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Exact sizing of battery capacity for photovoltaic systems

Yu Ru a, Jan Kleissl b, Sonia Martinez b,n

a GE Global Research at Shanghai, Chinab Mechanical and Aerospace Engineering Department, University of California, San Diego, United States

a r t i c l e i n f o

Article history:

Received 22 July 2013

Accepted 16 August 2013

Recommended by AstolAlessandroAvailable online 16 September 2013

Keywords:

PV

Grid

Battery

Optimization

a b s t r a c t

In this paper, we study battery sizing for grid-connected photovoltaic (PV) systems. In our setting, PV

generated electricity is used to supply the demand from loads: on one hand, if there is surplus PV

generation, it is stored in a battery (as long as the battery is not fully charged), which has a xed

maximum charging/discharging rate; on the other hand, if the PV generation and battery discharging

cannot meet the demand, electricity is purchased from the grid. Our objective is to choose an appropriate

battery size while minimizing the electricity purchase cost from the grid. More specically, we want to

nd a unique critical value (denoted as Ecmax) of the battery size such that the cost of electricity purchase

remains the same if the battery size is larger than or equal to Ecmax, and the cost is strictly larger

otherwise. We propose an upper bound on Ecmax , and show that the upper bound is achievable for certain

scenarios. For the case with ideal PV generation and constant loads, we characterize the exact value of

Ecmax, and also show how the storage size changes as the constant load changes; these results are

validated via simulations.

& 2013 European Control Association. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Installations of solar photovoltaic (PV) systems have been growing

at a rapid pace in recent years due to the advantages of PV such as

modest environmental impacts (clean energy), avoidance of fuel

price risks, coincidence with peak electrical demand, and the ability

to deploy PV at the point of use. In 2010, approximately 17,500 mega-

watts (MW) of PV were installed globally, up from approximately

7500 MW in 2009, consisting primarily of grid-connected applica-

tions[2]. Since PV generation tends to uctuate due to cloud cover

and the daily solar cycle, energy storage devices, e.g., batteries,

ultracapacitors, and compressed air, can be used to smooth out the

uctuation of the PV output fed into electric grids (capacity rming)

[14], discharge and augment the PV output during times of peak

energy usage (peak shaving)[16], or store energy for nighttime use,

for example in zero-energy buildings.In this paper, we study battery sizing for grid-connected PV

systems to store energy for nighttime use. Our setting is shown in

Fig. 1. PV generated electricity is used to supply loads: on one

hand, if there is surplus PV generation, it is stored in a battery for

later use or dumped (if the battery is fully charged); on the other

hand, if the PV generation and battery discharging cannot meet

the demand, electricity is purchased from the grid. The battery has

a xed maximum charging/discharging rate. Our objective is to

choose an appropriate battery size while minimizing the electri-city purchase cost from the grid. We show that there is a unique

critical value (denoted as Ecmax , refer toProblem 1) of the battery

capacity (under xed maximum charging and discharging rates)

such that the cost of electricity purchase remains the same if the

battery size is larger than or equal to Ecmax , and the cost is strictly

larger otherwise. We rst propose an upper bound on Ecmax given

the PV generation, loads, and the time period for minimizing the

costs, and show that the upper bound becomes exact for certain

scenarios. For the case of idealized PV generation (roughly, it refers

to PV output on clear days) and constant loads, we analytically

characterize the exact value ofEcmax, which is consistent with the

critical value obtained via simulations.

The storage sizing problem has been studied for both off-grid

and grid-connected applications. For example, the IEEE standard[11] provides sizing recommendations for lead-acid batteries in

stand-alone PV systems. In [18], the solar panel size and the

battery size have been selected via simulations to optimize the

operation of a stand-alone PV system. If the PV system is grid-

connected, batteries can reduce the uctuation of PV output or

provide economic benets such as demand charge reduction,

capacity rming, and power arbitrage. The work in [1] analyzes

the relation between available battery capacity and output

smoothing, and estimates the required battery capacity using

simulations. In addition, the battery sizing problem has been

studied for wind power applications [21,5,12] and hybrid wind/

solar power applications[4,8,20]. Most previous work completely

Contents lists available atScienceDirect

journal homepage: www.elsevier.com/locate/ejcon

European Journal of Control

0947-3580/$- see front matter & 2013 European Control Association. Published by Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.ejcon.2013.08.002

n Corresponding author. Tel.: 1 858 822 4243.

E-mail addresses:[email protected] (Y. Ru),[email protected] (J. Kleissl),

[email protected] (S. Martinez).

European Journal of Control 20 (2014) 2437
http://www.sciencedirect.com/science/journal/09473580http://www.elsevier.com/locate/ejconhttp://dx.doi.org/10.1016/j.ejcon.2013.08.002mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.ejcon.2013.08.002http://dx.doi.org/10.1016/j.ejcon.2013.08.002http://dx.doi.org/10.1016/j.ejcon.2013.08.002http://dx.doi.org/10.1016/j.ejcon.2013.08.002mailto:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.ejcon.2013.08.002http://dx.doi.org/10.1016/j.ejcon.2013.08.002http://dx.doi.org/10.1016/j.ejcon.2013.08.002http://www.elsevier.com/locate/ejconhttp://www.sciencedirect.com/science/journal/09473580


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relies on trial and error approaches to calculate the storage size.

Only very limited work has contributed to the theoretical analysis

of storage sizing, such as[13,9,17]. In[13], discrete Fourier trans-

forms are used to decompose the required balancing power into

different time-varying periodic components, each of which can be

used to quantify the physical maximum energy storage require-

ment. In [9], the storage sizing problem is cast as an innite

horizon stochastic optimization problem to minimize the long-

term average cost of electric bills in the presence of dynamic

pricing as well as investment in storage. In [17], we cast the

storage sizing problem as a nite horizon deterministic optimiza-

tion problem to minimize the cost associated with the net power

purchase from the electric grid and the battery capacity loss due to

aging while satisfying the load and reducing peak loads. Lower and

upper bounds on the battery size are proposed that facilitate the

efcient calculation of its value. The contribution of this work is

the following: exact values of battery size for the special case of

ideal PV generation and constant loads are characterized; in

contrast, in [17], only lower and upper bounds are obtained. In

addition, the setting in this work is different from that of [9] in

that a nite horizon deterministic optimization is formulated here.

These results can be generalized to more practical PV generation

and dynamic loads (as discussed in Remark 10).

We acknowledge that our analysis does not apply to the typical

scenario of net-metered systems,1 where feed-in of energy to

the grid is remunerated at the same rate as purchase of energy

from the grid. Consequently, the grid itself acts as a storage system

for the PV system (and Ecmax becomes 0). However, from a grid

operator standpoint it would be most desirable if the PV system

could just serve the local load and not export to the grid. This

motivates our choice of no revenue for dumping power to the grid.

Our scenario also has analogues at the level of a balancing area by

avoiding curtailment or intra-hour energy export. For load balan-

cing, in a balancing area (typically a utility grid) steady-stateconditions are set every hour. This means that the power imports

are constant over the hour. The balancing authority then has to

balance local generation with demand such that the steady state

will be preserved. This also corresponds to avoiding outow of

energy from the balancing area. In a grid with very high renewable

penetration, there may be more renewable production than load.

