Applied Energy 176 (2016) 245–257

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Applied Energy

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

A physics-based integer-linear battery modeling paradigm

http://dx.doi.org/10.1016/j.apenergy.2016.05.0230306-2619/� 2016 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: msciolet@mines.edu (M.S. Scioletti), gtpaddle@gmail.com

(J.K. Goodman), kohl@gatech.edu (P.A. Kohl), anewman@mines.edu (A.M. Newman).

Michael S. Scioletti a, Johanna K. Goodman b, Paul A. Kohl b, Alexandra M. Newman a,⇑aDepartment of Mechanical Engineering, Colorado School of Mines, Golden, CO, United Statesb School of Chemical and Biomolecular Engineering, Georgia Institute of Technology, Atlanta, GA, United States

h i g h l i g h t s

� We describe a dispatch optimization paradigm for a hybrid energy system.� We provide a detailed battery model.� We simplify the battery model for inclusion in an optimization model.� We measure the error between the simplified battery model and the detailed model.� We show that our simplified model yields minimal error.

a r t i c l e i n f o

Article history:Received 12 January 2016Received in revised form 2 May 2016Accepted 2 May 2016

Keywords:OptimizationMicrogrid designBattery dispatchHybrid powerRate-capacitySteady-state

a b s t r a c t

Optimal steady-state dispatch of a stand-alone hybrid power system determines a fuel-minimizing dis-tribution strategy while meeting a forecasted demand over six months to a year. Corresponding opti-mization models that integrate hybrid technologies such as batteries, diesel generators, andphotovoltaics with system interoperability requirements are often large, nonconvex, nonlinear, mixed-integer programming problems that are difficult to solve even using the most state-of-the-art algorithms.The rate-capacity effect of a battery causes capacity to vary nonlinearly with discharge current; omittingthis effect simplifies the model, but leads to over-estimation of discharge capabilities. We present aphysics-based set of integer-linear constraints to model batteries in a hybrid system for a steady-statedispatch optimization problem that minimizes fuel use. Starting with a nonlinear set of constraints,we empirically derive linearizations and then compare them to a commonly used set of constraints thatassumes a constant voltage and neglects rate-capacity. Numerical results demonstrate that assuming afixed voltage and capacity may lead to over-estimating discharge quantities by up to 16% compared toour overestimations of less than 1%.

� 2016 Elsevier Ltd. All rights reserved.

1. Introduction

Optimal steady-state dispatch of a stand-alone hybrid powersystem determines a fuel-minimizing power distribution strategywhile meeting a forecasted demand over six months to a year. Cor-responding optimizationmodels that integrate hybrid technologiessuch as batteries, diesel generators, and photovoltaics (PV) withsystem interoperability requirements are often large, nonconvex,nonlinear, mixed-integer programming (MINLP) problems thatare difficult to solve [5,29]. Mixed-integer programs (MIPs) thatemploy linearization and approximation techniques to boundand solve a simplified nonlinear problem frequently serve as tract-able modeling alternatives to MINLPs. Although a MIP does not

guarantee the feasibility and/or optimality of the solution to thecorresponding nonlinear problem, it can still be an effective mod-eling option.

Batteries are integral to hybrid systems because of their abilityto both store energy and provide on-demand, dispatchable power[10,21]. There are a variety of ways to effectively dispatch batteriesin hybrid systems such as load-leveling, peak-shaving, and/or load-following [7], all of which store energy to minimize the use of addi-tional technologies with start-up and ramping costs. Subject to apower loss of up to 10%, all batteries generate a direct-current(DC) power that requires conversion to alternating-current (AC)power to meet demand. Batteries store energy in electrochemicalform in ways that differ by type; common battery chemistriesassociated with hybrid systems are nickel cadmium, nickel-metalhydride, lead acid, and lithium-ion. Each has its own documentedadvantages [35,39], e.g., cost, safety, performance, and size. We

246 M.S. Scioletti et al. / Applied Energy 176 (2016) 245–257

use data from lithium-ion and lead-acid battery manufacturers torepresent baseline products within the industry.

Steady-state dispatch modeling for hybrid-power systems oftenassumes that the instantaneous fluctuations in frequency and volt-age over time do not affect power dispatch. When a battery dis-charges, the electro-active species oxidizes at the anode andreduces at the cathode. The chemistry of the electrodes determinesthe voltage of an individual cell. Because concentrations of theelectro-active species play a large role in determining the voltageand capability of a battery, detailed battery models often track con-centrations in various parts of the battery. These diffusion-basedmodels are accurate, because they take into account the transientvoltage behavior of the battery, such as the voltage profile whenswitching between charge and discharge. The time for diffusion,i.e., the dissipation of the concentration gradient, is estimated bythe quotient of the square of the battery electrolyte thickness anda coefficient that accounts for the diffusivity of the active species.The diffusivity of lithium ions in the electrolyte is on the order of10�10 m2=s, while the scale for different battery components isaround 100 lm. This gives diffusion-based models a timescale,

length2=diffusivity, on the order of 100 s, which, when compared

at the hourly timefidelity of our steady-state application, representstoo short of a time interval for it to impact the accuracy of themodel.

Although steady-state assumptions facilitate simplifying somecomplex relationships through linearization [8,13], modeling bat-tery performance often requires both nonlinear and integer consid-erations. We recognize that lifetime is also an importantcharacteristic in batterymodeling; however, lifetime data is limitedand difficult to properly test. The focus of this paper is on the set ofconstraints within a hybrid system optimizationmodel that dictatebattery performance, which are nonlinear. For more details, werefer the reader to [33]. Energy, which describes the total amountof electricity a source can provide, and power, which representsthe total energy consumed per unit time, are functions of voltage.Although voltage varies with time, our steady-state assumptionsrender this variance negligible. However, voltage also variesdepending on state of charge (SoC), i.e., the fraction of batterycapacity available for discharge, and current; failing to consider thisvariability causes an over-estimation of battery resource availabil-ity. By employing a physics-based definition, in which averagepower is the product of current and voltage, and Ohm’s law, whichrelates current and resistance to voltage, we canmodel the rate andassociated amount of electricity available from batteries in terms ofa single independent variable, i.e., current.

