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Energy storage models require binary variables to correctly model reserves and to ensure that the storage cannot charge and discharge simultaneously. This paper proposes a tight linear program (LP), i.e., convex hull, for the storage, which guarantees that there is no better LP approximation to its mixed-integer program (MIP) counterpart. Although the resulting LP formulation cannot guarantee that charge and discharge are mutually exclusive at all times, it does not affect the feasibility of providing reserves. By embedding the proposed LP formulation into large optimization problems, it helps to provide solutions equal to or very near to the exact integer feasible behaviour of the storage; and when used in its integer form, it speeds up MIP problems. Furthermore, the tight LP formulation is extended to include storage investment decisions, thus providing a very strong LP relaxation, opposite to the LP relaxation resulting from the big-M constraints commonly used in storage investment models.
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1
LP Formulation for Optimal Investment and
Operation of Storage Including Reserves
Germán Morales-España, Ricardo Hernandez-Serna, Niina Helistö, and Juha Kiviluoma
Abstract—Energy storage models require binary variables to
correctly model reserves and to ensure that the storage cannot
charge and discharge simultaneously. This paper proposes a tight
linear program (LP), i.e., convex hull, for the storage, which
guarantees that there is no better LP approximation to its mixed-
integer program (MIP) counterpart. Although the resulting LP
formulation cannot guarantee that charge and discharge are
mutually exclusive at all times, it does not affect the feasibility of
providing reserves. By embedding the proposed LP formulation
into large optimization problems, it helps to provide solutions
equal to or very near to the exact integer feasible behaviour of
the storage; and when used in its integer form, it speeds up MIP
problems. Furthermore, the tight LP formulation is extended to
include storage investment decisions, thus providing a very strong
LP relaxation, opposite to the LP relaxation resulting from the
big-M constraints commonly used in storage investment models.
Index Terms—Linear programming, mixed-integer program-
ming, optimal planning, storage, reserves.
I. INTRODUCTION
ENERGY storage systems have become a promising option
to increase power system flexibility and harness larger
shares of variable renewable energy. To get a full picture of
their potential operation and benefits, a realistic representation
of their characteristics is essential in power system models.
Existing storage models require binary variables to correctly
model reserves and to ensure that the storage cannot charge
and discharge simultaneously [1], [2], as further discussed
in Section II-A. Furthermore, to avoid this simultaneous
charge/discharge, investment models use big-M constraints [2]
which greatly damage the strength of the mixed-integer pro-
gram (MIP), thus significantly increasing their computational
complexity [3]. Since solving MIP problems are computa-
tionally demanding and can pose a significant impediment
in large-scale models, a common practice is to formulate
them as linear programs (LP), where the storage is allowed
to charge and discharge simultaneously, hoping that the the
optimal solution avoids this option since it increases losses
and potentially costs. An attempt to diminish simultaneous
charging and discharging was presented in [4], albeit without
reserves or investment decisions, where the authors argue that
fast storage units can actually charge and discharge within a
period, by charging one part of the period and discharging
the remaining part. Energy models can then greatly benefit
from a tight MIP representation of energy storage because
1) it accelerates solving times of MIP models, and 2) in its
relaxed LP form, the solution will be very near to the optimal
exact MIP solution. This LP formulation can then be used
as a good proxy of the original MIP model in large-scale
optimization models. Furthermore, an LP model allows to
directly apply decomposition algorithms that require convexity
of the operations (second-stage) problem [2].
The main contributions of this letter are threefold: 1) we
propose an LP formulation of storage that correctly models
reserves and better exploits the flexibility of fast storage
units; 2) the proposed LP formulation is the tightest possible,
i.e., convex hull, and consequently it naturally minimizes the
possibility of simultaneous charging and discharging, thus
providing the best linear approximation when used in LP
models, and speeding up solving times when used in MIP
models; and finally, 3) the tight LP formulation is extended
to include storage investment decisions, thus providing a
very good approximation (strong LP relaxation) to its MIP
counterpart, opposite to the LP relaxation resulting from the
big-M constraints commonly used in investment models.
II. MATHEMATICAL FORMULATION
A. Why an MIP for correct modelling of storage + reserves?
Generic storage models are used to represent many devices
capable of storing energy, such as pump hydro, electric ve-
hicles, thermal storage, and some forms of demand response.