In that case, the energy would be dumped or curtailed. However,

with demand response (e.g., loads with relatively exible sche-

dules) or battery storage, curtailment could be avoided.

The paper is organized as follows. In the next section, we

introduce our setting, and formulate the battery sizing problem.

An upper bound on Ecmax is proposed in Section 3, and the exact

value of Ecmax is obtained for ideal PV generation and constant

loads in Section 4. In Section 5, we validate the results via

simulations. Finally, conclusions and future directions are given

inSection 6.

2. Problem formulation

In this section, we formulate the problem of determining the

storage size for grid-connected PV system, as shown inFig. 1. Solar

panels are used to generate electricity, which can be used to

supply loads, e.g., lights, air conditioners, microwaves in a resi-

dential setting. On one hand, if there is surplus electricity, it can be

stored in a battery, or dumped to the grid if the battery is fully

charged. On the other hand, if there is not enough electricity to

power the loads, electricity can be drawn from the electric grid.

Before formalizing the battery sizing problem, we rst introduce

different components in our setting.

2.1. Photovoltaic generation

We use the following equation to calculate the electricity

generated from solar panels:

Ppvt GHIt S ; 1

where GHI (W m2) is the global horizontal irradiation at the

location of solar panels, S(m2) is the total area of solar panels, and

is the solar conversion efciency of the PV cells. The PV

generation model is a simplied version of the one used in [15]

and does not account for PV panel temperature effects.

2.2. Electric grid

Electricity can be drawn from (or dumped to) the grid. We

associate costs only with the electricity purchase from the grid,

and assume that there is no benet by dumping electricity to

the grid. The motivation is that, from a grid operator standpoint, it

would be most desirable if the PV system could just serve the

local load and not export to the grid. In a grid with very high

renewable penetration, there may be more renewable production

than load. In that case, the energy would have to be dumped (or

curtailed).

We use Cgpt (/kW h) to denote the electricity purchase rate,

Pgpt Wto denote the electricity purchased from the grid at time

t, and Pgdt W to denote the surplus electricity dumped to the

grid or curtailed at time t. For simplicity, we assume that Cgpt is

time independentand has the value Cgp. In other words, there is nodifference between the electricity purchase rates at different time

instants.

2.3. Battery

A battery has the following dynamic:

dEBt

dt PBt; 2

whereEBt W his the amount of electricity stored in the battery

at time t, and PBt W is the charging/discharging rate (more

specically, PBt40 if the battery is charging, and PBto0 if the

Solar Panel

Electric Grid

Load

Battery

Fig. 1. Grid-connected PV system with battery storage and loads.

1 Note that in [17], we study battery sizing for net-metered systems under

more relaxed assumptions compared with this work.

Y. Ru et al. / European Journal of Control 20 (2014) 2437 25


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battery is discharging). We impose the following constraints on

the battery:

(i) At any time, the battery charge EB(t) should satisfy EBminr

EBtrEBmax, where EBmin is the minimum battery charge, EBmaxis the maximum battery charge, and2 0oEBmin rEBmax .

(ii) The battery charging/discharging rate should satisfy PBminr

PBtrPBmax , wherePBmino0, PBmin is the maximum battery

discharging rate, and PBmax40 is the maximum batterycharging rate.

For lead-acid batteries, more complicated models exist (e.g., a

third order model is proposed in [7,3]).

2.4. Load

Ploadt W denotes the load at time t. We do not make explicit

assumptions on the load considered inSection 3except that Ploadt

is a (piecewise) continuous function. Loads could have a xed

schedule such as lights and TVs, or a relatively exible schedule

such as refrigerators and air conditioners. For example, air condi-

tioners can be turned on and off with different schedules as long as

the room temperature is within a comfortable range. InSection 4, weconsider constant loads, i.e.,Ploadt is independent of time t.

2.5. Minimization of electricity purchase cost

With all the components introduced earlier, now we can

formulate the following problem of minimizing the electricity

purchase cost from the electric grid while guaranteeing that the

demand from loads are satised:

minPB;Pgp;Pgd

Z t0 Tt0

CgpPgpd

s:t: Ppvt Pgpt Pgdt PBt Ploadt; 3

dEBt

dt PBt; EBt0 EBmin;EBminrEBtrEBmax;

PBminrPBtrPBmax;

PgptZ0; PgdtZ0; 4

wheret0is the initial time,T is the time period considered for the

cost minimization. Eq. (3) is the power balance requirement for

any time tAt0; t0T; in other words, the supply of electricity

(either from PV generation, grid purchase, or battery discharging)

must meet the demand.

2.6. Battery sizing

Based on Eq.(3), we obtain

Pgpt Ploadt Ppvt PBt Pgdt:

Then the optimization problem in Eq. (4) can be rewritten as

minPB;Pgd

Z t0 Tt0

CgpPload Ppv PB Pgdd

s:t: dEBt

dt PBt; EBt0 EBmin;

EBminrEBtrEBmax;

PBminrPBtrPBmax;

PgdtZ0:

Now there are two independent variables PB(t) and Pgdt. To

minimize the total electricity purchase cost, we have the following

key observations:

(A) If the battery is charging, i.e.,PBt40 andEBtoEBmax , then the

charged electricity should only come from surplus PV generation.

(B) If the battery is discharging, i.e., PBto0 and EBt4EBmin ,

then the discharged electricity should only be used to supply

loads. In other words, the dumped electric powerPgdtshould

only come from surplus PV generation.

In observation (A), the battery can be charged by purchasing

electricity from the grid at the current time and be used later on

when the PV generated electricity is insufcient to meet demands.

However, this incurs a cost at the current time, and the saving of

costs by discharging later on is the same as the cost of charging

(or could be less than the cost of charging if the discharged

electricity gets dumped to the grid) because the electricity purchase

rate is time independent. That is to say, there is no gain in terms of

costs by operating the battery charging in this way. Therefore, we

can restrict the battery charging to using only PV generated electric

power. In observation (B), if the battery charge is dumped to the

grid, potentially it could increase the total cost since extra electricity

might need to be purchased from the grid to meet the demand. In

summary, we have the following rule to operate the battery and

dump electricity to the grid (i.e., restricting the set of possible

control actions) without increasing the total cost.

Rule 1. The battery gets charged from the PV generation only when

there is surplus PV generated electric power and the battery can still

be charged, and gets discharged to supply the load only when the

load cannot be met by PV generated electric power and the battery

can still be discharged. PV generated electric power gets dumped to

the grid only when there is surplus PV generated electric power

other than supplying both the load and the battery charging.

With this operating rule, we can further eliminate the variable

Pgdt and obtain another equivalent optimization problem. On one

hand, ifPloadt Ppvt PBto

0, i.e., the electricity generated fromPV is more than the electricity consumed by the load and charging the

battery, we need to choose Pgdt40 to make Pgpt 0 so that the

cost is minimized; on the other hand, ifPloadt Ppvt PBt40, i.e.,

the electricity generated from PV and battery discharging is less than

the electricity consumed by the load, we need to choose Pgdt 0 to

minimize the electricity purchase costs, and we have Pgpt Ploadt

Ppvt PBt. Therefore,Pgpt can be written as

Pgpt max0; Ploadt Pvt PBt;

so that the integrand is minimized at each time.

