cleared demand in (2.24).

The required amounts of reserve capacity at each time

period are determined in Eqs. (2.25)–(2.26) and (2.29). The

required amount of up reserve capacity is the maximum of (i)

the difference between the day-ahead scheduled wind output

and the amount of wind output utilized at the balancing

stage (Eq. (2.25)) and (ii) zero (Eq. (2.29)). Similarly, the

required amount of down reserve capacity is determined via

6

Eqs. (2.26) and (2.29). Finally, constraint (2.27) is used to

limit wind curtailment, while (2.28) imposes non-negativity

on the remaining variables.

The dual variables associated with the day-ahead power

balance constraint (2.2), the upward reserve requirement (2.3),

the downward reserve requirement (2.4) and the balancing

power balance constraint (2.5) may be interpreted as energy,

up and down reserve capacity and balancing market prices.

C. Solution Strategy

To allow the use of off-the-shelf solvers, we reformulate the

mathematical problem with equilibrium constraints (MPEC)

above as a large-scale mixed-integer linear program (MILP).

To this end, the lower-level problem may be replaced by

its KKT optimality conditions so that the resulting single-

level equivalent problem contains non-linear complementary

slackness conditions, which may be recasted using the big M-

method as a set of inequalities, see, e.g., [5]. The remaining

non-linear terms are found in the right-hand side of Eq. (1.2),

which includes three multiplications of dual and primal vari-

ables. Based on the optimality conditions, the strong duality

theorem and some algebraic operations, one may obtain an

equivalent linear expression for the expected proﬁt (i.e., the

ﬁrst term in objective (1.1)):

X

p∈P

Πp·φp(Ξ) = X

t∈T

X

s∈S

CD

s·ds,t

−X

t∈T

X

i∈I

(CG

i·gi,t +CG↑

i·rg↑

i,t +CG↓

i·rg↓

i,t)

−X

p∈P

X

t∈T

X

i∈I

Πp·(CBG↑

i·rbg↑

p,i,t −CBG↓

i·rbg↓

p,i,t)

−X

t∈T

X

l∈L

Fl·γ−

l,t −X

t∈T

X

l∈L

Fl·γ+

l,t −X

t∈T

X

p∈P

X

l∈L

Fl·γb−

p,l,t

−X

t∈T

X

p∈P

X

l∈L

Fl·γb+

p,l,t −X

t∈T

X

s∈S

Ds,t ·δ+

s,t −X

t∈T

X

i∈I

Gi·ψ+

i,t

+X

t∈T

X

p∈P

X

w∈W

Pw

p,w,t ·τ↓

p,t −X

t∈T

X

p∈P

X

w∈W

Pw

p,w,t ·τ↑

p,t

−X

t∈T

X

p∈P

X

w∈W

Pw

p,w,t ·φ+

p,w,t −X

t∈T

X

p∈P

X

w∈W

Pw

p,w,t ·λb

p,s,t

−X

t∈T

X

w∈W

Pw

w,t ·µws

w,t −X

p∈P

X

t∈T

X

s∈S

Πp·Vlol

s·dshed

p,s,t (3.1)

Equation (1.2), which deﬁnes the proﬁt per scenario, can

be replaced by the following equivalent expression using

optimality conditions and the strong duality theorem:

φp(Ξ) = X

p

Πp·φp(Ξ)

−X

p∈P

X

t∈T

X

h∈H

Πp·(CBS↑

h·rbs↑

p,h,t −CBS↓

h·rbs↓

p,h,t)

+X

t∈T

X

h∈H

CBS↑

h·rbs↑

p,h,t −X

t∈T

X

h∈H

CBS↓

h·rbs↓

p,h,t

+X

t∈T

X

h∈H

1

Πp

·ζbs↑+

p,h,t ·rs↑

h,t +X

t∈T

X

h∈H

1

Πp

·ζbs↓+

p,h,t ·rs↓

h,t

−X

t∈T

X

h∈H

X

p∈P

ζbs↑+

p,h,t ·rs↑

h,t −X

t∈T

X

h∈H

X

p∈P

ζbs↓+

p,h,t ·rs↓

h,t (3.2)