A battery’s capacity varies based on the magnitude of the dis-charge current. The higher the current draw, the less total capacityavailable, which we refer to as the rate-capacity effect. Peukert’sequation [12], which accounts for the rate-capacity effect in lead-acid batteries by exponentially relating battery capacity to dis-charge current, could also be applied to other battery chemistries.The Kinetic Energy Battery Model (KiEBM), which portrays thechange in capacity as a nonlinear function of charge and dischargerates, models a battery’s capacity as two tanks, one of which isimmediately available for discharge and the other of which ischemically bound [22]. Alternately, the CIEMAT (Centre for Energy,Environment and Technology in Madrid, Spain) model presents anonlinear set of equations that accounts for dynamic and complexbattery operations [9]. The common theme among these batteryrepresentations is that neglecting rate-capacity leads to over-estimating the performance of the battery, which may yield aninfeasible dispatch solution, i.e., battery discharge power that isnot actually available. Researchers tend to use constraint sets ina hybrid model that neglect rate-capacity (see Section 2 for casesin which tractability is a concern). These large-scale MIPs minimizecosts in design and/or dispatch problems that determine optimal

technology procurement and/or operation to satisfy demand[6,16–20,25,27,30,31,34,37,38]; they may also approximate unitcommitment that schedules technologies to meet demand[15,23,36].

Recognizing the pitfalls of over-simplifying battery perfor-mance and the potential impact rate-capacity error has on thesolution, we first introduce simple energy constraint set (E), whichnot only neglects rate-capacity effects, but assumes a constantvoltage. As an alternative to (E), we present a nonlinear set of con-straints (N ). Next, we linearize the nonlinear, nonconvex relation-ships in (N ) to empirically derive a physics-based set of constraints(PB). We then determine the theoretical error associated with both(E) and (PB). Lastly, we present (PF ) (see Appendix), whichincludes a fuel-minimizing objective function and a set of mixed-integer constraints for system interoperability and PV and genera-tor technologies. The combination of (PB) with (PF ) and (E) with

(PF ) forms two optimization models, which we term (PBþ) and

(Eþ), respectively, for the hybrid power steady-state dispatch prob-lem. We solve 12 scenarios and compare battery dispatch solutionsof each model by quantifying the error in each.

The remainder of this paper is organized as follows: Section 2presents ðEÞ; Section 3 reveals our physics-based mixed-integer,nonlinear battery constraints (N ) presented in MINLP format.Section 4 details how we derive (PB) from (N ). Section 5 presentsscenarios and results, while Section 6 concludes.

2. A commonly used set of battery constraints ðEÞ

In this section, we present a commonly employed set of mixed-integer battery constraints (E) in which voltage is constant and SoCserves as an unrestricted proxy for the fraction of capacity avail-able for discharge. We use lower-case letters for parameters andupper-case letters for variables. We also use lower-case lettersfor indices and upper-case script letters for sets. Superscripts andaccents distinguish between parameters and variables that utilizethe same base letter, while subscripts identify elements of a set.The units of each parameter and variable are provided in bracketsafter its definition, where applicable.

Sets

t 2 T a single time period within the set of time periods

Parameters

s length of one time period [h] �e manufacturer energy maximum rated-capacity of the

battery [W h]

gþ;g� conversion efficiency of power flow into and out of

the battery, respectively

Variables

Bsoct SoC of the battery in time period t

Pþt ; P

�t

aggregate power into and out of the battery in time

period t, respectively [W]

Bþt ;B

�t

1 if the battery is charging or discharging,

respectively, in time period t, 0 otherwise

Constraints (E)

�eBsoct ¼ �eBsoc

t�1 þ sðgþPþt � P�

t Þ 8t 2 T : t > 1 ð1aÞsP�

t 6 �eBsoct�1 8t 2 T : t > 1 ð1bÞ

sPþt 6 �eð1� Bsoc

t�1Þ 8t 2 T : t > 1 ð1cÞsP�

t 6 �eB�t 8t 2 T ð1dÞ

sPþt 6 �eBþ

t 8t 2 T ð1eÞBsoct 6 1 8t 2 T ð1fÞ

M.S. Scioletti et al. / Applied Energy 176 (2016) 245–257 247

Bþt þ B�

t 6 1 8t 2 T ð1gÞBsoct ; Pþ

t ; P�t P 0 8t 2 T ð1hÞ

Bþt ;B

�t binary 8t 2 T ð1iÞ

Eq. (1a) updates the SoC of the battery based on the previoustime period’s SoC and the energy entering and leaving the battery.We apply gþ to account for energy lost converting AC power to DCpower that applies when storing energy. We include g� when weintegrate (E) with ðPF Þ, because the latter model contains systeminteroperability, specifically, power balance constraints.Constraints (1b) and (1c) restrict energy exiting and entering thebattery, respectively, while constraints (1d) and (1e) relate powerdischarge and charge variables to their respective binaries.Constraint (1f) restricts SoC operating levels, while constraint(1g) prevents simultaneous charge and discharge, but also allowsthe battery to be idle. Constraints (1h) and (1i) account for nonneg-ativity and binary variable restrictions, respectively.

3. A nonlinear set of battery constraints (N )

In this section, we present a set of nonlinear, mixed-integerconstraints (N ) for modeling battery dispatch in a hybrid systemto meet demand. For an algorithmic interpretation of these con-straints, we refer the reader to [32]. Throughout this section, weapply notational conventions similar to those in (E).

Sets

t 2 T a single time period within the set of time periods h 2 H a single iteration within the set of iterations

Parameters

s length of one time period [h] av ; bv voltage slope and intercept coefficients, respectively

[V]

cref manufacturer-specified capacity of the battery [A h] dt power demand in time period t [W] �ı maximum current allowed [A]

iref

reference current of the battery [A]

gþ;g�

conversion efficiency of power flow into and out ofthe battery, respectively

p

Peukert’s constant rint internal resistance of the battery [Ohm] �s maximum allowed SoC

Variables

Bsoct SoC of the battery in time period t

~Bsoct

adjusted SoC of the battery in time period t

~Ct adjusted capacity of the battery in time period t [A h]

Rintt

internal resistance of the battery in time period t[Ohm]

Iþt ; I�t

charge and discharge current of the battery in time

period t, respectively [A]

Pþt ; P

�t

aggregate power into and out of the battery in time

period t, respectively [W]

Vþt ;V

�t

charge and discharge terminal voltage of the battery

in time period t, respectively [V]

Bþt ;B

�t

1 if the battery is charging or discharging,

respectively, in time period t, 0 otherwise

3.1. Nonlinear battery modeling

We model average battery power output as a function of thephysics-based nonlinear, nonconvex relationship between currentand voltage, which, in general form for discharge, is:

P�t ¼ I�t V

�t 8t 2 T ð2aÞ

Here V�t represents the terminal voltage of the battery, which is

not constant throughout discharge. Open circuit voltage is afunction of SoC and is often modeled as a linear relationship[2,26]. Terminal voltage deviates from open circuit voltage, whichis a linear function of SoC, due to the internal resistance (fromOhm’s law) of the battery (positive for charge, negative for dis-charge) [3,9,22,28].

