Open Journal of Applied Sciences, 2013, 3, 5-11
doi:10.4236/ojapps.2013.32B002 Published Online June 2013 (http://www.scirp.org/journal/ojapps)
Supply Mix Optimization for Decentralized
Energy Systems
J. K. Gruber, J. L. Mínguez Fernández, M. Prodanovic
Electrical Systems Unit, IMDEA Energy Institute, Móstoles (Madrid), Spain
Received 2013
ABSTRACT
In recent years, the energy sector has undergone an important transformation as a result of technological progress and
socio-economic development. The continuous integration of renewable energy sources forces a gradual transition from
the traditional business model based on a reduced number of large power plants to a more decentralized energy produc-
tion. The decentralization and the increased number of energy sources lead to a series of new challenges in the energy
sector. This paper presents an approach to determine the optimal energy supply mix for small and medium sized build-
ings or installations. The optimization algorithm considers the electricity and heat demand and determines the optimal
combination of energy sources by minimizing an economic index. The optimization problem can be solved for multiple
demand profiles and takes into account the possibility to integrate accumulator systems. The proposed approach pro-
vides a high degree of flexibility and can be used to study the influence of the energy prices on the optimal energy sup-
ply mix. The performance of the proposed optimization approach is illustrated by the results obtained from a simulation
example.
Keywords: Energy Supply Mix; Optimal Configuration; Decentralized Generation
1. Introduction
The prosperity of modern societies is closely related to
the availability of energy and the continuous supply is an
important factor for the industrial development. In many
countries a significant part of the energy demand is satis-
fied with fossil fuels such as oil and gas or by means of
nuclear power production. The constantly growing de-
mand for fossil fuels and the shrinking reserves in com-
bination with the necessary and more expensive modern
extraction technologies lead to increased energy prices.
Today, many economies depend highly on energy im-
ports from a small group of countries with oil and gas
reserves. The environmental impact of fossil fuels as well
as the risks related to nuclear power generation gave rise
to a serious discussion about the effects of traditional
energy production. These drawbacks led in recent years
to an increased research and development in alternative
energy sources.
In recent years, the significant increase in the integra-
tion of renewable energy sources compensated, at least to
some extent, the problems related to the traditional en-
ergy production. The continuous increase of alternative
energy sources, especially wind turbines and photo-
voltaic panels, emphasizes the change from a centralized
energy production with few large power plants to a more
distributed generation. The power production near the
place of consumption reduces the transmission losses,
increases the energy efficiency and helps to ensure a high
quality in the energy supply.
Nowadays, buildings contribute strongly to the total
energy demand and account in some countries for up to
45% of the primary energy consumption [1, 2, 3]. A suit-
able energy mix, especially the use of renewable energy
sources, and an optimal supply system can improve the
energy efficiency of buildings and reduce costs. The op-
timal energy supply system for buildings in the tertiary
sector is determined in [4] solving an economic minimi-
zation problem by mixed integer linear programming
(MILP). The algorithm in [5] considers both centralized
and decentralized technologies and determines the opti-
mal technology mix for a neighbourhood or a small town
taking into account constraints such as a maximum emis-
sion of CO2. The integrated approach presented in [6]
improves the energy efficiency combining the computa-
tion of the optimal energy supply mix with a proactive
energy management. The multi-objective optimization of
an energy supply system for an industrial district [7] in-
cludes the costs of the energy supply system and the en-
vironmental impact of a positive or negative CO2 balance
with respect to traditional systems.
In [8] the optimal technology mix is determined with a

J. K. GRUBER ET AL.
6
special focus on the variability in energy production as a
result of an increased wind power deployment. The au-
thors underline the importance of storage systems which
provide certain flexibility in the integration of intermit-
tent energy sources. The prototype energy model pre-
sented in [9] deals with the complex problem to consider
social factors in the optimization of the energy supply
mix. The approach translates energy policy goals, both
quantitative and qualitative ones, into a set of mathe-
matical expressions for the posterior optimization.
This paper presents a method to determine the optimal
energy supply mix for small and medium sized buildings
under consideration of seasonal profiles for electricity
and heat demand. The optimization approach focuses on
distributed energy generation in combination with elec-
tric batteries and backup grid connection. The minimiza-
tion of the objective function, an economic index based
on the initial inversion and the generation costs, is car-
ried out with sequential quadratic programming (SQP).
The paper is organized as follows: Section 2 presents a
detailed problem description and defines the objective
function. The proposed approach for the minimization of
the cost is given in Section 3. The implementation of the
proposed optimization procedure and the obtained results
for an office building are given in Section 4. Finally, in
Section 5 the mayor conclusions are drawn.
2. Problem Description
This work presents an approach to determine the optimal
energy supply mix for a small or medium sized building
minimizing an economic index. The general objective
function to be minimized is given by:

