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Solar Energy 2005; 79(1): 33-46. (http://www.sciencedirect.com/science/journal/0038092X)
DOI: 10.1016/j.solener.2004.10.004
Design and Control Strategies of PV-Diesel Systems
Using Genetic Algorithms
Rodolfo Dufo-López†, José L. Bernal-Agustín*
Department of Electrical Engineering – University of Zaragoza. Calle María de Luna, 3.
E-50018 Zaragoza (Spain).
Abstract
Hybrid photovoltaic systems (PV-hybrid) use photovoltaic energy combined with other
sources of energy, like wind or Diesel. If these hybrid systems are optimally designed, they
can be more cost effective and reliable than PV-only systems. However, the design of
hybrid systems is complex because of the uncertain renewable energy supplies, load
demands and the non-linear characteristics of some components, so the design problem
cannot be solved easily by classical optimisation methods. When these methods are not
capable of solving the problem satisfactorily, the use of heuristic techniques, such as the
genetic algorithms, can give better results.
The authors have developed the HOGA program (Hybrid Optimisation by Genetic
Algorithms), a program that uses a Genetic Algorithm (GA) to design a PV-Diesel system
(sizing and operation control of a PV-Diesel system). The program has been developed in
C++.
In this paper a PV-Diesel system optimised by HOGA is compared with a stand-alone PV-
only system that has been dimensioned using a classical design method based on the
available energy under worst-case conditions. In both cases the demand and the solar
irradiation are the same. The computational results show the economical advantages of the
PV-hybrid system. HOGA is also compared with a commercial program for optimisation of
hybrid systems.
Furthermore, we show a number of results and conclusions about hybrid systems optimised
by HOGA.
Keywords: Hybrid Photovoltaic Systems, Genetic Algorithms.
1. Introduction
A PV-Diesel system has greater reliability for electricity production than a PV-only system (Diesel
engine production is independent of atmospheric conditions). This fact provides greater flexibility, higher
efficiency and lower costs for the same energy quantity produced (Muselli et al., 1999). Also, PV-Diesel
systems, compared with Diesel-only systems, provide a reduction of the operation costs and air pollutants
emitted to the atmosphere (Wies et al., 2004).
Hybrid energy systems are recognised as a viable alternative to reticulated grid supply or conventional,
fuel-based, remote area power supplies (Wichert, 1997).
2
The design and operation control (Ashari and Nayar, 1999) is not a linear problem due to non-linear
component characteristics with a large number of variables (Seeling-Hochmuth, 1997, 1998). The optimal
design of problems like this cannot be achieved easily using classical optimisation methods. This paper
presents a method of optimisation for PV-Diesel systems using a Genetic Algorithm (GA) (Goldberg,
1989). Genetic Algorithms are an adequate search technique for solving complex problems when other
techniques are not able to obtain an acceptable solution. The PV-hybrid system studied is an AC-only
system (no DC-loads) shown in Fig. 1.
Ich_AC
Iinv_AC
VDC
VAC Id IAC
Iinv_DC
Ibat
Ire
DIESEL
GENERATOR
Battery
Charger
Batteries
SOLAR
Inverter
A.C.
LOAD
Charge
Regulator
Ich_DC
Fig. 1. PV-Diesel AC-only system.
There are some programs that simulate hybrid systems, as HYBRID2 (Green and Manwell, 1995),
developed by the NREL (National Renewable Energy Laboratory, USA) and TRNSYS (Turcotte et al.,
2001), developed by the universities of Wisconsin and Colorado (USA). HYBRID2 simulates hybrid
systems with very high precision calculations, but it does not optimise the system. TRNSYS was initially
developed to simulate thermal systems but it has incorporated PV systems to simulate hybrid systems
such as those proposed here, however it cannot optimise them.
The NREL developed the program HOMER (Hybrid Optimisation Model for Electric Renewables)
(Turcotte et al., 2001), which optimises hybrid systems. This program uses the kinetic battery model
(Manwell and McGowan, 1993). The user must enter the parameters for the optimisation by choosing the
different combinations for PV array power, the battery power and the inverter power. HOMER does not
give the number of panels and their type as a solution, only a PV array power, from ones chosen by the
user. The user must select the type of battery, and no optimization between different types of battery is
made. There are three main dispatch strategies (Barley, 1995) but the SOC (State of Charge) set point
(described below in 3.2) is a user entered value and it is not optimised by the program (although different
cases may be compared by sensibility analysis).
3
Ohsawa et al.(1993) applied an artificial neural network to the operation control of PV-Diesel systems.
Ashari and Nayar (1999) proposed the optimisation of the dispatch strategy, based on Barley (1995), by
means of the Diesel generator stopping and starting set points.
In 1997, Kaiser et al. presented an article describing a new procedure for simultaneous optimisation of
operation control criteria and system design, and an on-line optimisation of operation control (the control
strategy is continually redefined during operation of the PV system), based on the “decision theory”.
In 1997 Seeling-Hochmuth presented an article about the optimisation of PV-hybrid energy systems. The
program described optimises the configuration of the system and the control strategy by means of GA.