V�t ¼ avBsoc

t�1 þ bv � Rintt I�t 8t 2 T : t > 1 ð2bÞ

A battery may be able to support a high-discharge current, butthe corresponding capacity achieved is less than the capacityachieved at a lower current. This effect can be empiricallydescribed by Peukert’s nonlinear equation, which gives an adjusted

capacity ~Ct based on an applied current I�t [12].

~Ct ¼ crefiref

I�t

!p�1

8t 2 T ð2cÞ

The rate-capacity and adjusted capacity are temporary effects.At the end of a high-discharge current, a battery may appear emptybut can actually be discharged further by reducing the current. Thisrecovery effect occurs primarily in lithium-ion batteries and is aresult of concentration gradients building up in the cell. Fig. 1shows that the final capacity achieved stays constant despitehigh-rate discharges giving lower capacities.

To monitor the capacity available, we introduce adjusted SoC,~Bsoct , which relates adjusted capacity to the battery’s SoC and dis-

charge current (capacity consumed).

~Bsoct ¼

~Ct � cref ð1� Bsoct�1Þ � sI�t

~Ct

8t 2 T : t > 1 ð2dÞ

If the adjusted SoC is greater than or equal to 0, then the dischargecurrent is feasible, which, when we use the right-hand side of (2d)to represent ~Bsoc

t , yields the following condition for feasibility:

~Ct � cref ð1� Bsoct�1Þ � sI�t

~Ct

P 0 8t 2 T : t > 1 ð2eÞ

We use an ampere-hour counting method to increase, at frac-tional rate gþ, or decrease the previous time period’s SoC by thequotient of the current passed during each time period and thecapacity [3,28]. Discharge efficiency g� applies to system interop-erability constraints, so we exclude it here.

Bsoct ¼ Bsoc

t�1 þ s gþIþt � I�tcref

� �8t 2 T : t > 1 ð2fÞ

3.2. A mixed-integer, nonlinear set of battery-only constraints ðN )

We now present (N ), a mixed-integer nonlinear set of battery-only constraints for a hybrid optimization model, which provides amore comprehensive dispatch methodology to solving this prob-lem as it considers the entire time horizon simultaneously.

Constraints (N )

Pþt ¼ ðavBsoc

t�1 þ bv þ Rintt Iþt ÞI

þt 8t 2 T : t > 1 ð3aÞ

P�t ¼ ðavBsoc

t�1 þ bv � Rintt I�t ÞI

�t 8t 2 T : t > 1 ð3bÞ

~Ct ¼ crefiref

I�t

!p�10@

1AB�

t 8t 2 T ð3cÞ

~Ct � cref ð1� Bsoct�1Þ � sI�t

~Ct

!B�t P 0 8t 2 T : t > 1 ð3dÞ

Fig. 1. Discharge data for a lithium-ion battery demonstrating the rate-capacity effect. We present three different discharge currents. In this case, when the batterydischarges at a higher current I�t (118 A and 47 A), it achieves only part of the reference capacity (425 A h and 439 A h, respectively), i.e., adjusted capacity ~Ct , while when itdischarges at a lower current of 24 A, it consumes all the capacity.

248 M.S. Scioletti et al. / Applied Energy 176 (2016) 245–257

Iþt 6 �ıBþt 8t 2 T ð3eÞ

I�t 6 �ıB�t 8t 2 T ð3fÞ

Bsoct 6 1 8t 2 T ð3gÞ

Bsoct ¼ Bsoc

t�1 þ s gþIþt � I�tcref

� �8t 2 T : t > 1 ð3hÞ

Bþt þ B�

t 6 1 8t 2 T ð3iÞBsoct ; ~Ct ; I

þt ; I

�t ; P

þt ; P

�t ;R

intt P 0 8t 2 T ð3jÞ

Bþt ;B

�t binary 8t 2 T ð3kÞ

Eqs. (3a) and (3b) determine the power entering and leaving thebattery as the product of current and voltage, respectively (see (2a)and (2b)). We do not explicitly model voltage in these constraintsas it is a function of current and SoC. Eq. (3c) adjusts the battery’scapacity to account for the discharge current (see (2c)). Constraint(3d) also only applies to discharge (see (2d) and (2e)); if theleft-hand side is greater than or equal to zero, then the dischargecurrent is feasible. Constraints (3e) and (3f) bound charge and dis-charge current, respectively, by relating them to their respectivebinary variables, while (3g) restricts SoC to a maximum value. InEq. (3h), we update the SoC (see (2f)). We include g� when we inte-grate (N ) with ðPF Þ, because the latter model contains systeminteroperability, specifically, power balance constraints. Constraint(3i) prevents the battery from simultaneously charging and dis-charging during a given time period; it also allows the battery tobe idle. Lastly, constraints (3j) and (3k) provide nonnegativityand binary restrictions, respectively.

We do not attempt to solve a hybrid optimization model with(N ), because its nonlinearity and nonconvexity lead to tractabilityissues even if the domain violations in constraints (3c) and (3d) areremoved. Although constraints (3a)–(3d) are nonlinear, these non-linearities are linearizable by employing known linearization tech-niques and empirical approximations, as we show in the nextsection.

4. A physics-based linear set of battery constraints (PB)

In this section, we demonstrate how to approximate the follow-ing operational characteristics present in (N ): (i) voltage thatvaries with current and the previous time period’s SoC, and (ii)capacity that varies exponentially with discharge current.

Throughout this section, we apply similar notational conventionsas we do to (E).