Jx
 
ehb
JJ JJxxxx
(1)
where
N
x denotes the vector with the installed
capacities and N is the total number of energy sources
considered in the cost function. The terms


e
Jx
e
,
h and represent the costs related to the
electricity sources, the heat sources and the battery, re-
spectively. The capacities are given by

Jx

b
J

x
N
ex for the
electricity sources, h
N
hx
,,
TT
eh
for the heat sources and
b for the battery. The vector of capacities used in
(1) is defined as
x
T
b
x

xxx

The cost related to the electricity sources is
defined as:
.
()
e
x
e
J
,,
()() ()
eeef eeve
JJ Jxxx (2)
being , the annual fixed costs and , the
variable costs. The fixed costs are given by:
()
ef e
Jx()
ev e
Jx
,() ef e
Jx
() ()
,()
1
()
eii
N
ee
ef ei
ie
x
I
Jy



x (3)
where ()i
e
x
, ()i
e
I
and ()i
e
y
are the installed capacity, the
necessary initial investment costs per installed capacity
and the technical lifetime of the i-th electricity source,
respectively. Note that the capacity ()i
e
x
corresponds to
the i-th element of the vector e. The variable electric-
ity costs are
defined as:
x
() ( ii
e
,() ev e
Jx
, ev
)
1
()
e
N
e ee
i
Jp
e
s






x (4)
where ()i
e

is the amount of energy of the i-th electric-
ity source consumed during a year and is the cor-
responding price per energy unit. The shortfall of electric
energy is considered as an additional term based on the
amount of missing energy e
()i
e
p
s
and the corresponding
penalization cost e

per unit.
In the case of heating, the cost is given by:
()
hh
x
hv
J
J
,
()
f h,
() ()
hh hh
JJ

xx
,hf
J
x (5)
with the annual fixed costs and the variable
costs defined as:
(x)
h
,() hv h
xJ
() ()
()
ii
hh
i
h
,() hf h
1
h
N
i
x
I
()ii
h
Jy x
s


()
h
(6)
,
1
()
h
N
hv hhh
i
Jp






x (7)
where ()i
h
x
, ()i
h
I
and h
()i
y
are the installed capacity, the
initial investment per installed capacity and the technical
life cycle of the i-th heat source, respectively. It is im-
portant to mention that ()i
h
x
is the i-th element of the
capacity vector The variable .
h
x()i
h

denotes the amount
of energy of the i-th heat source consumed during a year
and is the price per energy unit. Besides, the vari-
able costs consider the case of a shortfall where h
()i
h
p
s
de-
notes the amount of missing heat and h

is the penali-
zation price of each energy unit.
For the battery, the cost ()
bb
J
x
,
()
b
is defined as:
,
() )
bb bfb
(
bv
J
xJ xxJ


)
b
(8)
with the fixed costs ,(
bf
J
x and the variable costs
,( bv b
)
J
x given by:
,() b
b
b
bf b
x
I
y

()
1
e
N
b
Jx (9)
()ii
e,()
bv b
i
J
x 

p


 (10)
where b
x
denotes the installed battery capacity, b
I
represents the necessary initial investment per storage
capacity and b is the lifetime of the battery. The vari-
able costs depend on the amount of energy
y
()i
b

from
the electricity sources used for battery charging and the
corresponding price per energy unit.
()i
e
p
The amounts of consumed (()i
e

, ()i
h

and b
()i

) and
missing energy (e
s
and h
s
) are determined during the
optimization as these values depend directly on the ca-

J. K. GRUBER ET AL. 7
pacities of the energy sources. The initial investments
(()i
e
I
,()i
h
I
and b
I
), the technical lifetimes (()i
e
y
,()i
h
y
and
b), the prices per energy unit ( and ) as well as
the costs for each missing energy unit (
y()i
e
p()i
h
p
()i
e

and ()i
h

)
are constant parameters and their values are known.
3. Optimization Procedure
The objective of the optimization procedure is to mini-
mize the energy costs (1) for a given electricity and heat
demand. The use of multiple profiles allows considering
the variations in the demands throughout the year and
provides a more realistic estimation of the energy costs.
For a given set of capacities and given demand profiles,
the chosen approach calculates the amount of energy
consumed from each source and the amount of missing
energy to satisfy the demand. The following sections
explain in detail the use of the available energy sources
to cover the demands.
3.1. Initial Considerations
Consider a sampling time of and the installed
capacities
1h
s
t
()i
e
x
for , 1, , e iN
()i
h
x
for 1,i,h N


