Optimal Repartition of Hybrid Renewable Energy Pv and Wind Systems
Abstract-This optimal
Souad Belhour
Abdelouhab Zaatri
Department of Physics University of Constantine1 Constantine, Algeria
[email protected]
Department of Mechanics University of Constantine1 Constantine, Algeria
[email protected]
paper
repartition
proposes of
the
renewable
determination energies.
The
of
the
general
problem can be formulated as a problem of optimal allocation of limited resources constrained to meet specific demands. Thus,
it
is
expressed
in
terms
of
linear
programming.
Examples of estimation of the energetic potential of renewable energy resources that can be extracted from a given site are derived for Photovoltaic panels and for wind turbines. Two types of situations are considered. The first type considers situations resource
where is
the
installed
continuous.
The
energetic
Simplex
capacity
linear
of
each
programming
method is adopted to solve this problem. The second type considers situations where the installed energetic capacity is only available by means of specific units (discontinuous). For instance, we may need to combine photovoltaic panels and wind turbines with specific capacities to meet an energetic demand in a specific site with a lowest cost. Therefore, determining
the
optimal
energy
to
be
installed
leads
of
determining the number of units from each source. This problem is formulated as an integer linear program where the objective
functions to be minimized is the
initial capital
investment and the decision variables are the numbers of units which should be pure integer numbers.
Keywords-On/off control, pulse width modulation, chopper design, optimal control
I.
INTRODUCTION
The topic of optimizing the integration of renewable energy sources in a complementary way is a very interesting but a challenging one both scientifically and technologically. The general issue of combining renewable energy sources we are considering can be stated as follows. In a specific site, given the capacities of some renewable energy sources and an energetic demand, how to determine the optimal repartition of these energies that meets the demand. We will consider the combination of two or more renewable resources such as photovoltaic energy and wind turbine energy. We will firstly address the problem in case where the energy to be allocated can be continuous. Secondly, we will address the problem in case where the energy to be allocated is available as specific discrete units. For instance, manufacturers may provide photovoltaic panels as units with a specific capacity characterized by its generated pick power. Manufacturer also may provide wind turbines as units with specific size and then with a specific capacity according to the wind characteristics in the considered site. Therefore, determining the optimal energy to be installed leads to
determining the number of units from each source which are required to meet the specified energetic demand. There have been few proposed approaches to solving this problem with and without taking into account energy storage systems such as batteries, diesel engines, hydrogen, etc. Among these approaches, we can notice linear programming, dynamic programming, genetic algorithms techniques, etc.[I, 2, 3, 4].Some software for analysis and optimization of hybrid energetic systems has been developed and are actually largely used such as HOMER, SOMES, RAPSIM, SOSIM, etc .[5, 6, 7]. Revise some relevant papers concerning the simulation and optimization techniques, as well as the tools existing that are needed to simulate and design stand-alone hybrid systems for the generation of electricity. In this study, we will address the first issue which concerns the case where the energy is continuous. As proposed in reference [1], we modeled the problem of resource allocation in terms of linear program and solve it with simplex algorithm. The second issue concerning the case of discrete units of energy requires integer numbers of renewable energetic units has been formulated in many ways. But the most used approach seeks to minimize the Loss of Power Supply Probability (LPSP). [8] Consider the case where they have photovoltaic panels and wind turbines as specific units. Thus, the problem consists of determining the numbers of PV panels, wind turbines, and batteries. Their formulation leads to a non linear integer programming problem. They solve the problem by using the LPSP in the framework of ant systems by minimizing the initial capital investment. [9] Present an optimal sizing method for stand-alone hybrid solar-wind system with LPSP technology based on genetic algorithms. It consists of determining, among other elements, the number of PV and the number of wind turbines. [10] Uses a hybrid system and the formulation determine the number of micro hydro, PV, wind turbines, and batteries by minimizing the life cycle cost. [11] The LPSP model is used and the lowest Levelised Cost of Energy is considered as the economical optimal configuration. The optimal configuration determine the number of PV modules, the capacity of wind turbine, and the capacity of battery bank. Following the simple but attractive formulation using linear programming to model and solve the problem of sources repartition without taking into account the storage issue, we address the problem of optimal distribution of renewable energetic hybrid systems where the energies provided by each source are constituted of specific units. We formulate the problem as an integer linear programming one by minimizing the initial capital
978-1-4799-5627-2/14/$31.00 ©2014
IEEE
investment under constraints such as ensuring the annual required demand. The formulation requires the estimation of the annual energy that should be provided by each renewable energy unit as well as its cost. To illustrate our analysis, we provide examples combining PV and wind turbines units. II. A.
