A Multi-Objective Design Methodology for
Hybrid Renewable Energy Systems
R. Chedid, S. Karaki and A. Rifai
Abstract — This paper describes a methodology to design a
hybrid renewable energy system over a certain planning horizon.
Traditionally a system plan was developed to achieve a minimum
cost objective (MCO) while satisfying the energy demand,
reliability, stability and battery constraints. The minimum
emissions objective (MEO) is now an important target to achieve
subject to the above mentioned constraints. Each of the above
problems may be solved using linear programming, but
minimizing the two preceding objectives at the same time forms a
multi-objective problem which is solved by the ε-constraint and
the goal attainment methods. The ε-constraint method minimizes
the total cost while the emissions are less than a certain value ε
determined by the linear programming when minimizing
emissions only or by the designer. The goal attainment method
tries to balance all the objectives and make them as close as
possible to the initial goals determined by MCO and MEO. A
case study is presented to illustrate the applicability and the
usefulness of the proposed method.
Index Terms-- Renewable energy, Hybrid systems, Multiobjective optimization.
I. INTRODUCTION
S
olar and wind energy systems are among the most
developed renewable energy systems (RES), and have
been widely used in both autonomous and grid connected
applications. Much research has been carried out to optimize
their size and evaluate their performance. Chedid and Rahman
[1] developed a linear programming model to optimize the
size of a hybrid system with battery storage and diesel sets.
However, the solution provided did not consider system’s
expansion over a future horizon. Kellogg et al. [2] presented a
numerical algorithm to determine the optimum size of
system’s components for three different configurations: wind
alone, photovoltaic (PV) alone and hybrid wind/PV. Karaki et
al. [3] presented a probabilistic model of a stand-alone
wind/PV power system. The model takes into consideration
system stability, outages due to the primary energy
fluctuations and hardware failure. Gavanidou and Bakirtzis [4]
developed a multi-objective planning technique to design a
R. B. Chedid is with the Faculty of Engineering and Architecture,
American University of Beirut, PO Box 11-0236, Beirut, Lebanon, (e-mail:
rchedid@layla.aub.edu.lb).
S. Karaki is with the Faculty of Engineering and Architecture, American
University of Beirut, PO Box 11-0236, Beirut, Lebanon, (e-mail:
skaraki@layla.aub.edu.lb).
A. Rifai is a graduate student with the Faculty of Engineering and
Architecture, American University of Beirut, PO Box 11-0236, Beirut,
Lebanon.
hybrid system based on minimization of both capital
investment and loss of load probability (LOLP). They applied
the trade off/risk method which rejected inferior plans and
gave a set of robust scenarios to the designer. Protogeropoulos
et al. [5] tried to determine the optimum size of a hybrid
system and to assess its economical and technical merits
against single PV and wind standalone systems. Morgan et al.
[6] described the development of a simulation program that
enables the designer to find the reliable level of a renewable
energy system. When both load and meteorological data are
known, the program calculates the system autonomy level and
predicts the battery voltage.
II. PROBLEM DEFINITION AND MATHEMATICAL FORMULATION
The problem is to develop a multi-objective model to
design a hybrid PV-wind system with battery storage and
diesel generators taking into consideration future system
expansion. The design objectives are cost and gas emission
minimization, which include minimization of carbon dioxide
(CO2), sulfur dioxide (SO2) and nitrogen oxides (NOx) from
diesel units over the life-time of the project. The problem
constraints are:(1) reliability constraint which dictates that a
certain percentage of the peak demand must be secured as a
reserve, (2) energy balance constraint, (3) stability constraint
to limit the renewable capacity to a certain percentage of the
peak demand, and (4) battery constraint, which determines the
upper size of the batteries installed each year.
As mentioned above, the problem has 2 categories of
objectives: cost and emission minimization, and is solved
subject to a set of constraints:
A. Cost function: The cost objective can be written as follows:
min[T wlife +t −1]


X wt  (Cw − S wt )Wt +
∑ α w Ow LFn H Wn 

n =t


min[T slife +t −1]