Let

xt EBt EBmaxEBmin

2 ;

ut PBt, and

EmaxEBmaxEBmin

2 :

Note that 2Emax EBmax EBmin is the net (usable) battery capacity,

which is the maximum amount of electricity that can be stored in

the battery. Then the optimization problem can be rewritten as

J minu

Z t0 Tt0

Cgpmax0; Pload Ppv ud

s:t: dxt

dt ut; xt0 Emax;

jxtjrEmax; PBminrutrPBmax: 5

Now it is clear that only u(t) (or equivalently, PB(t)) is an

independent variable. As argued previously, we can restrict u(t)

2 Usually, EBmin is chosen to be larger than 0 to prevent fast battery aging. For

detailed modeling of the aging process, refer to [16].

Y. Ru et al. / European Journal of Control 20 (2014) 243726


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to satisfyingRule 1without increasing the minimum costJ. We dene

the set of feasible controls (denoted asUfeasible) as controls that satisfy

the constraintPBminrutrPBmax and do not violateRule 1.

If wexthe parameters t0; T; PBmin, andPBmax ,Jis a function of

Emax , which is denoted as JEmax. IfEmax 0, then ut 0, and J

reaches the largest value

Jmax Z t0 T

t0

Cgpmax0; Pload Ppvd: 6

If we increase Emax, intuitively J will decrease (though may not

strictly decrease) because the battery can be utilized to store extra

electricity generated from PV to be potentially used later on when

the load exceeds the PV generation. This is justied in the

following proposition.

Proposition 1. Given the optimization problem in Eq. (5), if

E1maxoE2max , then JE

1maxZJE

2max.

Proof. Refer toAppendix A.

In other words, Jis monotonically decreasing with respect to

the parameterEmax, and is lower bounded by 0. We are interested

in nding the smallest value ofEmax (denoted as Ecmax) such that J

remains the same for any EmaxZEc

max, and call it the battery sizing

problem.

Problem 1 (Battery sizing). Given the optimization problem in

Eq.(5) with xed t0; T; PBmin and PBmax , determine a critical value

EcmaxZ0 such that 8EmaxoEcmax , JEmax4JE

cmax, and 8EmaxZ

Ecmax , JEmax JEcmax.

Remark 1. In the battery sizing problem, we x the charging and

discharging rate of the battery while varying the battery capacity.

This is reasonable if the battery is charged with a xed charger,

which uses a constant charging voltage but can change the

charging current within a certain limit. In practice, the charging

and discharging rates could scale with Emax , which results in

challenging problems to solve and requires further study.

Note that the critical value Ecmax is unique as shown in thefollowing proposition, which can be proved via contradiction.

Proposition 2. Given the optimization problem in Eq.(5) withxed

t0; T; PBmin and PBmax , Ecmax is unique.

Remark 2. One idea to calculate the critical value Ecmax is that we

rst obtain an explicit expression for the function JEmax by

solving the optimization problem in Eq. (5) and then solve for

Ecmax based on the function J. However, the optimization problem

in Eq.(5)is difcult to solve due to the state constraint jxtjrEmaxand the fact that it is hard to obtain analytical expressions for

Ploadt and Ppvt in reality. Even though it might be possible to

nd the optimal control using the minimum principle[6], it is still

hard to get an explicit expression for the cost function J. Instead, in

the next section, we rst focus on bounding the critical value Ecmaxin general, and then study the problem for specic scenarios in

Section 4.

3. Upper bound onEcmax

In this section, we rst identify necessary assumptions to ensure a

nonzero Ecmax, then propose an upper bound on Ecmax, and nally

show that the upper bound is tight for certain scenarios.

Proposition 3. Given the optimization problem in Eq. (5) with xed

t0; T; PBmin and PBmax ,Ecmax 0if any of the following conditions holds:

(i) 8tAt0; t0T, Ppvt Ploadtr0,

(ii) 8tAt0; t0T, Ppvt PloadtZ0,

(iii) 8t1AS1 , 8 t2AS2,t2ot1, where

S1ftAt0; t0TjPpvt Ploadt40g; 7

S2ftAt0; t0TjPloadt Ppvt40g: 8

Proof. Refer toAppendix B.

Remark 3. The intuition of condition (i) inProposition 3is that if

8tAt0; t0T,Ppvt Ploadtr0, no extra electricity is generatedfrom PV and can be stored in the battery to strictly reduce the

cost. The intuition of condition (ii) in Proposition 3 is that if

8tAt0; t0T, Ppvt PloadtZ0, the electricity generated from

PV alone is enough to satisfy the load all the time, and extra

electricity can be simply dumped to the grid. Note thatJmax 0 for

this case. As dened in condition (iii), S1 \S2 because it is

impossible to have both Ppvt Ploadt40 and Ploadt Ppvt40

at the same time for any time t.

Based on the result in Proposition 3,we impose the following

assumption onProblem 1to make use of the battery.

Assumption 1. There exists t1 and t2 for t1; t2A t0; t0T, such

that t1ot2, Ppvt1 Ploadt140 and Ppvt2 Ploadt2o0.

Remark 4. Ppvt1 Ploadt140 implies that at time t1 there is

surplus electric power available from PV. Ppvt2 Ploadt2o0

implies that at time t2the electric power from PV is not sufcient

for the load. Ift1ot2, the electricity stored in the battery at time t1can be discharged to supply the load at time t2 to strictly reduce

the cost.


t0; T; PBmin and PBmax under Assumption 1, 0oEcmaxrminA; B=2,

where

A

Z t0 Tt0

minPBmax; max0; Ppvt Ploadtdt; 9

and

B

Z t0 T

t0

min PBmin; max0; Ploadt Ppvtdt: 10

Proof. Refer toAppendix C.

Remark 5. Note that if 8 tAt0; t0T we have Ppvt Ploadtr0,

thenA 0 following Eq.(9); therefore, the upper bound for Ecmax in

Proposition 4 becomes 0, which implies that Ecmax 0. If

8tAt0; t0T we have Ppvt PloadtZ0, then B0 following

Eq. (10); therefore, the upper bound for Ecmax in Proposition 4

becomes 0, which implies thatEcmax 0. Both results are consistent

with the results inProposition 3.


t0; T; PBmin and PBmax under Assumption 1, if 8t1AS1, 8t2AS2,

t1ot2, then Ecmax minA; B=2, where S1 (or S2, A, B, respectively)

is dened in Eq. (7) (or(8)(10), respectively). In addition, if Emax is

chosen to be minA; B=2, xt0T Emax (i.e., no battery charge

left at time t0T).

Proof. Refer toAppendix D.

4. Ideal PV generation and constant load

In this section, we study how to obtain the critical value for the

scenario in which the PV generation is ideal and the load is constant.