4.1. Voltage

Similar to constraint (2a), we model average battery power out-put as a function of the physics-based nonlinear, nonconvex rela-tionship between current and voltage (see (2b)). Voltage dropsthat occur during charge or discharge result from the internalresistance and concentration gradients that occur during use.Internal resistance is determined by the properties of the elec-trolyte and the distance between the two electrodes, and is oftenportrayed as a simple series resistance following Ohm’s law. Asthe battery charges or discharges, ions build up at the electrodeand electrolyte interfaces, which also causes the voltage to change.Both concentration gradient and internal resistance offsets dependon the applied current and usually remain constant throughout

discharge (Rintt ¼ rint). When the battery discharges, the terminal

voltage drops below the open circuit value. When the battery isrecharged, the terminal voltage is greater than the open circuitvalue. The hourly time fidelity of our application allows us toignore transient features, such as the build-up of concentrationgradients, and to model discharge voltage as a function of open cir-cuit voltage and resistance [9].

We approximate current in (2b) with iavg , which represents atypical current applied by the battery. The voltage offset due toOhm’s Law, then, simplifies to a single constant resistance termrintiavg , which we add to the open circuit voltage for charge and sub-tract for discharge. The error associated with this approximation isdiscussed in detail in Section 5. Fig. 2 displays this relationship.

Substituting voltage relationships into Eq. (2a) and the analo-gous equation for charge to determine power entering and exitingthe battery yields:

Pþt ¼ ðavBsoc

t�1 þ bv þ rintiavgÞIþt 8t 2 T : t > 1 ð4aÞP�t ¼ ðavBsoc

t�1 þ bv � rintiavgÞI�t 8t 2 T : t > 1 ð4bÞ

The bilinear relationships within Eqs. (4a) and (4b) between SoCand current are linearizable [33] using a convex-underestimationtechnique introduced by [4,24] to bound and solve the problem.

Fig. 2. Solid lines represent nonlinear voltage (V�t ) profiles for three discharge rates (24 A, 32 A, 80 A) of a lithium-ion battery plotted as a function of SoC (Bsoc

t�1). Dashed-linesrepresent the linear approximations of voltage embedded in (4b). In some cases, the depicted voltage contains small oscillations, which are normal and unpredictable, but ingeneral the slope is linear. We restrict the SoC range to both improve the accuracy of our linear approximation and to enforce an efficient operating policy.

M.S. Scioletti et al. / Applied Energy 176 (2016) 245–257 249

4.2. Capacity

A battery realizes its maximum capacity when discharges occurat low currents, because concentration gradients and the resultingvoltage shifts are small. At high currents, the capacity realized maybe less than half of that obtained at lower currents. We call thenonlinear relationship between capacity and discharge currentthe rate-capacity effect. Battery capacity as a function of currentis often approximated using an exponential relationship (see Peuk-ert’s equation (3c)). To facilitate tractability of the problem, we usea linear relationship, based on specific rate-capacity data, toapproximate capacity as a function of current in which the refer-ence capacity cref represents the y-intercept and c� representsthe slope of the line (c� 6 0).

~Ct ¼ cref þ c�I�t 8t 2 T ð4cÞı 6 I�t 6 �ı 8t 2 T ð4dÞ

Fig. 3 presents capacity data for a lead-acid battery. We fit Peuk-ert’s nonlinear empirical equation (3c) and our linear approxima-

Fig. 3. Capacity versus current data for a lead-acid battery. We fit Peukert’snonlinear empirical equation (3c) and a linear approximation (4c) employing leastsquares regression to a range of feasible current values given in constraint (4d).

tion (4c) employing least-squares regression to a range offeasible current values (see (4d)) and the range of current depictedin Fig. 3.

To determine a rate-capacity bound, i.e., to restrict current to a

feasible range, we substitute the linear approximation (4c) for ~Ct in(2e) and then solve for I�t , which represents an upper-bound, so wereplace the equality with an inequality:

I�t 6 cref

s� c�

� �Bsoct�1 8t 2 T : t > 1 ð4eÞ

4.3. Formulation

In this section, we present ðPBÞ as a mixed-integer set of con-straints derived in Sections 4.1 and 4.2. We apply similar nota-tional conventions to ðPBÞ as we applied to both ðEÞ and ðN Þ.

Additional parameters

c� battery discharge capacity slope coefficient [h] iavg predicted average current of the battery [A] ı minimum allowed current [A] s minimum allowed SoC

Constraints (PB)

Pþt ¼ ðavBsoc

t�1 þ bv þ rintiavgÞIþt 8t 2 T : t > 1 ð5aÞP�t ¼ ðavBsoc

t�1 þ bv � rintiavgÞI�t 8t 2 T : t > 1 ð5bÞ

I�t 6 cref

s� c�

� �Bsoct�1 8t 2 T : t > 1 ð5cÞ

s 6 Bsoct 6 �s 8t 2 T ð5dÞ

ıBþt 6 Iþt 6 �ıBþ

t 8t 2 T ð5eÞıB�

t 6 I�t 6 �ıB�t 8t 2 T ð5fÞ

Bsoct ¼ Bsoc

t�1 þ s gþIþt � I�tcref

� �8t 2 T : t > 1 ð5gÞ

Bþt þ B�

t 6 1 8t 2 T ð5hÞBsoct ; Iþt ; I

�t ; P

þt ; P

�t P 0 8t 2 T ð5iÞ

Table 1Battery parameters for c� and available capacity as a percentage of capacity in 1 hðsc%100%Þ based on manufacturer data. Rate-capacity effects are present in all sets ofbattery data analyzed.

Manufacturer Chemistry c� Available capacity (%)

K2 18650 Lithium-Ion �0.009 99.1A123 12-Volt Lithium-Ion �0.04 96.1Panasonic NCR 18650 Lithium-Ion �0.08 92.5Panasonic GCR 18650 Lithium-Ion �0.27 78.7Panasonic 18650 Lithium-Ion �0.47 68.0Hardy 12-Volt Lead-Acid �0.62 61.7

250 M.S. Scioletti et al. / Applied Energy 176 (2016) 245–257

Bþt ;B

�t binary 8t 2 T ð5jÞ

Eqs. (5a) and (5b) represent power entering and leaving thebattery, respectively (see (4a) and (4b)). Bilinear terms withinthese constraints are linearizable as in [33]. We quantify the errorassociated with this linearization in Section 5. Constraint (5c)provides rate-capacity bounds for discharge current (see (4e)).Constraint (5d) restricts the SoC of the battery to a range that sup-ports our linear assumptions (see Fig. 2), while constraints (5e) and(5f) do the same for charge and discharge current, respectively (seeconstraint (4d)). Constraints (5g)–(5j) follow identically from (N )(see constraints (3h)–(3k)).

