Hybrid Stand-Alone Power Supply Using PEMFC,PV and Battery — Modelling and Optimization

Jérémy Lagorse, Damien Paire and Abdellatif MiraouiSystem and Transport Laboratory (SeT),

University of Technology of Belfort Montbeliard (UTBM),Rue Thierry Mieg, 90000 BELFORT, FRANCE

jeremy.lagorse@utbm.fr

Abstract—Hybrid Stand-Alone Power Supplies (HSAPS), cou-pling several renewable energy sources and means of storage,are interesting for supplying small communities in remote areas.During the conception, the sizing of the elements in such systems,in terms of power and capacity, is a difficult and importantaspect. This design must ensure the lowest cost of energy andmust depend on the meteorological characteristics of the sitewhere the system is installed as well as the consumption profile.To answer this question, a model of the system and a methodto optimise the energy cost are proposed. Through a case studywith a photovoltaic generator, a battery and a fuel cell, it isshown that an optimal sizing is able to deliver energy with a costaround 0.5EUR/kWh. The model and the results are presentedand discussed in this paper.

Index Terms—Genetic algorithms, Hydrogen, Photovoltaicpower systems, Fuel cells

I. INTRODUCTION

The depletion of fossil energy coupled with the climatechange require to find alternative energy production. Thus,alternative energies know a fast expansion. However, whenthey are used in a centralized network, renewable energiesare more expensive than other sources like nuclear or oil.Nevertheless, in the case of stand-alone systems, it should benoted that renewable energies are already interesting from aneconomic standpoint. In the case of people living in isolatedareas with a difficult access (mountains, islands, deserts, etc..),construction of power lines to connect the grid involves a verysignificant additional cost and a Hybrid Stand-Alone PowerSupply (HSAPS) is often preferred.

When HSAPS uses energy sources like photovoltaic orwind power, it is always associated with energy storagesystems such as batteries in order to ensure load supply whenweather conditions are unpropitious. The HSAPS can alsoinclude an auxiliary generator that allows to supply the load,even when the batteries are empty. However, the sizing ofsuch a system is difficult. It should take into considerationweather characteristics of the installation place and also theconsumption profile. Indeed, how to size each element, interms of power and capacity, to ensure a continuous supplyto the load, while obtaining the lowest cost of energy? Forexample, on a simple system with only battery and solar panel,what should be preferred: a large area of photovoltaic panelswith a small battery bank or the contrary, with the risk to

damage the battery because of deep discharges?To address this question, a model of the system and an

optimization from the economic standpoint are proposed. Theproposed model relies on real weather data and takes intoaccount the PV tilt angle and the losses in power converters.It is also able to estimate the lifetimes of some elements(battery and fuel cell) and to calculate the amount of hydrogenconsumed by the fuel cell. Then, based on this model, a geneticalgorithm is used to find the optimal sizing of the power supplythat leads to the lowest cost of energy. Through the study of apractical case, it is shown that an optimal design of a HSAPSprovides an energy cost around 0.5 EUR/kWh. The results ofthe optimization are also presented.

II. HSAPS PRESENTATION

The system under study consists of a photovoltaic generator,a battery and a proton exchange membrane fuel cell (PEMFC)supplied by a hydrogen tank as shown in Fig. 1 [1] [2].

Fig. 1. System structure

This stand-alone system has to supply around fifteen peoplein a remote area closed to the region of Belfort (France).According to the estimates based on the consumption of anindustrialized nation, this community consumes nearly 4 kWon average which is about 35 MWh per year. In Fig. 2, theconsumption profile is presented, the database used is availableon the website [3]. It should be noted that the load profile isnot the same all the year round and is changing depending onthe season.

135978-1-4244-2544-0/08/$20.00 ©2009 IEEE

Fig. 2. Load profile

III. ENERGETIC MODEL OF HSAPS

To assess the cost of energy delivered by the system,expressed in EUR/kWh, a model must be determined on theenergy point of view. Thus, the battery power is determinedfrom the power of the other elements (see (1)).

PBAT = PPV + PFC − PLOAD (1)

A. Photovoltaic generator

The model of the photovoltaic panel (PV) is based onmeteorological data recorded during one year: direct insolation(Idir) and horizontal diffuse insolation (Idifh

). Then, thepower produced by the PV generator is calculated by takinginto account the incidence angle between the sun and the PVpanel, θ, as shown in equations (2) to (4).

