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When participating in electricity markets, owners of battery energy storage systems must bid in such a way that their revenues will at least cover their true cost of operation. Since cycle aging of battery cells represents a substantial part of this operating cost, the cost of battery degradation must be factored in these bids. However, existing models of battery degradation either do not fit market clearing software or do not reflect the actual battery aging mechanism. In this paper we model battery cycle aging using a piecewise linear cost function, an approach that provides a close approximation of the cycle aging mechanism of electrochemical batteries and can be incorporated easily into existing market dispatch programs. By defining the marginal aging cost of each battery cycle, we can assess the actual operating profitability of batteries. A case study demonstrates the effectiveness of the proposed model in maximizing the operating profit of a battery energy storage system taking part in the ISO New England energy and reserve markets.
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1
Factoring the Cycle Aging Cost of Batteries
Participating in Electricity Markets
Bolun Xu, Student Member, IEEE, Jinye Zhao, Member, IEEE, Tongxin Zheng, Senior Member, IEEE,
Eugene Litvinov, Fellow, IEEE, Daniel S. Kirschen, Fellow, IEEE
Abstract—When participating in electricity markets, owners of
battery energy storage systems must bid in such a way that their
revenues will at least cover their true cost of operation. Since
cycle aging of battery cells represents a substantial part of this
operating cost, the cost of battery degradation must be factored
in these bids. However, existing models of battery degradation
either do not fit market clearing software or do not reflect
the actual battery aging mechanism. In this paper we model
battery cycle aging using a piecewise linear cost function, an
approach that provides a close approximation of the cycle aging
mechanism of electrochemical batteries and can be incorporated
easily into existing market dispatch programs. By defining the
marginal aging cost of each battery cycle, we can assess the actual
operating profitability of batteries. A case study demonstrates the
effectiveness of the proposed model in maximizing the operating
profit of a battery energy storage system taking part in the ISO
New England energy and reserve markets.
Index Terms—Energy storage, battery aging mechanism, arbi-
trage, ancillary services, economic dispatch
I. INT ROD UC TI ON
In 2016, about 200 MW of stationary lithium-ion batteries
were operating in grid-connected installations worldwide [1],
and more deployments have been proposed [2], [3]. To ac-
commodate this rapid growth in installed energy storage ca-
pacity, system operators and regulatory authorities are revising
operating practices and market rules to take advantage of the
value that energy storage can provide to the grid. In particular,
the U.S. Federal Energy Regulatory Commission (FERC) has
required independent system operators (ISO) and regional
transmission organizations (RTO) to propose market rules that
account for the physical and operational characteristics of
storage resources [4]. For example, the California Independent
System Operator (CAISO) has already designed a market
model that supports the participation of energy-limited storage
resources and considers constraints on their state of charge
(SoC) as well as on their maximum charge and discharge
capacity [5].
As electricity markets evolve to facilitate participation by
battery energy storage (BES), owners of these systems must
develop bidding strategies which ensure that they will at
least recover their operating cost. Battery degradation must
be factored in the operating cost of a BES because the life of
electrochemical battery cells is very sensitive to the charge and
discharge cycles that the battery performs and is thus directly
B. Xu and D.S. Kirschen are with the University of Washington, USA
(emails: {xubolun, kirschen}@uw.edu).
J. Zhao, T. Zheng, and E. Litvinov are with ISO New England Inc., USA
(emails: {jzhao, tzheng, elitvinov}@iso-ne.com).
affected by the way it is operated [6], [7]. Existing models
of battery degradation either do not fit dispatch calculations,
or do not reflect the actual battery degradation mechanism.
In particular, traditional generator dispatch models based on
heat-rate curves cannot be used to represent the cycle aging
characteristic of electrochemical batteries.
This paper proposes a new and accurate way to model of
the cost of battery cycle aging, which can be integrated easily
in economic dispatch calculations. The main contributions of
this paper can be summarized as follows:
It proposes a piecewise linear cost function that provides
a close approximation of the cost of cycle aging in
electrochemical batteries.
System operators can incorporate this model in market
clearing calculations to facilitate the participation of BES
in wholesale markets by allowing them to properly reflect
their operating cost.
Since this approach defines the marginal cost of battery
cycle aging, it makes it possible for BES owners to design
market offers and bids that recover at least the cost of
battery life lost due to market dispatch.
The effectiveness of the proposed model is demonstrated
using a full year of price data from the ISO New England
energy markets.
The accuracy of the proposed model in predicting the
battery cycle aging cost is demonstrated using an ex-post
calculation based on a benchmark model.
The model accuracy increases with the number of lin-
earization segments, and the error compared to the bench-
mark model approaches zero with sufficient linearization
segments.
Section II reviews the existing literature on battery cycle
aging modeling. Section III describes the proposed predictive
battery cycle aging cost model. Section IV shows how this
model is incorporated in the economic dispatch. Section V
describes and discusses case studies performed using ISO New
England market data. Section VI draws conclusions.
II. LI TE RATU RE RE VI EW
A. Battery Operating Cost
Previous BES economic studies typically assume that bat-
tery cells have a fixed lifetime and do not include the cost
of replacing the battery in the BES variable operating and
maintenance (O&M) cost [8]. The Electricity Storage Hand-
book from Sandia National Laboratories assumes that a BES
performs only one charge/discharge cycle per day, and that
2
the variable O&M cost of a lithium-ion BES is constant and
about 2 $/MWh [9]. Similarly, Zakeri et al. [10] assume that
battery cells in lithium-ion BES are replaced every five years,
and assume the same 2 $/MWh O&M cost. Other energy
storage planning and operation studies also assume that the
operating cost of BES is negligible and that they have a fixed
lifespan [11]–[13]. These assumptions are not valid if the BES
is cycled multiple times per day because more frequent cycling
increases the rate at which battery cells degrade and hasten the
time at which they need to be replaced. To secure the battery
lifespan, Mohsenian-Rad [14] caps the number of cycles a
battery can operate per day. However, artificially limiting the
cycling frequency prevents operators from taking advantage
of a BES’s operational flexibility and significantly lessens its
profitability. To take full advantage of the ability of a BES to
take part in energy and ancillary markets, its owner must be
able to cycle it multiple times per day and to follow irregular
cycles. Under these conditions, its lifetime can no longer be
considered as being fixed and its replacement cost can no
longer be treated as a capital expense. Instead, the significant
part of the battery degradation cost that is driven by cycling
should be treated as an operating expense.
A BES performs temporal arbitrage in an electricity market
by charging with energy purchased at a low price, and dis-
charging this stored energy when it can be sold at a higher
price. The profitability of this form of arbitrage depends not
only on the price difference but also on the cost of the battery
cycle aging caused by these charge/discharge cycles. When
market prices are stable, the expected arbitrage revenue is
small and the BES owner may therefore opt to forgo cycling
to prolong the battery lifetime and reduce its cycle aging cost.