+ X st  (Cs − S st )Wt +
(1)
∑ α s Os LFn H Wn 

n =t


T
min[T dlife +t −1]
∑


t =1 + X dt  (Cd − Sdt )Wt +
∑ α d Od LFn H Wn 
n =t


min[T blife +t −1]  η − 1 


+ X bt  (Cb − Sbt ) Wt +
 dod α b Ob LFn H Wn 

∑


2


n =t


+ X rt (Cr − Srt )Wt
Where:
- X’s represent ratings of system’s units installed in year t.
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- C’s, O’s and S’s are the capital costs, operation and
maintenance costs and salvage values of system’s units
respectively.
- wlife, slife, dlife, blife and rlife are the life cycles of
system’s units.
α w , α s , α d and α b
-
are the resource availability of
system’s units.
- LFn is the load factor in year n. H is the number of hours
per year.
- dod and η are the deep discharge limit and the battery
efficiency respectively.
- Wt is the present worth factor of year t.
B. Emission functions: The emissions of CO2, SO2 and NOx
are of interest in this work. For example, the emission
function for CO2 can be expressed as follows:
 min[T dlife+ j −1]


X
H LFi FCO2 Dc 
∑
∑
dj 
j =1
i= j


T
where Et is the energy demand.
For example, the energy produced by the wind turbine in
the first year is X w1 α w LF1 H . The energy extracted from
the battery is modeled as follows. The battery will be charged
during light load and discharged during peak load. Assuming
that the battery is cycled each 12 hours, the energy supplied to
the demand during the peak period, from 0 to H / 2 , is
written as follows:
E w1 + E s1 + E d 1 + E bd = D '1
H
LF1
2
H
η dod
2
Ebd = X b
(5)
(6)
where
- E w1 , E s1 and E d 1 are the energy generated by WT, PV and
DU respectively during the peak period.
- E bd is the discharge energy from the battery.
'
(2)
- D 1 is the peak demand in the peak period and LF1 is the
load factor during the same period.
Where FCO2 is the weight of CO2 emitted due to burning
1kg fuel; Dc is the fuel consumption of the Diesel unit in kg/
kWh. Similar factors are used to calculate the quantities of
SO2 and NOx emitted.
During the light load period, from H / 2 to H , the energy
supplied to the demand can be written as below:
[kg]
C. Planning and Operational Constraints
Constraints are needed to ensure the stability of the system,
avoid over-sizing of system components and guarantee
demand supply at any time.
i. Reliability Constraint:
E w 2 + E s 2 + E d 2 − Ebc = D ' 2
H
LF2
2
H
dod
2
Ebc = X b
where
- Ew2 , Es 2
(7)
(8)
and Ed 2 are the energy generated by WT, PV
and DU during light load period.
'
These constraints ensure that the available capacity is
higher than the load thus allowing for a given reserve margin
R:
t
∑X
n =1
wn
t
t
t
n =1
n =1
n =1
α w + ∑ X snα s + ∑ X dnα d + ∑ X bn α b
t
+ ∑ X rn α r ≥ D0 × (1 + inc ) (1 + R )
t
- D 2 is the peak demand in this period and LF2 is the load
factor during the same period.
- E bc is the energy charged to the battery during the same
period.
(3)
t = 1, T
n =1
where D0 is the present peak demand, inc is the rate of load
increase per year and R is the reserve.
From the above equations, we can get the energy flow
from the battery as below
η − 1
Eb = Ebd - Ebc = X b H 
 dod
 2 
(9)
The energy produced by PV panels and diesel units is
calculated in the same way.
ii. Energy Constraint:
The energy generated by all system units has to
demand at all times.
t
t
 t
×
+
+
α
α
X
LF
X
LF
X dn α d LFt
∑
∑
t
sn
s
t
∑ wn w
n =1
n =1
 n =1
 t
η −1 
 dod LFt
∑ X bn α b 
 2 
 n =1
t = 1, T
equal the
iii. Renewable energy contribution limit:
For reliability purpose, the whole system at any time will
not operate with more than 40% of renewable contribution.