IdirC= Idir · cos θ (2)

IdifC= Idifh

·(

1 + cos θ

2

)(3)

PPV = (IdirC+ IdifC

) ·(

PPVp

In

)(4)

This incidence angle results from the different angles fromthe earth-sun geometry as shown in Fig. 3(a) and Fig. 3(b). Lis the latitude of the place, δ the solar declination (see (5)), Hthe hour angle (see (6)), Σ the inclination angle of the solarcollector, β the sun altitude and ΦS its azimut (see (7) and (8)).This approach is primarily based on explanations available in[4].

δ = 23.45 · sin(

360365

· (n − 81))

(5)

H = 15 · (12 − HS) (6)

sin β = cos L · cos δ · cos H + sinL · sin δ (7)

sin ΦS =cos δ · sin H

cos β(8)

cos θ = cos β · cos(ΦS − ΦC) · sin Σ (9)

+ sin β · cos Σ

(a) Sun-Earth geometry

(b) Sun position according to the azimut ΦS and the altitude β

Fig. 3. Angles from the Sun-Earth geometry

B. Fuel Cell

Regarding FC, it is assumed that, once activated, it operateson its nominal power. So when the FC is activated, itspower and its efficiency are constants [5] [6]. The hydrogenconsumption, EH2 , can be deduced by (10).

EH2 =∫

PFC · dt (10)

The activation of the FC depends on the battery state ofcharge (SOC) based on a hysteresis function as illustratedin Fig. 4. When the battery reaches a low level of charge(SOClow), the FC is activated. When the state of chargebecomes higher (SOChigh), the FC is stopped. The modelalso allows to calculate the number of starts of the FC and itsnumber of operating hours to determine its lifetime. Indeed,a PEM FC can usually undergo about 500 starts or 5000operating hours in the case of stationary applications.

C. Battery and lack of energy

The power required to the battery (PBAT) is deduced from(1). The model of the battery is based on the kinetic modelproposed in [7]. This model, mainly used for lead-acid batter-ies, allows to determine the battery state of charge (SOC) aswell as the maximum power (PBATMAX) which can be deliveredwithout damaging the battery. Through a fluid analogy, the

136

Fig. 4. Starting conditions of the fuel cell

kinetic model can be illustrated in Fig. 5. The battery consistsin two tanks of energy. The tank 1 contains a quantity ofenergy Q1 directly available at the battery output. On thecontrary the quantity Q2 in the tank 2 is not directly available.The maximum size of the tank 1, Q1max

, is expressed in termsof battery capacity through the constant c (11). Conversely, themaximum size of tank 2 is given by (12).

Fig. 5. Kinetic battery model

Q1max= c ·QBAT (11)

Q2max= (1 − c)QBAT (12)

A transfer of energy between these two reservoirs occurswith a debit k. It is important to remark that k and c are twoconstants that depend on the battery characteristics. With asimulation step ΔT , the energy contained in both tanks at thetime i is given by (13) and (14).

Q1i= Q1i−1e

−kΔT (13)

+(Qi−1kc − PBAT)(1 − e−kΔT )

k

+PBATc

(kΔT − 1 + e−kΔT

)k

Q2i= Q2i−1e

−kΔT (14)

+ Qi−1(1 − c)(1 − e−kΔT

)

+PBAT(1 − c)

(kΔT − 1 + e−kΔT

)k

The total amount of energy stored in the battery is given by(15) and the state of charge charge is then deducted by (16).

Qi = Q1i+ Q2i

(15)

SOC =Qi

QBAT(16)

If the tank 1 is deeply discharged, the battery cannot supplypower without being damaged. Thus, the maximum power thatcan deliver the battery is given by (17).

PBATMAX =kQ1i−1e

−kΔT + Qi−1kc(1 − e−kΔT )1 − e−kΔT + c(kΔT − 1 + e−kΔT )

(17)

If the power demand is greater than the maximum power(PBAT > PBATMAX), for example when the battery is discharged,there is a lack of power, Plack (see (18)). Based on this power,the lacking energy, Elack, is inferred, which allows to assess thecapacity of the HSAPS compared to the power consumption.Of course, the optimization will always try to reduce thisenergy lack in order to obtain a configuration able to supplythe load without interruption.