5. Theoretical error analysis

To effectively compare the two sets of battery constraints, wequantify the approximation error associated with linearizing (PB)and compare it to the rate-capacity and constant voltage assump-tion error in solutions to (E). We define the following variables tofacilitate this analysis:

Variables

dEt magnitude of error in time period t per feasible battery

dispatch according to (E) [W]

dPB

t

magnitude of error in time period t per feasible batterydispatch according to (PB) [W]

5.1. Rate-capacity and constant voltage assumption error in ðEÞ

In ðEÞ, the capacity of the battery, as given by the product of itsmaximum energy �e rating and SoC, is 100% available for discharge(see constraint (1b)). Based on our discussion of capacity and cur-rent in Sections 1 and 3, only a fraction of the battery’s capacity isactually available for discharge. Given that ðEÞ assumes a constantvoltage, and given our use of the physics-based definition of poweras a function of voltage and current, any bound on power in ðEÞ alsorestricts current. In (4e), we derive a rate-capacity boundcref Bsoc

t�1=ðs� c�Þ that relates current to SoC and capacity. The differ-ence between this bound and constraint (1b) is the rate-capacitymultiplier 1=ðs� c�Þ, which we label as c% [h�1]. If we divide bothsides of constraint (1b) by s and then replace the multiplier (1/s),which assumes no rate-capacity effects, on the right-hand sidewith c%, rate-capacity bounds on discharge power follow as:

P�t 6 c%�eBsoc

t�1 8t 2 T : t > 1 ð6aÞ

Table 1 provides values of the parameter c� and availablecapacity percentages for one lead-acid battery and five lithium-ion batteries. Even within the same chemistry, batteries canbehave differently based on cell construction and active materials.As c� decreases, the available capacity for discharge decreases. Forexample, for a one-hour time interval, the percentage of capacityavailable from a Hardy lead-acid battery (c� ¼ �0:062) is 61.7%.For a A123 lithium-ion battery (c� ¼ �0:04) this value is 96.1%,demonstrating that lead-acid batteries suffer much more fromthe rate-capacity effect than lithium-ion batteries.

Constraint (6a) provides an upper bound for discharge power in(E); if power discharged exceeds this bound, then there is a rate-capacity violation, which we quantify as:

max 0; P�t � c%�eBsoc

t�1

� �8t 2 T : t > 1 ð6bÞ

Additionally, ðEÞ assumes voltage is constant, which is not accu-rate, because voltage is a function of the battery’s current and SoC(see (2b)). Constant voltage assumption error compares the decrease

in SoC in ðEÞ for a given discharge quantity to that of ðN Þ. Wedetermine the SoC decrease as a function of power discharged inðEÞ by multiplying power by s=�e (see (1a)), yielding sP�

t =�e.To determine the SoC decrease for P�

t in ðN ), we first solve

Eq. (3b) with Rintt ¼ rint for discharge current, subtracting, rather

than adding, the radical in the numerator to remain within allow-able bounds:

I�t ¼ðavBsoc

t�1 þ bvÞ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðavBsoc

t�1 þ bvÞ2 � 4rintP�t

q2rint

8t 2 T ð6cÞ

Then, using I�t from (6c), we determine that the decrease in SoCfor (N ), which is the quotient of current and capacity, is I�t =c

ref (see(3h)); the corresponding decrease in SOC for (E) is given in (1a).The difference between SOC for (E) and (N ) multiplied by energygives the error in watts:

I�tcref

� P�t

�e

� ��e 8t 2 T ð6dÞ

The rate-capacity and constant voltage assumption error in ðEÞ areadditive, but because the constant voltage assumption error mayover- or under-estimate power, error in ðEÞ is not strictly anover-estimation.

dEt ¼ max 0; P�t � c%�eBsoc

t�1

� �þ I�t

cref� P�

t

�e

� ��e 8t 2 T ð6eÞ

The error associated with charge, rather than discharge, followssimilarly.

5.2. Approximation error in ðPBÞ

In Section 3.4 of [33], the authors indicate that the dischargeapproximation error in ðPBÞ stems from the upper bounds imposedby the linearization of current and SoC (Bsoc

t�1I�t ) in constraints (5b);

the upper bounds resulting from this linearization follow:

Bsoct�1I

�t 6 �ıBsoc

t�1 þ sI�t � s�ı 8t 2 T : t > 1 ð7aÞBsoct�1I

�t 6 ıBsoc

t�1 þ �sI�t � �sı 8t 2 T : t > 1 ð7bÞ

Because we use s ¼ 0 and ı ¼ 0 for the lithium-ion batteries inour scenarios, these constraints simplify to the following:

Bsoct�1I

�t 6 �ıBsoc

t�1 8t 2 T : t > 1 ð7cÞBsoct�1I

�t 6 �sI�t 8t 2 T : t > 1 ð7dÞ

Constraint (7c) in the approximation is dominated by (5c) inðPBÞ, because cref =ðs� c�Þ ¼ �ı; therefore, we only use constraint(7d) to determine the error. Moving the left-hand side to theright-hand side of (7d) and factoring out current yields the erroras the product of current and the difference between SoC and itsupper bound ð�s� Bsoc

t�1ÞI�t , which we multiply by av from (5b) to

convert from amperes to watts. Discharge approximation error fol-lows as:

avð�s� Bsoct�1ÞI

�t 8t 2 T : t > 1 ð7eÞ

M.S. Scioletti et al. / Applied Energy 176 (2016) 245–257 251

Additionally, we under-estimate current by approximating it inEq. (5b) with iavg ¼ cref =s

� �, which is the maximum possible

discharge quantity per time period. The resulting iavg error is theproduct of the difference between iavg and current, which, in unitsof power (W), follows from the last term in Eq. (5b):

rintcref

s� I�t

� �I�t 8t 2 T ð7fÞ

Given cref =s� �

is greater than or equal to I�t , we subtract the iavg errorfrom the discharge error. The combined discharge approximationand iavg error in ðPBÞ for discharge 8t 2 T : t > 1 is:

dPB

t ¼ avð�s� Bsoct�1Þ � rint

cref

s� I�t

� �� �I�t ð7gÞ

A similar analysis can be done using the lower bounds oncharge.