+
The concern here is to prevent system collapse due to stability
 H = E t considerations when suddenly loosing the renewable resource.


t
t

t
X
+
X sn ≤ Rl D0 (1 + inc ) , t = 1, T
(10)
wt
(4)
∑
n =1
∑
n =1
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where Rl is the maximum limit for renewable contribution in
the whole system. In [7] Rl is taken from 30% to 40% of the
the following load curve and increasing by 7% each year as
shown in Fig.2 and Fig. 3.
demand. In this thesis, it is taken equal to 40%. If the load
does not require quality supply, renewable resources should
be utilized as possible and therefore this constraint should be
removed. However, if the load requires quality supply as in
this thesis, Rl must be set to the mentioned values. After
setting Rl to 40%, and referring to the energy constraint, it is
found that renewable units supply around 10 % of the load
which is acceptable and the role of renewables is fuel saver.
Fig.1. Ksara Wind Speed Distribution.
iv. The Battery constraint:
It is assumed that the battery capacities must be less than
the difference between the maximum load and the minimum
load during the worst day in the year. The ratio of minimum
load over maximum load Min / Max is constant during all
the project period because the load is increasing at a constant
rate. Taking into account the deep discharge limit dod for
lead-acid battery and its efficiency, the size of batteries is
limited as follows:
t
∑X
n =1
≤ (1 + dod ) × D0 × (1 + inc ) (1 − Min / Max ) / η ,
t
bn
(11)
t = 1, T
III. SOLVING APPROACH
The practical approach followed to reach a solution is
based on three steps: First, LP is used to find a system with
minimum cost but with no consideration for the emissions.
Then LP is used again to find the system with minimum
emissions without any limit on cost. Since these objectives are
conflicting in nature, any improvement in one objective will
lead to a degradation of the other objective. Many techniques
were developed to solve multi-objective problems including
the ε-constraint and the goal attainment methods.
The ε-constraint method is used where the objective is to
minimize the total cost but the emission objectives are now
added to the above described constraints. There is a difficulty
in choosing the best εi, i=1,2,3, for the emission objectives in
order to get the best solution. Applying goal attainment
method to this problem will give a solution more tolerant than
the ε-constraint method.
Fig. 2. Load Duration Curve.
Demand
1200
1000
80
K
60
40
20
0
1
The above described methodology was applied to a
hypothetical site in Lebanon whose wind speed distribution is
shown below.
The availability of solar resource is assumed to be equal to
25 percent with maximum solar irradiance equal to 611.5
Wh/m2. The load to be met is 500 kW varying according to
5
7
yea
9
Fig. 3. A 10 year demand forecast.
The capital and operation and maintenance costs for all
system components are listed in the appendix A. The
operation and maintenance cost for diesel generators is
calculated according to the following equation:
O&M =
IV. SIMULATION RESULTS
3
FixedO & M * Rated + VariableO & M * Rated * CF * Hours
Rated * CF * hours
(12)
Where:- Rated is the rated power for the generator.
- CF is the capacity factor of the generator and it is assumed
to be equal to 0.6.
- Hours is the number of hours per year.
The fuel cost is added to the above O & M to get the
total operation and maintenance cost.
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The wind resource availability is calculated in the usual
way using the Weibull distribution of wind speed in
conjunction with the turbine power characteristics.
A. Minimizing Cost Only (MCO) Using LP
The results of minimizing cost only using LP, without any
constraints on the emissions are shown in Fig4.
The emissions in this case are, as expected lower than in the
MCO problem, at 28.3 thousand Tons.
C. ε-Constraint Method
In order to minimize the total cost and the emissions in the
same time, the ε-constraint method is applied. First of all ε,
our emission target, has to be determined by plotting the curve
of cost versus ε as shown in Fig. 6.
Installed Units Each Year
900
800
700
Reserve
KW
600
Battery
500
Diesel
400
Solar
300
Wind
200
100
0
1
2
3
4
5
6
Year
7
8
9
10
Fig.4. MCO Base Case.
As seen, no PV units are added because wind power is less