Estimation ofannual photovoltaic Energy
ho : Reference height which is usually 10m
(1)
�.: Performance ratio (average value 0.72 -0.75) with 0.75 for an optimal orientation of the panel. Fo : factor that accounts for losses
(1000
Irradiance
under
standard
conditions
provided by the manufacturer. We can estimate the annual power produced by one photovoltaic panel according to (1) but we can also use an empirical rule, in a sunny zone like our site, by considering that one Watt of peak power produces an annual energy of about 2 KWh .Since PV modules which are available at our laboratory are made of Silicon crystalline cells and have a peak power Pc = 33 W; So, a single photovoltaic panel (one KWh / year.
Vo : is the known wind speed at the reference height ( ho
=10 m )
a:
is the power law exponent, which is usually taken as 117
In the location where the measures have been obtained, the average value of the wind speed over 27 years is about 2.5 m I s which means that the site is not very interesting for wind turbine exploitation. Nevertheless, there are some elevated sites which are more windy. Thus, if we consider our experimental small wind turbine built up in our laboratory with a blade diameter of 2 m and an operating speed of 4
�.: is the nominal power under standard conditions as
unit) produces about: 66
V : The wind speed at hub height h
efficiency factor Cp
Getl : Effective annual incident Irradiance.
W I m2 ).
(3)
h : The hub height
ENERGY RESOURCES ESTIMATION
The energy produced by a photovoltaic panel over a period of time depends on many factors but mainly on the surface of the panel, its pick power, the incident irradiance on its location and on season, hour of the day, weather conditions, shadowing, etc. The energy Eph produced by a photovoltaic panel over a period of time can be estimated by the following expression [12]:
Go
increasing the wind speed according to the following expression:
about: 83 kWh / year. The swept area of the wind turbine is 2 3.14m • However, since the power is varying with respect to the environment conditions (solar radiation, wind speed, temperature ...), there is no way to avoid energy storage or energy generation to compensate the lack of energy for certain moments. III. A.
To estimate the power generated by a wind turbine, we use the following expression [13]: PTurh;ne
1
="2 C
p
2
p;rR V 3
Where V is the wind speed, R
(2) the blade radius, Cp
the operating efficiency factor, p the density of air at sea level, which is about 1.2 kg I m 3 • The height of the turbine may improve the performances of the wind turbine by
Souad Belhour
then elevated only at 30 m height, we
estimated power of about: PTurb;ne = 38 Watts. If we assume that the wind turbine produces an expected power of 2190 hours per year , then it would produce
The surface of a panel
Estimation ofannual wind turbine Energy
20% at a reference average wind
obtain a wind speed of about 4.67 ml s. This gives an
is: 0. 42 m2 B.
m Is,
=
MODELING HYBRID SYSTEMS FOR CONTINUOUS CASES
Continuous Energy Case
Given the characteristics and the energy demand for a specific site, the problem under consideration is to determine the capacity of renewable resources to be installed in order to meet the demand. Such an approach was made by [I] to solve a problem of rural electrification based on the integration and distribution of renewable resources in India. We consider the renewable energy resourcesE; with the available capacity limit L; . We need to provide also the unit costs of each resource C; as well as the global demand D to be met. The general form may be that of a linear programming problem of minimizing a cost function
Zr under given constraints. The problem can be expressed as follows:
L Ej. Nj = D
minZ1 = ICi·Ei
IEi=D
B.