On the other hand, if the market exhibits frequent large price
fluctuations, the BES owner could cycle the BES multiple
times a day to maximize its profits. Fig. 1 shows that the
price profile in a given market can change significantly from
day to day. Although the average market price is higher in
Fig. 1a, arbitrage is not profitable in this case because the price
fluctuations are small, and the aging cost from cycling is likely
to be higher than the revenue from arbitrage. On the other
hand, a BES owner is likely to perform three arbitrage cycles
if the price profile is similar to the one shown on Fig. 1b,
because the revenue opportunities arising from the large price
fluctuations are likely to be larger than the associated cycle
aging cost. It is thus crucial to accurately incorporate the cost
of cycle aging into the optimal operation of a BES.
B. Electrochemical Battery Degradation Mechanisms
Electrochemical batteries have limited cycle life [15] be-
cause of the fading of active materials caused by the charging
and discharging cycles. This cycle aging is caused by the
growth of cracks in the active materials, a process similar to
fatigue in materials subjected to cyclic mechanical loading [6],
[16]–[19]. Chemists describe this process using partial differ-
ential equations [20]. These models have good accuracy but
cannot be incorporated in dispatch calculations. On the other
hand, heuristic battery lifetime assessment models assume
that degradation is caused by a set of stress factors, each
0 5 10 15 20
-200
-100
0
100
200
LMP[$/MWh]
[hr]
(a) An example of stable market prices (Jan 7, 2015).
0 5 10 15 20
-200
-100
0
100
200
LMP[$/MWh]
[hr]
(b) An example of highly variable market prices (Jan 2, 2015).
Fig. 1. Market price daily variation examples (Source: ISO New England).
of which can be represented by a stress model derived from
experimental data. The effect of these stress factors varies with
the type of battery technology. In this paper, we focus on
lithium-ion batteries because they are widely considered as
having the highest potential for grid-scale applications. For
our purposes, it is convenient to divide these stress factors
into two groups depending on whether or not they are directly
affected by the way a grid-connected battery is operated:
Non-operational factors: ambient temperature, ambient
humidity, battery state of life, calendar time [21].
Operational factors: Cycle depth, over charge, over dis-
charge, current rate, and average state of charge (SoC) [6].
1) Cycle depth: Cycle depth is an important factor in a
battery’s degradation, and is the most critical component in
the BES market dispatch model. A 7 Wh Lithium Nickel
Manganese Cobalt Oxide (NMC) battery cell can perform
over 50,000 cycles at 10% cycle depth, yielding a lifetime
energy throughput (i.e. the total amount of energy charged
and discharged from the cell) of 35 kWh. If the same cell is
cycled at 100% cycle depth, it can only perform 500 cycles,
yielding an lifetime energy throughput of only 3.5 kWh [22].
This nonlinear aging property with respect to cycle depth is
observed in most static electrochemical batteries [7], [23]–
[25]. Section III explains in details our modeling of the cycle
depth stress.
2) Current rate: While high charging and discharging cur-
rents accelerate the degradation rate, grid-scale BES normally
have capacities greater than 15 minutes. The effect of current
rate on degradation is therefore small in energy markets
according to results of laboratory tests [25]. We will therefore
not consider the current rate in our model. If necessary, a
piecewise linear cost curve can be used to model the current
rate stress function as a function of the battery’s power output.
3) Over charge and over discharge: In addition to the
cycle depth effect, extreme SoC levels significantly reduce
battery life [6]. However, over-charging and over-discharging
are avoided by enforcing upper and lower limits on the SoC
3
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0
10
20
30
40
50
60
SoC [%]
[h]
s2
s3
s4
s5
s6
s7
s1
s0 s8
(a) An example of SoC profile.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0
10
20
30
40
50
60
SoC [%]
[h]
δ1 = 10%
δ3 = 10%
remaining
profile...
δ2 = 40%
(b) The cycle counting result.
Fig. 2. Using the rainflow algorithm to identify battery cycle depths.
either in the dispatch or by the battery controller.
4) Average state of charge: The average SoC level in each
cycle has a highly non-linear but slight effect on the cycle
aging rate [22], [26]. Therefore we do not consider this stress
factor in the proposed model.
C. The Rainflow Counting Algorithm
The rainflow counting algorithm is used extensively in
materials stress analysis to count cycles and quantify their
cumulative impact. It has also been applied to battery life
assessment [7], [27]. Given a SoC profile with a series of
local extrema (i.e. points where the current direction changed)
s0,s1,..., etc, the rainflow method identifies cycles as [28]:
1) Start from the beginning of the profile (as in Fig. 2a).
2) Calculate s1=|s0s1|,s2=|s1s2|,s3=
|s2s3|.
3) If s2s1and s2s3, then a full cycle of
depth s2associated with s1and s2has been identified.
Remove s1and s2from the profile, and repeat the
identification using points s0,s1,s4,s5...
4) If a cycle has not been identified, shift the identification
forward and repeat the identification using points s1,s2,
s3,s4...
5) The identification is repeated until no more full cycles
can be identified throughout the remaining profile.
The remainder of the profile is called the rainflow residue and
contains only half cycles [29]. A half cycle links each pair of
adjoining local extrema in the rainflow residue profile. A half
cycle with decreasing SoC is a discharging half cycle, while
a half cycle with increasing SoC is a charging half cycle. For
example, the SoC profile shown on Fig. 2b has two full cycles
of depth 10% and one full cycle of depth 40%, as well as a
discharging half cycle of depth 50% and charging half cycle
of depth 50%.
The rainflow algorithm does not have an analytical math-
ematical expression [30] and cannot be integrated directly
0 10 20 30 40 50 60 70 80 90 100
Cycle depth [%]
0
0.02
0.04
0.06
Cycle life loss [%]
cycle life curve
1-segment
2-segment
4-segment
Fig. 3. Upper-approximation to the cycle depth aging stress function.
within an optimization problem. Nevertheless, several efforts
have been made to optimize battery operation by simplifying
the rainflow algorithm. Abdulla et al. [31] and Tran et al. [32]
simplify the cycle depth as the BES energy output within each
control time interval. Koller et al. [33] define a cycle as the
period between battery charging and discharging transitions.
These model simplifications enable the incorporation of cycle
depth in the optimization of BES operation, but introduce
additional errors in the degradation model. He et al. [34]
decompose the battery degradation model and optimize BES
market offers iteratively. This method yields more accurate
dispatch results, but is too complicated to be incorporated in
an economic dispatch calculation.
We will use the rainflow algorithm as the basis for an ex-
post benchmark method for assessing battery cycle life. In this
model, the total life lost Lfrom a SoC profile is assumed to be
the sum of the life loss from all Inumber of cycles identified
by the rainflow algorithm. If the life loss from a cycle of depth
δis given by a cycle depth stress function Φ(δ)of polynomial
form, we have:
L=PI
i=1 Φ(δi).(1)
III. MAR GI NAL COST OF BATTERY CYCLIN G
In order to participate fully in electricity markets, owners
of batteries must be able to submit offers and bids that
reflect their marginal operating cost. As we argued above,
this marginal cost curve should reflect the cost of battery
degradation caused by each cycle. In order to keep the model
simple, and to obtain a cost function similar to those used
in existing market dispatch programs, we assume that battery
cycle aging only occurs during the discharge stage of a cycle,
so that a discharging half cycle causes the same cycle aging
as a full cycle of the same depth, while a charging half cycles
causes no cycle aging. This is a reasonable assumption because
the amounts of energy charged and discharged from a battery
are almost identical when assessed on a daily basis.