The control of the system is coded as a vector whose components are 5 decision variables for every hour
of the year. It is not clear how the optimal vector would be implemented physically in the system, and
how the variation of weather would change the running of the system. Well-defined dispatch strategies
would be easier to implement physically.
The PhD thesis presented in 1998 by Seeling-Hochmuth covers the optimisation of PV-hybrid energy
systems. The hybrid control algorithm is very simple, where the SOC set point is the only parameter
considered. There is no detailed description of the GA, with the results being compared with those of a
simulation program (HYBRID2), for this reason this work can be considered to be in the area of
simulations and not in optimisation of hybrid systems.
El-Hefnawi in 1998 presented a method to design PV-Diesel systems. The optimization procedure starts
by the definition of the model of the Diesel generator, and then optimising the PV and battery sizes,
determining the minimum number of storage days and the minimum PV array area.
HOGA, the program described in this article, optimises the hybrid PV-Diesel system using Genetic
Algorithms. The program calculates the optimal configuration of the system. This optimal configuration
is described very precisely: the number of PV panels and the type of PV panels, the number of batteries
and the type of battery, the inverter power, the Diesel generator power, the optimal control strategy of the
system with its parameters, the Total Net Present Value† of the system and the different relative costs
such as the fuel cost, and finally, the number of running hours for the Diesel generator per year. The
program also optimises the dispatch strategy, as does HOMER, but it also optimises the SOC set point,
that is an important variable.
2. Hybrid system mathematical model
The PV-Diesel system will be studied using an hourly time step (∆t = 1h), during one year. Every hour
the following input data must be estimated: the current from the PV generator (Ire), which depends on the
solar irradiation, the AC load current (IAC), which depends on the predicted load, and the battery State Of
Charge (SOC). With this data it is possible to calculate the currents that circulate in the hybrid system for
each hour.
2.1. Current from the PV generator
† Cost of the investments plus the discounted present values of all future costs.
4
The incident radiation in a typical year must be known. The data entered in the program must be either
the average clearness index or the daily radiation on the horizontal surface or the peak sun hours, over
each month of the year. The latitude of the location and the slope, azimuth and albedo of PV panels are
also needed. The PV panel tracking system can be: No Tracking, Horizontal Axis, Vertical Axis or Two
Axis.
Firstly, if the data entered is the radiation on the horizontal surface or the peak sun hours, the program
converts it into the average clearness index for each month of the year using the Rietveld equation
(Rietveld, 1978).
Secondly, the program obtains the clearness index for each day of the year and calculates the global
hourly irradiation G (kWh/m2) according to the Graham model (Graham and Hollands, 1990). We have
considered the Graham method as suitable because it takes into account the uncertainty associated with
the available irradiation data.
The current supplied by the panels, during the hour i, is calculated by the following equation:
piire IGI ⋅= (1)
where Ip is the peak current of the PV generator. The PV generator is connected to the batteries via a
charge regulator, no DC/DC converter has been considered. The battery is what actually maintains the
fixed voltage on the DC side. The voltage on the PV generator would in fact be obtained depending on
the point of intersection of the Thevenin equivalent load line seen by it (batteries and cables) and the I =
f(V) curve of the generator. However, the current supplied by the generator is approximately Imax (for
irradiance of 1 kW/m2) in a high range of voltages (I = f(V) curve of the generator). The parameter used in
the calculations is Imax and not the generator power.
2.2. Load profiles
Five different load profiles have been considered, Low Load (23W·24h = 552 Wh in one day), Domestic
Load (3450 Wh in one day), Farm Load (40.9 kWh in one day), High Load (120 kWh in one day) and
High Continuous Load (6.9 kW·24h = 165.6 kWh in one day). The hourly distribution is shown in Fig. 2
and 3.
0
50
100
150
200
250
300
350
1 3 5 7 9 11 13 15 17 19 21 23
Low load
Domestic load
hour
load (W)
Fig. 2. Daily load profiles used in optimisation.
5
0
2000
4000
6000
8000
10000
12000
1 3 5 7 9 11 13 15 17 19 21 23
High cont. load
High load
Farm
load (W)
hour
Fig. 3. Daily load profiles used in optimisation.
2.3. Battery State Of Charge
The maximum current that the battery can provide in one time step, Ibat,max, depends on its State of
Charge, SOC (Schuhmacher, 1993):
( ) ( )( ) ( )( )
∆
−⋅−+−⋅
∆=∆+
t
cSOCtSOCtSOCSOC
t
c,I,ttI
1min0max minmaxmaxmaxbat,
(2)
where SOCmin = Nbat_p CN ·(1 - DODmax) is the minimum SOC and SOCmax = Nbat_p CN is the maximum
SOC of the batteries bank, c is a binary variable where c = 1 means the battery is charging, and c = 0
means the battery is discharging and Imax is the maximum charge current.
CN is the nominal capacity of one battery (Ah), Nbat_p is the number of batteries in parallel and DODmax is
the maximum depth of discharge of the batteries.