and b
x
. Furthermore, consider the electricity and heat
profiles for a certain day given by the hourly demand
and with .
(
e
d)j
j
()
()j
, 2
()ii
ee
h
The amount of energy which can be produced in each
sample by a source is given by the expres-
sions
d0,3, 2
s
j
0, 3
() () ()e
i
g
jf
()ii
hh
jxt with 1,,e iN
and ()()
() ()h
i
s
g
jfjxt with 1,,h
iN
where the parameters () ()
i
e
f
j and ()i
h()
f
j are used to
consider external or internal influences on the energy
production. For controllable energy sources, e.g. gas tur-
bines, grid connection or biomass boiler, the possible
energy production is assumed to be ()
()
i
e
fj1

and
. In the case of the non-controllable energy
sources, i.e. renewable sources such as wind turbines,
photovoltaic panels or solar thermal collectors, the en-
ergy production also depends on some environmental
conditions leading to and .
For controllable energy sources with simultaneous heat
and energy production, e.g. combined heat and power
(CHP), the parameters e
()i
h()fj1
()
0
i
e
()
()fj
()fj
()ii
1() 0(
i
fj)1
h

 and ()
h
()
()
i
fj i


with are used.
() (
 
)1 ii
The operation modes of the considered battery does
not admit simultaneous charging and discharging, i.e. in
a given moment the battery represents either an energy
source or an energy sink. Another limitation arises from
the power rate capabilities c and d for charging
and discharging, respectively. The amount of energy that
can be drawn from or stored to the battery in each sample,
without taking into account the current state of the bat-
tery, must not exceed vj
m
m
m
cs
()
ct

and ()
dt ds
vj m

with 0,, 23j

.
The considered capacities ()i
e
x
, ()i
h
x
and b
x
allow
computing the fixed costs of the energy sources given by
(3), (6) and (9). The variable costs (4), (7) and (10) have
to be calculated after determining the amount of energy
supplied by each source.
3.2. Assignment of Electricity Sources
The satisfaction of the energy demand using different
sources is based on economic rules with the objective to
minimize the costs. For a given set of installed capacities
x, priority is given to sources with a cheaper generation.
The optimization satisfies the electricity demand using an
iterative procedure with the following assignment order:
renewable sources, CHP and controllable sources. For
simplicity of the notation and without loss of generality it
is assumed that the electricity sources with the capacities
()i
e
x
for e i1,, N


(0) () ()
e
rjd
are already sorted according to the
sequence of assignment.
With the initially unsatisfied electricity demand given
by for
e
j0,, 23j

, in the i-th iteration
with 1,, e
iN

 the amount of energy drawn from the
corresponding source is given by:


1) ()
()(),0,0, ,23
i
e
r jgjj

 
() (
() max
ii
ee
cj
() (
()
ee
rj
()
()
ee
uj
()

0,, 23j

e
(11)
Then, the remaining unsatisfied demand becomes:
1) ()
()(),0, ,23
ii i
e
rjcj j

  (12)
and the unused amount of energy which could be pro-
duced by the i-th energy source is:
() ()
()(),0, ,23
iii
e
gjcjj  (13)
Finally, finishing the iterative procedure, the total
amount of energy generated by each source throughout
an entire year can be calculated by:
23 ()
0
365( ),1,,
ii
ee
j
cj iN

 
 (14)
After using the energy sources, the remaining
electric demand for has to be
satisfied by the battery.
e
N
()
(
e
N
e
r)j0,, 23j
3.3. Assignment of Battery
The battery is charged by means of the electricity sources
using an iterative procedure and the previously described
assignment order. Taking into account only the power
rate limitations, the theoretical amount of energy that can
be stored to the battery in a sample is
for
(0) ()()
cc
wjvj

.
With the known initial state of charge and the
still unsatisfied demand for
(0)
c
q
j
() (0) () ()
e
N
be
rjr j0,, 23,


the amount of energy charged to the battery in the i-th
iteration with 1,i,e N

 (the charged energy is drawn

J. K. GRUBER ET AL.
8
from the i-th source) is calculated as:

()()( )(1)
() min(),,
iik
bebc
cj ujxqw


i
c
(15)
if and
(0) ()0
b
rj() () 0
i
b
cj

if with
and being a counter.
Evidently, charging will take place only in samples
without demand, i.e.
(0) ()0
b
rj
0,, 23j24 (kji
(0) ()0
b
rj
1)

