Statistical modeling and optimal energy distribution of cogeneration units by genetic algorithms

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 12, No. 2, April 2022, pp. 1131~1138
ISSN: 2088-8708, DOI: 10.11591/ijece.v12i2.pp1131-1138  1131
Journal homepage: http://ijece.iaescore.com
Statistical modeling and optimal energy distribution of
cogeneration units by genetic algorithms
Othmane Maakoul1
, Hamid El Omari1
, Aziza Abid2
1
Laboratory of Renewable Energies Environment and Development, Faculty of Science and Technology, Hassan 1st
University,
Settat City, Morocco
2
Laboratory LaSTI, National School of Applied Sciences, Sultan Moulay Slimane University, Beni Mellal City, Morocco
Article Info ABSTRACT
Article history:
Received Mar 1, 2021
Revised Oct 29, 2021
Accepted Nov 14, 2021
Our main objective is to evaluate the performance of a new method to
optimize the energy management of a production system composed of six
cogeneration units using artificial intelligence. The optimization criterion is
economic and environmental in order to minimize the total fuel cost, as well
as the reduction of polluting gas emissions such as COx, NOx and SOx.
First, a statistical model has been developed to determine the power that the
cogeneration units can provide. Then, an economic model of operation was
developed: fuel consumption and pollutant gas emissions as a function of the
power produced. Finally, we studied the energy optimization of the system
using genetic algorithms (GA), and contribute to the research on improving
the efficiency of the studied power system. The GA has a better optimization
performance, it can easily choose satisfactory solutions according to the
optimization objectives, and compensate for these defects using its own
characteristics. These characteristics make GA have outstanding advantages
in iterative optimization. The robustness of the proposed algorithm is
validated by testing six cogeneration units, and the obtained simulation
results of the proposed system prove the value and effectiveness of GA for
efficiency improvement as well as operating cost minimization.
Keywords:
Cogeneration unit
Cost of operation
Gas emissions
Genetic algorithms
Optimization
Statistical model
This is an open access article under the CC BY-SA license.
Corresponding Author:
Othmane Maakoul
Laboratory of Renewable Energies Environment and Development, Faculty of Science and Technology,
Hassan 1st
University
Settat City, Morocco
Email: othmane.1992.mkl@gmail.com
1. INTRODUCTION
The global demand for energy has increased strongly in the last decades due to the industrial
revolution and the change of lifestyle. On the one hand, the reserves of fossil resources (oil, natural gas, coal)
are limited, and on the other hand, the use of these resources is responsible for the increase of greenhouse gas
(GHG) concentrations in the atmosphere, leading to global warming [1]. Among the solutions adopted to
ensure an energy transition to non-polluting and reliable energies, we find the cogeneration units that help to
approve the demand for electricity and heat during peak periods with a total efficiency that exceeds 80% [2],
[3], which will significantly reduce the electricity purchase bill following a decrease in the amount of
electricity purchased from the supplier. It is a technology that can work with several types of fuel and is
independent of climatic conditions [4], [5], unlike other renewable energies that are intermittent, such as solar
and wind power [6].
Optimization of power generation sources is becoming a critical necessity in order to increase the
efficiency of the generation system while minimizing losses and emissions of pollutants [7]. An optimized

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Int J Elec & Comp Eng, Vol. 12, No. 2, April 2022: 1131-1138
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generation system is one that allows for a shift from a demand-driven generation system to one based on
electricity supply, feed-in tariffs and electrical energy storage [8]–[10].
This work is structured by two parts: the first part proposes a statistical modeling of a cogeneration
system, which allows the determination of the net heat requirement, determination of the thermal and
electrical power according to different criteria: combination of a minimum cost of the system for a requested
power in order to better meet the requirements, while taking into account the place where they are installed
and the construction costs according to a worst case mission profile. The second part is dedicated to the
adaptation of an intelligent optimization strategy to manage the energy distribution of the different
cogenerators installed in an electrical grid, which must satisfy the needs of consumers while respecting the
various constraints related to the operation of the production sources (maximum power, total efficiency,
energy losses) [11].
There are several optimization methods of energy systems applied for the objectives of minimizing
the operating cost and the emissions of pollutants [12]–[14], but these methods may have some limitations
when solving an objective function and may not be applicable to some complex functions. A new solving
method has been developed that can overcome these limitations and is more robust, called genetic algorithm
(GA) optimization, it is a powerful method with strong adaptation in artificial systems, flexible, well-
balanced mechanism and is suitable for solving a single objective optimization problem (production cost
optimization, optimization gas emissions) in which the power grid is fed by the cogeneration plant.
2. RESEARCH METHOD
2.1. Studied energy production grid
The objective of the expected statistical model is to have a tool to help the dimensioning and the
evaluation of the efficiency of a cogeneration unit from an energetic, environmental and economic point of
view. This model should ideally be able to predict the behavior of any type of cogenerator, and be easily
parameterized, depending on the optimization algorithms developed. The Figure 1 shows the structure of the
energy production system constituted by cogenerations fed by different types of fuel.
Figure 1. Overview diagram of the production system
2.2. Statistical model of a cogeneration group
A statistical model is based on field measurements (daily, monthly, and annual energy consumption
profile). These models are empirical and do not require a precise knowledge of the system components or
their interactions [15]. For a cogeneration unit this model takes as input the probability densities of fuel
supply (fuel) and power demanded by the consumer load. They provide probabilities of fuel savings and
other data useful for economic-financial evaluations (maintenance costs, price per KWh produced,