The ﬁrst term PpΠp·φp(Ξ) may be replaced by the

linear expression in Eq. (3.1). However, the terms containing

ζbs↑+

p,h,t ·rs↑

h,t and ζbs↓+

p,h,t ·rs↓

h,t remain non-linear. Note that this

non-linearity originates from the bilevel model structure, not

from the CVaR metric. To avoid solving an NP-hard MINLP,

one may opt to employ the binary expansion technique, see [9],

to recast ζbs↑+

p,h,t ·rs↑

h,t and ζbs↓+

p,h,t ·rs↓

h,t as a set of MILP inequality

constraints. Note that the binary expansion – inducing an

approximation error – is only required if one considers a

risk metric, which requires computing proﬁts per scenario.

However, as will be discussed in the computational tractabil-

ity analysis presented in Section IV, the binary expansion

technique results in intolerable computational times. Thus, we

construct an iterative procedure, where dual variables ζbs↑+

p,h,t

and ζbs↓+

p,h,t are considered as parameters in eq. (3.2) (but

not in the relevant stationarity and complementarity slackness

conditions, as they do not contain nonlinear products). These

parameters are updated with the values from the previous

iteration until stable values of φp(Ξ) are achieved. The step-

by-step procedure is described below:

1) Replace nonlinear terms ζbs↑+

p,h,t ·rs↑

h,t and ζbs↓+

p,h,t ·rs↓

h,t in

eq. (3.2) with linear terms ˆ

ζbs↑+

p,h,t ·rs↑

h,t and ˆ

ζbs↓+

p,h,t ·rs↓

h,t,

where ˆ

ζbs↑+

p,h,t and ˆ

ζbs↓+

p,h,t are constants.

2) Set ˆ

ζbs↑+

p,h,t =ˆ

ζbs↓+

p,h,t = 0 ∀p, h, t.

3) Solve the resulting MILP problem. Denote the resulting

estimate for the proﬁt per scenario as ˆ

φp(Ξ). Calculate the

optimal values ζbs↑+

p,h,t and ζbs↓+

p,h,t . Based on these values,

compute the actual proﬁt per scenario φp(Ξ).

4) If Pp∈P |ˆ

φp(Ξ) −φp(Ξ)| ≤ E · Pp∈P Πp·φp(Ξ), with

Ebeing a small real number, then stop the process.

Otherwise, update ˆ

ζbs↑+

p,h,t =ζbs↑+

p,h,t and ˆ

ζbs↓+

p,h,t =ζbs↓+

p,h,t

and return to step 3.

The computational efﬁciency of the proposed solving tech-

nique is discussed in Section IV.

To study a risk-neutral BES owner, maximizing expected

proﬁt, this iteration technique is not required, since Eq. (3.1)

contains only linear terms. In contrast, the approach from [9]

employs the binary expansion to solve the decision problem

of a risk-neutral BES owner.

III. CAS E ST UDY

The proposed formulation is tested on an 8-zone model of

the ISO New England system with 76 generators described

in [23]. Annual wind generation proﬁles with an hourly

resolution were taken from [24] for the 30% wind penetration

level in terms of the annual electricity produced. The peak

load on the chosen day is 11.1 GW, while the maximum

available wind power is over 3 GW. Ten equiprobable wind

scenarios are considered at the balancing stage, which should

be sufﬁcient to capture the uncertainty distribution of wind

power plant output, see e.g. [25], [26]. To test the proposed

formulation, we use a 100 MWh/100 MW battery storage with

0.9 discharge efﬁciency connected to different buses of the

network to identify the characteristics of the proposed model

and show relevant results. The BES bids to sell energy at

e0/MWh and purchase it at e100/MWh both in the day-

ahead energy market and reserve activation stages to ensure

7

TABLE I

PROFI T (e)IN T HE DAY-AHEAD ENERGY (DA), CAPACI TY R ES ERVATIO N (CA P.) AN D BA LAN CI NG M AR KE T AS W EL L AS EX PE CT ED P ROFI T (E XP.).

