 Open Access
 Total Downloads : 21
 Authors : Nilesh Kumar Srivastava, Ramesh Kumar Thakur, Amarjeet K. Panday, Amresh Kumar
 Paper ID : IJERTCONV4IS02012
 Volume & Issue : CMRAES – 2016 (Volume 4 – Issue 02)
 Published (First Online): 24042018
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Mathematical Modeling of Power Generation by Solar and Wind
Nilesh kumar Srivastava, Ramesh Kumar Thakur, Amarjeet K. Panday and Amresh Kumar
Dept. of Electrical and Electronics Engg., RVS College of Engineering and Technology, Jamshedpur831012, INDIA
Abstract Todays demand of electricity goes on increasing day by day, but to meet such demand we have limited energy resources. So, we need to find or search for alternatives and finally we turn towards renewable or nonconventional energy to fulfill our electricity demand. In this paper, we present the mathematical models of power generation using solar and wind energies.
Keywords Solar power; wind power; mathematical model

INTRODUCTION
Need of energy plays an important role in human life. Energy in nature is in many forms like heat, light, kinetic, potential and electrical. In spite of all these we are mainly concerned about electrical energy. Demand of electrical power is increasing day by day. If we use coal for generation, then it is nonrenewable and it is also costly. The same applies for nuclear power. So, we choose an alternative source among the renewable resources such as solar, wind, tidal, geothermal etc. for the production of electricity. From all these renewable sources, solar and wind is available everywhere and so the power generation using them is better than other resources.
In this paper mathematical models of power generation using solar and wind are presented. The organization of the paper is as follows. The next section describes the mathematical modelling of power generation using wind energy. The mathematical model of power generation using solar energy is presented in the third section and finally the conclusions are presented in the fourth section.

MATHEMATICAL MODELLING OF WIND ENERGY
The factors on which production of electricity through wind is dependent are:

Output curve of power

Velocity of wind

Height of hub
The most suitable model for wind turbine power is:
Pwind = PRE*(Vw Vwci ) / (VWR Vwci) if Vwci< Vw< VWR Pwind = PRE if VWR< Vw,<VWEF
Pwind = 0 if VW< VWEF & Vw> VWEF
Where PRE = rated electrical power
Vwci = cutin wind velocity or speed
VWR = rated wind speed
VWEF = cut off wind speed
Cut in wind speed is relatively small for small scale wind turbines. So, even when wind speed is not very high the turbine will operate.
Speed of wind changes with the height. We have two laws for determining the wind speed at some vertical height. They are:

Log law

Power law
Here, we use power law for determining the vertical height which is as
Vh/Vrh = (Zh/Zrh)
Vh=wind speed at vertical height Vrh=wind speed at reference height Vrh= vertical height of tower
Zrh=reference height
=power law exponent generally it is taken as 1/7 when there are no specific site data



MATHEMATICAL MODELLING OF SOLAR ENERGY CONVERSION SYSTEM
The three main parts that composed Photovoltaic (PV) system is:

PV modules

PV array

Solar radiation absorbed by PV modules
We describe each term one by one for mathematical modelling:

PV modules
Performance of PV modules is a function of PV cell silicon, the temperature of solar cell and solar irradiances exposed on the solar cell. Regression parameter for maximum power output of PV module is
Fig. 1. Wind power vs. Wind speed
Fig. 2 Increase in Wind power vs. Tower Height
The fundamental equation governing the mechanical power capture of the wind turbine rotor blades, which drives an electrical generator is:
Pw=1/2*{air density(kg/m3)}*{area swept of rotor(m3)}3*{power coefficient(e)}*efficiency of AC/DC converter
The maximum value of e is 0.59. Its value generally depends on rotor speed to wind speed ratio denoted by
= w x r/v
where w is the angular speed of the rotor and r is turbine radius.
P= (G +)*(T +0.3375G) + G +
where
G = Total solar radiation absorbed by PV module in w/m2
T = Temperature around PV module
,, and are constant from result of PV modules

PV array
To meet the demand, a number of PV modules are connected in series and parallel connection. Series connection determines the DC output voltage and parallel connection determine the capacity of PV array output.
Vt = n x vp
Pt = m x vp x ip Where Vt= Total voltage output
n= no. of PVs connected in series vp= single PV voltage
Pt= total capacity of PV system
m= no. of PV connected in parallel ip= single PV current

Solar radiation absorbed by PV modules:
Solar radiation absorbed by PV modules is dependent upon the tilted angles between solar panel and solar radiation. Perez model is utilized to determine the diffused solar radiation on any tilted angles. Modern Perez gives better
result including all parameters like isotropic diffused radiation, horizon brightening etc.
We have to calculate sky clearance K1 and sky brightness K2:
K1= [(G1h + G2h)/G1h + 1.0413]/[1 + 1.0413]
K2= (G1h x ma)/G =(G1h x ma)/(G/cos) = G1h/G
Where ma is mass of air
K1 and K2 are used for reduction of brightness coefficient called Perez coefficient.
K1 and K2 helps to calculate brightness coefficient B1 and B2.
B1 = B11(k1) + B12(K1)K2 + B13(K1) B2 = B21(K1) + B22(K1)K2 + B23(K1)
Now we calculate solar diffuse radiation on the tilted surface. G1t= G1h x cos2(/2) x (1B1) + G1h x B1(a/c) + G1h x B2 x sin
Where (a/c) is used to determine angular location of circumsolar and is given by:
(a/c)=max[0,cos]/max[cos85, cos]


CONCLUSIONS
Power generation using renewable sources of energy is becoming increasingly important in the modern era. Mathematical models for power generation using these renewable sources would be of great importance for engineers. Two mathematical models, one for power generation using wind energy and another for power generation using solar panels was presented in this paper. The author intends to provide the mathematical models of other renewable sources in his future work.
REFERENCES
[1.] Sandeep k., Viay K. Garg., A Hybrid Model of Solar Wind Power Generation System, International Journal of Advanced Research in Electrical Electronics and Instrumentation Engineering, Vol. 2(8), 2013. [2.] Hongxing Y, Lin L, Wei Z., A novel optimization sizing model for hybrid solarwind power generation system, Solar Energy, Vol. 81, 2007.