The SOC for the next step can be calculated as follows:
( ) ( ) ( ) ( )( )tIηttIδtSOCttSOC batbat)1( ⋅∆⋅+−⋅=∆+ (3)
where δ and η are the self-discharge coefficient and the efficiency of the batteries, and Ibat is the battery’s
current in the previous step. We have considered the battery and the DC/AC converter efficiencies to be
constant. We have not considered the possibility of using more complex models for the battery given that
our objective is to design a hybrid system and not to simulate, in detail, the working of the said system.
2.4. System currents calculation
Every hour the net load in DC will be calculated as:
re
invDC
ACACnet_DC I
ηV
VII −
⋅⋅= (4)
where VDC and VAC are the DC and AC voltages and ηinv is the inverter efficiency.
Depending on the Inet_DC value, the following cases apply:
a) If Inet_DC ≤ 0: The remaining current will be used to charge the batteries:
)( net_DCmaxbat,bat I,ImaxI = (5)
6
b) If Inet_DC > 0: The remaining current will be given by the Diesel generator or by the battery or by
both of them, depending on the dispatch strategy (see section 3):
b1) If the batteries are able to give Inet_DC and the strategy allows: Batteries discharging, Diesel
generator Off
net_DCbat II = (6)
b2) If the batteries are not able to supply such a current, or the strategy does not allow so: Diesel
generator On, Batteries will neither be charged nor discharged
inv
AC
DCreACd η
V
VIII −= (7)
b3) If the strategy requires the Diesel generator to run at full power (provided there are no energy
losses), the batteries will be charged with the remaining current:
+−=
DC
chAC
AC
invDCreAC
AC
Ngenmaxbat,bat min
V
η·V·
V
η·VII
V
P,II (8)
−+=
AC
invDCre
chAC
DCbatAC
AC
Ngend max
V
η·VI
η·V
VII,
V
PI (9)
where PNgen is the Diesel generator rated capacity and ηch is the battery charger efficiency.
b4) If the net load exceeds the Diesel generator rated capacity, the Diesel generator will run at full
power and the batteries will attempt to make up the difference:
AC
Ngend
V
PI = (10)
−−=
invDC
AC
AC
invDCre
AC
NgenACmaxbat,bat min
η·V
V·
V
η·VI
V
PI,II (11)
2.5. Objective function
The objective function to be minimized includes the following costs:
• Costs of the acquisition of the PV panels, the batteries, the inverter, the charge regulator and the
Diesel generator.
• Costs of replacing the battery charger throughout the life of the system (it does not depend on the
strategy because we assume it has fixed initial cost and life).
• Costs of maintenance of the PV panels and the batteries (they do not depend on the strategy).
• Costs of replacing the batteries, the inverter, the charge regulator and the Diesel generator
throughout the life of the system.
• Costs of operation and maintenance of the Diesel generator throughout the life of the system.
• Cost of the fuel consumed throughout the life of the system.
In the following sections the calculation of these costs and the objective function used by the developed
program are described.
We assume that the system life is the life of the PV panels that are the elements that have a higher
lifetime.
7
3. Dispatch strategies.
The dispatch strategies used in HOGA are based on the strategies described by Barley in 1995 and used
by the HOMER program.
3.1. Load Following Strategy.
If the batteries cannot meet the net load the Diesel generator runs at a rate that produces only enough
power to meet the net load. The batteries will be charged whenever the renewable power exceeds the
primary load, but they will not be charged by the Diesel generator.
There is a “Frugal” option that can be applied in all the strategies. The Critical Discharge Load (Ld) is the
net load above which the marginal cost of generating energy with the Diesel generator is less than the cost
of drawing energy out of the batteries. If the Frugal option is applied, then the Diesel generator meets the
net load whenever the net load is above the critical discharge load, regardless of whether or not the
battery bank is capable of meeting the net load.
The cost of generating energy with the Diesel generator and the cost of drawing energy out of the
batteries are equal when the net load is Ld:
inv
dtcycling_badfuelrep_gen_hMgen&OfuelNgen
η
L·CL··PrACC·PrP·B =+++ (12)
then, Ld can be calculated as follows:
fuelinvtcycling_ba
rep_gen_hMgen&OfuelNgeninvd
)(
·PrA·ηC
CC·PrP·B·ηL
−
++= (13)
where:
CO&Mgen is the Diesel generator’s hourly operation and maintenance cost (€/h)
Prfuel is the fuel price (€/l)
A = 0,246 l/kWh and B = 0,08415 l/kWh are the fuel curve coefficients (Skarstein and Ullen, 1989). The
fuel cost of 1h Diesel running, Cfuel (€) is:
( )genNgenfuelfuel PAPBPrC ⋅+⋅⋅= (14)
Pgen is the Diesel generator output power in this hour (kW).