Int J Elec & Comp Eng ISSN: 2088-8708 
Statistical modeling and optimal energy distribution of … (Othmane Maakoul)
1133
contribution to CO2 emission). Noting that these models are difficult to transport from one site to another and
do not allow for specificities such as energy storage or energy flows through the system to be taken into
account [16].
2.2.1. Determination of the net heat requirement
In the majority of cases, to obtain a cogeneration that makes the most of the energy produced, the
unit is sized according to the heat requirements, as the electricity produced can always be either consumed
internally or sold to the grid via a device for synchronization to the electrical grid [17]. At this stage, the net
heat requirement (BNeC) must be deducted, which is expressed by (1):
BNeC = ηboiler room. (Q − 𝐿𝑜𝑠𝑠𝑒𝑠) (𝐾𝑤ℎ/𝑦𝑒𝑎𝑟) (1)
with ηboiler room: boiler room energy efficiency, Q: annual fuel consumption, and Losses: energy wastage due
to poor insulation or poor regulation.
2.2.2. Determination of the electrical power of the cogeneration unit
Each "typical profile" of heat consumption corresponds to an optimal size to meet this essential
constraint "valorizing all the heat produced". The energy profiles can be influenced by different parameters:
seasonality, geographical location, consumer behavior, technical restrictions [11], [17]. This optimal size is
expressed by the factor Partcogen, which represents the share of the maximum heat consumption that can be
provided by the cogeneration unit. The thermal capacity of the cogeneration unit:
𝑃𝑄𝑐𝑜𝑔𝑒𝑛 =
𝐵𝑁𝑒𝐶 . 𝑃𝑎𝑟𝑡𝑐𝑜𝑔𝑒𝑛
𝑈𝑄
(KWq)
(2)
where UQ: the number of hours that a properly sized conventional heating system should operate at rated
capacity to produce the net heat demand.
The amount of heat supplied by the cogeneration unit is:
𝑄𝑐𝑜𝑔𝑒𝑛 = 𝑃𝑄𝑐𝑜𝑔𝑒𝑛. 𝑈𝑐𝑜𝑔𝑒𝑛 (KWhq) (3)
the two parameters of the pre-dimensioning, Ucogen and Partcogen, are thus known, with the Ucogen is the base of
the rectangle, represents the operating time of the cogeneration unit at full power. The annual quantity of
electricity produced by the cogeneration becomes:
𝐸𝑐𝑜𝑔𝑒𝑛 = 𝑃𝐸𝑐𝑜𝑔𝑒𝑛 . 𝑈𝑐𝑜𝑔𝑒𝑛 (KWhe) (4)
2.3. Optimization of the energy distribution of the studied grid
In the literature a number of methods for the optimization of energy flows [18]–[20]. For all
methods it is necessary to have a consumption profile during a reference period in order to make a calculation
of energy balance at all levels of the system. The energy produced by the grid must adjust in real time to the
consumption of users, depending on the power capacities, characteristics and potentials of the technologies
used [21], [22]. The optimization problem of our case study "the objective function" is the constrained
minimization of the total discounted operating cost and the minimization of the toxic gas emissions.
2.3.1. Problem statement and constraints
− Cogeneration system (CHP): The electrical energy produced by the cogenerator set noted PCHP' (t) will be
used to meet the demand and e−CHP
is the electrical efficiency of the conversion of the fuel into usable
electricity, and PCHP (t) is the total energy produced including electricity, heat and losses.
PCHP(t) = e−CHP
. PCHP′(t) (5)
− Supply/demand energy flow balance: The energy production system is designed to supply a complete
electrical load noted Pc(t), with Pc'(t) the total electrical power supplied by the studied grid. At each time
interval t the balance is expressed as in (6):
∑ P(t) = 0 ; Pc′(t) − Pc(t) = 0 (6)