During a cycle, if the BES is discharged from a starting
SoC eup to an end SoC edn and later charged back (or vice-
versa), the depth of this cycle is the relative SoC difference
(eup edn)/Erate, where Erate is the energy capacity of the
BES. Let a battery be discharged from a cycle depth δt1at
time interval t1. This battery’s cycle depth at time tcan be
calculated from its output power gtover time (assuming the
time interval duration is one hour):
δt=1
ηdisErate gt+δt1,(2)
4
where ηdis is the BES discharge efficiency, and gthas non-
negative values because we ignore charging for now. The
incremental aging resulting from this cycle is Φ(δt), and the
marginal cycle aging can be calculated by taking the derivative
of Φ(δt)with respect to gtand substituting from (2):
Φ(δi)
∂gt
=dΦ(δi)
i
∂δi
∂gt
=1
ηdisErate
dΦ(δi)
i
,(3)
To define the marginal cost of cycle aging, we prorate the
battery cell replacement cost R($) to the marginal cycle aging,
and construct a piecewise linear upper-approximation function
c. This function consists of Jsegments that evenly divide the
cycle depth range (from 0 to 100%)
c(δt) =
c1if δt[0,1
J)
.
.
.
cjif δt[j1
J,j
J)
.
.
.
cJif δt[J1
J,1]
,(4)
where
cj=R
ηdisErate JΦ( j
J)Φ(j1
J),(5)
and δtis the cycle depth of the battery at time t. Fig. 3
illustrates the cycle depth stress function and its piecewise
linearization with different numbers of segments.
IV. OPT IM IZ IN G TH E BE S DI SPATC H
Having established a marginal cost function for a BES, we
are now able to optimize how it should be dispatched assuming
that it acts as a price-taker on the basis of perfect forecasts of
the market prices for energy and reserve. A formal description
of this optimization requires the definitions of the following
parameters:
T: Number of time intervals in the optimization horizon,
indexed by t
J: Number of segments in the cycle aging cost function,
indexed by j
M: Duration of a market dispatch time interval
S: Sustainability time requirement for reserve provision
E0: Initial amount of energy stored in the BES
Efinal: Amount of energy that must be stored at the end
of the optimization horizon
Emin and Emax: Minimum and maximum energy stored
in the BES
D,G: Discharging and charging power ratings
cj: Marginal aging cost of cycle depth segment j
ej: Maximum amount of energy that can be stored in
cycle depth segment j
e0
j: Initial amount of energy of cycle depth segment j
ηch,ηdis: Charge and discharge efficiencies
λe
t,λq
t: Forecasts of the energy and reserve prices at t
This optimization uses the following decision variables:
pch
t,j ,pdis
t,j : Charge and discharge power for cycle depth
segment jat time t
et,j : Energy stored in marginal cost segment jat time t
dt,gt: Charging and discharging power at time t
dq
t,gq
t: BES baseline charging and discharging power at
time tfor reserve provision
qt: Reserve capacity provided by the BES at time t
vt: Operating mode of the BES: if at time tthe BES is
charging then vt= 0; if it is discharging then vt= 1. If
the BES is idling, this variable can take either value. If
some sufficient conditions are satisfied, for example the
market clearing prices should not be negative, the binary
variable vtcan be relaxed [35]
ut: If at time tthe BES provides reserve then ut= 1,
else ut= 0
The objective of this optimization is to maximize the
operating profit of the BES. This profit is defined as the
difference between the revenues from the energy and reserve
markets and the cycle aging cost C
max
p,g,d,qΩ := PT
t=1 Mhλe
t(gtdt) + λq
tqtiC . (6)
Depending on the discharge power, the depth of discharge
during each time interval extends over one or more segments.
To model the cycle depth in multi-interval operation, we
assign a charge power component pch
t,j and an energy level
et,j to each cycle depth segment, so that we can track the
energy level of each segment independently and identify the
current cycle depth. For example, assume we divide the cycle
depth of a 1 MWh BES into 10 segments of 0.1 MWh. If
a cycle of 10% depth starts with a discharge, as between s2
and s3in Fig. 2a, the BES must have previously undergone a
charge event which stored more than 0.1 MWh according to
the definition from the rainflow method. Because the marginal
cost curve is convex, the BES always discharges from the
cheapest (shallowest) available cycle depth segment towards
the more expensive (deeper) segments. So that the proposed
model provides a close approximation to the rainflow cycle
counting algorithm, detailed proofs and a numerical example
are included in the appendix.
The cycle aging cost Cis the sum of the cycle aging costs
associated with each segment over the horizon:
C=PT
t=1 PJ
j=1 Mcjpdis
t,j .(7)
This optimization is subject to the following constraints
dt=PJ
j=1 pch
t,j (8)
gt=PJ
j=1 pdis
t,j (9)
dtD(1 vt)(10)
gtGvt(11)
et,j et1,j =M(pch
t,j ηch pdis
t,j dis )(12)
et,j ej(13)
Emin PJ
j=1 et,j Emax (14)
e1,j =e0
j(15)
PJ
j=1 eT,j Efinal ,(16)
Eq. (8) states that the BES charging power drawn from
the grid is the sum of the charging powers associated with
each cycle depth segment. Eq. (9) is the equivalent for the
5
discharging power. Eqs. (10)–(11) enforce the BES power
rating, with the binary variable vtpreventing simultaneous
charging and discharging [36]. Eq. (12) tracks the evolution
of the energy stored in each cycle depth segment, factoring
in the charging and discharging efficiency. Eq. (13) enforces
the upper limit on each segment while Eq. (14) enforces the
minimum and maximum SoC of the BES. Eq. (15) sets the
initial energy level in each cycle depth segment, and the final
storage energy level is enforced by Eq. (16).
Because the revenues that a BES collects from providing
reserve capacity are co-optimized with the revenues from the
energy market, it must abide by the requirements that the North
American Electric Reliability Corporation (NERC) imposes on
the provision of reserve by energy storage. In particular, NERC
requires that a BES must have enough energy stored to sustain
its committed reserve capacity and baseline power dispatch
for at least one hour [37]. This requirement is automatically
satisfied when market resources are cleared over an hourly
interval, as is the case for the ISO New England day-ahead
market. If the dispatch interval is shorter than one hour (e.g.for
the five-minute ISO New England real-time market), this one-
hour sustainability requirement has significant implications on
the dispatch of a BES because of the interactions between its
power and energy capacities. For example, let us consider a
36 MW BES with 3 MWh of stored energy. If this BES is not
scheduled to provide reserve, it can dispatch up to 36 MW
of generation for the next 5-minute market period. On the
other hand, if it is scheduled to provide 1 MW of reserve,
its generation capacity is also constrained by the one hour
sustainability requirement, therefore it can only provide up
to 2 MW baseline generation for the next 5-minute market
period.