Crep_gen_h (€/h) is the Diesel hourly replacement cost:
gen
genrep_gen_h
Life
CC = (15)
Cgen is the Diesel generator acquisition cost plus O&M cost throughout Diesel generator lifetime (€) and
Lifegen is the Diesel generator lifetime (h)
Ccycling_bat (€/kWh) is the cost of cycling energy through the batteries:
1000cycles_eqDCbat_pN
battcycling_ba
/N·U·N·C
CC = (16)
Cbat is the batteries bank acquisition cost plus O&M cost throughout batteries lifetime (€), CN is the
nominal capacity of one battery (Ah), Nbat_p is the number of batteries in parallel, and Ncycles_eq is the
number of full cycles of battery life. We have assumed that the batteries can cycle a certain amount of
8
energy, which divided by its nominal capacity, gives the equivalent cycles (full cycles). It is true that the
energy that a battery can cycle depends on the depth of discharge, but is almost constant if the discharge
is never allowed to fall below SOCmin, this being greater than 20%.
3.2. Cycle Charging strategy.
If the batteries cannot meet the net load, the Diesel generator runs at full power (or at a rate not exceeding
the maximum energy that batteries are capable of absorbing) and charges the batteries with any surplus
power. If a SOC set point is applied, the Diesel generator will continue running until the batteries reach
this SOC set point.
The Frugal option also can be applied in this strategy.
3.3. Combined strategy.
This strategy combines both strategies. If the net load is lower than the Critical Charge Load, Lc (kW), the
Cycle Charging strategy is applied. If the net load is higher than Lc, the Load Following strategy is
applied.
The Critical Charge Load is the net load where the cost of generating this load with the Diesel generator
(exactly this load and no more) for 1 hour is the same as the cost of supplying this load, for 1 hour, with
the batteries that have been previously charged by the Diesel generator. Mathematically this is:
inv
ctcycling_ba
invbatch
cfuelcfuelrep_gen_hMgen&OfuelNgen
η
L·C
η·η·η
L··PrAL··PrACC·PrP·B +=+++ (17)
where Lc is:
fuelinvbatchtcycling_babatch
rep_gen_hMgen&OfuelNgeninvbatchc
)1(
)(··
·PrA·η·η·ηC·η·η
CC·PrP·B·ηηηL
−+
++= (18)
where ηbat is the battery efficiency in the charging process.
The Frugal option also can be applied in this strategy.
4. Developed Algorithm
The problem to solve has a great number of possible solutions (combinations of solar generator, batteries,
Diesel generator and strategy variables), for this reason it is difficult to solve this problem with classical
mathematical techniques (for example with mixed-integer programming).
The Genetic Algorithms technique works with individuals (possible solutions). An individual can be
represented by a vector whose components represent the parameters of the system using an integer code.
The GA developed in HOGA is divided in two parts: main and secondary algorithm.
4.1. Main Algorithm
The main algorithm works with an integer vector with the number of PV panels in parallel (a), the solar
generator type code (PV panel) (b), the battery type code (c), the number of batteries in parallel (d) and
the Diesel generator type code (e): | a | b | c | d | e |
9
Each solar generator is from a different manufacturer and their characteristics are: power, voltage, Imax
and acquisition cost.
Each battery is from a different manufacturer and their characteristics are: rated capacity, voltage,
acquisition cost, DODmax, number of equivalent cycles and efficiency.
Each Diesel generator is from a different manufacturer and their characteristics are: power, voltage,
acquisition cost, lifespan, minimum output power, and O&M hourly cost.
The algorithm simultaneously uses Nm vectors such as the one described beforehand.
The main algorithm obtains the optimal configuration of PV panels, batteries and Diesel generator,
minimizing the Total Net Present Cost of the system (CTOT), which includes all the costs throughout the
useful lifetime of the system, which are translated to the initial moment of the investment using the
effective interest rate, according to standard economical procedures.
M_B&OM_PV&OREP_BCHACQ_GEN
ACQ_BCHACQ_BACQ_PVSECTOT
CCCC
CCCCC
++++
++++= (19)
where,
CSEC includes the costs that depend on the optimal strategy. It is evaluated in the secondary algorithm,
explained in 4.2.
CACQ_PV, CACQ_B, CACQ_BCH, CACQ_GEN are the costs of the acquisition of the PV panels, the batteries, the
battery charger and the Diesel generator
CREP_BCH is the cost of replacing the battery charger throughout the life of the system (it does not depend
on the strategy because we assume it has fixed initial cost and life)
CO&M_PV, CO&M_B are, respectively, the costs of maintenance of the PV panels and the batteries (they do
not depend on the strategy).
CTOT must be calculated for each combination, represented by one of the Nm vectors which constitute the
population.
The fitness function of the combination i of the main algorithm is assigned according to its rank in the
population (rank 1 for the best individual considering the objective function, and rank Nm for the worst
solution):
∑ −+
−+=
j
jN
iNi
fitness]1)[(
1)(
m
mMAIN
j = 1……..Nm (20)
4.2. Secondary Algorithm
The secondary algorithm works with a Boolean vector with the dispatch strategies (Cycle charging or
Combined), the “Frugal” option, and 5 bits that represent the SOC set point in Gray code (better than
binary code for GA): | Strategy | Frugal | g0 | g1 | g2 | g3 | g4 |
The Load Following strategy is evaluated at the end, as this strategy has no SOC set point.
