OPTIMIZATION OF THE SIZING OF HYBRID WIND, SOLAR PHOTOVOLTAIC, BIODIESEL AND STORAGE SYSTEMS USING THE INTEGER LINEAR PROGRAMMING METHOD.

This article presents a program for optimizing the sizing of hybrid energy systems, composed of wind turbines, photovoltaic modules and biodiesel generators, with battery storage, for the electrification of isolated sites. Technical-economic optimization is obtained using the integer linear programming (ILP) method. In order to validate and demonstrate the performance of the model developed and programmed in python language, a comparative analysis is performed by doing the same sizing with the HOMER software. The simulations have given, for an isolated site around the city of Lomé in Togo whose peak amounts to 65 kW: 162 photovoltaic modules of 340 W each, 152 batteries of 1 kWh,


ISSN: 2320-5407
Int. J. Adv. Res. 7 (6), 871-882 872 In addition, the ecological constraints due to global warming, leading to all the undesirable phenomena of climate change, require us to take an exploratory step in this direction so that our electricity production systems meet the desired ecological standards.
In the literature, we can find two methods used for the combination of renewable energies including (wind and photovoltaic) in the electrical networks, the deterministic method which is based on the analysis of a reduced quantity of positions estimated a priori as uncertain for which one examines the behavior of the electrical system and the probabilistic method which considers all the possible cases with their probability of circumstance in order to appreciate the danger of not respecting a constraint of the system (Seydou Ouedraogo and al., 2015; KANSARA B. and PAREKH B.R, 2011 ). It should be noted that currently, connection studies are based on deterministic cases by introducing a probabilistic determination of the means of production and consumption.
The optimization of hybrid renewable energy systems involves modeling the various energy sources of the system, defining the various technical and economic constraints of the system and the objective function of minimizing the total cost of the system. This leads to a mathematical model whose resolution leads to the sizing of the system.

Methodology:-
The sizing of the installation to be studied will be organized as follows: the definition of the optimization method adopted, the modeling of all the production sources of the hybrid system and the explanation of the objective function reflecting the economic optimization.

Definition of the optimization method:-
Integer Linear Programming (ILP) is a field of mathematics and theoretical computer science in which problems of optimization of a particular form are considered. These problems are described by a cost function with linear constraints and integer variables (LIBERTI L. and RUSLAN S., 2006).
An integer linear program corresponds to a system of linear equations and inequations (constraints) whose unknowns are positive or zero integer values and the coefficients are integers, with a function to be optimized (minimize), which is linear to real coefficients. An integer linear optimization program can be in matrix form according to the following formulation (BOUHARCHOUCHE A. and al. Values are the decision variables to which negativity constraints are applied; , and are constants whose different equations (inequalities) constitute the linear constraints. The determination of all the constants and decision variables is done by considering the energy consumption of the site and the modeling of the various energy generators of the system.

Architecture of the Hybrid Energy System:-
The hybrid energy system to be sized consists of photovoltaic panels, wind turbines, biodiesel generators and a battery storage system. Figure 1 shows the configuration of the power generation system and the connection of the generators to the different buses.
The system consists of two connection bus: AC and DC. The photovoltaic panels and the storage system are connected to the DC bus. Wind turbines and generators producing alternating current and voltage are connected to the AC bus. The two bus are connected via a bidirectional AC / DC converter. All loads at the isolated site are connected to the AC bus assuming that all electrical equipment is AC powered and that all non-AC equipment has its own transformer-rectifier.

Figure 1:-Configuration of the Hybrid Energy System
Modeling and sizing of different energy sources:-Any sizing of a hybrid system is required to satisfy the energy demand at each moment of the site. As the latter varies over time, it must be covered by a combination of different energy sources.