(i
=
Nt
Application
As an example, let's consider 4 sources of renewable energies: the photovoltaic energy Eph with the limit L ph and the wind turbine energy EWi with the limit Lwi' Let's also provide the unit costs Ci of each energy as well as the global demand D .According to the data provided in Tab. I, the problem can be expressed as follows: min Zr = 0.3 Eph+ 0 .08 EWi Eph+ Ew,+=3000 E h � 1650 p E � 1700 '"
(kWh /year)
(5)
(kWh /year) (kWh /year)
The optimal repartition of renewable energies and their percentage to participate to satisfy the global demand is presented in the "Tab I". TABLE 1. Energy resource Ei Eph E1Vi
OPTIMAL ENERGYREPARTlTlON
Data Unit Costs €lkWh 0.30 0.08 D =3000
Capacity Limit
Results Energy repartition
kWh/year
kWh/year
in %
1650 1700
1300 1700
43.3 56.7
(KWh/year)
MinZ =526
Knowing the characteristics of the energy and of the site, this technique enables to sizing the appropriate renewable energy system to be used. IV. A.
MODELING HYBRID SYSTEMS FOR DISCRETE CASES
Discrete Energy Case
The problem consists of determining the number of photovoltaic panel as well as the number of wind turbines. Therefore, it can be converted into an optimization program known as integer linear programming problem that requires pure integer solutions for the decision variablesN) andN2. A basic general formulation consists of minimizing a cost function Z1 while satisfying the demand D. The unit costs of each renewable unit are Ci and its annual energy production is Ei . The problem can be expressed as follows:
(6)
Nj � 0
(4)
ph, wi, ... )
i = 1,2,3,00'
Integers
Compared to linear programming problems, integer linear programming problems are more difficult to solve and have specific techniques for their resolution. In the last twenty years, the most effective technique has been based on dividing the problem into a number of smaller problems in a method called branch and bound. For simpler case with two decision variables, graphical representation can help to analyze and solve the problem if a solution exists. B.
A Case Study
Let's consider the following example of a small PV/Wind hybrid system. The estimated energy produced by a photovoltaic panel (one unit) iSE h =66 KWHyear and the p estimated energy produced by a wind turbine (one unit) iSE",;=83 KWHyear. If we assume a cost of 4$ per Watt of peak power, then the unit cost of a photovoltaic panel is around Cph=130 $ . On the other hand, the reduced cost of the type of wind turbine which has been built up (except the electrical part) at our laboratory can be estimated at about C2=100 $ . This leads to 2$ / Watt . For simplicity, we may only consider in this analysis the investment for capital cost of the hybrid system which may involve the number of PV panels (N) ) and the number of wind turbines (N2). This capital cost which is the objective function has to be minimized, therefore: min Z1 = C) .N) + C 2N2 ($ ) (7) If we assume an electrical demand D( KWH year); thus, to meet this demand, we need to use a certain number of PV panels and a certain number of wind turbines. This leads to satisfy the following constraint: (8) Ep/,N., + E,vi.N2=D (KWHyear) Note that, in integer linear programming problems, it may be difficult to find out a feasible solution that meets this equality constraint. So, one need to extend the field of feasible solutions by replacing the equality (8) by the inequality constraint such as: EphN1 + Ew;.N2 2': D (KWh / year ) (9) Moreover, since the number of PV panels (N1 ) and the number of wind turbines (N2) should be integers, therefore,
the problem can be basically expressed as a pure integer linear program as follows: minZj =Cj.Nj +C2.N2 Subject to
(10)
Eph.N [+Ewi.N2 >=D Nj ,N2 >=0
Search ofthe Solution Let's consider C1 = 130 $ and C2 = 100 $
the objective function Zr min Zr = l30.NI
+
,then
can be expressed as follows: IOO.N2
and E h =66 KWH year p EWi =83 KWH year and an electrical demand D of about3000 KWh! year; thus, the problem can be expressed assume
as a pure integer linear program as follows: minZr =130.N[ +100.N2 Subject to
(12)
66.N[ +83.N2 =3000 N[ ,N2 ;::::0 N[,N2 Integers
Within this form, the problem has no solution. But if we replace the equality constraint by an inequality as follows: min Z1 =130.Nj +100.N2 Subject to
(13)
66.N[ +83.N2 ;:::3000 : N[ ,N2 ;::::0 NpN2 Integers
The solution gives an objective function value =37 00 $ with Nj =0 photovoltaic panel and N[ 37 wind turbines. According to this result, the system will be only composed of wind turbines because a wind turbine is cheaper and produces more energy than a photovoltaic panel. =
D.