Furthermore, the model of storage output is similar to that of
transmission lines with losses and reserves [5]. A typical MIP
model of storage [6] including reserves is
et=et1+ctηCtdt
ηDtt(1)
EetEt(2)
ct+rc
ttt(3)
dt+rd+
tD(1 δt)t(4)
ctrc+
t0t(5)
dtrd
t0t(6)
r+
t=rc+
t+rd+
tt(7)
r
t=rc
t+rd
tt(8)
rc+
t, rd+
t, rc
t, rd
t,ct, dt0t(9)
δt {0,1}(10)
where the index tstands for time periods and the parameter t
represents the time duration (e.g., t=1h). Parameters C/D
are charge/discharge capacities, ηCDare their corresponding
efficiencies, and E/E are the min/max storage capacities.
Variables ct, dt, etare charge, discharge, and state of charge,
respectively. The binary variable δtindicates when the storage
is 1) charging (δt= 1) and it can provide up rc+
tand down
rc
treserves, or 2) discharging (δt= 0) where it can also
2
Fig. 1. Feasible regions (shadow area) for the maximum up reserves that
storage can provide by using an LP relaxed (left) or an integer (right) model.
provide up rd+
tand down rd
treserves. The state of charge
of the storage is tracked in (1) and its capacity limits in (2).
The upper bounds for charge and discharge are imposed in
(3) and (4), and their corresponding lower bounds in (5) and
(6). The total up and down reserves r+
t, r
tare obtained in
(7) and (8). Finally, (9) defines the continuous variables as
non-negative, and (10) defines the variable δtas binary.
1) Binary variable to model reserves: The following ex-
ample illustrates why δthas to be integer to correctly model
reserves. Suppose C= 10MW and D= 20MW, if the storage
is charging ct= 8MW, then the maximum up reserves the
model guarantee is rc+
t8MW from (5). Also, since the
unit is charging δt= 1 forces rd+
t, xd
t, dt= 0. On the other
hand, in the LP relaxation, when ct= 8MW, δtcan take
the value of 0.8from (3), then rc+
t8MW from (5), and
rd+
t4MW from (4), see Fig. 1. The total up reserves are
now r+
t= 12MW from (7), and even though the storage unit
can be fast enough to provide them (see Section II-B), these
reserves are 50% higher than what the MIP model (1)-(10)
intended to be feasible. There are many possible combinations
where the relaxed model can maximize the amount of reserves
outside the feasible region imposed by the integer model.
2) Binary variable to avoid simultaneous charging and
discharging: If there is an incentive to maximize demand, e.g.,
in the event of negative prices, the model in its LP relaxed
form can choose to charge and discharge simultaneously to
virtually increase losses [2], which could be unfeasible in
practice. Therefore, defining δtas integer together with (3) and
(4) imposes that charge and discharge are mutually exclusive.
B. LP (re)formulation
The capacity limits of the storage output (ctdt)with
reserves could be modelled as
(ctdt) + r
tCt(11)
(ctdt)r+
t Dt(12)
where the only difference compared with a model of traditional
generating units, which do not need binary variables to include
reserves, is that the storage output (ctdt)can take either
positive (charge) or negative (discharge) values.
The capacity limits for the storage output are defined as
ctC, dtDt(13)
r+
tR+, r
tRt(14)
r+
t, r
t, ct, dt0t(15)
Although the resulting LP formulation from (11)-(15) does
not guarantee that variables ctand dtare mutually exclusive,
it does not affect the feasibility of providing reserves, unlike
traditional storage formulations (Section II-A1). Therefore, the
LP formulation (11)-(15) together with (1) and (2) is sufficient
to model lossless (ηC=ηD= 1) storage providing reserves.
This formulation better exploits the flexibility of storage
units: notice that the maximum possible reserves r+
t, r
tare
min (R+
, C +D)and min (R
, C +D), respectively. That is,
if the storage unit is fast enough, i.e., R+
, RC+D,
then, within a period, the storage can go from fully charging
(discharging) and deploy the maximum up (down) reserves
to go fully discharging (charging), which can be feasible in
practice, e.g., within an hour by using an intra-hour model.
C. Avoiding simultaneous charging and discharging
The set of constraints (16)-(22) is the tightest possible
representation, i.e., convex hull, for the mutually exclusive
constraints for charging and discharging of storage including
reserves, as proven in the Appendix.
ctdt+r
tCt(16)
ctdtr+
t Dt(17)
ct t(18)
dtD(1 δ)t(19)
r+
tR+, r
tRt(20)
r+
t, r
t, ct, dt0t(21)
0δt1t(22)
This convex hull guarantees that there is no better LP
approximation for this set of constraints; consequently, by
embedding it into larger optimization problems, 1) it helps
to provide solutions equal to or very near to the exact integer
feasible solution of the storage, and 2) in MIP problems, where
δtis defined as binary, it speeds up solving times.