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− Type profiles of an electrical load: In our case, we have adopted the data of electrical energy
consumption of a collective building, considered has a surface of 500 m² during a day (24 hours) of basic
test, with a step of one hour. Figure 2 shows the daily curve of electricity demand during the winter
season.
Figure 2. Demand energy data in KW
2.3.2. Optimization of the fuel cost function
The operating cost for each cogeneration unit for the different fuels used is usually modeled by an
objective function in second degree polynomial form in power output [23].
𝐶𝑡𝑜𝑡𝑎𝑙 = ∑(𝐶𝑖) = ∑(𝛼𝑖 + 𝛽𝑖. 𝑃𝑖 + 𝛾𝑖. 𝑃𝑖
2
) [$/ℎ]
6
𝑖=1
6
𝑖=1 (7)
Where 𝛼𝑖, 𝛽𝑖, 𝛾𝑖 are constant coefficients specific to cogenerator i.
To solve the problem of economic optimization of the production of electrical energy, i.e. to
minimize the total cost of the fuel necessary for this production, which is presented in the form of a non-
linear function, taking into account certain constraints of type equality and inequality. For the six
cogeneration groups, the power output (PCHPi) is bounded by an upper limit (PCHPi- max) and a lower limit
(PCHPi- min), which gives the inequality (9):
min 𝐶𝑡𝑜𝑡𝑎𝑙 = 𝑚𝑖𝑛 ∑(𝛼𝑖 + 𝛽𝑖. 𝑃𝑖 + 𝛾𝑖. 𝑃𝑖
2
)[$/ℎ]
6
𝑖=1 (8)
𝑝𝐶𝐻𝑃−𝑚𝑖𝑛 ≤ 𝑝𝐶𝐻𝑃−𝑖 ≤ 𝑝𝐶𝐻𝑃−𝑚𝑎𝑥 (9)
with i=1, 2, …6.
2.3.3. Optimization of the toxic gas emission function
In the first case study, we dealt with the problem of optimizing the emission function of toxic gases
to the environment, on a first test grid of 6 production units. This first case is devoted to the analysis of the
problem of the emission of toxic gases based on quadratic emission functions for the gases (NOx, COx, SOx)
treated individually and non-linear type for the case treated together, the technical characteristics of the
production units are presented in Table 1. For this problem, the formulation is the same as the fuel cost but
one needs to replace the cost coefficients with emission coefficients.
𝐸𝑡𝑜𝑡𝑎𝑙 = ∑(𝐸𝑖) = ∑(𝐴𝑖 + 𝐵𝑖. 𝑃𝑖 + 𝐶𝑖. 𝑃𝑖
2
)
4
𝑖=1
4
𝑖=1 (10)
0
20
40
60
80
100
120
140
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Hour
Demand energy (KW)