Bus DA Cap. Balancing scenarios Exp.

12345678910

1 263 0 1,319 526 225 998 2,898 334 2,454 2,565 1,115 1,319 1,638

2 -99 0 3,269 2,681 3,139 0 670 1,272 1,684 1,571 1,272 1,275 1,584

3 171 0 2,565 1,610 2,457 0 403 998 1,246 1,178 998 1,000 1,416

4 1,709 0 0 0 0 0 0 0 0 0 0 0 1,709

5 471 0 1,483 794 334 998 3,864 192 2,469 2,565 1,174 1,483 2,006

6 63 0 2,565 435 2,442 0 109 998 1,067 1,048 998 1,000 1,129

7 89 34 2,090 1,122 1,466 240 998 1,923 825 1,001 161 288 1,134

8 1,612 51 2,565 1,616 4,298 3,364 2,565 219 1,607 3,915 3,830 1,387 4,200

its bids are accepted. Both up and down BES reserve capacity

is offered at e0/MW. The parameters related to the accurate

battery charging model (1C maximum charging power) are

available in [19]. Thermal generators offer their up and down

reserve capacity at 30% of their generation costs.

First part of the case study, presented in Section III-A,

focuses on a risk-neutral market participant, where parameter

αin Eq. (1.1) is set to 1. The second part of the case study,

Section III-B, shows the results for different values of CVaR

parameters αand . The ﬁnal part of the case study in Section

III-C quantiﬁes the importance of using the accurate battery

charging model in strategic battery models.

In all simulations, the optimality gap is set to 0.1%. For the

risk-averse cases, Eis set to 0.01. The models are implemented

in GAMS V33.2 and solved using CPLEX V12.1 on Intel Core

i7-6600 CPU clocking at 2.6 GHz with 8 GB of RAM.

A. Risk-neutral BES

The results for a risk-neutral BES at all eight locations, i.e.

buses, are shown in Table I. The proﬁts are broken down into

three parts: i) the day-ahead proﬁt in the energy-only market;

ii) the day-ahead capacity reservation payment; and iii) the

balancing market proﬁt. In most cases, the day-ahead proﬁt

is rather low, indicating that the BES relies on the balancing

market to monetize (a portion of the) charging actions in the

day-ahead stage. Generally, the BES capacity is divided in two

categories, the ﬁrst devoted to the day-ahead arbitrage (both

charging and discharging in the day-ahead market), and the

second one to providing up reserve from the energy charged

in the day-ahead market (charging in the day-ahead market

and discharging in the balancing market). When the BES is

connected to bus 2, the BES capacity devoted to the day-

ahead market arbitrage is much lower, as compared to the

capacity devoted to providing the up reserve, resulting in a

negative day-ahead proﬁt. On the other hand, when the BES is

connected to bus 4, it takes part only in the day-ahead market

and ignores the balancing market. Low capacity reservation

proﬁt in all cases is a result of low up capacity reservation

prices in hours in which the BES is scheduled to provide

reserves. However, up reserve capacity reservation enables the

BES to provide up reserve in balancing scenarios, where the

BES makes majority of its revenue when connected to any bus

but bus 4. Especially proﬁtable is balancing scenario 3 when

the BES is connected to bus 8, yielding e4,298 revenue. On

the other hand, balancing scenario 6 brings only e219, and

if this balancing scenario materializes the BES will receive

only the day-ahead and capacity reservation revenue, i.e.

e1,663. The ﬁnal column in Table I shows the expected proﬁt

0 4 8 12 16 20 24

Timesteps

0

2

Reserve capacity

Upward

Downward

0 4 8 12 16 20 24

Timesteps

0

100

200

Upward reserve

provision (MW)

