References
environments: i. Irradiation on the solar panel = 800W/m2. ii. Wind velocity = 1 m/s. iii. Air temperature = 20°C. iv. Inclination angle (open rack mounted) = 45° from the horizontal. Equation (2.3) gives the relationship between the temperature of solar, the ambient temperature, the nominal cell operating temperature and the global irradiation; Ts= Tamb + ( NT ? 20 800 ) Girr (2.3) where Ts is the temperature of solar, Tamb is the ambient temperature, NT is the nominal operating cell temperature (20oC) and Girr is the global solar irradiance. 2.2.2 Mathematical Modelling of Wind Turbine The instantaneous output power Pwt(t) from the wind turbine depends principally on wind speed vw, which is dependent on weather conditions, and is expressed in Equation (2.4) as: Pwt(t) = { 0 if vw < vci 1 2 Cp (?,?) ?air Awt vw3 (t) if vci?vw?vrw Prated if vrw< vw< vco 0 if vw ? vco (2.4) where Pwt(t) is the instantaneous power output, vw is the wind velocity, vci is the cut-in wind velocity, Cp is the coefficient of power (related to ? and ?), ? is the tip speed ratio,? is the blade pitch angle, ?air is the air density, Awt is the swept blade area, vrw is the rated wind velocity and vco is the cut-out wind velocity. If the wind speed is above the rated speed of 9 m/s, the wind turbine operates at a constant output power however when the wind speed is below 3 m/s or above 20 m/s, the wind turbine stops operation [11]. 2.2.3 Mathematical modelling of Batteries Equation (2.5) expresses the output energy of the hybrid generator consisting of the photovoltaic array and wind turbine as: Ehyb(t) = Epvg(t) + Ewt(t) (2.5) where Ehyb(t) is the output energy from the hybrid generator, Epvg(t) is the output energy from the photovoltaic generator and Ewt(t) is the output energy from the wind turbine. The battery is said to be said to be in a charging process when output energy of the generator Ehyb(t) is greater than the energy consumption of the load Eld(t). This state of charge at the time t is expressed in Equation (2.6) as: SOC(t) = SOC( t?1 ) ?( 1 ?? ) + [ Ehyb(t) ? Eld(t) ?inv ] ? ?b_c (2.6) where SOC is the state of charge, ? is the hourly self-discharge rate, Ehyb(t) is the output energy from the hybrid generator, Eld(t) is the energy consumption of the load, ?inv is the efficiency of the Inverter (0.93) and ?b_c is the efficiency of the battery charge (0.65 - 0.85). SOC(t) and SOC(t?1) represent the charge level at time t and t-1 respectively. The battery is in discharge state, expressed in Equation (3.7), when the output energy of the generator Ehyb(t) is lesser than the energy consumption of the load Eld(t). During this discharge process, its state of charge at the time t is given as: SOC(t) = Soc( t?1 ) ? ( 1 ?? ) + [ Eld(t) ?inv ? Ehyb(t)] ? ?b_dis (2.7) where SOC is the state of charge, ? is the hourly self-discharge rate (0.13% daily),Ehyb(t) is the output energy from the hybrid generator,Eld(t) is the energy consumption of the load,?inv is the efficiency of the Inverter (0.93) and ?b_dis is the efficiency of the battery discharge. The batteries capacities Cbatt, modeled in equation 2.8, is required to satisfy the ELD during a determined number of days Nd. Nd is taken for one day in this study. DoD represents the maximum depth of discharge, which generally makes 70% [11]. Cbatt = Eld(t) ? Nd DoD ? ?inv ? ?b_dis (2.8) where Cbatt is the battery capacity, Eld(t) is the energy consumption of the load, Nd is the number of days, DoD is the depth of discharge, ELD is the electric load demand, ?inv is the efficiency of the inverter (0.93) and ?b_dis is the efficiency of the battery discharge. 