1135
Table 1. Specifications for the CHPS units
Fuel cost NOx emission parameters Cox emission parameters Sox emission parameters
CHP N i i i Ai Bi Ci Ai Bi Ci Ai Bi Ci
CHP1
CHP2
CHP3
CHP4
CHP5
CHP6
200
190
150
170
180
100
7.95
7.82
7.97
7.75
7.85
7.92
0.0016
0.0025
0.0031
0.048
0.040
0.050
81
30
300
550
320
50
-0.39
-0.81
-1.42
-2.41
-1.35
-0.38
0.0064
0.0065
0.0032
0.0070
0.0032
0.0062
5000
3900
1350
1900
1350
1200
-62
-30
-10
-13
-10
-120
0.27
0.15
0.11
0.11
0.12
0.40
52
185
510
170
515
125
6
4
5
6
5
5
0.0013
0.0025
0.0015
0.0082
0.0014
0.0041
2.4. Optimization by a genetic algorithm
When solving an objective function by traditional optimization methods one can be limited to local
maxima (or minima), so these methods are not applicable to some complex functions. To solve these
problems a new optimization method has been developed that can overcome these limitations; called
population-based genetic algorithms find near-optimal solutions to difficult optimization problems [24], [25].
Genetic algorithm (GA) is an optimization technique based on the principle of natural selection for
reproduction and for various other operations such as crossover and mutation [26]. Genetic algorithms aim to
define an approximate solution to an optimization problem in a reasonable time, when there is no exact
method to solve it or the duration of the computational phase is too long on the human life scale [27].
In order to optimize a problem with an unknown solution, a set of possible solutions is randomly
generated. This composition is called "the population" [28]. The variables in a GA are then used in gene
sequences that will be combined with other genes to form chromosomes, and then individuals [29]. Each
solution is represented by an individual; this individual is evaluated and then classified according to its
similarity to the best solution to the problem. As in biological systems, the best individuals in the population
are those that have a better chance of reproducing and giving some of their genes to the next generation. A
new population (generation) will be created by arranging the genes of the parents [30]. Some individuals in
the new generation have the best characteristics of both parents, so they will be better and present a better
solution to the problem.
3. RESULTS AND DISCUSSION
In order to evaluate the performance of the algorithmic optimization method, we adopted a test
system with 6 cogeneration units powered by different fuel types. The objective functions of our system
(operating cost, toxic gas emission) of each cogeneration unit are chosen as a quadratic function. The
proposed method for optimization by an AG was implemented in MATLAB. The code represented by the
binary format is 16 bits long for each generator. The distribution of the energies of the CHPs: in this case, we
applied our optimization program to a mono-objective minimization of the objective functions: operating cost
and gas emissions.
Figure 3 shows the optimal energy distribution of the six CHP units to meet a specified demand.
This distribution corresponds to the lowest possible operating cost of the system. At this point the simulation
results obtained under ToolBox/MATLAB concerning the minimization of the cost function are optimal
values respecting the equality/inequality constraints of the powers generated by the six CHPs. Figures 4-6
give an acceptable result of the minimization of emission functions as a function of the power generated by
the cogeneration units.
Figure 3. Repartition with minimization of the
operating cost
Figure 4. Repartition with minimization of NOx
0
5
10
15
20
25
1 3 5 7 9 11 13 15 17 19 21 23
PCHP1 PCHP2 PCHP3
PCHP4 PCHP5 PCHP6
Power (KW)
hour 0
5
10
15
20
25
1 3 5 7 9 11 13 15 17 19 21 23
PCHP1 PCHP2 PCHP3
PCHP4 PCHP5 PCHP6
Power (KW)
hour

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Figure 5. Repartition with minimization
of COx emissions
Figure 6. Repartition with minimization
of SOx emissions
The graph in Figure 7 gives the final value of the optimal operating cost optimized for a power
demand of 130 KW is obtained equal to 990 dollars (minimum in this case). And the 2nd one at the bottom
shows the number of the best individuals corresponding to each variable of the function. For NOx, COx and
SOx emissions are shown in Figures 8, Figure 9, and Figure 10, the optimal values obtained are 1331, 14700,
1557 (Kg/h) respectively, which are acceptable results of the minimization of the emission functions.
Figure 7. Cost function optimization Figure 8. NOx emission optimization
Figure 9. COx emission optimization Figure 10. SOx emission optimization
0
5
10
15
20
25
1 3 5 7 9 11 13 15 17 19 21 23
PCHP1 PCHP2
PCHP3 PCHP4
Power (KW)
hour 0
5
10
15
20
25
1 3 5 7 9 11 13 15 17 19 21 23
PCHP1 PCHP2 PCHP3
PCHP4 PCHP5 PCHP6
Power (KW)
hour

1137
4. CONCLUSION
The present work is a contribution to the optimization of the energy distribution of an electrical
network constituted by cogeneration groups by applying a new meta-heuristic algorithm, called genetic
algorithm, whose objective functions is to minimize the total cost of operation of the system as well as the
toxic gas emissions. The results obtained by simulation show economically acceptable performances, with a
strict respect of the technical conditions at the active power produced.
In the continuation of this research work, it is necessary to improve the studied power generation
system either from the point of view of modeling the cogeneration units and from the point of view of
optimization by using genetic algorithms. These improvements must also take into account the load losses in
the CHPs. In order for the algorithm to be fully self-adaptive, it is essential to incorporate an economic model
to account for fuel costs, energy sold into the power grid, and maintenance costs. In this way, it might not
even be simply static, as is currently the case, but evolve with research progress.
ACKNOWLEDGEMENTS
The research team would like to thank the National Center for Scientific and Technical Research, for
its funding of the project "Study of appropriate technologies for the conversion of organic waste and biomass
into renewable energy and sustainable bio-fertilization".
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