expensive. Also, the addition of wind turbines is limited by
the wind resource contribution constraint, and no batteries are
adopted because of their high cost. The minimum present
worth cost is $4.7 millions, which corresponds to an energy
price of $0.13/ kWh. The CO2 emissions were at 28.9
thousand Tons. Decreasing the cost of wind turbine by 20%
does not cause an increase in the added capacity of WT
because the stability constraint (8) is biding in this case
preventing any new renewable resource from being added.
B. Minimizing Emission Only (MEO) Using LP
The results of minimizing emissions only using LP,
without any constraint on the cost, are shown in Fig 5.
Fig. 6. Cost Versus Emission Rates.
As seen, we have higher cost with a lower discount rate
and vice-versa. Also the program starts to give a feasible
solution when ε is equal to 0.80894 kg of CO2 per Kwh,
0.0097 for SO2 and 0.001166 for NOx. At this limit, the base
case total cost is $49.591m.
If the renewable contribution limit is relaxed to 75%, the
model produces feasible solutions after 0.7305 kg of CO2 per
kWh (Fig. 7). But as a result, the total cost increases as
shown in Fig. 7.
Installed Units Each Year
1200
1000
Reserve
KW
800
Battery
Diesel
600
Solar
400
Wind
200
0
1
2
3
4
5
6
Year
7
8
9
10
Fig. 5. MEO Base Case.
Here, the model takes less diesel units in order to reduce
emissions. To compensate for the energy lost, solar panels
rather than wind turbines are adopted . This is because the
availability of the solar resource is 25% whereas the
availability of wind is around 21%. As expected the cost here
is higher at $5.8 millions, which corresponds to $0.17/ kWh.
Fig. 7. Relaxing Renewable contribution limit.
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From Fig. 6, it is obvious that minimum cost and minimum
emissions are obtained when ε is equal to 0.80894 kg of CO2
per kWh. The base case capacity additions are shown in Fig.
8.
lower. This is indicated in the cost of the GA method being
lower than the cost in the ε-constraint. Also the model
installs more PVs than WTs because the availability of solar
power is higher than that of wind power. In contrast to the
LP and ε-constraint methods, the goal attainment adds
batteries.
V. CONCLUSION
Fig. 8. ε-Constraint Base Case.
It is to be noted that the total cost is higher than that of
minimizing cost only objective due to higher installation of
PVs, and is less than that of minimizing emissions only. The
emissions are the same as those resulting from the base case
of minimizing emissions only because ε is set at its minimum
value that gives feasible solution.
D. Goal Attainment Method
The goal attainment method (GA) tries to balance
between the emissions and the cost objectives. The results of
goal attainment show that more emissions are obtained as
compared with MEO and ε-constraint method by a certain
percentage and the cost is increased as compared with MCO
by the same percentage which is the attainment factor. The
results of the GA method in the base case are shown in Fig.
9.
This paper has presented a methodology to design a
hybrid wind photovoltaic system with diesel generators and
battery banks over a certain planning horizon. First, linear
programming is used to minimize the total system cost while
satisfying the energy demand, reliability, stability and battery
constraints. The results of this technique are an optimum set
of wind turbines, photovoltaic panels, diesel generators and
battery banks. Second, linear programming is used to
minimize the total emissions of carbon dioxide (CO2), sulfur
dioxide (SO2) and nitrogen dioxide (NOx) subject to the
constraints used in the first exercise. The results make
another optimal hybrid renewable energy system.
Minimizing all the precedent objectives at the same time
forms a multi-objective problem which is solved by the εconstraint and the goal attainment methods. The ε-constraint
method minimizes the total cost while meeting the same
constraints but with total emissions being less than a certain
value ε determined by the linear programming when
minimizing emissions only or by the designer. The goal
attainment method tries to balance all the objectives and
make them as close as possible to the initial goals determined
by the minimum cost and minimum emissions sub-problems.
VI.
Installed Units Each Year
900
800
700
Reserve
KW
600
Battery
500
Diesel
400
Solar
300
Wind
200
100
0
1
2
3
4
5 6
Year
7
8