. After storing
in the battery, the remaining amount of energy that could
be stored in a sample is given by:
()
()
i
b
cj
()( 1)()
()()(),0, ,23
ii i
cc b
wjwjcj j

  (16)
and the state of charge of the battery can be written as:
(1)()()
(),0, ,23
kki
ccb
qqcjj
  (17)
The state of charge depends on the charging in
previous samples due to the recursive character of (17).
As a direct consequence, (15)-(17) have to be evaluated
together for each sample in the i-th iteration.
(1)k
c
q
After charging the battery using the different energy
sources, the state of charge is given by (24 ) (0) e
N
dc
qq
and represents the initial state considered in the dis-
charging procedure. The amount of energy drawn from
the battery in each sample is defined as:

(0)
() min(),(), (),0, ,23
bbdd
zjr jqjvjj
e
(18)
The resulting unsatisfied demand after using the bat-
tery can be written as:
(1) (0)
()()(),0, ,23
bb b
rjrjzj j  (19)
and the amount of energy stored in the battery is:
(1)() (),0,,23
ddb
qjqjzjj (20)
Finally, after the charging and discharging procedures,
the amount of energy taken from each source throughout
the year and stored in the battery is given by:
23
() ()
0
365( ),1,,
ii
bb
j
cj iN 

 
 (21)
and the overall unsatisfied demand becomes:
23 (1)
0
365( )
eb
j
rj 


 (22)
3.4. Assignment of Heat Sources
The heat demand is covered step by step using an itera-
tive procedure that gives priority to sources with a
cheaper energy production. Based on this simple eco-
nomic rule, the assignment order applied by the proce-
dure is: renewable sources, CHP and controllable sources.
For simplicity of the notation and without loss of gener-
ality it is assumed that the considered heat sources with
the capacities ()i
h
x
with are already sorted
in the given assignment order.
1,, h i
Defining the initial heat demand as
for
(0) () ()
hh
rjdj
0,, 23j

, the amount of energy generated in the
i-th iteration by the corresponding source is given by:


()( 1)()
() max()(),0,0, ,23
iii
hhh
cjr jgjj


h
j
(23)
with the generated energy , the remaining unsat-
isfied demand becomes:
()
()
i
h
cj
()(1)()
()()(),0, ,23
ii i
hh h
rjrjcj j

  (24)
Finally, after finishing the iterative procedure the total
amount of heat generated by each source during an entire
year is given by:
23
() ()
0
365( ),1,,
ii
hh
j
cj iN 

 
 (25)
and the overall unsatisfied heat demand becomes:
23 ()
0
365( )
h
N
hh
j
r 


 (26)
3.5. Cost minimization
The generated energy (()i
e

,()i
b

,()i
h

) and the missing
energy (e

,h

) determined with the assignment proce-
dures are used to calculate the variable costs (4), (7) and
(10). Furthermore, the fixed costs are given by (3), (6)
and (9) for a given set of capacities . Finally, the fixed
and variable costs allow computing the overall cost (1)
related to the energy supply.
x
Now, the optimal capacities of the energy sources
are calculated solving the following minimization prob-
lem under consideration of constraints:
*
x
*argmin()
s.t.
J
A


x
xx
xb
(27)
with c
N
N
A


 and c
N
b where c denotes the
number of constraints. The optimization problem based
on several iterative assignment procedures can then be
solved with nonlinear programming (NLP).
N
4. Implementation & Results
The proposed approach for the minimization of the cost
related to the energy supply of small and medium sized
buildings has been implemented in Matlab. The optimal
capacities are computed solving the optimization
problem (27) with Matlab’s built-in function for sequen-
tial quadratic programming (fmincon).
*
x
4.1. Implementation
For the satisfaction of the electric demand, wind turbines,
photovoltaic systems, gas turbines, CHP plants, grid
connection and batteries have been considered. Heat
pumps, oil boilers, CHP plants, solar thermal collectors
N

J. K. GRUBER ET AL. 9
and biomass boilers have been taken into account for
heating. The technical lifetimes (()i
e
y
,()i
h
y
,), the initial
investments per capacity unit (
b
y
()i
e
I
,()i
h
I
,b
I
) and the
prices per energy unit (,) of the energy sources
are given in Table 1. In the case of an energy shortfall, a
penalization of
()i
e
p
10
()i
h
p
00 €/kWh
eh

  is used for elec-
tricity and heat.
The parameters () ()
i
e
f
j and () ()
i
h
f
j
0.3
for photovoltaic
and solar thermal systems located in Madrid (Spain) have
been taken from the Photovoltaic Geographical Informa-
tion System (PVGIS) [10], see Figure 1. For wind tur-
bines an average performance of 30%, i.e.,
has been assumed and for CHP systems the electric-
ity/heat ratio is given by and . For
all other electricity and heat sources,
()
() 0.3
i
e
fj
() 0.7 i
 