The algorithm use Nsec vectors such as the one previously described.
For each vector of the main algorithm, the optimal strategy is obtained (minimizing the non-initial costs,
including operation and maintenance costs, CSEC) by means of the secondary algorithm.
10
FUELM_GEN&OREP_GENREP_REGREP_INV
REP_BACQ_REGACQ_INVSEC
CCCCC
CCCC
+++++
+++= (21)
where,
CACQ_INV, CACQ_REG are the acquisition costs of the inverter and the charge regulator respectively (the
inverter maximum power and the charge regulator current depend on the strategy, so their cost must be
here)
CREP_B, CREP_INV, CREP_REG, CREP_GEN are the costs of replacing the batteries, the inverter, the charge
regulator and the Diesel generator throughout the life of the system.
CO&M_GEN is the cost of operation and maintenance of the Diesel generator throughout the life of the
system.
CFUEL is the cost of the fuel consumed throughout the life of the system.
We assume that the system life is the life of the PV panels which are the elements that have a greater
lifetime.
The fitness function of the combination i of the secondary algorithm is:
∑ −+
−+=
j
jN
iNi
fitness]1)[(
1)(
sec
secSEC
j = 1……..Nsec (22)
4.3. Implementation of the GA developed (HOGA)
HOGA has been implemented in the following way:
1. Initially, Nm vectors are obtained randomly from the main algorithm. These vectors have been
described in 4.1, each one representing a possible configuration of PV panels, batteries and
Diesel generator.
2. For each vector Nm of the main algorithm, the secondary algorithm is executed, obtaining the
optimal dispatch strategy for each Nm vector:
2.1. Nsec vectors are obtained randomly from the secondary algorithm. These vectors have been
described in 4.2, each one representing a possible dispatch strategy.
2.2. The Nsec vectors are evaluated by means of their aptitude (equation 22).
2.3. The best vectors (fittest) have a greater probability of reproducing themselves, crossing
with other vectors. In each cross of two vectors, two new vectors are obtained
(descendents). The descendents are evaluated and the best of them replace the worst
individuals of the previous generation (iteration).
2.4. To find the optimal solution and not to stay in local minimal, some solutions randomly
change some of their components (mutation). The mutations can effect the change of the
control strategy or the change of a bit of SOC set point
2.5. The individuals (vectors) obtained from reproduction and mutation are evaluated, making
the next generation.
2.6. The process continues (from 2.2 to 2.5) until a determined number of generations
(Ngen_sec_max) have been evaluated.
11
3. Nm solutions will have been obtained (vectors of the main algorithm with their optimal dispatch
strategies). The Nm solutions are evaluated by means of their aptitude (equation 20).
4. Reproduction, crossing and mutation are carried out on the obtained solutions, making the next
generation.
5. The process continues (from 2 to 4) until a determined number of generations (Ngen_main_max) have
been evaluated. The best solution obtained is that which has the lowest value of CTOT.
The flow diagram of the algorithm is represented in Fig. 4.
Fig. 4. Flowchart of HOGA.
YES
Evaluation of the control strategy for
each of the Nm vectors of the main
algorithm. The secondary algorithm is executed Nm times.
i =1…. Nm
Calculation of CTOT for
each of the Nm vectors and
evaluation of the Nm
solutions obtained (Equation 20).
Random generation of Nm
vectors from the main algorithm
Ngen_main = 1
¿Ngen_main < Ngen_main_max?
NO
Calculate CTOT for the Nm vectors. The best solution is
the lowest value of CTOT.
END
Reproduction, crossing and
mutation of the main
algorithm vectors.
Ngen_main = Ngen_main+1
Evaluation of the NSEC vectors of the
secondary algorithm (Equation 22)
Reproduction, crossing and mutation
of the secondary algorithm vectors.
Ngen_sec = Ngen_sec+1
¿Ngen_sec < Ngen_sec_max?
NO
Calculate Csec for the Nsec vectors. The best solution is that
represented by the vector with lowest value of NSEC.
i = i+1
¿i < Nm?
NO
YES
Random generation of NSEC vectors of the
secondary algorithm from the vector i of the
main algorithm.
Ngen_sec = 1
i = 1
12
The crossing between two individuals can be done by different methods. The “One crossing point”
method has been used in the HOGA program.
The crossing rate (0 < CR < 1) is a parameter that defines how many descendants will be produced in
each generation. If the size of the population is N, the number of descendants will be N·CR.
The mutation rate (0 < MR <1) is a parameter that defines how many individuals will be mutated in each
generation. If N is the size of the population, and NP is the number of parameters that each individual has,
the total number of mutations in each generation will be N·NP·MR.
The size of the population (number of individuals), the number of generations, and the crossing and
mutation rates for both algorithms, are parameters that can be modified by the user of the program.