Ideal PV generation occurs on clear days; for a typical south-facing PV

array on a clear day, the PV output is zero before about sunrise, rises

continuously and monotonically to its maximum around solar noon,

then decreases continuously and monotonically to zero around



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sunset, as shown inFig. 2(a). In other words, there is essentially no

short time uctuation (at the scale of seconds to minutes) due to

atmospheric effects such as clouds or precipitation. By constant load,

we mean Ploadt is a constant for tAt0; t0T. A typical constant

load is plotted in Fig. 2(b). To further simplify the problem, we

assume thatt0is 0000 h Local Standard Time (LST), and T t0k

24 h where k is a nonnegative integer, i.e., T is a duration of

multiple days.Fig. 2plots the ideal PV generation and the constant

load for T 24 h. Now we summarize these conditions in the

following assumption.

Assumption 2. The initial time t0 is 0000 h LST, T k 24 h

where k is a positive integer, Ppvt is periodic on a timescale of

24 h, and satises the following property for tA0; 24 h: there

exist three time instants 0otsunriseotmaxotsunseto24 h such

that

1. Ppvt 0 for tA0; tsunrise [ tsunset; 24 h;

2. Ppvt is continuous and strictly increasing for tAtsunrise; tmax;

3. Ppvt achieves its maximum Pmaxpv at tmax;

4. Ppvt is continuous and strictly decreasing for tA tmax; tsunset,

and Ploadt Pload for tAt0; t0T, where Pload is a constant

satisfying 0oPloadoPmaxpv .

It can be veried thatAssumption 2impliesAssumption 1.


t0 ; T; PBmin and PBmax under Assumption 2 and T 24 h, Ecmax

minA1; B1=2, where

A1

Z t2t1

minPBmax; max0; Ppvt Ploaddt;

and

B1

Z 24t2

min PBmin; max0; PloadPpvtdt;

in which t1ot2 and Ppvt1 Ppvt2 Pload.

Proof. Refer toAppendix E.

Remark 6. A1 and B1 in Proposition 6 are shown in Fig. 3. In

words, A1 is the amount of extra PV generated electricity that can

be stored in a battery, and B1 is the amount of electricity that is

necessary to supply the load and can be provided by battery

discharging. Note that t1 and t2 depend on the value of Pload. To

eliminate this dependency, we can rewrite A1 as

A1

Z 240

minPBmax; max0; Ppvt Ploaddt; 11

and rewrite B1as

B1

Z 24tmax

min PBmin; max0; Pload Ppvtdt; 12

wheretmax is dened inAssumption 2.

Remark 7. If the PV generation is not ideal, i.e., there are

uctuations due to clouds or precipitation, the Ecmax value in

Proposition 6 based on ideal PV generation naturally serves as

an upper bound on Ecmax for the case with the non-ideal PV

generation. Similarly, if the load varies with time but is bounded

by a constant Pload, the Ecmax values inProposition 6based on the

constant loadPload naturally serves as an upper bound on Ecmax for

the case with the time varying load.

Now we examine how Ecmax changes as Pload varies from 0 to

Pmaxpv .


t0; T; PBmin and PBmax under Assumption 2 and T 24 h, then

(a) there exists a unique critical value of PloadA0; Pmaxpv (denoted as

Pcload) such that Ecmax achieves its maximum;

(b) if Pload increases from 0 to Pcload,E

cmax increases continuously and

monotonically from 0 to its maximum;

(c) if Pload increases from Pcload to P

maxpv , E

cmax decreases continuously

and monotonically from its maximum to0.

Proof. Refer toAppendix F.

Remark 8. A typical plot ofEcmax as a function ofPload is given in

Fig. 4. Note that the slopes at 0 andPmaxpv are both 0, which can be

derived from the expressions of dA1=dPload and dB1=dPload. The

result has the implication that there is a (nite) unique battery

capacity that minimizes the grid electricity purchase cost for any

Pload40.Fig. 10(a) veries the plot via simulations.

Now we focus on the case with multiple days.


t0; T; PBmin and PBmax under Assumption 2 and T k 24 h where

k41 is a positive integer, Ecmax minA2; B2=2, where

A2 Z t2

t1


0 tsunrise tsunset 24

Ppv (W)

t(h)

0 24

Pload (W)

t(h)

Fig. 2. Ideal PV generation and constant load. (a) Ideal PV generation for a clear day. (b) Constant load.

0 t1 t2 24

Ppv(W)

t(h)

PBmax

PBmin

A1

B1

Fig. 3. PV generation and load, where PBmax (or PBmin) is the maximum charging

(or discharging) rate, and A1; B1; t1; t2 are dened inProposition 6.



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and

B2

Z t3t2


in which tiATcrossingftA0; k 24jPpvt Ploadg, t1; t2; t3 are the

smallest three time instants in Tcrossingand satisfy t1ot2ot3 .

Proof. Refer toAppendix G.

Remark 9. A2 and B2 in Proposition 8 are shown in Fig. 5. In

words, A2 is the amount of extra PV generated electricity that can

be stored in a battery in the time interval t1; t3, and B2 is theamount of electricity that is necessary to supply the load and can

be provided by battery discharging in the time interval t1; t3. Note

that t1; t2 ; and t3 depend on the value of Pload. To eliminate this

dependency, we can rewrite A2as

A2

Z 240

minPBmax; max0; Ppvt Ploaddt; 13

and rewrite B2 as

B2

Z tmax 24tmax

min PBmin; max0; PloadPpvtdt; 14

wheretmax is dened inAssumption 2.

It can be veried that the following result on how Ecmax changesholds based on an analysis similar to the one in Proposition 7

using Eqs.(13)and(14), andFig.10(b) and (c) veries the trend via

simulations.


t0; T; PBmin and PBmax under Assumption 2 and T k 24 h where

k41 is a positive integer, then

(a) there exists a unique critical value of PloadA0; Pmaxpv (denoted as

Pcload) such that Ecmax achieves its maximum;

(b) if Pload increases from 0 to Pcload,E

cmax increases continuously and

monotonically from 0 to its maximum;

(c) if Pload increases from Pcload to P

maxpv ,E

cmax decreases continuously

and monotonically from its maximum to 0.

Remark 10. Note thatAssumption 2can be relaxed. Given T24,

if Ploadt and Ppvt are piecewise continuous functions, and

intersect at two time instants t1; t2, in addition S1 (as dened in

Eq.(7)) is the same as the open interval t1 ; t2, then the result in

Proposition 6also holds, which can be proved similarly based on

the argument inProposition 6. Besides these conditions, ifPloadt

and Ppvt are periodic with period 24 h, then the result in

Proposition 8also holds. However, with these relaxed conditions,

the results inPropositions 7 and 9do not hold any more since theload might not be constant.

5. Simulations

In this section, we verify the results in Sections 3 and 4 via

simulations. The parameters used inSection 2are chosen based on

a typical residential home setting.

The GHI data is the measured GHI between July 13 and July 16,

2010 at La Jolla, California. In our simulations, we use 0:15 and

S 10 m2 . Thus Ppvt 1:5 GHIt W. We choose t0 as 0000 h

LST on July 13, 2010, and the hourly PV output is given inFig. 6for the

following four days starting fromt0. Except the small variation on July

15, 2010 and being not exactly periodic for every 24 h, the PV

generation roughly satises Assumption 2, which implies thatAssumption 1holds. Note that 0rPpvto1500 W fortAt0; t096.