Table 2Constant voltage assumption error in dEt for a range of feasible discharge power and SoC valuin (6a) on power and SoC combinations. Negative values indicate an under-estimation ofreflect an over-estimation. Maximum and minimum values are denoted by bold font and

Fig. 4. Rate-capacity error in dEt for all the batteries listed in Table 1 at a 200 kW h capcapacity.

5.3. Comparative error analysis of dEt and dPB

t

Although dEt and dPB

t are indexed on t; t is not necessary for thetheoretical analysis. Without loss of generality, we simply addressdischarge error in this subsection. We compare error as a functionof the maximum feasible discharge quantities, which for ðEÞ isP�t ¼ �eBsoc

t�1=s and for (PB) is I�t ¼ cref Bsoct�1=ðs� c�Þ (see Eqs. (1d)

and (5c), respectively).Fig. 4 depicts rate-capacity error in dEt for each battery listed in

Table 1 with a 200 kW h capacity as a function of SoC. Regardless ofbattery capacity, we find that the greatest errors in solutions to ðEÞoccur when the battery is fully charged (Bsoc

t ¼ �s), but, in general,error increases linearly with SoC. Also, results indicate that as theparameter c� increases, rate-capacity error declines, but as c�

decreases, rate-capacity error increases, i.e., the higher the capacityavailable per Table 1, the less chance of a rate-capacity violation.

Constant voltage assumption error in (E) is a function of dis-charge power P�

t and SoC Bsoct�1 (see (6d)). Table 2 displays dEt for

es for a 200 kW h battery. Feasibility is determined by imposing rate-capacity boundsdischarge power for the combination of P�

t and Bsoct�1, while positive shaded numbers

are underlined. Empty cells depict infeasible combinations of P�t and Bsoc

t�1.

acity when dispatched at maximum feasible quantities. Results are proportional to

Fig. 5. Discharge approximation error for all the batteries listed in Table 1 for a200 kW h capacity as a function of SoC. We assume maximum feasible dischargegiven SoC. Results are proportional to capacity.

Fig. 6. Discharge iavg error for all the batteries listed in Table 1 for a 200 kW hcapacity as a function of SoC. We assume maximum feasible discharge given SoC.Results are proportional to capacity.

252 M.S. Scioletti et al. / Applied Energy 176 (2016) 245–257

the constant voltage assumption error only, which, for the dis-charge case, ranges from over-estimating P�

t by up to 0.64 kWper time period to under-estimating it by 2.48 kW.

Fig. 5 depicts analysis analogous to that in Fig. 4, but for dPB

t ; it

reveals that dPB

t and SoC Bsoct�1 have a quadratic relationship that

obtains a maximum when the battery is half charged. As c�

decreases, dPB

t decreases as well, causing a ‘‘reversal” in orderingof the battery curves relative to that for dEt . Rate-capacity boundsincrease as c� decreases, which enables higher discharge currentsand, with that, increased error.

Similar to discharge approximation error, Fig. 6 presents iavg

error, which is also quadratic and achieves a maximum of 0.6 kWgiven SoC for all batteries listed in Table 1 at 200 kW h capacity.The maximum discharge error, when we subtract iavg error fromthe approximation error, is less than 2 kW.

Fig. 7 represents total discharge error dEt and dPB

t as a function ofSoC for each battery listed in Table 1. This analysis indicates high-rate capability lithium-ion batteries such as K2 and A123 can bemodeled with ðE), because the total error is similar to that ofðPB); however, as the rate capability decreases, as it does withthe remaining batteries listed in Table 1, the potential for largeerrors increases. Total discharge error, which is dominated byrate-capacity error in ðE), ranges from 2 kW to 76 kW comparedto less than 2 kW in ðPB). This wide range in error in the latter

cases implies that ðE) in its current form is only useful for a smallsubset of batteries.

6. Computational study

Optimal steady-state dispatch for a hybrid power systemchallenges modelers to find an acceptable relationship betweenaccuracy and solvability. In this section, we present (PF ) (seeAppendix A), a mixed-integer, linear optimization model that min-imizes fuel consumption subject to a set of constraints thatincludes system interoperability and bounds for PV and generatortechnologies. Because the focus of our study lies in battery perfor-mance, we eliminate system procurement considerations, includ-ing the related aspect of battery lifecycles, from a morecomprehensive model [33]. The combination of (PB) with (PF )

and (E) with (PF ) forms two optimization models, (PBþ) and (Eþ),

respectively, for the hybrid power steady-state dispatch problem.We solve these models and compare their solutions by quanti-

fying the error present in each. We employ the following variables:

Variables

PEþ

t

power discharged by the battery in time period t per adispatch solution to (Eþ) [W]

PPBþ

t

power discharged by the battery in time period t per a

dispatch solution to (PBþ) [W]

dEþ

t

magnitude of error as defined by dEt (see (6e)) in timeperiod t per battery dispatch solution to (Eþ) [W]

dPBþ

t

magnitude of error as defined by dPB

t (see (7g)) in time

period t per battery dispatch solution to (PBþ) [W]

6.1. Model parameters and scenarios

We focus on hybrid systems comprised of diesel generators,batteries (energy storage), and PV (renewable energy). Alltechnologies represent baseline industry products in terms of per-formance and capabilities. We consider three sizes of solar arrays:(i) a 50 kW PV array, which represents roughly half of the maxi-mum demand, (ii) a 100 kW PV array, which represents roughlythe same magnitude as the maximum demand, and (iii) a200 kW PV array, which corresponds to twice the maximumdemand. We consider two sizes of generators (see Table 3) andtwo sizes of Panasonic 18650 batteries, which we assume fullycharged and new, i.e., off the shelf. We do not consider terminalconditions of the battery, because our focus is on battery perfor-mance constraints for a six-month time horizon, which is not longenough to impact battery lifetime. Correspondingly, we don’t con-sider battery lifecycles in our analysis. We choose the lithium-ionchemistry because it is the most applicable to our situation in thatit has long cycle life, the best energy density, i.e., capacity perweight, and has received the most attention. Without loss of gen-erality, we examine the Panasonic 18650 whose characteristicsalso help emphasize the effects of the rate-capacity error. Table 4presents the associated parameters for this battery, which resultfrom empirically derived equations in Section 4, and are notdirectly available from manufacturers or from other empiricalstudies to our knowledge. Our battery constraint sets necessitatecareful tailoring of existing data, which we consider one of the con-tributions of our work.