0dtdq
tD(1 ut)(17)
0gtgq
tG(1 ut)(18)
dq
tDut(19)
gq
tGut(20)
gq
t+qtdq
tGut(21)
qtεut(22)
S(gq
t+qtdq
t)PJ
j=1 ej,(23)
Equations (17)–(23), enforce the constraints related to the
provision of reserve by a BES. In particular, Eq. (23) enforces
the one-hour reserve sustainability requirement. Depending on
the requirements of the reserve market, the binary variable ut
and constraints (17)–(23) can be simplified or relaxed.
The optimization model described above can be used by
the BES owner to design bids and offers or self-schedule
based on price forecasts. The ISO can also incorporate this
model into the market clearing program to better incorporate
the aging characteristic of BES. In this case, the cycle aging
cost function should be included in the welfare maximization
while constraints (8) - (23) should be added to the market
clearing program constraints. A BES owner should include
cycle aging parameters cjand ejin its market offers, and
parameters D,G,Emin,Efinal ,ηch,ηdis for ISO to manage
its SoC and its upper/lower charge limits.
V. CASE STUDY
The proposed model has been tested using data from ISO
New England to demonstrate that it improves the profitability
and longevity of a BES participating in this market. All simu-
lations were carried out in GAMS using CPLEX solver [38],
and the optimization period is 24 hours for all simulations.
A. BES Test Parameters
The BES simulated in this case study has the following
parameters:
Charging and discharging power rating: 20 MW
Energy capacity: 12.5 MWh
Charging and discharging efficiency: 95%
Maximum state of charge: 95%
Minimum state of charge: 15%
Battery cycle life: 3000 cycles at at 80% cycle depth
Battery shelf life: 10 years
Cell temperature: maintained at 25C
Battery pack replacement cost: 300,000 $/MWh
Li(NiMnCo)O2-based 18650 lithium-ion battery cells
These cells have a near-quadratic cycle depth stress func-
tion [19]:
Φ(δ) = (5.24E-4)δ2.03 .(24)
Fig. 3 shows this stress function along with several possible
piecewise linearizations. We assume that all battery cells are
identically manufactured, that the battery management system
is ideal, and thus that all battery cells in the BES age at
the same rate. Since the BES dispatch is performed based on
perfectly accurate price forecasts, our results provide an upper
bound of its profitability in this market.
B. Market Data
BES dispatch simulations were performed using zonal price
for Southeast Massachusetts (SE-MASS) region of ISO New
England market price data for 2015 [39] because energy
storage has the highest profit potential in this price zone [40].
Three market scenarios were simulated:
Day-Ahead Market (DAM): Generations and demands
are settled using hourly day-ahead prices in this energy
market. The DAM does not clear operating reserve ca-
pacities. DAM is a purely financial market, and is used in
this study to demonstrate the BES dispatch under stable
energy prices.
Real-Time Market (RTM) with 1-hour settlement period:
The real-time energy market clears every five minutes and
generates 5-minute real-time energy and reserve prices.
Generations, demands, and reserves are settled hourly
using an average of these 5-minute prices. The reserve
sustainability requirement is one hour.
RTM with 5-minute settlement periods: ISO New England
plans to launch the 5-minute subhourly settlement on
March 1, 2017 [41]. The reserve sustainability require-
ment remains one hour.
Fig. 4a compares the energy prices in these different markets
and shows that the 5-minute real-time prices fluctuate the
6
5 10 15 20 25 30 35 40 45
-200
-100
0
100
200
LMP[$/MWh]
[hr]
DA hourly
RT hourly
RT 5-min
(a) Locational marginal price in the day-ahead market (DAM), in an hourly and a 5-minute real-time market (RTM).
0 5 10 15 20 25 30 35 40 45
0
20
40
60
80
100
SoC [%]
[hr]
no cost 1-segment 2-segment 4-segment 16-segment
(b) BES SoC profile in RTM with 5-minute settlement.
0 5 10 15 20 25 30 35 40 45
-1
-0.5
0
0.5
1
Power [normalized]
[hr]
no cost 1-segment 16-segment
(c) BES output power profile in RTM with 5-minute settlement.
Fig. 4. BES dispatch for different cycle aging cost models (ISO New England SE-MASS Zone, Jan 5th & 6th, 2015).
most, while the day-ahead prices are more stable than real-
time prices.
C. Accuracy of the Predictive Aging Model
Fig. 4b and 4c compare the BES dispatches for piecewise-
linear cycle aging cost functions with different numbers of
cycle depth segments. A cost curve with more segments is a
closer approximation of the actual cycle aging function. The
price signal for these examples is the 5-minute RTM price
curve shown in Fig. 4a. Fig. 4b shows the SoC profile while
Fig. 4c shows the corresponding output power profile, where
positive values correspond to discharging periods, and negative
values to charging periods.
The gray curve in Fig. 4 shows the dispatch of the BES
assuming zero operating cost. This is the most aggressive
dispatch, and the BES assigns full power to arbitrage as long
as there are price fluctuations, regardless of the magnitude of
the price differences. Fig. 4c shows that the BES frequently
switches between charging and discharging, and Fig. 4b that
it ramps aggressively. This dispatch maximizes the market
revenue for the BES, but not the maximum lifetime profit,
because the arbitrage decisions ignore the cost of cycle aging.
We will show in Section V-D that this dispatch actually results
in negative profits for all market scenarios.
The yellow curve in Fig. 4 illustrates the dispatch of the
BES when the cycle aging cost curve is approximated by a
single cycle depth segment. In this case, the marginal cost of
cycle aging is constant and, as shown in Fig. 3, it overestimates
the marginal cost of aging over a wide range of cycle depths.
Therefore, this dispatch yields the most conservative arbitrage
response, and the BES remains idle unless price deviations are
very large, as demonstrated in Fig. 4c. Consequently, the BES
collects the smallest market revenues, but the BES never loses
money from market dispatch because the actual cycle aging is
always smaller than the value predicted by the model.
As the number of segments increases, the BES dispatch
becomes more sensitive to the magnitude of the price fluc-
tuations, and a tighter correlation can be observed between
the market price in Fig. 4a and the BES SoC in Fig. 4b. The
red curve shows the dispatch of the BES using a 16-segment
linearization of the cycle aging cost curve. When small price
fluctuation occurs, the BES only dispatches at a fraction of
its power rating, even though it has sufficient energy capacity.
This ensures that the marginal cost of cycle aging does not
exceed the marginal market arbitrage income.