Requested power P d (t):-
The power required at a given time (relation 1) corresponds to the sum of all the powers of the electrical loads to be supplied on the site at that time. It leads to the first constraint, which is the response to demand.
where at a given time t considered, ( ) ≥ 0 is the wind power which depends on the wind speed and the number of wind turbines installed, ( ) ≥ 0 is the power of the photovoltaic field depending on the torque (irradiance, temperature) and the number of photovoltaic modules installed, ( ) ≥ 0 is the power of biodiesel generators according to the number of generators in operation, ( ) is the power exchanged between the batteries and the whole system depending on the number of batteries installed and ( ) ≥ 0 is the excess power that cannot be stored and will therefore have to be evacuated.

Modeling and sizing of the photovoltaic generator:-
The single diode model is the most classic model. It uses a current generator for modelling the incident light flux, a diode for cell polarization phenomena, two resistors (series and shunt) for losses ( Figure 2). This model allows us to calculate by equation (2) the instantaneous power produced by the modules from the temperature and irradiance data.
(integer value) is the number of photovoltaic modules to be installed, ( )(W/m²) is the irradiation at time t, (m²) represents the module surface and is the efficiency characterizing the photo-electron conversion rate of the module given by the relation (3): In this equation, is the reference efficiency of the photovoltaic module and represents the efficiency characterizing the influence of the load, the latter is equal to 1 in the case where the photovoltaic system operates in MPPT (Maximum Power Point Tracking) mode. β is the assumed constant temperature coefficient of the photovoltaic module, the cell reference temperature is represented by (°C), similarly, (t) is the cell temperature given by the relationship (4) : , , , and respectively represent the number of wind turbines to be installed, the rated power, the starting speed, the rated speed and the maximum speed of a wind generator.

Modeling and sizing of the biodiesel generator:-
The type of generator chosen is the one that provides variable power according to demand. This power is closely related to consumption as modelled in Figure 4.

Figure 4:-Linear modelling of biodiesel generator consumption
With a slight curvature, it is often replaced by a linear interpolation corresponding to a line. The generator utilization rate must always be higher than µ = 30 % in order to avoid excessive generator wear and tear in addition to excess fuel consumption (AZIB T. and al., 2016; KASERA J., and al., 2012) The equations for generator sizing are respectively the relations (6) and (7): is the number of generators potentially installed, ( ) the number of generators started at a time t and is the nominal power of a generator set. The relation (6) reflects the ability of groups to satisfy all demand in the event of total unavailability of other sources in the system.

Sizing of energy storage by battery:-
Storage capacity is the maximum amount of energy that can be stored in the batteries. It depends on the maximum energy and the number of batteries installed which is an integer variable. A battery cannot be charged or discharged at any power only because of its own characteristics and representing respectively the maximum discharge and charging power of the batteries expressed by (8).
With 0 the percentage of initial energy of the battery in relation to its maximum capacity ; 0 ≤ ( ) ≤ 1 is the pressure drop coefficient representing the percentage of energy automatically lost by the battery every hour ; 0 ≤ ≤ 1 is the charge / discharge efficiency coefficient.

Sizing of the converter:-
The converter chosen is bidirectional. It can therefore transmit energy in both directions (from the continuous bus to the alternative bus and vice versa). Its sizing amounts to finding the maximum power passed through it. The constraint associated with the converter is relation (10): , is not time-dependent because it is a fixed value and the number of converters is 1 because only one converter is used for the whole system. Initial cost of the system :-It corresponds to the purchase of the various elements (generators, supports, accessories), their transport and their installation and is expressed by the relation (12).

The cost of biodiesel fuel :-
Considering the hypothesis that the fuel consumption of the group is linear between µg and 100% of its nominal power as modelled in Figure 4, we obtain a straight line with a coefficient director and the ordinate at origin . being the price per liter of fuel, the total cost is given by the relation (16):  (19):

Simulation of the hybrid system:-
The optimal sizing model presented is now used with the manufacturers' technical data sheets to implement the algorithm using computer tools. To simulate the hybrid system, we used Python language using the Spyder interpreter (ANACONDA.ORG, 2018) and the PuLP library for constraint resolution. Spyder is extensible with plugins, includes support for interactive tools for data inspection and incorporates Python code-specific quality assurance and introspection tools, such as Pyflakes, Pylint and Rope.