If the model is always used in its LP form, we can reduce its
dimension by eliminating the variable δtthrough the Fourier-
Motzkin elimination procedure, then the constraints (18), (19)
and (22) can be replaced by their exact equivalent:
ctC, dtDt(23)
ct
C+dt
D1t(24)
where (24) dominates both constraints in (23), then only (24)
is needed and consequently (23) can be removed.
Notice that although (24) cannot fully guarantee that ctand
dtare mutually exclusive, it diminishes its effect: e.g., if ct=
C(or dt=D), then (24) forces dt= 0 (or ct= 0).
D. LP formulation for investment
For investment problems, the capacity parameters
C, D, E , E become investment variables, then (18) and
(19) become bilinear constraints. These bilinear constraints
are commonly reformulated as big-M MIP constraints [2],
which greatly damage the tightness of the formulation, that is,
the LP relaxation provides a very bad (weak) approximation
3
to its MIP counterpart. Here, we then extend the tight
formulation above including investment decisions:
et=et1+ctηCtdt
ηDtt(25)
ctdt+r
tC0+ct(26)
ctdtr+
t α(C0+c)t(27)
ct+dt
αC0+ct(28)
r+
tβ(C0+c)t(29)
r
tθ(C0+c)t(30)
φE0+eetE0+et(31)
r+
t, r
t, ct, dt0t(32)
e, c 0(33)
where parameters C0and E0are the initial capacities for
charging and storage, and variables cand eare their corre-
sponding new investments. Since all storage capacity values
are proportional between one another [2], the investment
decisions for discharge and up/down reserve capacities are
defined as a proportion αand β/θof the charge capacity,
respectively; and the minimum capacity of the storage is
defined as a proportion φof the maximum capacity. Notice
that (28) is the equivalent of (24) for investment, thus limiting
the possibility of simultaneous charging and discharging in an
LP investment model.
III. CONCLUSIONS
This article provides the tightest possible LP formulation
for the storage operation, thus guaranteeing that there is no
better LP approximation to its MIP counterpart. Although
it cannot guarantee that charge and discharge are mutually
exclusive at all times, it does not affect the feasibility of
providing reserves. Finally, this tight formulation is extended
to investment problems. The formulation for the storage output
can also be used for transmission with losses and reserves,
since transmission could be modelled in the same manner. The
proposed tight formulation can benefit large-scale optimization
problems by providing a good approximation of the storage
operation in LP problems and speeding up solving times in
MIP problems.
ACKNOWLEDGEMENT
This work has received funding from the EU Horizon 2020
research and innovation program under project TradeRES
(grant agreement No 864276)
APPENDIX
The disjunctive set of constraints (34) for a storage unit
activates only the left side of the formulation if the unit is
charging, or only the right side if discharging. To simplify the
notation, here we do not report the time index t.
c+rC
cr+ D
cC
d= 0
r+R+, rR
r+, r, c 0
or
d+rC
dr+ D
c= 0
dD
r+R+, rR
r+, r, d 0
(34)
The convex hull of disjunctive constraints is given by [7]
and expressed in a higher dimensional polyhedron:
c1+r11,d2+r2Cδ2(35)
c1r1+ 1,d2r2+ Dδ2(36)
c11, c2= 0 (37)
d1= 0, d22(38)
r1+ R+δ1, r2+ R+δ2(39)
r1Rδ1, r2
tRδ2(40)
r1+, r1, c1, d1, δ10, r2+ , r2, c2, d2, δ 20(41)
r+=r1+ +r2+ (42)
r=r1+r2(43)
c=c1+c2(44)
d=d1+d2(45)
δ1+δ2= 1 (46)
and the projection of this convex hull onto variables r+, r, c
and dalso results in a convex hull [3].
To reduce the dimensionality of (35)-(46), i.e., projecting
it onto variables r+, r, c and d, we start by eliminating
c1, c2, d1, d2: since c2= 0 (37) and d1= 0 (38) then c1=c
(44) and d2=d(45). Now we rename the variable δ1as
δ=δ1, and obtain δ2= 1 δ(46). We also replace r2+ , r2
by r2+ =r+r1+ (42) and by r2=rr1(43),
respectively. Then the polyhedron (35)-(46) becomes
c+r1, d+rr1C(1 δ)(47)
cr1+ Dδ, dr++r1+ D(1 δ)(48)
c, d D(1 δ)(49)
r1+ R+δ, r+r1+ R+(1 δ)(50)
r1Rδ, rr1R(1 δ)(51)
r1+, r1, c 0, r+, r, d 0(52)
0δ1(53)
Now we apply the Fourier-Motzkin elimination procedure
to variables r1and r1+ , and after removing all redundant
constraints, the projection of the convex hull (35)-(46) onto
variables r+, r, c and dresults in the convex hull (16)-(22).
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