N1(PV Panels)
N2(Wind Turbines)
80 81 82 83 84 85
4700 4480 5540 No solution 3880 5100
20 16 38 II 6 30
21 24 6 II 31 12
To illustrate the search of solutions in integer linear programming, we present and discuss the graphical solution of the following program with the equality constraint andEwi =84. minZr =130.N[ +100.N2
(11)
Let's
Z($):objective function
a wind turbine)
From Tab.2, we may notice how much the integer program is sensitive to a small change with respect to E,Vi .
N[,N2 Integers C.
Ewi (cost of
Subject to
(14)
66.N[ +84.N2 =3000 N[ ,N2 ;::::0 N[,N2 Integers
The graphical �-�lysis is presented in "Fig. 1" where the N x-axis represents j and the y-axis representsN2. The red line represents the equality constraint ( 66 .N1 + 84.N2 = 3000 ). The blue lines represent the objective function ( Z = 130.N1 +100N2) which is narameterized by the valueZ. So, by increasing the value of Z while moving the blue line in the direction of the arrows, we seek to interest the red line in a point where bothN1 and N2 should be integers. If there are many points that meet this condition of integrality, we should select the point which corresponds to the minimal value of Z. In the example under consideration, the optimal value corresponds to the point M whereZ=3880$ withN1 =6 andN2 = 31 . If we consider the integer linear program given in (10), there will be no intersection between the equality constraint and the objective function that corresponds to a point with integer coordinates. 40
--1-
---.j--+--�--I-- �--+---�
Sensitivity o{the solution with respect to a unit cost
Let's analyze the sensitivity of the solution (if it exists) of the integer program with the equality constraint (12) with respect to a small change in the unit cost characterizing the annual energy production of a wind turbine unit. The table "Tab II". Shows the sensitivity of the solution with respect toE,,! TABLE II.
SENSITIVITY OF THE SOLUTION W.R.T E,,,
10
15 20 25 N1 (prctovoltaic panels)
30
35
40
Figure I: Graphical analysis of a solution in Integer Linear programming
V.
CONCLUSIONS
We have presented a formulation for optimally combining renewable energy sources in order to build up economical hybrid energetic systems in case where these energies are available in a continuous form or as specific units. The proposed models directly relate the group of input parameters which are the estimated power and annual energy production of each renewable energy source. These parameters depend on environmental conditions (wind velocity, solar irradiation, temperature), on geographical characteristics of the site (latitude, longitude, altitude), and on the design characteristics of each energetic system (rotor diameter and hub height of the wind turbine, peak power, panel orientation, etc.). In case of continuous energy, given an estimation of the annual energy production and a cost of each unit of the renewable energy source as well as an annual demand, the simplex algorithm outputs (if possible) the optimal repartition of each renewable energy resource. Knowing the energy and the characteristics of the site, this enable to sizing the renewable energy system to be used. In case of discrete units, given an estimation of the annual energy production and the cost of each unit of the renewable energy source as well as an annual demand, the program outputs (if possible) the number of units from each source to meet the demand while satisfying all the constraints. Both approaches are generic since they are adaptive to various sites, demands, and different energy sources. Illustrative examples have been given combining PV and wind turbines sources. However, solving integer programs is much difficult than solving linear programs. REFERENCES [ I]
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