9
10
Fig. 9. Goal Attainment Base Case.
First, the model installs more diesel units than those of
the ε-constraint method which increases the emissions and
reduces the total cost as expected. The DUs added in all
years according in the ε-constraint method are the maximum
allowable for the specified emissions. In the GA, the
specified emission constraints are slightly relaxed and more
DUs are added in this case. The rest of demand is supplied
from WTs and PVs rather than PVs alone as in the εconstraint method because the cost of such a combination is
APPENDIX
Parameter
Wind turbine price
Solar panel price
Diesel engine price
Battery price
Wind O&M
Solar O&M
Diesel total O&M
Battery O&M
Project life span
Wind turbine life span
Solar panel life span
Diesel life span
Battery life span
Battery efficiency
Battery depth of discharge
Fuel cost
Discount rate
Escalation rate
Carbon % in diesel
Sulfur % in diesel
Nitrogen % in diesel
Value
1015 $/KW
4401 $/KW
813 $/KW
1600 $/KW
0.01 $/KWh
0.0046 $/KWh
0.1402 $/KWh
0.02 $/KWh
10 years
20 years
20 years
7 years
5 years
0.85
0.45
1.82 $/ gallon
0.1
0.03
86.4
2
0.1
VII. REFERENCES
[1]
[2]
R. Chedid and S. Rahman:’Unit Sizing And Control Of Hybrid WindSolar Power Systems’, IEEE Transactions On Energy Conversion,
Vol.12, No.1, March 1997.
W. Kellogg, M. Nehrir, G. Venkataramanan and V.Gerez:’Generation
Unit Sizing And Cost Analysis For Stand- Alone Wind, Photovoltaic,
And Hybrid Wind/PV Systems’, IEEE Transactions On Energy
Conversion, Vol.13, No.1, March 1998.
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[3]
[4]
[5]
[6]
[7]
S. Karaki, R. Chedid and R. Ramadan:’ Probabilistic Performance
Assessment Of Autonomous Solar-Wind Energy Conversion Systems’,
IEEE Transactions On Energy Conversion,
Vol.14, No.3, March
1999.
E. Gavanidou and A. Bakirtzis; Design of a Stand Alone System with
Renewable Energy Sources Using Trade-Off
Methods. IEEE
Transactions On Energy Conversion, Vol.7, No.1, March 1992.
C. Protogeropoulos, B. Brinkworth and R. Marshall:’ Sizing and
Techno-Economical Optimization For Hybrid Solar Photovoltaic/Wind
Power Systems With Battery Storage’, International Journal Of Energy
Research, Vol.21, 1997.
T. Morgan, R. Marshall and B. Brinkworth:’ “ARES”- A Refined
Simulation Program For The Sizing And optimization Of Autonomous
Hybrid Energy Systems’, Solar energy, Vol.59, Nos.4-6, 1997.
G. Seeling-Hochmuth,:’ A Combined Optimization Concept For The
Design And Operation Strategy Of Hybrid-PV Energy Systems’, Solar
Energy, Vol.61, No.2, 1997.
VIII. BIOGRAPHIES
Riad Chedid was born in Lebanon in 1960. He got his M.Sc degree (with
distinction) in electrical engineering from Moscow Power Engineering
Institute in 1986, and his Ph.D from Imperial College of Science,
Technology and Medicine, U.K. in 1992 He joined the American University
of Beirut in 1992 where he is currently a Professor of Electrical and
Computer Engineering. His research interests include energy planning and
policy, modeling of renewable energy systems, and numerical simulation of
electromagnetic fields.
Sami H. Karaki is professor of electrical engineering at the American
University of Beirut (AUB), Lebanon. He joined AUB in 1991 and
contributed to the development of its
Electric Power Engineering program. From
1981 to 1990 he was with Kuwait Institute
for Scientific Research, Kuwait where he
contributed to two regionally leading
projects
on
the
power
system
interconnection of Arabic countries. He
obtained his BE from AUB in 1975 and his
Ph.D. from the University of Manchester
Institute of Science and Technology, UK, in
1980. He is presently teaching courses in
digital
systems
design,
computer
programming, power electronics, and power
systems. His main research interests are in
modeling and analysis of renewable energy systems, power system planning
under competition, short-term load forecasting, and artificial intelligence
applications in electric power systems.
Ahmad Al-Rifai was born in Lebanon (Arsal) in June 20, 1977. He
obtained his Bachelor of Electrical Engineering from Damascus University
in 2001. He graduated with Honors and was awarded the Bassel Assad Prize
in 1999-2000. He obtained his Master of Engineering degree: major
Electrical Engineering in June 2004 from AUB. He is currently working with
Saudi Oger in the high-current design office in Riyadh, Saudi Arabia.
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