Presentation of the IDE Spyder:-
The IDE Spyder integrates many libraries for scientific use: Matplotlib, NumPy, SciPy, IPython and PuLP. It is the latter that we used for the optimization of the sizing whose objective function and constraints have been detailed previously.

Presentation of the PuLP library:-
PuLP is a library for the Python programming language that describes mathematical programs by providing objects that can represent optimization problems and decision variables. Easily deployed on any system that has a Python interpreter because it is not dependent on any other software package, this library can be easily extended to additional solvers and is very useful for projects that require linear optimization capabilities. The PuLP library has many mathematical solvers and has been designed to solve many optimization problems by reducing them to linear functions with integer, real, boolean or mixed variables. It then fits well with the optimization problem posed and performs mathematical operations that other solvers do, such as CPLEX and Gurobi, which are not free.

ILP resolution flowchart:-
The most commonly used methods or algorithms in ILP resolution (LIBERTI L. and RUSLAN S., 2006; FEDDAOUI Omar, 2014) are : 1. the Gomory slice algorithm or secant plane method which consists in adding constraints to the pre-established problem and which depend on their structure, 2. the Branch and Bound method, which consists in splitting the solution domain into feasible regions bounded by borders aligned on integers: this separates the domains, and evaluates which region to explore first, 3. the method that combines the last two (Branch and Cut), 4. the Knapsack algorithm (backpack) used to solve problems with logical conditions whose decision variables are Boolean (0 or 1).

Presentation of the selected isolated site:-
The sizing was carried out for a locality near the city of Lomé in Togo with 100 households with five inhabitants per household excluding cooking and heating. The consumption profile of a household (working day and weekend) is shown in Figure 6. Weather data were collected on the SoDaPro website (Solar radiation Data, 2018). A study of the complementarity of solar and wind potentials has enabled us to judge that the chosen locality is favorable to the installation of a hybrid electrical system.

Sizing results:-
The sizing was carried out for a total duration of 25 years. The simulation was done over a week and then extended to the rest of the moments. Figure 7 shows the results of the sizing of the hybrid system by the ILP.

Analysis of the results:-
No wind turbine has been retained by the optimal sizing because it is expensive and the wind potential is relatively low. The photovoltaic field and the generator set each share almost half of the energy production. Since the fuel chosen is biodiesel and is inexpensive, the generator set remains the most important component of the hybrid system in both cases, as shown in Figure 8. . The same data were simulated to compare the two methods. HOMER's results are given in Figure 9 summarized in Table 1.  The main difference is in the use of generators in energy production. The dimensioning of the batteries is closely linked to this because the more the photovoltaic panels are used, the more consequent the storage device is and vice versa. As the fuel cost is the most significant among those of the project, the one found by HOMER is slightly higher than that of the ILP (Figure 10). The cost difference between the sizing by the ILP and that of the HOMER software was at the generator set level, more precisely at the fuel cost. The initial costs are relatively identical. There is a certain complementarity between maintenance costs and replacement costs. Indeed, the first is high for ILP and low for HOMER and vice versa in the second case. The grouped costs by component indicated by the two methods are similar except for the cost of the generator set, which we could have expected given the operating time and the energy share of this source in both optimization methods.

Conclusion:-
From comparative studies of the two sizing optimization methods, we can affirm that the linear integer programming method is effective. With a lower total project cost than HOMER software, this method has significantly reduced the economic weight, ecological impact and dependence of hybrid energy systems for autonomous production of generators. The second advantage of the ILP is its ability to find at each moment considered by the simulation, the most appropriate distribution of the different sources to satisfy the demand at that precise moment. A data mining algorithm in the rest of our future work will make it possible to identify the strategy for distributing the energy sources to be installed by an intelligent controller who will be responsible for choosing the sources to be combined at any time according to certain parameters. References:-