Besides considering the impact of the piecewise lineariza-
tion on the BES dispatch, it is also important to compare the
cycle aging cost used by the predictive model incorporated
in the dispatch calculation with an ex-post calculation of this
cost using the benchmark rainflow-counting algorithm. Using
the ˆet,j calculated using the optimal dispatch model (6), we
generate a percentage SoC series:
σt=PJ
j=1 ˆet,j /Erate ,(25)
7
TABLE I
DIS PATCH OF A 20MW / 1 2.5MW H BES IN IS O-NE EN ERG Y MAR KET S (FU LL-Y EA R 2015).
Market DAM RTM with hourly settlement RTM with 5-minute settlement
Cycle aging cost model no cost 1-seg. 16-seg. no cost 1-seg. 16-seg. no cost 1-seg. 16-seg.
Annual market revenue [k$] 138.8 0 21.3 382.5 197.5 212.5 789.3 303.8 372.3
Revenue from reserve [%] No price for reserve in DAM 29.6 74.1 73.6 13.8 34.9 29.8
Annual life loss from cycling [%] 24.4 0 0.3 43.6 1.0 1.1 77.0 2.2 2.6
Annual prorated cycle aging cost [k$] 913.8 0 11.3 1626.3 36.3 38.8 2887.5 81.3 96.3
Annual prorated profit [k$] -775.0 0 10 -1243.8 161.3 173.8 -2101.3 222.5 276.3
Profit from reserve [%] No price for reserve in DAM - 90.7 90.0 - 47.7 40.2
Battery life expectancy [year] 2.9 10.0 9.7 1.9 9.1 9.1 1.1 8.2 8.0
0 5 10 15 20 25 30 35 40
0
10
20
30
40
Fig. 5. Difference between the cycle aging cost calculated using the predictive
model and an ex-post calculation using the benchmark rainflow method for a
full-year 5-minute RTM dispatch simulation.
This SoC series is fed into the rainflow method as described
in Section II-C, and the cycle life loss Lis calculated as in
(1) with the cycle stress function (24). The relative error ǫon
the cycle aging cost is calculated as:
ǫ=|ˆ
CRL|/(RL),(26)
where ˆ
Cis the cycle aging cost from (7). Fig. 5 shows the
difference between the predicted and ex-post calculations for
the simulations based on the RTM with a 5-minute settlement.
As the number of segments increases to 16, the error becomes
negligible.
D. BES Market Profitability Analysis
Table I summarizes the economics of BES operation under
the three markets described in Section V-B and for three
cycle aging cost models: no operating cost;single segment
cycle aging cost; and 16-segment cycle aging cost. The market
revenue, profit, and battery life expectancy calculations are
based on dispatch simulations using market data spanning all
of 2015. On the fifth row, the life loss due to market dispatch
is calculated using the benchmark cycle life loss model of
Eqs. (1), and (24). In the sixth row, we calculate the cycle
aging cost by prorating the battery cell replacement cost to
the dispatch life loss. In the seventh row, the cost of cycle
aging is subtracted from the market revenue to calculate the
operating profit. In the last row, we estimate the battery cell
life expectancy assuming the BES repeats the same operating
pattern in future years. The life estimation Lexp includes shelf
(calendar) aging and cycle aging
Lexp = (100%)/(∆Lcal + ∆Lcycle),(27)
where Lcal is the 10% annual self life loss as listed in
Section V-A, and Lcycle is the annual life loss due to cycle
aging as shown in the sixth row in Table I.
The 16-segment model generates the largest profit in all
market scenarios. Compared to the 16-segment model, the no
cost model results in a more aggressive operation of the BES,
while the 1-segment model is more conservative. Because the
no-cost model encourages arbitrage in response to all price
differences, it results in a very large negative profit and a very
short battery life expectancy in all market scenarios. The 1-
segment model only arbitrages during large price deviations.
In particular, the BES is never dispatched in the day-ahead
because these market prices are very stable.
The BES achieves the largest profits in the 5-minute RTM
because this market has the largest price fluctuations. The
revenue from reserve is lower in the 5-minute RTM than the
hourly RTM. This result shows that the proposed approach
is able to switch the focus of BES operation from reserve to
arbitrage when market price fluctuations become high. In the
RTM, the BES collects a substantial portion of its profits from
the provision of reserve, especially in the hourly RTM. A BES
is more flexible than generators at providing reserves because
it does not have a minimum stable generation, it can start
immediately, and can remain idle until called. Therefore, the
provision of reserve causes no cycle aging. In the hourly RTM,
the provision reserve represents about 74% of the market
revenue and 90% of the prorated profits for this BES.
VI. CONCLUSION
This paper proposes a method for incorporating the cost
of battery cycle aging in economic dispatch, market clearing
or the development of bids and offers . This approach takes
advantage of the flexibility that a battery can provide to the
power system while ensuring that its operation remains prof-
itable in a market environment. The cycle aging model closely
approximates the actual electrochemical battery cycle aging
mechanism, while being simple enough to be incorporated
into market models such as economic dispatch. Based on
simulations performed using a full year of actual market price
data, we demonstrated the effectiveness and accuracy of the
proposed model. These simulation results show that modeling
battery degradation using the proposed model significantly
improves the actual BES profitability and life expectancy.
8
APP EN DI X
In this appendix we prove that the proposed piecewise linear
model of the battery cycle aging cost is a close approxi-
mation of the benchmark rainflow-based battery cycle aging
model, and that the accuracy of the model increases with
the number of linearization segments. The proposed model
produces the same aging cost as to the benchmark aging
model for the same battery operation profile with an adequate
number of linearization segments. To prove this, we first
explicitly characterize the cycle aging cost result calculated
using the proposed model (Theorem 1). We then show that
this cost approaches the benchmark result when the number
of linearization segments approaches infinity (Theorem 2).
We consider the operation of a battery over the period T=
{1,2,...,T}, the physical battery operation constraints are
(t∈ T )
dtD(1 vt)(28)
gtGvt(29)
etet1=M(dtηch gtdis)(30)
We denote d={d1, d2,...,dT}as the set of all battery charge
powers, and g={g1, g2,...,gT}as the set of all discharge
powers. Hence, a set in the form of (d,g)is sufficient to
describe the dispatch of a battery over T. Let P(e0)denote
the set of all feasible battery dispatches that satisfy the physical
battery operation constraints (28)–(30) given an battery initial
energy level e0.
Since we are only interested in characterizing the aging
cost calculated by the proposed model for a certain battery
operation profile, we will regard the battery operation profile
as known variables in this proof. It is easy to see that once
the dispatch profile (d,g)is determined, any battery dispatch
problem that involves the proposed model with a linearization
segment set J={1,2,...,J}, such as the one formulated
in Section IV, can be reduced to the following problem if we
neglect any operation prior to the operation interval T
ˆ
parg min
pR+PT
t=1 PJ
j=1 Mcjpdis
t,j ,(31)
s.t.
dt=PJ
j=1 pch
t,j (32)
gt=PJ
j=1 pdis
t,j (33)
et,j et1,j =M(pch
t,j ηch pdis
t,j dis )(34)
0et,j ej(35)
PJ
j=1 e0,j =e0(36)
where (d,g)∈ P(e0)is a feasible battery dispatch set,
and p={pch
t,j , pdis
t,j |t T , j J } denotes a set of the
battery charge and discharge powers for all segments during
all dispatch intervals. Although the objective is still cost
minimization, the problem in (31)–(36) does not optimize
battery dispatch, instead it simulates cycle operations pand
calculates the cycle aging cost with respect to a dispatch profile
(d,g). Hence, the evaluation criteria to this problem is its
accuracy compared to the benchmark aging cost model.