The electricity purchase rate Cgp is chosen to be 7.8 /kW h,

which is the semipeak rate for the summer season proposed by

0 t1 t2 24

Ppv(W)

t(h)

PBmax

PBmin

A2

B2

t3 48

Fig. 5. PV generation and load (two days), where PBmax (or PBmin) is the

maximum charging (or discharging) rate, and A2; B2; t1; t2; t3 are de

ned inProposition 8.

0

500

1000

1500

Jul 13, 2010 Jul 14, 2010 Jul 15, 2010 Jul 16, 2010

Ppv

(w)

PBmax

PBmin

Fig. 6. PV output for July 1316, 2010, at La Jolla, California. For reference, a constant

load 200 W is also shown (solid line) along with PBmax PBmin 200 W. The shaded

area above the 200 W horizontal line corresponds to the amount of electricity that can

be potentially charged to a battery, while the shaded area below the 200 W line

corresponds to the amount of electricity that can potentially be provided by

discharging the battery.

0 Pcload Pmax

pv Pload (W)

Ecmax (Wh)

B1

2

A1

2

Fig. 4. Ecmax as a function ofPload for 0rPloadrPmaxpv .

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25100

200

300

400

500

600

700

800

900

1000

Time (h)

Load(w)

Fig. 7. A typical residential load prole.



7/14

SDG&E (San Diego Gas & Electric)[19]. For the battery, we choose

EBmin 0:4 EBmax , and then

Emax EBmax EBmin

2 0:3 EBmax:

The maximum charging rate is chosen to be PBmax 200 W, and

PBmin PBmax . Note that the battery dynamic is characterized by

a continuous ordinary differential equation. To run simulations, we

use 1 h as the sampling interval, and discretize Eq. (2) as

EBk 1 EBk PBk:

5.1. Dynamic loads

We rst examine the upper bound in Proposition 4 usingdynamic loads. The load prole for one day is given in Fig. 7,

which resembles the residential load prole in Fig. 8(b) in[15].3

Note that one load peak appears in the early morning, and the

other in the evening. For multiple day simulations, the load is

periodic based on the load prole inFig. 7. We study how the cost

function J of the optimization problem in Eq. (5) changes as a

function ofEmax by increasing the battery capacity Emax from 0 to

1500 W h with the step size 10 W h. We solve the optimization

problem in Eq. (5) via linear programming using the CPlex solver

[10]. IfT 24 h (or T 48 h, T 96 h, respectively), the plot

of the minimum costs versus Emax is given inFig. 8(a) (or (b), (c),

respectively). The plots conrm the result inProposition 1, i.e., the

minimum cost is a decreasing function ofEmax , and also show the

existence of the uniqueEcmax . IfT 24 h, Ecmax is 700 W h, which

can be identied fromFig. 8(a). The upper bound in Proposition 4

is calculated to be 900 W h. Similarly, ifT 48 h,Ecmax is 900 W h

while the upper bound in Proposition 4 is calculated to be

1800 W h; if T 96 h, Ecmax is 900 W h while the upper bound

in Proposition 4 is calculated to be 3402 W h. The upper bound

holds for these three cases though the difference between the

upper bound and Ecmax increases when Tincreases. This is due to

the fact that during multiple days battery can be repeatedly

charged and discharged; however, this fact is not taken into

account in the upper bound inProposition 4. Since the load prole

roughly satises the conditions imposed in Remark 10, we can

calculate the theoretical value for Ecmax based on the results inPropositions 7 and 9 even for this dynamic load. IfT 24 h, the

theoretical value is 690 W h which is obtained from Proposition 6

by evaluating the integral in A1 and B1 using the sum of the

integrand for every hour fromt0to t0T. Similarly, ifT 48 hor

T 96 h, then the theoretical value is 900 W h which is obtained

fromProposition 8. Due to the step size 10 W h used in the choice

ofEmax , these theoretical values are quite consistent with results

obtained via simulations.

5.2. Constant loads

We now study how the cost function J of the optimization

problem in Eq. (5) changes as a function ofEmax with a constant

0 500 1000 15001.8

1.9

2

2.1

2.2

2.3

2.4

2.5

Emax

(wh)

MinimumCost($)

0 500 1000 15000.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

Emax(wh)

MinimumCost($)

0 500 1000 15000.85

0.9

0.95

1

1.05

1.1

1.15

1.2

Emax

(wh)

MinimumCost($)

Fig. 8. Plots of minimum costs versus Emax obtained via simulations given the load prole inFig. 7 and PBmax 200 W. (a) One day. (b) Two days. (c) Four days.

3 However, simulations in[15]start at 7 AM soFig. 7is a shifted version of the

load prole in Fig. 8(b) in [15].



8/14

load, and the load is used from t0to t0Tto satisfyAssumption 2.

We x the load to be Pload 200 W, and increase the battery

capacity Emax from 0 to 1500 W h with the step size 10 W h. We

solve the optimization problem in Eq. (5) via linear programming

using the CPlex solver[10]. IfT 24 h (or T 48 h, T 96 h,

respectively), the plot of the minimum costs versus Emax is given in

Fig. 9(a) (or (b), (c), respectively). The plots conrm the result in

Proposition 1, i.e., the minimum cost is a decreasing function of

Emax , and also show the existence of the unique Ecmax .

Now we validate the results in Propositions 69. We vary the

load from 0 to 1500 W with the step size of 100 W, and for each

Pload we calculate the Ecmax and the minimum cost corresponding

to theEcmax . In Fig. 10(a), the left gure shows howEcmax changes as

a function of the load for T 24 h, and the right gure shows the

corresponding minimum costs. The plot in the left gure isconsistent with the result in Proposition 7except that the max-

imum ofEcmax is not unique. This is due to the fact that the load is

chosen to be discrete with step size 100 W. The right gure is

consistent with the intuition that when the load is increasing,

more electricity needs to be purchased from the grid (resulting in

a higher cost). Note that the blue solid curve corresponds to the

costs with Ecmax , while the red dotted curve corresponds toJmax, i.

e., the costs without battery. The plots forEcmax and the minimum

cost for T 48 h and T 96 h are shown inFig. 10(b) and (c).

The plots in the left gures ofFig. 10(b) and (c) are consistent with

the result in Proposition 9. Note that as Tincreases, the critical

load Pcload decreases as shown in the left gures of Fig. 10. One

observation on the left gures of Fig. 10 is that Ecmax increases

roughly linearly with respect to the load when the load is small.

The justication is that when the load is small,Ecmax is determined

by B1 in Fig. 3 (or B2 in Fig. 5 for multiple days) and B1 (or B2)

increases roughly linearly with respect to the load as can be seen

fromFig. 3(or Fig. 5).