5. Computational results
By using the developed program (HOGA), a system located in Zaragoza (Spain) has been designed and
optimised. Five different load profiles have been considered (Fig. 2 and 3), as mentioned in 2.2. The daily
load profiles are represented by a sequence of powers, each considered as constant over a time-step of 1
hour. The parameters used in this case are the following (they are entered by the user):
The crossover rate is 0.7. The mutation rate is 0.01. Number of possible different PV panel types is 9.
Maximum number of panels is 25. Number of different batteries types is 12. Maximum number of
batteries is 10. Number of possible Diesel generator types is 11 (commercial Diesel generators from 3 kW
to 13 kW, with prices according to Schmid and Hoffman, 2004: 0.55 €/W). The costs of the different PV
panels and batteries are shown in Tables 1 and 2. The irradiation for Zaragoza is shown in Table 3. The
PV panel lifetime is 25 years. The batteries cycle life is 500 full cycles, and their SOCmin is 40%. The
Diesel generator lifetime is 7000 h, and its minimum Diesel generator output power is 30%.
Table 1
Investment costs of 12V PV panels
Peak Power (Wp) 20 36 50 55 75 90 100 110 125
Cost (€) 278 297 385 413 525 676 744 812 884
Table 2
Investment costs of 12V batteries
Nominal
Capacity
(Ah)
43 64 69 96 144 160 187 200 308 385 462 524
Cost (€) 155 202 207 258 288 357 433 565 843 971 1017 1054
Table 3
Average daily irradiation
Month Jan. Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec.
Wh/m2
2108 2688 4150 4931 6318 6941 6644 5593 4830 3456 2555 2138
The effective interest rate considered is 2%. The fuel price is 0.8 € /l. The O&M cost of battery bank is 50
€/year, and the O&M cost of the PV array is 40 €/year. The O&M cost of the Diesel generator is 0.2 €/h.
The efficiencies are 80% for the batteries, 90% for the inverter and 90% for the battery charger.
13
Number of total possible combinations for the vector used by the Main Algorithm is: 9·25·12·10·11 =
297000. Number of total possible combinations for the vector used by the Secondary Algorithm is: 27 =
128.
The number of combinations of the components (PV panels, batteries, Diesel generators) and strategy
variables is almost 38 million (297000·128). If we had to evaluate all of the possible combinations, it
would take approximately 15 hours for the calculation times. By means of the GA, a good solution can be
obtained examining only a tiny part of the 38 million possible combinations. The number of combinations
examined is obtained by the product of the Main Algorithm Generations, the Main Algorithm Population,
the Secondary Algorithm Generations and the Secondary Algorithm Population.
In Fig. 5 the Total Net Present Cost of the best combination found versus the number of combinations
examined is shown. The Farm load profile has been used. The computer is a Pentium III 1.1 GHz, 256
MB RAM, with Windows 98 operating system. The program has been implemented in C++. About 700
evaluations can be performed per second.
162000
164000
166000
168000
170000
172000
174000
176000
178000
0 50000 100000 150000 200000 250000
Examined combinations
To
tal
Net
Pre
sen
t C
ost
(€)
Fig. 5. Total Net Present Cost of the best combination found versus the number of combinations examined.
The results for HOGA in Table 4 have been obtained with the following values: Main Algorithm
Generations are 50. Main Algorithm Population is 20. Secondary Algorithm Generations are 25.
Secondary Algorithm Population is 10. So the number of combinations examined is: 50·20·25·10 =
250000. The optimisation has taken 350 seconds, compared with 15 hours without the use of GA. The
results are shown further on.
5.1. Comparison between different methods
Five methods have been applied:
1) The HOGA program, explained above, and developed by the authors, for a PV-Diesel system
with 0% Unmet Load allowed, where % Unmet Load is defined as:
100r)Load(kWh/y electrical Annual Total
r)Load(kWh/y Unmet TotalLoadUnmet% ×=
14
2) The HOGA program, for a Diesel-only system with 0% non Unmet Load allowed.
3) The HOGA program, for a PV-only system with 0% Unmet Load allowed.
4) The method used by HOMER program for a PV-Diesel system with 0% Unmet Load allowed.
5) The method based on the available energy under worst-case conditions of the year (4 days of
battery range), for a PV-only system.
The output design results, using the Farm load profile, are shown in Table 4. The best configuration found
by the GA program developed by the authors, the best found by HOMER, and the configuration for the
Worst Case (PV-only) are shown. The Total Net Present Cost of the PV-Diesel system is lower than the
PV-only system or the Diesel-only system.
Table 4
Comparison between different methods (Farm Load, VDC = 48 V) HOGA
(PV-Diesel)
HOGA
(Diesel-only)
HOGA
(PV-only)
HOMER
(PV-Diesel)
Worst case
(PV-only)
Peak power of PV
panels (kW)
5.6
(14x4 panels of
100 W,12V)
- 17.6
(44x4 panels of
100 W,12V)
6 14.52
(66x4 panels of 55
W, 12V)
Nominal capacity
of Batteries (kWh)
13.8
(2x4 batteries of
144Ah, 12V)
13.8
(2x4 batteries of
144 Ah, 12V)
62.2
(9x4 batteries of
144 Ah, 12V)
13.8
(2x1 batteries of
144 Ah, 48V)
345.6
(50x4 batteries of
144 Ah, 12V)
Rated capacity of Diesel generator
(kW)
3 4 - 4 -
Charge Regulator Current (A)
107 63 339 - 301
Dispatch Strategy Cycle Charging.