Let c={cj|j J } denote a set of piecewise linear battery
aging cost segments derived as in equation (4), so that cjis
associated with the cycle depth range [(j1)/J, j/J )and
J=|J | is the number of segments. We say that a battery has
aconvex aging cost curve (i.e., non-decreasing marginal cycle
aging cost) if a shallower cycle depth segment (i.e., indexed
with smaller j) is associated with a cheaper marginal aging
cost such that c1c2...cJ, and let Cdenote the set of
all convex battery aging cost linearizations.
Theorem 1. Let ˆ
p={ˆpch
t,j ,ˆpdis
t,j |t∈ T , j ∈ J } and
ˆpch
t,j = min dtPj1
ζ=1 ˆpch
t,ζ ,(ejˆet1,j )/(ηchM)(37)
ˆpdis
t,j = min gtPj1
ζ=1 ˆpdis
t,ζ , ηdis ˆet1,j /M(38)
ˆe0,j = min ej,max(0, e0Pj1
ζ=1 ˆe0)(39)
ˆet,j = ˆet1,j +M(pch
t,j ηch pdis
t,j dis ).(40)
Then ˆ
pis a minimizer of the problem (31)–(36) as long as
the battery dispatch is feasible and the cycle aging cost curve
is convex, i.e.,
ˆ
parg min
pR+(31)–(36) ,
(d,g)∈ P(e0),e0[Emin, Emax],c∈ C.(41)
Proof. Equations (37)–(40) describe a battery operating policy
over the proposed piecewise linear model. To calculate this
policy, we start from (39) which calculates the initial segment
energy level from the battery initial SoC e0. (39) is evaluated
in the order of j= 0,1,2,3,...,J such as (note that
P0
ζ=1 ˆe0= 0)
ˆe0,1= min e1,max(0, e0)
ˆe0,2= min e2,max(0, e0ˆe0,1)
ˆe0,3= min e3,max(0, e0ˆe0,1ˆe0,2)
. . . ,
so that energy in e0is first assigned to ˆe0,1which corresponds
to the shallowest cycle depth range [0,1/J], the remaining
energy is then assigned to the second shallowest segment ˆe0,2,
and the procedure repeats until all the energy in e0has been
assigned.
We then calculate all battery segment charge power at t= 1
in the order of j= 0,1,2,3,...,J as
ˆpch
1,1= min dt,(e1ˆe0,1)/(ηchM)
ˆpch
1,2= min dtˆpch
1,1,(e2ˆe0,2)/(ηchM)
ˆpch
1,3= min dtˆpch
1,1ˆpch
1,2,(e3ˆe0,3)/(ηchM)
. . . ,
and the procedure is similar for segment discharge power
ˆpdis
1,j . We calculate the segment energy level ˆe1,j at the end
of t= 1 using (40), and move the calculation to t= 2. This
procedure repeats until all values in ˆ
phave been calculated.
Therefore in this policy, the battery always prioritizes energy
in shallower segments for charge or discharge dispatch. For
example, if the battery is required to discharge a certain
amount of energy, it will first dispatch segment 1, then the
9
remaining discharge requirement (if any) is dispatched from
segment 2, then segment 3, etc.
Given this policy, this theorem stands if the battery cycle
aging cost curve cis convex, i.e., c∈ C, which means a
shallower segment is associated with a cheaper marginal oper-
ating cost. Since the objective function (31) is to minimize the
battery aging cost and the problem involves no market price,
then a minimizer for the problem (31)–(36) will give a cheaper
segment a higher operation priority, which is equivalent to the
policy described in (37)–(40).
Following Theorem 1, the cycle aging cost calculated by the
proposed piecewise linear model Cpwl for a battery dispatch
profile (d,g)can be written as a function of this profile and
the linearization cost set as
Cpwl(c,d,g) = PT
t=1 PJ
j=1 Mcjˆpdis
t,j ,(42)
where ˆ
pis calculated as in (37)–(40).
Let Φ(δ)be a convex battery cycle aging stress function, and
c(Φ) be a set of piecewise linearizations of Φ(δ)determined
using the method described in equation (4). Let |c(Φ)|denote
the cardinality of c(Φ), i.e. the number of segments in this
piecewise linearization.
For a feasible battery dispatch profile (d,g)∈ P(e0), let
be the set of all full cycles identified from this operation
profile using the rainflow method, dis for all discharge half
cycles, and ch for all charge half cycles. The benchmark
cycle aging cost Cben resulting from (d,g)can be written
as a function of the profile and the cycle aging function Φ
(recall that a full cycle has symmetric depths for charge and
discharge)
Cben,d,g) = RP||
i=1 Φ(δi) + RP|dis|
i=1 Φ(δdis
i).(43)
Theorem 2. When the number of linearization segments
approaches infinity, the proposed piecewise linear cost model
yields the same result as the benchmark rainflow-based cost
model:
lim
|c(Φ)|→∞ Cpwlc(Φ),d,g=Cben Φ,d,g.(44)
Proof. First we rewrite equation (42) as
PJ
j=1 cjPT
t=1 Mˆpdis
t,j =PJ
j=1 cjΘj,(45)
where Θj=PT
t=1 Mˆpdis
t,j is the total amount of energy
discharged at a cycle depth range between (j1)/J and j/J.
Once the number of segments |c(Φ)|=Japproaches infinity,
we can rewrite Θjinto a function Θ(δ)indicating the energy
discharged at a specific cycle depth δ, where δ[0 1]. With an
infinite number of segments, we substitute (3) in and rewrite
the cycle aging function in (42) in a continuous form
Cpwl,d,g) = Z1
0
R
ηdisErate Θ(δ)dΦ(δ)
dδ . (46)
We define a new function Ndis
T(δ)the number of discharge
cycles of depths equal or greater than δduring the operation
period from t= 0 to t=T, accounting all discharge half
cycles and the discharge stage of all full cycles. Ndis
T(δ)can be
calculated by normalizing Θ(δ)with the discharge efficiency
and the energy rating of the battery
Ndis
T(δ) = 1
ηdisErate Θ(δ),(47)
recall that Θ(δ)is the amount of energy discharged from the
cycle depth δ. This relationship is proved in Lemma 1 after
this theorem.