2.3 Technical Analysis of the Photovoltaic-Wind-Battery Hybrid Energy System Equations (2.9) and (2.10) give the energy generated from the hybrid system as: Ehyb = Epvg + Ewt (2.9) { Epvg(t) = Ppvg(t) ??t Ewt(t) = Pwt(t) ??t Eld(t) = Pl(t) ??t (2.10) where Epvg is the energy generated from the photovoltaic system, Ewt is the energy generated from the wind turbine,Ppvg is the power generated from the photovoltaic system,Pwt is the power generated from the wind turbine, Eldis the energy consumption of the load,Pld(t)is the power consumption of the load and ?t is the step time of simulation. The maximum energy to be stored in the battery, known as the maximum state of charge (SOCmax), is necessary to be accounted and is as expressed in Equation (2.11) below; SOC(t) ? SOCmax (2.11) The maximum allowed energy to be extracted from the battery is known as the minimum state of charge (SOCmin) and is expressed below in Equation (2.12) as; SOC(t) ? SOCmin (2.12) For the battery system, if SOC(t) ? SOCmin then a percentage of the ELD will not be supplied. Modeled below in equation 3.13 is the loss of power supply . The loss of power supply represents the energy deficit which occurs when the energy generated from the renewable energy system and the energy stored in the battery system is not sufficient to supply the electric load demand [11]. It is expressed as; LPS(t) = Eld(t) ?inv?Ehyb(t) ?[SOC( t?1 ) ?(1 ??) ?SOCmin] ? ?b_dis (2.13) The LPS is given as a ratio, between 0 and 1, of all the LPS values over the ELD consumption values during a specific period of time. The loss of power supply probability, LPSP, is the measure of the reliability of the power supply system. It quantifies the probability or frequency with which the power supply is unable to meet the energy demand. Its function is given below in Equation (2.14) as: LPSP= ? LPS(t) T t=1 ? Pld(t) ?t T t=1 ? (2.14) An LPSP of 0 indicates perfect reliability, while an LPSP of 1 indicates complete unreliability. 2.4 Economic Analysis of the Photovoltaic-Wind-Battery Hybrid Energy System The total present costs for the photovoltaic, wind turbine and battery systems are expressed in Equations (2.15), (2.16) and (2.17) respectively. The total present cost is a summation of the initial capital, operational and maintenance costs associated with each of the individual systems. Cpv= Ccap pv + Co&m pv (2.15) Cwt= Ccap wt + Co&m wt (2.16) Cbat= Ccap bat + Co&m bat (2.17) where Cpv, Cwt, Cbat represent the total present cost for the photovoltaic, wind turbine and battery systems respectively. Ccap pv, Ccap wt, Ccap bat represent the initial capital cost for the photovoltaic, wind turbine and battery systems respectively.Co&m pv, Co&m wt, Co&m bat represent the operational and maintenance costs for the photovoltaic, wind turbine and battery systems respectively. For a specific year, the total present value of the system components, TPV, which is a summation of the total present cost for each of the individual system components (cost is country specific) is expressed in Equation (2.18) as: TPV= Cpv+ Cwt+ Cbat (2.18) Equation (2.19) expresses the capital recovery factor, CRF, which enables the conversion of the initial investment cost into an annual capital cost. The equation is as follows; CRF= i ( 1+ i ) t ( 1+ i ) t ? 1 (2.19) where t is the period of economic valuation and i is the interest rate. Equation (2.20) expresses the levelized cost of electricity . The levelized cost of electricity facilitates the selection of the best system architecture from among various possible hybrid system configurations based on the lowest cost of energy i.e. the lowest total investments costs in a possible renewable energy plant. The LCE is expressed as; LCE= TPV ? CRF Ehyb (2.20) 2.5 Modelling for Sizing optimization The main objective of the sizing optimization is the minimization of the loss of power supply probability and Levelized cost of energy of the hybrid photovoltaic-wind energy system. Therefore the 2 primary purposes of the proposed function are : i. Maximizing power reliability (minimizing LPSP) ii. Minimizing produced energy cost and directly the total present cost of the PWHS (minimizing LCE) Ascertaining the most optimal design size and energy flow management which are: photovoltaic, wind turbine and battery capacity, is the ultimate objective