Frugal No. SOC
set point 94%
Combined.
Frugal Yes. SOC
set point 78%
- Cycle Charging.
Frugal No. SOC
set point 90%
-
Ld (W) 2,768 3,391 - - -
Lc (W) 1,186 1,452 - - -
Unmet Load (%) 0 % 0 % 0 % 0 % 4 %
Annual Battery
Throughput Energy (kWh/yr)
3,121 4,737 7,680 1,796 7,286
Batteries
replacement cycle (yr)
2.21 1.45 4.05 3.85 23.7
Annual Energy
delivered by PV generator
(kWh/yr)
5,689 - 14,927 8,067 14,330
Annual Overall
load Energy (kWh/yr)
14,927 14,927 14,927 14,927 14,927
Solar Fraction (%) 38 % - 100 % 54 % -
Annual Excess
energy (kWh/yr) 6.3 0 1665 985 -
Annual Hours of
Diesel operation 3,469 4,374 - 5,047 -
Annual Fuel Cost
(€/yr) 2,638 4,481 - 4,040 -
Diesel gen.
replacement cycle
(yr)
2.65 1.6 - 1.37 -
Annual O&M cost (€/yr)
784 965 90 1034 90
Total Net Present
Cost of the system (€)
162,388 179,938 186,934 168,239 173,052
15
The best configuration found by HOGA is similar to the one found by HOMER, keeping in mind the
differences between both programs (different battery model and small differences in economic
calculations: in HOMER the PV generators considered are not made up of individual panels but taken as a
one PV generator; HOMER does not consider the battery charger; HOGA considers that at the end of the
useful life of the installation the components that have not reached the end of their useful life still have an
economic value, where as in HOMER this residual value for each component is data introduced by the
user)
The optimal dispatch strategy obtained by HOGA for this profile is “Cycle Charging” without the
“Frugal” option and 94% SOC set point.
Fig. 6 shows the evolution of the best Total Net Present Cost found with HOGA as a function of the main
algorithm generations in an optimisation, where the number of generations in the main algorithm is 100.
150000
160000
170000
180000
190000
200000
210000
220000
0 20 40 60 80 100
Main Algorithm Generations
To
tal N
et
Pre
sen
t C
ost
(€)
Fig. 6. Evolution of the Total Net Present Cost in an optimisation where Main Algorithm Generations = 100.
5.2. Influence of some parameters of the system.
Some cases have been studied and optimised by HOGA. The influence of the most important parameters
of the system is shown below.
5.2.1. Influence of the minimum Diesel generator output power
The Diesel generator manufacturers usually recommend that their generators should not be run below a
certain load, expressed as a percentage of its rated capacity.
The optimisation of the hybrid system, using the Farm load profile, shows that the total net present value
depends on the minimum Diesel output power allowed. For example, if the manufacturer of the Diesel
generator used in the system optimised in Table 5 recommends not to run under 15% of its rated capacity,
if run at this capacity, the total cost of the system throughout its life will be higher than if we had built the
system preventing the generator running under 30% of its rated capacity. So the optimal minimum
generator output power allowed can be higher than the one given by the manufacturer. In future
developments we will add this as a parameter for the control strategy.
16
Table 5
Influence of the minimum Diesel generator output power (Farm Load, VDC = 48V. Panels and batteries 12V) PV PANNELS BATTERIES DIESEL
Minimum power
(% of rated) Number in
parallel Power (W)
Number in parallel
Cn (Ah)
Rated Power
(kW)
Unmet Load
(%)
Dispatch Strategy
Total net
present
cost (€)
15 14 4x110 1 4x144 4 0 Combined Frugal No
SOC set point 53%
181841
30 14 4x100 2 4x144 4 0 Cycle Charging Frugal No
SOC set point 94% 162388
45 15 4x75 5 4x144 4 0 Cycle Charging.
Frugal Yes
SOC set point 70%
168617
60 24 4x55 5 4x144 3 0 Cycle Charging Frugal No
SOC set point 75% 174671
Table 6
Influence of the minimum batteries SOC (Farm Load, VDC = 48V. Panels and batteries 12V) PV PANNELS BATTERIES
Minimum SOC (% of Cn)
Number in
parallel Power (W)
Number in
parallel Cn (Ah)
DIESEL Rated
Power
(kW)
Unmet Load
(%)
Dispatch Strategy
Total net
present
cost (€)
20 13 4x110 5 4x144 4 0 Combined
Frugal Yes
SOC set point 48%
172925
40 14 4x100 2 4x144 4 0 Cycle Charging Frugal No
SOC set point 94% 162388
60 13 4x110 2 4x144 5 0 Cycle Charging.