Now we examine the results inProposition 6. ForT 24 h, we

evaluate the integral in A1 and B1 using the sum of the integrand

for every hour from t0to t0Tgiven a xed load, and then obtain

Ecmax; this value is denoted as the theoretical value. The theoretical

value is plotted as the red curve (with the circle marker) in the left

plot ofFig. 11(a). The valueEcmax calculated based on simulations is

plotted as the blue curve (with the square marker) in the left plot

ofFig. 11(a). In the right plot ofFig. 11(b), we plot the difference

between the value obtained via simulations and the theoretical

value. Note that the value obtained via simulations is always larger

than or equal to the theoretical value because Emax is chosen to bediscrete with step size 10 W h. The differences are always smaller

than 9 W h,4 which conrms that the theoretical value is very

consistent with the value obtained via simulations. The same

conclusion holds for T 48 h, as shown in Fig. 11(b). For

T 96 h, the largest difference is around 70 W h as shown in

Fig. 11(c); this is more likely due to the slight variation in the PV

generation for different days. Note that the differences for 1016 58of

the load values (which range from 0 to 1500 W with the step size

100 W) are within 10 W h.

0 500 1000 15000.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

Emax(wh)

Minim

umCost($)

0 200 400 600 800 1000 1200 1400 16000.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Emax(wh)

MinimumCost($)

0 500 1000 15000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Emax(wh)

MinimumCost($)

Fig. 9. Plots of minimum costs versus Emax obtained via simulations given Pload 200 W andPBmax 200 W. (a) One day. (b) Two days. (c) Four days.

4 The step size forEmax is 10 W h, so the difference between the value obtained

via simulations and the theoretical value is expected to be within 10 assuming the

theoretical value is correct.



9/14

0 500 1000 15000

100

200

300

400

500

600

700

Load (w)

Emaxc(wh)

0 500 1000 15000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Load (w)

Cost

($)

Minimum cost

Jmax

0 500 1000 15000

200

400

600

800

1000

1200

Load (w)

Emaxc(wh)

0 500 1000 15000

0.5

1

1.5

2

2.5

3

3.5

4

Load (w)

Cost($)

Minimum cost

Jmax

0 500 1000 15000

200

400

600

800

1000

1200

Load (w)

Em

axc(

wh)

0 500 1000 15000

1

2

3

4

5

6

7

8

Load (w)

C

ost($)

Minimum cost

Jmax

Fig.10. Ecmax(left), and costs (both Jmaxand the cost corresponding toEcmax, right) versus thexed load obtained via simulations forPBmax 200 W. (a) One day. (b) Two days.

(c) Four days.



10/14

0 500 1000 15000

100

200

300

400

500

600

700

Load (w)

Emax

c(wh)

Theoretical

Simulation

0 500 1000 15000

1

2

3

4

5

6

7

8

9

Load (w)

Error(wh)

0 500 1000 15000

200

400

600

800

1000

1200

Load (w)

Emax

c(wh)

Theoretical

Simulation

0 500 1000 15000

1

2

3

4

5

6

7

8

9

Load (w)

Error(wh)

0 500 1000 15000

200

400

600

800

1000

1200

Load (w)

Em

ax

c(wh)

Theoretical

Simulation

0 500 1000 15000

10

20

30

40

50

60

70

Load (w)

Error(wh)

Fig. 11. Plots ofEcmax versus the xed load for the theoretical value (the curve with the circle marker) and the value obtained via simulations (the curve with the square

marker). (a) One day. (b) Two days. (c) Four days.



11/14

6. Conclusions

In this paper, we studied the problem of determining the size of

battery storage for grid-connected PV systems. We proposed an

upper bound on the storage size, and showed that the upper

bound is achievable for certain scenarios. For the case with ideal

PV generation and constant load, we characterized the exact

storage size, and also showed how the storage size changes as

the constant load changes; these results are consistent with theresults obtained via simulations.

There are several directions for future research. First, the dynamic

time-of-use pricing of the electricity purchase from the grid could be

taken into account. Large businesses usually pay time-of-use elec-

tricity rates, but with increased deployment of smart meters and

electric vehicles some utility companies are moving towards different

prices for residential electricity purchase at different times of the day

(for example, SDG&E has the peak, semipeak, offpeak prices for a day

in the summer season [19]). New results (and probably new

techniques) are necessary to deal with dynamic pricing. Second, we

would like to study how batteries with a xed capacity can be

utilized (e.g., via serial or parallel connections) to implement the

critical battery capacity for practical applications. Last, we would also

like to extend the results to wind energy storage systems, and

consider battery parameters such as round-trip charging efciency,

degradation, and costs.

Acknowledgments

This work was funded by NSF grant ECCS-1232271.

Appendix A. Proof toProposition 1

Given E1max , suppose a feasible control u1t achieves the

minimum electricity purchase cost JE1max and the corresponding

state x is x1t. Since jx1tjrE1maxoE2max , u

1t is also a feasible

control for problem(5) with the state constraint E2max and satisfy-

ing Rule 1, and results in the cost JE1max. Since JE2max is the

minimal cost over the set of all feasible controls which include

u1t, we must have JE1maxZJE2max.

Appendix B. Proof toProposition 3

Condition(i) holds: Since 8 tAt0; t0T, Ppvt Ploadtr0, we

have Ploadt PpvtZ0. Denote the integrand inJof Eq.(5)as , i.

e., t Cgpmax0; Ploadt Ppvt ut. If Ploadt Ppvt 0,

then we could choose ut 0 to make to be 0. If

Ploadt Ppvt40, we could choose Ppvt Ploadtruto0 to

decrease , i.e., by discharging the battery. However, since

xt0 Emax, there is no electricity stored in the battery at theinitial time. To be able to discharge the battery, it must have been

charged previously. FollowingRule 1, the electricity stored in the

battery should only come from surplus PV generation. However,

there is no surplus PV generation at any time because

8tAt0; t0T, Ppvt Ploadtr0. Therefore, the cost is not

reduced by choosing Ppvt Ploadtruto0. In other words, u

can be chosen to be 0. In both cases, u(t) can be 0 for any

tAt0; t0T without increasing the cost, and thus, no battery is

necessary. Therefore, Ecmax 0.

Condition (ii) holds: Since 8 tAt0; t0T, Ppvt PloadtZ0, we

have Ploadt Ppvtr0. Denote the integrand in J as , i.e., t

Cgpmax0; Ploadt Ppvt ut. If Ploadt Ppvtr0, we could

choose ut 0, and then max0; Ploadt Ppvt ut

max0; Ploadt Ppvt 0. Since u(t) can be 0 for any tAt0; t0T

without increasing the cost, no battery is necessary. Therefore,

Ecmax 0.

Condition (iii)holds:S1is the set of time instants at which there is

extra amount of electric power that is generated from PV after

supplying the load, while S2is the set of time instants at which the

PV generated power is insufcient to supply the load. According to

Rule 1, at time t, the battery could get charged only if tAS1, and

could get discharged only iftAS2. If8 t1AS1, 8 t2AS2,t2ot1 implies

that even if the extra amount of electricity generated from PV isstored in a battery, there is no way to use the stored electricity to

supply the load. This is because the electricity is stored after the time

instants at which battery discharging can be used to strictly decrease

the cost and initially there is no electricity stored in the battery.

Therefore, the costs are the same for the scenario with battery and

the scenario without battery, andEcmax 0.

Appendix C. Proof toProposition 4

It can be shown, via contradiction, that under Assumption 1,

A40 and B40, which implies that minA; B=240.