Now the proposed cost function becomes
Cpwl,d,g) = RZ1
0
dΦ(δ)
Ndis
T(δ)dδ , (48)
which is a standard formulation for calculating rainflow fatigue
damage [42], and the function Ndis
T(δ)is an alternative way
of representing a rainflow cycle counting result. We substitute
(51) from Lemma 1 into (48)
Cpwl,d,g)
=RZ1
0
dΦ(δ)
||
X
i=1
1[δδi]+
|dis|
X
i=1
1[δδdis
i]!
=R
||
X
i=1 Z1
0
dΦ(δ)
1[δδi]+R
|dis|
X
i=1 Z1
0
dΦ(δ)
1[δδdis
i]
=R
||
X
i=1
Φ(δi) + R
|dis|
X
i=1
Φ(δdis
i)
=Cben,d,g),(49)
then it is trivial to see that this theorem stands if the proposed
model yields the same counting result Ndis
T(δ)as the rainflow
algorithm. This relationship is proved in Lemma 1.
Lemma 1. We assume that the proposed model has an infinite
number of segments, then Ndis
T(δ), as defined in Theorem 2,
is the number of discharge cycles of depths equal or greater
than δduring the operation period from t= 0 to t=T,
accounting all discharge half cycles and the discharge stage
of all full cycles, hence
Ndis
T(δ) = Θ(δ)/(ηdisErate )(50)
=P||
i=1 1[δδi]+P|dis|
i=1 1[δδdis
i],(51)
where 1[x]has a value of one if xis true, and zero otherwise.
Proof. (50) defines Ndis
T(δ)as the number fo times that energy
is discharged from the cycle depth δ, while (51) means the
number of cycles with depths at least δ. Therefore in this
lemma we prove that these two definitions are equivalent,
hence the proposed model has the same cycle counting result
as the rainflow method.
Let Ndis
t(δ)be the number of times energy is discharged
from the depth δduring the operation period [0, t], accounting
all discharge half cycles and the discharge stage of all full cy-
cles. Similarly, define Nch
t(δ)accounting all charge half cycles
and the charge stage of all full cycles. Because we assume
charge dispatches cause no aging cost, we can alternatively
model battery initial energy level e0as an empty battery being
10
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0
10
20
30
40
50
60
SoC [%]
[h]
δ1 = 10%
δ3 = 10%
remaining
profile...
δ2 = 40%
initial SoC
δdis = 50%
Fig. 6. Cycle counting example.
charged to e0at the beginning of operation (such as in Fig. 6),
hence at t= 0 we have
Nch
0(δ) = (1δe0
0δ > e0
, Ndis
0(δ) = 0 .(52)
Now assume at time t1the battery is switched from charging
to discharging, and eventually resulted in a cycle of depth x
that ends at t2, regardless whether it is a half cycle or a full
cycle. We also assume that there is no other cycles occuring
from t1tp t2, since in the rainflow method The battery must
have been previously charged at least δdepth worth of energy
since we now assume the battery starts from empty. Therefore
according to Theorem 1, segments in the range [0, x]must be
full at t1, hence
Nch
t1(δ)Ndis
t1(δ) = 1 δx , (53)
which is a sufficient condition for all discharge energy in this
cycle being dispatched from segments in the depth range [0, x],
according to Theorem 1. After performing this cycle, all and
only segments within the range [0, x]are discharged one more
time, in other words, all and only cycle depths in the range
[0, x]have one more count at end of this cycle t2compared
to t1when the discharge begins, hence
Ndis
t2(δ)Ndis
t1(δ) = 1[δx].(54)
Therefore the proposed model has the same counting result
as to the rainflow method for any cycles, which proves this
lemma.
A. Numerical example
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0
10
20
30
40
50
60
SoC [%]
[h]
s2
s3
s4
s5
s6
s7
s1
s0 s8
Fig. 7. An example of SoC profile.
We include a step-by-step example to illustrate how the
proposed model is a close approximation of the benchmark
rainflow cost model using the battery operation profile shown
in Fig. 7. To simplify this example, we assume a perfect
TABLE II
BATTE RY OPER ATIO N EXAM PL E.
t SoC energy segments discharge power cost
[et] [pdis
t]Ct
- - deeper depth deeper depth -
0 60 1,1,1,1,1,1,0,0,0,0 0,0,0,0,0,0,0,0,0,0 0
1 10 0,0,0,0,0,1,0,0,0,0 1,1,1,1,1,0,0,0,0,0 25
2 20 1,0,0,0,0,1,0,0,0,0 1,1,1,1,1,0,0,0,0,0 0
3 30 1,1,0,0,0,1,0,0,0,0 0,0,0,0,0,0,0,0,0,0 0
4 20 0,1,0,0,0,1,0,0,0,0 1,0,0,0,0,0,0,0,0,0 1
5 30 1,1,0,0,0,1,0,0,0,0 0,0,0,0,0,0,0,0,0,0 0
6 40 1,1,1,0,0,1,0,0,0,0 0,0,0,0,0,0,0,0,0,0 0
7 50 1,1,1,1,0,1,0,0,0,0 0,0,0,0,0,0,0,0,0,0 0
8 40 0,1,1,1,0,1,0,0,0,0 1,0,0,0,0,0,0,0,0,0 1
9 30 0,0,1,1,0,1,0,0,0,0 0,1,0,0,0,0,0,0,0,0 3
10 40 1,0,1,1,0,1,0,0,0,0 0,0,0,0,0,0,0,0,0,0 0
11 30 0,0,1,1,0,1,0,0,0,0 1,0,0,0,0,0,0,0,0,0 1
12 20 0,0,0,1,0,1,0,0,0,0 0,0,1,0,0,0,0,0,0,0 5
13 10 0,0,0,0,0,1,0,0,0,0 0,0,0,1,0,0,0,0,0,0 7
14 60 1,1,1,1,1,1,0,0,0,0 0,0,0,0,0,0,0,0,0,0 0
all - - - 43
efficiency of 1and that the cycle aging cost function is 100δ2.
We consider 10 linearization segments, with each segment
representing a 10% cycle depth range. The proposed model
therefore has the following cycle aging cost curve
c={1,3,5,7,9,11,13,15,17,19}.(55)
According to the rainflow method demostrated in Fig. 2 , this
example profile has the following cycle counting results
Two full cycles of depth 10%, each costs 1
One full cycle of depth 40% that costs 16
One discharge half cycle of depth 50% that costs 25
One charge half cycle that costs zero,
hence the total aging cost identified by the benchmark
rainflow-based model is 43.
We implement this operation profile using the policy in The-
orem 1 and record the marginal cost during each time interval.
The results are shown in Table II. In this table, the first two
columns are the time step and SoC. The third column shows
the energy level of each linearization segment represented in
a vector from et.etis a 10 ×1vector, and its energy level
segments are sorted from shallower to deeper depths. Segment
energy levels are normalized so that one means the segment is
full, and zero means the segment is empty. The fourth column
shows how much energy is discharged from each segment
during a time interval, represented by a discharge power vector
pdis
tand is calculated as (the discharge efficiency is 1)
pdis
t= [et1et]+.(56)
The last column shows the operating cost that arises from each
time interval, which is calculated as
Ct=cpdis
t.(57)
11
This example profile results in the same cost of 43 in both
the proposed model and the benchmark model, as proved in
Theorem 2.