Generators + BES

BES

0 4 8 12 16 20 24

Timesteps

0

50

100

Downward reserve

provision (MW)

Generators + BES

BES

Fig. 2. Up and down reserve capacity prices and volumes at the day-ahead

stage when the BES is connected to bus 8.

calculated as the sum of the day-ahead payment, capacity

reservation payment and weighed average of the balancing

scenario revenues. This proﬁt is the lowest when the BES

is connected to bus 6 and highest when connected to bus

8, which is the assumed BES location in the remainder of

this case study. The presented results stress the importance

of an adequate choice of the BES location, depending on the

congestion patterns in the network, conﬁrming the results of,

i.a., [4].

Figure 2 shows reserve capacity prices as well as up and

down reserved quantities at the day-ahead stage when the BES

is connected to bus 8. The up reserve price is often zero with

spikes up to e3.2/MW at certain hours. However, only in hour

19 the non-zero up reserve capacity price coincides with the

BES providing up reserve capacity (compare the upper two

graphs in Figure 2). The BES up capacity is reserved in hour

10, 12.0 MW, followed by hours 12, 9.2 MW, and 16, 7.9

MW, while the highest reserved volume is 15.9 MW in hour

19. The BES does not provide any down reserve. Reserve

activation per balancing scenario is analyzed in Figure 3. The

scheduled 12.0 MW of up reserve capacity in hour 10 is fully

activated in all balancing scenarios but scenarios 7 and 10.

In hour 12, the up reserve capacity is not activated at all in

scenario 6, while in scenarios 2 and 3 the up reserve is only

partially activated (5.1 and 8.2 MWh). In scenario 6 the up

reserve capacity is activated only in hour 10, while in hours 12,

16 and 19 the BES remains inactive. The balancing prices in

Figure 4 reveal that this scenario has very low balancing price

(red line), thus this inactivity has a minor effect on the BES

proﬁtability. On the other hand, scenarios 9, 8 and 4 reach an

8

0 4 8 12 16 20 24

-50

0

50

Total balancing energy

Generators

Load shedding

BES

0 4 8 12 16 20 24

-50

0

50

0 4 8 12 16 20 24

-50

0

50

0 4 8 12 16 20 24

-50

0

50

0 4 8 12 16 20 24

-50

0

50

Reserve activation (MWh)

0 4 8 12 16 20 24

-50

0

50

0 4 8 12 16 20 24

-50

0

50

0 4 8 12 16 20 24

-50

0

50

0 4 8 12 16 20 24

-50

0

50

0 4 8 12 16 20 24

Timesteps

-50

0

50

Fig. 3. Reserve activation across ten balancing scenarios when the BES is

connected to bus 8. The y-axis range is limited to better observe the relatively

low BES reserve activation.

extremely high balancing price e228/MWh in hours 10, 12

and 16, respectively, which is the main reason for the high

proﬁtability of these scenarios exhibited in Table I. However,

the most proﬁtable is scenario 3 for two reasons. First, in

this scenario the BES fully activates the up reserve in all

four relevant hours, and, second, it achieves the highest up

0 4 8 12 16 20 24

Timesteps

0

50

100

150

200

250

Day-ahead

Scenario 1

Scenario 2

Scenario 3

Scenario 4

Scenario 5

Scenario 6

Scenario 7

Scenario 8

Scenario 9

Scenario 10

Fig. 4. Day-ahead and balancing prices across ten balancing scenarios when

the BES is connected to bus 8 (balancing prices in hours 10, 12, 16 and 19,

in which the BES provides up reserve, are marked with squares).

reserve price, e192/MWh, in hour 19, which is the hour with

the highest volume of the activated BES up reserve. In our

case study, since up reserve provision reserves a portion of its

capacity, the BES will provide up reserve only if its activation

is very likely (in our case at least eight out of ten scenarios)

and balancing prices in average exceed the day-ahead energy

prices, given the low reserve capacity prices.

