Mathematical Modeling of Power Generation by Solar and Wind

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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, Jamshedpur-831012, 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 non-conventional 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


    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 non-renewable 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.


    The factors on which production of electricity through wind is dependent are:-

    1. Output curve of power

    2. Velocity of wind

    3. 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 = cut-in 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:

      1. Log law

      2. 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


    The three main parts that composed Photovoltaic (PV) system is:

    1. PV modules

    2. PV array

    3. Solar radiation absorbed by PV modules

    We describe each term one by one for mathematical modelling:-

    1. 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 +


      G = Total solar radiation absorbed by PV module in w/m2

      T = Temperature around PV module

      ,, and are constant from result of PV modules

    2. 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

    3. 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 co-efficient called Perez co-efficient.

    K1 and K2 helps to calculate brightness co-efficient 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 (1-B1) + 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]


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.


[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 solar-wind power generation system, Solar Energy, Vol. 81, 2007.

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