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12
Bolun Xu (S’14) received B.S. degrees in Electrical and Computer Engineer-
ing from Shanghai Jiaotong University, Shanghai, China in 2011, and the M.Sc
degree in Electrical Engineering from Swiss Federal Institute of Technology,
Zurich, Switzerland in 2014.
He is currently pursuing the Ph.D. degree in Electrical Engineering at the
University of Washington, Seattle, WA, USA. His research interests include
energy storage, power system operations, and power system economics.
Jinye Zhao (M’11) received the B.S. degree from East China Normal
University, Shanghai, China, in 2002 and the M.S. degree in mathematics
from National University of Singapore in 2004. She received the M.E. degree
in operations research and statistics and the Ph.D. degree in mathematics from
Rensselaer Polytechnic Institute, Troy, NY, in 2007.
She is a lead analyst at ISO New England, Holyoke, MA. Her main
interests are game theory, mathematical programming, and electricity market
modeling.
Tongxin Zheng (SM’08) received the B.S. degree in electrical engineering
from North China University of Electric Power, Baoding, China, in 1993,
the M.S. degree in electrical engineering from Tsinghua University, Beijing,
China, in 1996, and the Ph.D. degree in electrical engineering from Clemson
University, Clemson, SC, USA, in 1999.
Currently, he is a Technical Manager with the ISO New England, Holyoke,
MA, USA. His main interests are power system optimization and electricity
market design.
Eugene Litvinov (SM’06-F’13) received the B.S. and M.S. degrees from
the Technical University, Kiev, Ukraine, and the Ph.D. degree from Urals
Polytechnic Institute, Sverdlovsk, Russia.
Currently, he is the Chief Technologist at the ISO New England, Holyoke,
MA. His main interests include power system market-clearing models, system
security, computer applications in power systems, and information technology.
Daniel S. Kirschen (M’86-SM’91-F’07) received his electrical and mechan-
ical engineering degree from the Universite Libre de Bruxelles, Brussels,
Belgium, in 1979 and his M.S. and Ph.D. degrees from the University of
Wisconsin, Madison, WI, USA, in 1980, and 1985, respectively.
He is currently the Donald W. and Ruth Mary Close Professor of Electrical
Engineering at the University of Washington, Seattle, WA, USA. His research
interests include smart grids, the integration of renewable energy sources in
the grid, power system economics, and power system security.
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... Constraints (3a)-(3g) for battery degradation and nonnegativity are added as described in [5], [11]. ...
... case, degradation is considered. The model is equal to the DC case, except the objective function is replaced with (5). ...
... Time, and, often, ambient temperature, are uncontrollable, external parameters, while the SOC is affected by operational decisions [7]- [9]. Prolonged,high SOC levels are devastating to batteries [8], [10]. ...
... To represent CD-induced degradation, the piecewise linear approach has been adopted as an accurate approximation for the Rainflow model to count cycles [8], [29]- [34]. This allows the penalization of discharges more than proportional to their CD. ...
... First, it can be based on the Arrhenius equation, but this results, however, in a concave stress function [30], so we disregard this option. Second, it can be based on an amplitude function for physical stress (e.g., Eq. (1) [8], [15]). Parameter a displays the maximum capacity loss per cycle, while m is the fatigue strength exponent. ...
Preprint
p>Batteries are crucial to manage the rising share of intermittent energy sources and variability in demand. ost techno-economic models in the literature oversimplify battery degradation representation. Accounting properly for battery degradation allows for better cost tradeoffs and optimal battery usage, especially in dynamic settings. We propose a highly accurate and scalable formulation for battery degradation that considers the combined impact of cycle depth and state of charge on calendar and cycle aging. We test the consequences of battery degradation in a stylized price arbitrage model on battery operation and solution times. When ignoring battery degradation, ex-post calculations reveal hidden degradation costs that exceed revenues and hence turn seemingly profitable trades into losing trades. Considering battery degradation leads to smaller cycle depths and lower average states of charge. Overall, we show that a much-improved representation of battery degradation is possible at modest computational cost.</p
... LFP yields the highest valuation due to having a significantly higher cycle life, while NMC produced higher values than the NCA. Given that the costs of all three battery technologies are similar [54] , the results recommend LFP as the best technology for building grid-scale energy storage. ...
... When assuming 15 years of calendar life, the average LFP valuation results at AESO and ERCOT surpassed $400/kW, followed by CAISO, which falls slightly below $400/kW. Given that the current system cost for building 4-hour LFP utility-scale battery storage is around $400/kW [54] , our result suggests that LFP battery projects provide positive investment return only through energy markets in these three markets, and the profit potential will likely to increase in the future as renewable penetration deepens. While we may not generalize the findings from this study, we believe it can help battery operators and investors participating in North American electricity markets make future economic decisions on which battery technology is most appropriate for a particular market, given the storage capacity and project lifetime. ...
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Lithium-ion batteries are becoming critical flexibility assets in future electric power systems. Batteries can arbitrage price differences in wholesale electricity markets to make a profit while at the same time reducing total system operating costs and improving renewable energy integration. However, lithium-ion batteries have a limited lifetime due to capacity degradation, and one battery pack can only make a limited profit before reaching its end-of-life. In this paper, we screen the profit potential of Lithium iron phosphate (LFP), nickel manganese cobalt (NMC), and lithium nickel cobalt aluminum oxides (NCA) batteries in all nine wholesale electricity markets in North America. We apply a systematic dynamic valuation framework that finds the highest revenue potential for the considered lithium-ion battery project subjecting to its degradation mechanism, while the degradation model used in the valuation is derived based on real lab test data over varying cycle conditions. The study found that battery valuation depends largely on battery technology and storage duration and varies across operational locations. Moreover, the study revealed that calendar life has a greater impact on battery valuation than cycle life for an 8-years calendar life scenario while cycle life shows greater impact for a 15-year calendar life scenario for all battery technologies. This impact is more pronounced in LFP than in NMC and NCA. The study recommends battery operators consider strategies that would maximize a longer cycle life or calendar life usage of a battery as this would accumulate higher profits over its lifetime.
... As such, many studies on EV charging optimization completely ignore battery degradation [3][4][5]. Some studies include degradation with a fixed degradation rate in a single objective function [6,7], and others use a degradation model for stationary batteries or EVs using only one or two of the impacting degradation parameters: battery temperature, depth of discharge (DOD), state-of-charge (SOC), and charging rate (C-rate) [8,9]. In this context, battery temperature, as an important influencing parameter, was often not considered in battery charging optimizations, or simplifications were made, and battery temperature was considered constant. ...
... The maximum charge/discharge power of the EV fleet is limited by (9) and (10), respectively: ...
... where i  is the charging and discharging loss of the energy storage device, which can be expressed by a linear function [24], [25]. e  is the sharing price. ...
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