B. Risk-averse BES

We study a risk-averse BES connected to bus 8, assuming

an equal weighting between the expected proﬁt and the CVaR.

We set αto 0.5 and vary from 0.9 to 0.1 in increments of

0.1. The results are summarized in Table II.

In the risk-neutral case, the BES owner is indifferent w.r.t.

the variability in proﬁts between balancing scenarios (ﬁrst row

in Table II). As such, it opens itself up to a signiﬁcant ﬁnancial

risk – its balancing proﬁt may vary between e219 and e4,298.

The introduction of the CVaR metric, however, strongly re-

duces this variability in the balancing market outcomes. The

maximum difference in proﬁt among the considered balancing

scenarios decreases from e4,079 in the risk-neutral case to

e826 for -values between 0.9 and 0.6. This comes at the

expense of a decrease in expected proﬁt of at most e23

or 0.55%. The variations in the expected proﬁt for -values

between 0.9 and 0.6 are within the optimality gap. The risk-

averse BES owner maintains the same offers in the day-ahead

energy market as the risk-neutral BES owner, but changes the

timing of its reserve capacity offers. Since the BES owner’s

reserve offers are constrained by its battery capacity, it may

offer the same reserve capacity, but at different times, limiting

the variability in proﬁts per scenario, hence, its risk.

As the CVaR metric spans a smaller part of the proﬁt

distribution (-values between 0.5 and 0.1), the BES owner

foregoes any proﬁt in the balancing scenarios.. This results in

a drop in expected proﬁt of e95 or 2.3%. As such, in this

particular case study, a BES operator is able to eliminate its

ﬁnancial risk – associated with participating in the balancing

market – entirely by limiting itself to arbitrage in the day-

ahead energy market.

If the risk-averse BES owner does not seek a trade-off

between CVaR and expected proﬁts (i.e., sets α= 0), the

9

TABLE II

PROFI T OF A R IS K-AVER SE BES O WNE R (e)IN T HE DAY-AHEAD ENERGY (DA), CAPAC IT Y RES ERVATIO N (C AP.) AND BALANCING MARKET AS WELL AS

EX PEC TE D PRO FIT ( EX P.) AND CONDITIONAL VALUE-AT-RI SK (CVAR), DEPE ND IN G ON I TS R ISK -ATTI TU DE (αI S SE T TO 0.5 I N ALL C AS ES ).

DA Cap. Balancing scenarios Exp. CVaR

12345678910

Risk-neutral 1,612 51 2,565 1,616 4,298 3,364 2,565 219 1,607 3,915 3,830 1,387 4,200 -

0.9 1,612 0 2,106 2,773 2,773 2,773 2,773 2,773 1,955 2,773 2,773 2,175 4,177 4,154

0.8 1,612 0 2,166 2,765 2,773 2,765 2,765 2,765 1,950 2,765 2,765 2,175 4,178 4,127

0.7 1,612 0 2,166 2,773 2,767 2,765 2,765 2,765 1,950 2,765 2,789 2,175 4,181 4,091

0.6 1,612 0 2,166 2,766 2,758 2,766 2,766 2,766 1,950 2,766 2,766 2,175 4,177 4,043

0.5 →0.1 4,105 0 0 0 0 0 0 0 0 0 0 0 4,105 4,105

outcomes discussed above are observed as well, albeit at

different -values. For equal to 0.8 or 0.9, the BES owner still

participates in the balancing market, but ensures the variability

in proﬁts per balancing scenario is limited at the expense of

a 0.55% drop in expected proﬁt. If is equal or less than 0.7,

the BES owner does not participate in the balancing market,

eliminating its ﬁnancial risk, but reducing its expected proﬁt

to e4,105.