In [33], the authors present 14 year-long demand profiles athourly-fidelity. We employ the first six months of the San Salvador,El Salvador scenario [14] scaled by 1.15, which, given the cyclicalnature of the data, represents a typical steady-state forecast. Theminimum demand is 19 kW and the maximum demand is 100 kW.

Fig. 7. Total discharge error dPB

t and dEt associated with a maximum feasible discharge current for each battery listed in Table 1 at a 200 kW h capacity as a function of SoC.

Table 4Battery parameter values employed in each scenario, in which rint ; iavg ; av , and bv

derive from relationships defined in Section 4.1, and cref ; c� from linearizations inSection 4.2.

Parameter 100 kW h 200 kW h

rint 0.00587 0.00293

iavg 448 897av 10.76 10.76bv 217.6 217.6

cref 448 897c� 0.4783 0.4783

Table 3Generator parameter values from [33] employed in each scenario.

Parameter 60 kW 100 kW

b f 0.0645 0.0644

c f 0.59 0.95

M.S. Scioletti et al. / Applied Energy 176 (2016) 245–257 253

We model PV power by first determining the AC power outputof a 1 kW mono-crystalline panel using a PVWatts simulation forthe location from which the demand originates [11]. We then takethe product of the PVWatts results and the respective sized PVarray (50 kW, 100 kW, or 200 kW) as optimization model inputto determine the hourly PV power output. Fig. 8 displays the rela-tionship between demand and PVWatts solar radiation output forthe three sizes of PV arrays over a one-week (jT j = 168) horizon.

Table 5 presents our 12 scenarios, in which each scenario repre-sents a different power-rated combination of PV, generator, andlithium-ion battery.

Fig. 8. A week-long (jT j = 168) example from the six-month long (jT j = 4380) demand profile. The three curves associated with PV power represent PVWatts simulationoutput for PV arrays of size: (i) 50 kW, (ii) 100 kW, and (iii) 200 kW.

Table 6Mathematical characteristics, which include number of constraints and variables inðEþÞ and ðPBþ Þ for all scenarios.

Constraints Variables

Linear Continuous Binary

ðEþÞ 87,600 61,326 35,040

ðPBþ Þ 170,819 65,709 13,140

Table 5Each scenario has a diesel generator, PV array, and lithium-ion battery; however,scenarios differ by the power rating of each technology type.

Scenario Generator (kW) Solar array (kW) Battery (kW h)

1 60 50 2002 60 50 1003 60 100 2004 60 100 1005 60 200 2006 60 200 1007 100 50 2008 100 50 1009 100 100 200

10 100 100 10011 100 200 20012 100 200 100

254 M.S. Scioletti et al. / Applied Energy 176 (2016) 245–257

Given the differences in power capabilities between eachscenario, we expect to see a variety of ways in which to dispatchtechnologies to minimize fuel use. Fig. 9 provides three likely dis-patch strategies, which are related to the size of the PV array, for a24-h period. Fig. 9a displays a likely solution if PV is less thandemand (scenarios 1, 2, 7, 8), i.e., PV powers only a fraction ofthe demand. In these scenarios, the limited availability of PV notonly mandates the dispatch of additional technologies to meetdemand, but it implies the generator is necessary to charge the bat-tery. Fig. 9b indicates that the increase in PV availability (scenarios3, 4, 9, 10) allows PV to independently power the load and providesome power to assist the generator in charging the batteries. Fig. 9cdemonstrates a dispatch solution for scenarios with an even

(a) (b)

Fig. 9. Anticipated technology dispatch for the scenarios that consider PV arrays sized a(scenarios 5, 6, 11, 12), respectively. Positive values represent power dispatched from a tepower discharge. The demand is the same for all scenarios.

greater PV availability (scenarios 5, 6, 11, 12) than that exhibitedin Fig. 9a and b.

6.2. Results

We solve ðEþÞ and ðPBþÞ on a Sun Fire x2270 m2 with 24 proces-sors (2.93 GHz each), 48 GB RAM, 1 TB HDD, using GAMS 24.1.3,which employs CPLEX version 12.5.1.0 [1], a commercial state-of-the-art solver that uses the branch-and-bound algorithm coupledwith heuristics to improve the best integer solution and cuts toimprove bounds. Each MIP solves to an optimality gap of threepercent in fewer than two hours. Table 6 reports the size of the dis-patch problem with each set of battery-only constraints.

Fig. 10 presents the maximum and average of dEt and dPB

t byscenario. In all but one scenario (Scenario 3), both the maximum

and average error of dPBþ

t are notably less than that of dEþ

t . In this

solution, there are no rate-capacity violations so dEþ

t equals justthe constant voltage assumption error (see Table 2), which is

(c)

t (a) 50 kW (scenarios 1, 2, 7, 8), (b) 100 kW (scenarios 3, 4, 9, 10), and (c) 200 kWchnology, except batteries, in which positive power represents charge and negative

Fig. 10. The maximum and average errors over the time horizon for solutions to ðEþÞ and ðPBþ Þ. Odd-numbered scenarios (1,3,5,7,9,11) correspond to the 200 kW h battery,while even-numbered scenarios (2,4,6,8,10,12) involve the 100 kW h battery.