Plack = PLOAD − PFC − PPV − PBATMAX (18)

The battery lifetime is also determined by the simulationmodel. This prediction is based on the evolution in the batterystate of charge. Indeed, the battery lifetime depends mainly onthe number of cycles and on their depths. A characteristic ofthe battery lifetime is given in Fig. 6 and (19) (CF=Cycle tofailure) where ai are fitting constants and DOD is the depthof discharge.

CF = a1 + a2 · ea3 ·DOD + a4ea5 ·DOD (19)

Fig. 6. Battery lifetime versus the depth of discharge

However, the evolution of the battery state of charge consistsin many cycles with different depths (see Fig. 7). That iswhy, the “rainflow” counting method is used [8] [9]. Thismethod, commonly used in material fatigue analysis, dividesthe evolution of the SOC in partial cycles. Then, each partialcycle is associated with a partial damage.

According to the Miner’s rule, the partial damage is definedby the ratio between the number of cycles performed at acertain depth of discharge and the number of cycles thatwould cause the rupture of the battery at the same depth ofdischarge (obtained by (19)). The Miner’s rule also stipulatesthat the damages are additive. Therefore, the rupture occurswhen the sum of the damages reaches the unit [10]. It is thenpossible to predict the battery lifetime based on (20), whereNcy represents the number of partial cycles.

137

Fig. 7. Extract of the SOC evolution

LTBAT =1∑Ncy

i=11

CFi

(20)

D. DC-DC converters

Power losses are due to the converters that link the elementsto the DC bus. These losses are not constants and depend onthe power flowing through the converter. The efficiency curvelooks like the one presented in Fig. 8. From this curve, theconversion losses are taken into account in order to obtain anaccurate modelling of the HSAPS.

Fig. 8. Efficiency of a DC-DC converter versus the input power

E. Simulation diagram

To sum up, the model can be represented by the blockdiagram Fig. 9. It has been implemented in C++ language.This model computes the possible lack of energy, the hydrogenconsumption and the lifetimes of the battery and of the FC.

IV. SIZING OPTIMIZATION

A. Cost function

In the previous section, it has been shown that the pro-posed model is able to determine the elements lifetime (LT ,expressed in years) and so, the number of elements requiredover the system lifetime imposed by the designer. Based onthese simulation results, the cost of one element is expressed asin (21). Nelement represents the number of elements required,Pelement represents the element power (for the battery, it isthe capacity instead), UCelement represents the unit cost (forinstance expressed in EUR/W), Creplacement represents theinstallation cost or the replacement cost of the element and

Fig. 9. Simulation bloc diagram

Cmaintenance is for the annualized maintenance cost (for instanceexpressed in EUR/Wyear).

Celement = DV ·Cmaintenance + Nelement ·Pelement

· (UCelement + Creplacement) (21)

If the total cost is defined as only the sum of the cost ofeach element, the cheapest is the one with the parameters setto zero. However, in this case, the system will not be ableto supply the load. Therefore, a penalty cost has to be added.This penalty cost depends directly on the lack of energy (Elack)and it reflects the adaptation of the power supply to the energyconsumption. Thus, the system total cost is given by (22) andthe penalty cost is expressed in (23).

Ctotal = CPV + CFC + CH2 + CBAT + CPenalty (22)

CPenalty = UCPenalty ·Elack (23)

B. Optimization and result

The purpose of the optimization is to obtain a system thatsupplies correctly the load with the lowest cost. It returnsto find the minimum of the function expressed in (22). Inthe system under study, the following parameters have to bedetermined during the design of the system:

• PV power (PPV)• PV tilt angle (β)• FC power (PFC)• battery capacity (QBAT)• battery state of charge starting the FC (SOClow)• battery state of charge stopping the FC (SOChigh)

Moreover, these parameters can modify the total cost. Forinstance, the variation of SOClow can lead the FC to start moreoften. In this way, the ageing is more rapid and can increasethe energy cost delivered to the user. Thus, to find the optimalcombination of the six variables, a genetic algorithm has beenemployed. The overall structure of the program is shown inFig. 10.