C. Effect of the Accurate Battery Charging Model

To quantify the importance of using the accurate battery

charging model, we compare its performance to that of an

equivalent constant-power charging model, as used in a vast

majority of battery-related energy economics studies, i.a., [9],

[10]. The difference between the models is that the basic,

constant-power charging model does not include constraints

(1.12)–(1.15) and on the right-hand side of constraints (1.6)–

(1.8) the term ∆soeh,t

ηch

h

is replaced with Qch

h,t/ηch

h.

We run both models for a BES connected to bus 8. The

expected proﬁt resulting from both models is within the

optimality gap, indicating that the more rigorous accurate

battery charging model, although it essentially increases the

number of time periods needed to fully charge the BES,

does not deteriorate the objective function value. This is

because the majority of electricity is still being charged in

the lowest-price time period and only a fraction is charged in

the surrounding time periods. However, the obtained charging

and discharging schedules are different and we analyze the

real-world feasibility of the basic battery charging model,

i.e., the ability of an actual battery to follow the obtained

schedule in reality. The accurate battery charging model is

considered to accurately describe the battery charging process,

as proven in [19], thus the battery schedule obtained by using

the basic battery charging model is run against the battery

charging constraints of accurate battery charging model. Any

deviations from the planned state-of-energy are transferred

to subsequent time periods incurring further inconsistencies.

The higher the inconsistencies in late time periods of the

optimization horizon, the higher risk the use of the basic

battery charging model imposes to the BES owner.

Figure 5 shows the scheduled evolution of the BES SoE over

time for all ten balancing scenarios for (a) the accurate battery

charging model; (b) the basic battery charging model; (c) the

re-evaluation of the BES’ schedule obtained with the basic

battery charging model. When using the accurate charging

model (Figure 5(a)), the BES is charged from the initial 50

MWh to 92.8 MWh in hour 6, then to 99.1 MWh in hour

7 and slowly topped in the following hours until it reaches

full charge in hour 9, recognizing the fact that the charging

capacity reduces as the battery state-of-energy increases. The

charged energy is used to offer upward balancing services

from hour 10 to hour 19. Whenever the BES provides up

reserve, this reserve is fully deployed in at least eight out of

ten balancing scenarios. Scenario 6 activates the least reserve

and thus ﬁnishes the day at the highest state-of-energy level,

36.7 MWh. On the other hand, scenarios 1, 4, 5, 8 and 9

activate the entire BES up reserved capacity, resulting in a

fully depleted BES at the end of the day.

As opposed to the accurate one, the basic battery charging

model assumes it may charge the BES to 100 MWh in a single

hour (hour 6), as shown in Figure 5(b). However, this is not

possible in reality as the BES at 50 MWh state-of-energy can

charge only to 92.75 MWh in one hour. Hence, the actual state-

of-energy at the end of hour 6 will be 92.75 MWh instead

of 100 MWh (Figure 5(c)). This means that the generators

providing down reserve will be activated. Figure 6 shows the

scheduled up and down generators’ reserve capacity available

for activation during hour 6. The BES is not scheduled to

0 5 10 15 20 25

0

20

40

60

80

100

(a) Accurate charging model

0 5 10 15 20 25

0

20

40

60

80

100

(b) Basic charging model

0 5 10 15 20 25

-20

0

20

40

60

80

100

(c) Basic charging model – actual realization

Fig. 5. The battery system’s state-of-energy (SoE) for different balancing scenarios: (a) the accurate charging model; (b) the basic charging model; (c)

re-evaluation of the basic charging model’s schedule via the accurate charging model.