We showEcmax40 via contradiction. SinceEcmaxZ0, we need to

exclude the case Ecmax 0. Suppose E

cmax 0. If we choose

Emax4Ecmax 0, JEmaxoJE

cmax because under Assumption 1 a

battery can store the extra PV generated electricity rst and then

use it later on to strictly reduce the cost compared with the case

without a battery (i.e., the case with Emax 0). A contradiction to

the denition ofEcmax .

To show EcmaxrminA; B=2, it is sufcient to show that if

EmaxZminA; B=2, then JEmax JminA; B=2. There are two

cases depending on ifArB or not:

1. ArB. Then minA; B A. At time t, max0; Ppvt Ploadt is

the extra amount of electric power that is generated from PV

after supplying the load, and

minPBmax; max0; Ppvt Ploadt

is the extra amount of electric power that is generated from PV

after supplying the load and can be stored in a battery subject

to the maximum charging rate. Then

A

Z t0 Tt0

minPBmax; max0; Ppvt Ploadtdt

is the maximum total amount of extra electricity that can be

stored in a battery while taking the battery charging rate into

account. Even if 2EmaxZA, i.e., EmaxZA=2, the amount of

electricity that can be stored in the battery cannot exceed A.

Therefore, any control that is feasible with jxtjrEmax is

also feasible with jxtjrA=2. Therefore, JEmax JA=2

JminA; B=2.

2. A4B. Then minA; B B. At time t, max0; Ploadt Ppvt is

the amount of electric power that is necessary to satisfy the

load (and could be supplied by either battery power or grid

purchase), and

min PBmin; max0; Ploadt Ppvt

is the amount of electric power that can potentially be discharged

from a battery to supply the load subject to the maximum

discharging rate (in other words, if Ploadt Ppvt4PBmin,

electricity must be purchased from the grid). Then

B

Z t0 Tt0

min PBmin; max0; Ploadt Ppvtdt

is the maximum total amount of electricity that is necessary to be

discharged from the battery to satisfy the load while taking the



12/14

battery discharging rate into account. When 2EmaxZB, i.e.,

EmaxZB=2, the amount of electricity that can be charged can

exceedBbecauseA4B; however, the amount of electricity that is

strictly necessary to be (and, at the same time, can be) discharged

does not exceed B. In other words, if the stored electricity in the

battery exceeds this amount B, the extra electricity cannot help

reduce the cost because it either cannot be discharged or is not

necessary. Therefore, any control that minimizes the total cost

with the battery capacity being B also minimizes the total costwith the battery capacity being 2Emax. Therefore,JEmax JB=2

JminA; B=2.

Appendix D. Proof toProposition 5

From Proposition 4, we have EcmaxrminA; B=2. To prove

Ecmax minA; B=2, we show that EcmaxominA; B=2 is impossible

via contradiction. Suppose EcmaxominA; B=2. If 8 t1AS1, 8t2AS2,

t1ot2, then during the time interval t0; t0T, the battery is rst

charged, and then discharged following Rule 1. In other words,

there is no charging after discharging. There are two cases

depending on A and B:

1. ArB. In this case, EcmaxoA=2, i.e., 2EcmaxoA. If the battery

capacity is 2Ecmax , then the amount of electricity A 2Ecmax40

(which is generated from PV) cannot be stored in the battery. If

we choose the battery capacity to be A , this extra amount can

be stored and used later on to strictly decrease the cost because

ArB. Therefore, JA=2oJEcmax. A contradiction to the deni-

tion ofEcmax . In this case, ifEmax is chosen to be A=2, then the

battery is rst charged with A amount of electricity, and then

completely discharged before (or at) t0T because ArB.

Therefore, we have xt0T Emax .

2. A4B. In this case, EcmaxoB=2, i.e., 2EcmaxoB. If the battery

capacity is 2Ecmax , at most 2EcmaxoBoA amount of PV gener-

ated electricity can be stored in the battery. Therefore, the

amount of electricityB 2Ecmax40 must be purchased from thegrid to supply the load. If we choose the battery capacity to be

B, the amount of electricity B 2Ecmax purchased from the grid

can be provided by the battery because the battery can be

charged with the amount of electricity B (sinceA4B), and thus

the cost can be strictly decreased. Therefore, JB=2oJEcmax. A

contradiction to the denition of Ecmax . In this case, if Emax is

chosen to be B=2, then the battery is rst charged with B

amount of electricity (that is to say, not all extra electricity

generated from PV is stored in the battery since A4B), and

then completely discharged at time t0T. Therefore, we also

have xt0T Emax .

Appendix E. Proof toProposition 6

Due toAssumption 2,Ploadtintersects with Ppvtat two time

instants for T 24 h; the smaller time instant is denoted as t1,

and the larger is denoted ast2, as shown inFig. 3. It can be veried

that Ppvt4Pload for tAt1; t2 and PpvtoPload for tA0; t1 [

t2; 24 followingAssumption 2. FortA0; t1, a battery could only

get discharged followingRule 1; however, it cannot be discharged

becausex0 Emax. Therefore,u(t) can be 0 while achieving the

lowest cost for the time period 0; t1. Then the objective function

of the optimization problem in Eq.(5) can be rewritten as

J min Z 24

0

Cgpmax0; PloadPpv udJ0J1;

whereJ0Rt1

0 CgpPloadPpvdis a constant which is indepen-

dent of the control u, and

J1 min

Z 24t1

Cgpmax0; PloadPpv ud:

In other words, the optimization problem is essentially the same

as minimizing J1for tAt1; 24; accordingly, the critical value Ecmax

will be the same since the battery is not used for the time interval

0; t1. For the optimization problem with the cost functionJ1underAssumption 2, S1 t1; t2 and S2 t2; 24 according to Eqs.

(7)and (8). Since 8t1AS1, 8 t

2AS2, t

1ot2ot

2, the conditions in

Proposition 5 are satised. Thus, we have Ecmax minA; B=2,

where

A

Z 24t1

minPBmax; max0; Ppvt Ploaddt

Z t2t1


which is essentially A1, and

B

Z 24t1

min PBmin; max0; PloadPpvtdt

Z 24

t2


which is essentially B1. Thus the result holds.

Appendix F. Proof toProposition 7

LetfA1B1, whereA1and B1are dened in Eqs.(11)and(12).

Note that fis a function ofPload. IfPload 0, then

A1

Z 240

minPBmax; max0; Ppvtdt40;

according to Eq.(11), and

B1Z 24

tmaxmin PBmin; max0; Ppvtdt 0;

according to Eq. (12). Therefore, f0 A10 B1040. If

Pload Pmaxpv , then

A1

Z 240

minPBmax; max0; Ppvt Pmaxpv dt 0;

and

B1

Z 24tmax

min PBmin; max0; Pmaxpv Ppvtdt40:

Therefore, fPmaxpv A1Pmaxpv B1P

maxpv o0. In addition, since f is

an integral of a continuous function ofPload,fis differentiable with

respect to Pload, and the derivative is given as

df

dPload

dA1dPload

dB1dPload

:

Since for PloadA0; Pmaxpv ,

dA1dPload

Z 240

1 I 0oPpvt PloadrPBmax

dt

o0;

and

dB1dPload

Z 24tmax

1I 0oPloadPpvtrPBmin

dt

40;

we have df=dPloado0, where If0oPpvt PloadrPBmaxg is the

indicator function (i.e., if 0oPpvt PloadrPBmax , the function



13/14

has value 1, 0 otherwise). Therefore, f is continuous and strictly

decreasing forPloadA 0; Pmaxpv . Since f040 and fP

maxpv o0, there

is one and only one value ofPload such that fis 0. We denote this

value as Pcload and have A1Pcload B1P

cload.