System settings, such as the unit costs and the lifetimes,are set through an user interface which an extract is given

138

Fig. 10. General structure of the program

in Fig. 11. The program was developed in C++ instead ofa more conventional simulation software as Matlab to obtaina better performance on the optimization time. Generally, anoptimization requires less than 2 hours of calculation on acomputer fitted with a processor “Celeron 1.5 GHz”.

Fig. 11. Extract of the user interface

V. RESULTS

In this study, the cost parameters and lifetimes have beenretained from various sources including [11] and [12]. Theyare given in Table I.

Element Cost Unit Lifetime

PV 5 EUR/W 25 yearsFC 5 EUR/W 5000 h or 500 starts

Battery 70 EUR/kWh Deduced by simulationHydrogen 0.14 EUR/kWh -Penalty 70 EUR/kWh -

TABLE IUNIT COSTS AND LIFETIMES OF THE ELEMENTS

The result of the optimization led to a total cost of454114 EUR for a system with a 25 year lifetime. This systemconsists in the following elements:

• a 48.6 kW photovoltaic generator south-oriented with aninclination of 68◦

• a battery bank capacity of 217 kWh replaced 4 times

• a 5 kW fuel cell activated when battery SOC reaches33 % and desactivated when it gets back to 38 %. It willbe replaced 3 times.

• The annual consumption of hydrogen in these conditionsis a 15 m3 tank under a pressure of 200 bar, that is10 kWh/year.

The most expensive device is the photovoltaic generator(243 kEUR), followed by the FC (100.3 kEUR), the battery(75.9 kEUR) and the hydrogen (35 kEUR) (see Fig. 12).

Fig. 12. Distribution of the costs

The energy price is 0.524 EUR/kWh. That price, aboutthree times higher than the price from EDF (the Frenchelectricity company), is acceptable for a stand-alone system.Moreover, it is interesting to note that, by applying the sameprinciple of optimization to a system without FC, the energyprice is doubled. This shows the interest of coupling auxiliarygenerators with photovoltaic-battery systems.

VI. CONCLUSION AND FUTURE WORK

This article has developed an accurate modelling of anHSAPS from an energy point of view. Then, based on theproposed model, a genetic algorithm optimization has shownthat it is possible to obtain a system supplying energy with aminimum cost and already interesting for isolated systems. Inthe proposed study, the HSAPS consists in photovoltaic panels,a battery and a fuel cell. Using the same method, it would bepossible to realize the optimization of more complex systemsby adding, for instance, a wind turbine and an electrolyser toproduce hydrogen directly on the site. It is also important tonote that, even if the use of fuel cells and hydrogen have yet tobe developed, the proposed method can be directly applied toa system such as a diesel-fueled generator. The completion ofthis work should lead to the production of a program directlyusable by HSAPS designers.

APPENDIX

Notations:β : Sun altitudeδ : Solar declinationΔT : Simulation step timeη : EfficiencyΦC : Collector azimuth angleΦS : Sun azimuth angleΣ : Collector inclinationBAT : Subscript for "Battery"

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LOAD : Subscript for "Load"FC : Subscript for "Fuel Cell"Penalty : Subscript for "Penalty"PV : Subscript for "Photovoltaic"Θ : Incident angle between the sun beam and the collectorai : Fitting coefficient for the battery lifetimeC : Costc : Parameter of the batteryCmaintenance : Maintenance cost of an elementCreplacement : Replacement cost of an elementCF : Battery lifetime in cyclesDOD : Battery depth of dischargeLT : Element lifetimeIlack : Lack of energyIC : Total insolation on the collectorIdifh

: Diffuse horizontal insolationIdifC

: Diffuse insolation on the collectorIdirC

: Direct insolation on the collectorIdir : Direct insolationEH2 : Quantity of hydrogen consumedIn : PV nominal insolationH : Hour angleH2 : Subscript for "hydrogen"HS : Solar hourk : Parameter of the batteryL : Latituden : Day of the yearNelement : Number of elementsNcy : Number of partial cyclesP : PowerPlack : Lack of powerPBATMAX : Battery maximum powerPPVc

: PV power peakQ : Total energy stored in the batteryQBAT : Battery capacity

Q1 : Battery available energyQ2 : Battery bound energySOC : Battery state of chargeSOClow : State of charge starting the fuel cellSOChigh : State of charge stopping the fuel cellUCelement : Unit cost of an element

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