10

12345678910

Balancing scenario

-40

-20

0

20

40

60

Activated reserve (MWh)

Up reserve capacity

Down reserve capacity

Activated reserve without BES imbalance

Activated reserve with BES imbalance

Fig. 6. Sum of the generators’ deployed up and down reserve capacity per

balancing scenario in hour 6 with and without imbalance caused by the BES.

provide any reserve at this hour. The blue bars indicate the

reserve activation per scenario if the BES would not cause any

imbalance. However, as the BES cannot charge the scheduled

amount based on the basic BES model, the required down

reserve to be activated in all scenarios is increased by 7.25

MWh (=50 MWh – 42.25 MWh), as shown by orange bars

in Figure 6. This reduces the required up reserve activation in

scenarios 7, 8 and 9, causes down reserve activation within

the scheduled down reserve capacity in scenarios 3, 4 and

10, but in scenarios 1, 2, 5 and 6 requires activation of the

generators’ down reserve not scheduled at the day-ahead stage.

Hence, emergency down reserve capacity needs to be activated

or wind production needs to be curtailed to stabilize the system

in these four scenarios.

As presented in Figure 5(c), the reduced amount of stored

energy in the BES at the end of hour 6 is sufﬁcient to

provide up reserve in the subsequent hours until 21. In hour

21 the BES needs to provide 31.36 MWh of reserve in all

ten scenarios. However, in eight scenarios (1–3, 5–9) that

would result in a negative state-of-energy. To balance the

system, the generators scheduled to provide up reserve in

hour 21 need to counteract this imbalance. In most scenarios

the scheduled generators’ up reserve capacity is sufﬁcient

(Figure 7). However, in scenario 5, the generators’ up reserve

is already fully deployed to balance the wind deviation and

no additional up reserve capacity is scheduled to balance

the BES’s inability to provide up reserve. Again, emergency

measures such as load shedding would be required to stabilize

the system. This analysis clearly demonstrates the importance

of using the accurate BES charging model instead of the

generic one, which is highly unsuitable for the BES market

scheduling purposes.

IV. COM PUTATIONA L EFFICI EN CY

Solving the day-ahead decision problem of the risk-neutral

BES owner requires between 170 and 416 seconds, with an

average of 305 seconds. Recall that no iterations are required

to solve this problem.

The iterative procedure proposed in Section II-C to solve

the decision problem of the risk-averse BES owner terminates

in 2 to 5 iterations (on average 3). Each iteration requires

12345678910

Balancing scenario

0

20

40

60

80

Activated reserve (MWh)

Up reserve capacity

Activated reserve without BES imbalance

Activated reserve with BES imbalance

Fig. 7. Sum of the generators’ deployed up reserve capacity per balancing

scenario in hour 21 with and without imbalance caused by the BES.

between 240 and 549 seconds, with an average of 366 seconds

per iteration. Total calculation times range from 530 to 2,407

seconds, with an average of 1,057 seconds. As a benchmark,

we implemented the equivalent MILP problem based on the

binary expansion technique. In order to ensure solutions with

a similar accuracy as those obtained based on the iterative pro-

cedure, we discretized variables rs↑

h,t and rs↓

h,t with a resolution

of 1 MW. Solving these NP-hard MILP problems, in our case

study, requires more than 20,000 seconds.

V. CONCLUSION

The paper presented a novel formulation of the decision

problem faced by a strategic BES owner in the joint day-ahead

energy-reserve and balancing markets, which allows managing

a variety of risks. First, the ﬁnancial risk is addressed by

using the CVaR, which enables the BES to evenly distribute its

proﬁt expectations over the possible realizations of uncertainty

without the reduction in the expected proﬁt. This is achieved

by changing the timing of reserve capacity offers to ﬂatten

the proﬁt curve across all scenarios. Second, the risk of

the inability to deliver the scheduled reserves is addressed

embedding the worst-case reserve activation constraints in the

formulation, i.e., by assuming that all reserves scheduled to

the BES may be consecutively activated in the up and/or down

direction. Last, the risk of the inability to follow the day-ahead

schedule resulting from an inaccurate battery model should

be mitigated by adopting accurate battery models, while the

generic energy storage models are ill-suited for BES. In the

case study, we illustrated that these three model features allow

risk-averse BES owners to hedge their day-ahead position

without jeopardizing their expected proﬁt, while ensuring the

feasibility of their day-ahead schedule.

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