Frugal No SOC set point 85%
183922
80 13 4x110 1 4x200 4 0 Cycle Charging Frugal No
SOC set point 80% 173573
5.2.2. Influence of the minimum SOC of the batteries
The battery manufacturers usually recommend that the SOC should not fall under a certain percentage of
their Capacity Cn (A·h).
The optimisation of the hybrid system shows that the total net present cost depends on the minimum SOC
allowed. For example, if the manufacturer of the batteries used in the system optimised in Table 6
recommends that they should not be run lower than 20% of Cn, it would be better to build the system to
prevent the batteries from going lower than 40%. The optimal minimum SOC allowed can be a higher
than the one given by the manufacturer (Table 6, Farm Load Profile). In future developments we will add
this as a parameter for the control strategy (at the moment the SOC set point is not a parameter for the
load following strategy). We presume that the maximum SOC could be a parameter to optimise the
control strategy (at the moment it is 100% of Cn), so it will be included in future developments.
5.2.3. Influence of the load profile
In Fig. 7 the total net present value of the five systems designed by the GA (considering the five load
profiles described above in section 2.2) are compared. For each system we compare the cost of PV-only
configuration (Worst Case Method, non-served energy calculated between 4 and 7%), the cost of PV-
Diesel and the cost of Diesel-only (HOGA, 0% non-served energy allowed). We can see that only for low
load systems the cost is lower in the case of PV-only. For peak load profiles, the PV-Diesel option is
more advantageous. The optimal in High Continuous load profiles is Diesel-only.
17
0
100000
200000
300000
400000
500000
600000
700000
800000
900000
Low Domestic Farm High High cont.
PV-only (Worst case). Non-served 4 - 7%
PV-Diesel. Non-served 0%
Diesel-only. Non-served 0%
LOAD
Total Cost (€)
Fig. 7. Comparison of design cost using different load profiles.
5.3. Costs of the different elements of the system
Fig. 8 shows the cost of the different elements of the system optimised by HOGA, for the Farm load
configuration as a percentage of the Total Net Present Cost. The auxiliary elements (Charge Regulator,
Inverter, Battery Charger and Other) only have a total of less than 1% of the Total Net Present Cost of the
system. Although the Diesel generator (acquisition and replacing) only costs 9% of the total, the cost of
the Fuel plus the operation and maintenance throughout the life of the system is 45% (O&M make up the
most important portion of the costs of the Diesel generator. O&M costs depend on operating hours, but
they are fairly independent of the generator type. The labour costs are the greatest part of the costs). The
PV panels are the most expensive elements at 29%.
Fuel 36%
O&M Diesel 9%Batteries 16%
Diesel Generator
9%
PV panels 29%
Other 1%
Fig. 8. Costs of the different elements throughout the life of the system for the Farm load system
optimised by HOGA, in percentage of the Total Net Present Cost.
18
6. Conclusions
The following conclusions have been reached:
Our program gives the best solution of all possible combinations, finding the best solution with the help
of GA. The advantage of HOGA is the precision: it gives the number of PV panels and its type, and the
number of batteries in parallel and their type.
HOGA finds the optimal SOC set point whether the cycle charging or combined strategies are the optimal
ones. HOMER does not optimise the SOC set point, although in the sensibility analysis the user can
introduce distinct values of SOC set point so that HOMER can compare them. A precise calculation of the
optimum value of the SOC set point in this way would suppose multitudes of combinations in the
sensibility analysis, and consequently high calculation times. HOGA optimises the SOC set point in small
calculation times, due to the GA.
0% non-served energy can be reached with PV-Diesel systems with a low total cost, but it is very difficult
to reach it with PV-only systems, unless the total cost is high.
With high loads the “Frugal” option is usually optimal.
The minimum output power of the Diesel generator and the minimum SOC of the batteries have influence
on the Total Net Present Cost and in the optimal dispatch strategy.
PV-Diesel systems are economically better than PV-only or Diesel-only systems for peak load profiles.
For low load profiles PV-only system present lower costs, and for high continuous or semi-continuous
load profiles Diesel-only is the best configuration.
7. Future Developments
A new general control strategy is being developed. All the strategies explained in this article will be
particular cases of this general strategy. This strategy will consider the minimum Diesel generator power
and the minimum and the maximum SOC as variables.
The battery’s efficiency will be modelled depending on the SOC, and the lifetime of the battery will
depend on the DOD. The fuel consumption when starting the Diesel generator will be considered. The
inverter efficiency will also be modelled depending on the load.
A methodology based on Pareto’s optimality will be applied for taking into account various objectives
simultaneously. For example, costs and Unmet Load.
New sources, such as wind energy and fuel cells, will be considered. Also, the uncertainty associated with
solar irradiation data and with the wind data will be taken into account in the mathematical model.
Acknowledgements
This work has been supported by IBERCAJA and the University of Zaragoza under the program “Ayudas
a la Investigación Científica y al Desarrollo Tecnológico”.
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