M.S. Scioletti et al. / Applied Energy 176 (2016) 245–257 255

comparable in magnitude to discharge approximation error in

ðPBþ Þ (see Fig. 5).If we separate the 12 scenarios into odd and even sets, in which

odd sets correspond to the 200 kW h battery and even sets the100 kW h battery (see Table 5), and examine the results by set,we see a common theme. Rate-capacity errors occur more fre-quently and at higher magnitudes when demand is greater thanthe upper bound on battery discharge. Because the capacity ofthe battery is twice that of the maximum demand (100 kW) forodd scenarios, rate-capacity errors are small. Fig. 10 reports thatthe largest violation within the odd set is 9 kW. On the contrary,within the even set, because battery capacity is similar to the max-imum demand, the likelihood of a rate-capacity violation increases.In five of the six even scenarios, rate-capacity error is nearly 16 kWand the average of this error across all scenarios is twice that of theodd set. Over-estimating performance by 16% in even scenariosallows a 100 kW h battery to operate as a 116 kW h battery with-out penalty. However, the procurement cost differential at $500per kW h between a 100 kW h and 116 kW h battery is $8000.Additionally, a 100 kW generator consumes 2 gallons of fuel in pro-ducing 16 kW in an hour, which, depending on the cost of fuel, maybe significant.

In general, because the objective of our problem is to minimizefuel use, and battery discharge does not consume fuel, an optimalsolution seeks to maximize battery dispatch; therefore, whendemand is greater than the rate-capacity bounds, which occursmost often in the even-numbered scenarios, the likelihood and sizeof rate-capacity violations increases. If the opposite is true, whichis the case for the odd-numbered scenarios, rate-capacity boundsmay not be as restrictive, thereby reducing the likelihood and sizeof rate-capacity violations. Although there are situations in which

choosing to use ðEþÞ versus ðPBþÞ is inconsequential, the scenarioresults indicate that failure to account for rate-capacity effectswhen considering four of the six batteries in Table 1 is nontrivialand it may invalidate the solution.

7. Conclusion

We present a detailed set of battery constraints to account forvariable voltage and rate-capacity effects associated with steady-

state dispatch for a hybrid power optimization problem. We pro-vide a nonlinear physics-based set of constraints (N ) in MINLP for-mat, then derive a tractable, linear approximation (PB) that limitsover-estimation error. A theoretical analysis examining dischargeerror relative to (N ) as a function of SoC indicates that error in(E) increases linearly with SoC and over-estimates performanceby as much as 34% for a given battery type, while error in (PB) isminimal and quadratic. To validate this analysis, we solve 12 sce-

narios and compare the resulting error in solutions to (PBþ) against

that in (Eþ) for a 6-month horizon at hourly fidelity. Results indi-cate that rate-capacity violations are most likely to occur when abattery’s capacity is less than demand, leading to increased overes-timation errors. Although rate-capacity errors in solutions to (Eþ)dominate constant voltage assumption errors in magnitude,assuming a constant voltage presents up to two percent error inmagnitude per discharge. The maximum and average error in solu-

tions to (PBþ) compared to the error in solutions to (Eþ) is signifi-

cantly less for 11 of 12 scenarios and similar in one. In particular,(Eþ) over-estimates discharge by as much as 16% in a number ofscenarios compared to less than one percent across all scenarios

in solutions to (PBþ), but, more importantly, (Eþ) is exposed as a

model that lacks detail, consistency, and has limited application.

Acknowledgements

The authors would like to thank Dr. Mark Spector, Office ofNaval Research (ONR) for full support of this research effort undercontract award #N000141310839. We also would like to acknowl-edge the support of the National Renewable Energy Lab (NREL) forits involvement in this project. We appreciate the meticulous com-ments of our two anonymous referees and of our associate editor.

Appendix A

We present the mathematical formulation of the objective func-tion and system interoperability and technology constraints, whichwe call (PF ), necessary to minimize fuel-use for the steady statedispatch problem (P) presented in [33]. The combination of (PF )with a set of battery constraints forms a hybrid power model

256 M.S. Scioletti et al. / Applied Energy 176 (2016) 245–257

(see Section 6). For each scenario, we fix procurement, which issimilarly modeled in both (E) and (PB), to the technologies andsizes depicted in Table 5, because our objective is to compare dis-patch error between two sets of battery constraints in a hybridpower optimization model. Additionally, we do not model lifetime,because it is outside the scope of this paper, so all such associatedparameters, variables, and constraints are omitted. We apply thesame formulation characteristics to (PF ) as referenced in thepaper.

Sets

t 2 T a single time period within the set of time periods

Parameters

s length of one time period [h] dt steady-state power demand in time period t [W] g� power conversion efficiency of power exiting the

battery

�pb; �pg maximum power rating of the battery and the

generator, respectively [W]

b f; c f

fuel consumption coefficients for generator power

[gal/W h, gal/h]

ct power output of a PV panel in time period t

[W/system]

ks fraction of PV power necessary to meet spinning

reserve requirements

xPV integer number of PV systems [systems]

Variables

Pþt ; P

�t

aggregate power into and out of the battery in time

period t, respectively [W]

P gt aggregate power out of the generator in time period t

[W]

PPVt

aggregate power out of PV in time period t [W]

~Ft amount of fuel used in time period t [gal] Gt 1 if generator is operating in time period t, 0

otherwise

Steady-state dispatch problem (PF)Objective functionMinimizeX

t2T

~Ft ð8aÞ

subject toConstraints

P gt þ g�P�

t � Pþt þ PPV

t ¼ dt 8t 2 T ð8bÞ�pbBsoc

t þ �pgGt � P gt

� �P ksPPV

t 8t 2 T ð8cÞPgt 6 �pgGt t 2 T ð8dÞ

~Ft P sðbf P gt þ c f GtÞ 8t 2 T ð8eÞ

PPVt 6 ctx

PV t 2 T ð8fÞ~Ft ; P

þt ; P

�t ; P

PVt ; P g

t P 0 t 2 T ð8gÞGt binary t 2 T ð8hÞ

A.1. Detailed discussion of formulation

The objective function (8a) minimizes fuel use. Constraint (8b)ensures that the hourly dispatch strategy of generator, battery,and solar technologies meets demand. We apply a parameter g�

to account for conversion associated with battery discharge powerfrom DC to AC. Due to the intermittence of solar power, constraint

(8c) enforces ‘‘spinning reserve,” which ensures that a backuppower source, either batteries and/or generators, is available tomeet a fraction of the load supplied by PV. If a generator is running,constraint (8d) bounds output power to be less than amanufacturer-specified level. Constraint (8e) determines theamount of fuel used during time period t. We limit the PV outputpower to the product of ct and size of the PV array in constraint(8f). Finally, constraints (8g) and (8h) enforce nonnegativity andbinary restrictions, respectively.

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