IfPloadA0; Pcload,f40, i.e.,A14B1. Therefore,E

cmax B1=2. Since

dB1=dPload40, Ecmax increases continuously (since B1 is differentiable

with respect to Pload) and monotonically from 0 to the value

B1Pcload=2. On the other hand, if PloadAP

cload; P

maxpv , fo0, i.e.,

A1oB1. Therefore, EcmaxA1=2. Since dA1=dPloado0,E

cmax decreases

continuously (since A1 is differentiable with respect to Pload) and

monotonically from the valueA1Pcload=2 B1P

cload=2 to 0. Therefore,

Ecmaxachieves its maximum at Pcload. This completes the proof.

Appendix G. Proof toProposition 8

Due toAssumption 2, Ploadt intersects with Ppvt at 2k time

instants forT k 24 h; we denote the set of these time instants

as TcrossingftA 0; k 24jPpvt Ploadg. We sort the time instants

in an ascending order and denote them as t1; t2; t3;;

t2i 1; t2i;; t2k 1 ; t2k, where 2rirk. FollowingRule 1, at time t,

a battery could get charged only if tAt1; t2 [ t3; t4

[t2k 1; t2k, and could get discharged only if tA0; t1 [t2; t3 [ t4; t5 [ t2k; k 24. As shown in the proof to

Proposition 6, u(t) can be zero for tA0; t1, and results in the

lowest cost J0Rt1

0 CgpPloadPpvd, which is a constant. At

time t1, there is no charge in the battery. Then the battery is

operated repeatedly by charging rst if tAt2i 1; t2i and then

discharging if tAt2i; t2i 1 for i 1; 2;; k and t2k 1 k 24.

Naturally, we could group the charging interval t2i 1; t2i with

the discharging interval tAt2i; t2i 1 to form a complete battery

operating cycle in the interval t2i 1; t2i 1.

Now the objective function of the optimization problem in Eq.

(5)satises

J minu Z

k24

0

Cgpmax0; PloadPpv ud

J0minu

k 1

i 1

LiLk

!;

where

Li

Z t2i 1t2i 1

Cgpmax0; PloadPpv ud;

for i 1; 2;; k1, and

Lk

Z t2k 1t2k 1

Cgpmax0; PloadPpv ud:

Note that given a certain Emax , if the battery charge at the end

of the rst battery operating cycle is larger than 0 (i.e.,

xt34Emax), then Emax4Ecmax . This can be argued as follows. If

the battery charge at the end of the rst cycle is larger than 0 (thisalso implies that the battery charge at the end of the ith cycle is

also larger than 0 due to periodic PV generations and loads), i.e.,

there is more PV generation than demand in the time interval

t1; t3, then Emax can be strictly reduced to a smaller capacity so

thatxt3 Emax without increasing the electricity purchase cost

in the interval t1; t3. Due to periodicity of the PV generation and

the load, the smallerEmax can be used for the interval t2i 1; t2i 1

for i 2;; k 1 without increasing the electricity purchase cost.

Therefore, this Emax must be larger than Ecmax . In other words, if

Ecmax is used, thenxt2i 1for i 1; 2;; k 1 has to be Ecmax , i.e.,

no charge left at the end of each operating cycle. Now we only

consider Emax such that at the end of each operating cycle

xt2i 1 Emax for i 1; 2;; k1 (necessarily, Ecmax is smaller

than or equal to any such Emax). For such Emax , the control actions

for each operating cycle are completely decoupled.5 Therefore, the

total cost Jcan be rewritten as

JJ0 k 1

i 1

JiJk;

whereJi minuL i fori 1; 2;; k.

Now we focus on J1. For the optimization problem with the cost

function J1 under Assumption 2, S1 t1; t2 and S2 t2; t3

according to Eqs. (7) and(8). Since 8t

1AS1 , 8 t

2AS2, t

1ot2ot

2,the conditions in Proposition 5 are satised. Thus, we have

Ecmax1 minA; B=2, where Ecmax1 is the E

cmax when we only

consider the cost function J1,

A

Z t3t1

minPBmax; max0; Ppvt Ploaddt

Z t2t1


which is essentially A2, and

B

Z t3t1

min PBmin; max0; PloadPpvtdt

Z t3

t2 min PBmin; max0; PloadPpvtdt;

which is essentially B2. Thus we have Ecmax1 minA2 ; B2=2.

Based onProposition 5, we also know that xt3 Ecmax1. Thus,

this Ecmax1 satises the requirement that at the end of the

operating cycle there is no charge left.

For the cost functionJ2, the optimization problem is essentially

the same as the problem with the cost function J1because

1. Ppvt Ppvt24 for tAt3; t5 because Ppvt is periodic with

period 24 h. Note that t 24At1; t3;

2. Ploadt is a constant; and

3. there is no charge left at t3.

In other words, there is no difference between the optimizationproblem with the cost functionJ2 and the one with J1 other than

the shifting of time tby 24 h. Therefore, the Ecmax2 will be the

same as Ecmax1. The same reasoning applies to the optimization

problem with the cost function Ji for i 3;; k1. Therefore, we

have Ecmaxi Ecmax1 for i 2;; k 1.

For the optimization problemJk, there is no charge left at time

2k 1. This problem is essentially the same as the problem studied

in the proof to Proposition 6with the cost function J1 except the

shifting of time tby k1 24 h. The solution Ecmaxk is given as

minAk; Bk=2, where

Ak

Z t2kt2k 1


and

Bk

Z t2k 1t2k

min PBmin; max0; PloadPpvtdt:

Note that AkA2, and BkoB2. If we choose Emax minA2; B2=2

which is larger than or equal to Ecmaxk, we have JkEmax

JkEcmaxk.

5 Note that the control action for tAt2i 1; t2i and the control action for

tA t2i ; t2i 1 for i 1; 2;; k are coupled in the sense that battery cannot be

discharged if at time t2i there is no charge in the battery. In general, the control

action fortAt2i ; t2i 1 and the control action for tAt2i 1; t2i 2for i 1; k 1 can

also be coupled if at time t2i 1 there is extra charge left in the battery because the

extra charge will affect the charging action in the interval tAt2i 1; t2i 2. Here,

there is no such coupling for the latter case when we restrictEmaxso that at the end

of each operating cycle there is no charge left.



14/14

Now we claim that Ecmax when considering the cost function J is

exactly minA2; B2=2. If we choose EmaxominA2; B2=2, then

JEmax4JminA2; B2=2 by an argument similar to the one in

Proposition 5. On the other hand, if we choose EmaxZminA2; B2=2,

then JEmax JminA2; B2=2. Therefore, Ecmax to the optimization

problem with the cost function Jis minA2; B2=2.

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