()
() 1
i
e
fj
()i


and
have been used.
()
() 1
i
h
fj
Two demand profiles (see Figure 2), one for winter
and one for summer, have been used for the considered
medium sized office building in Madrid. The optimiza-
tion procedure of the energy supply mix considers half
year of summer and half year of winter.
Table 1. Technical lifetimes, investments per capacity and
energy prices of the considered energy sources.
Source Investment
[€/kW]
Price
[€/kWh]
Lifetime
[a]
wind turbine 2400 0.04 20
photovoltaic system 4000 0.02 25
gas turbine 1200 0.08 10
CHP plant 1300 0.06 10
grid connection 15.97 0.12 1
battery 500 €/kWh − 5
geothermal heat pump 1500 0.07 20
oil boiler 600 0.08 10
solar thermal collector 1000 0.03 25
biomass boiler 350 0.08 20
Figure 1. Efficiency of the photovoltaic and solar thermal
systems in summer (dash-dotted line) and in winter (solid
line) and for the wind turbine in all seasons (dashed line).
4.2. Results
The implemented procedure was used to determine the
optimal energy supply mix for the office building. The
costs were optimized using current market prices (see
Table 1) for the different energy sources.
The obtained results of the energy consumption are
given for the summer profile in Figure 3 and for the
winter profile in Figure 4. It can be observed that that
the major part of the electricity and heat demands are
covered by fossil energy sources and only a reduced
proportion is satisfied by renewable energy. Furthermore,
some of the available technologies are not included in the
optimal energy mix (batteries, geothermal heat pumps,
solar thermal collectors and oil boilers) due to economic
reasons. The capacities of the energy sources and the
amount of energy produced by each source are given in
Table 2. With the current prices, 82.7% of the energy
demand is covered with fossil fuels, 5.2% is drawn from
the grid and 12.1% is taken from renewable energy
sources. In spite of the high capital costs of wind turbines
Figure 2. The considered electricity and heat demands for
winter (solid line) and summer season (dashed line).
Figure 3. Satisfaction of the electricity (top) and heat de-
mand (bottom) in summer.

J. K. GRUBER ET AL.
10
and photovoltaic systems, these sources satisfy 14.1% of
the yearly electricity demand.
With the continuously decreasing prices of renewable
energy sources and the rising fossil fuel prices, decen-
tralized generation will be more important in the future.
Furthermore, it is expected that the improvements in
storage technologies will reinforce the use of batteries
and other accumulator systems.
Figure 4. Satisfaction of the electricity (top) and heat de-
mand (bottom) in winter.
Table 2. Capacities of the energy sources and amounts of
energy generated during the year.
Source Capacity [kW] Annual gen. [MWh]
wind turbine 98.2 (6.0%) 258.0 (4.5%)
photovoltaic system 191.3 (11.6%) 392.2 (6.8%)
gas turbine 501.3 (30.4%) 2304.0 (39.9%)
CHP plant 593.2 (36.0%) 2469.1 (42.8%)
grid connection 191.2 (11.6%) 301.0 (5.2%)
battery 0 0
geothermal heat pump 0 0
oil boiler 0 0
solar thermal collector 0 0
biomass boiler 73.7 (4.5%) 46.4 (0.8%)
5. Conclusions
A procedure for the computation of the optimal supply
mix for decentralized energy systems has been developed
and implemented. The presented approach minimizes an
objective function for a given energy demand using basic
economic rules. The use of multiple profiles allows con-
sidering seasonal variations in the electricity and heat
demand. The described algorithm has been implemented
in Matlab considering ten energy sources, including bat-
teries and CHP systems. The optimization algorithm re-
gards capital costs and variable costs resulting from the
energy generation. Additional costs resulting from de-
commissioning, waste management, CO2 transport and
storage, fixed operating costs and others can be included
easily in the minimization problem for a differentiated
optimization and analysis of the supply mix.
The optimal energy supply mix for a medium sized of-
fice building located in Madrid (Spain) was determined
using the proposed optimization procedure. The obtained
results showed a predominance of energy production
from fossil fuels for current market prices and only a
reduced use of renewable energy. The high flexibility of
the procedure allows studying changes in the optimal
energy supply in function of the investment costs and
energy prices.
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