Smart Grids and Micro-Grids. Umashankar Subramaniam

      Where

      (1.12)

      (1.13)

The above transcendental equations of V_t, R_se and R_sh need to be solved using either analytical or numerical approaches. Using the derived results, the parameters of I_LG and I_sat can be obtained from equations (1.4) and (1.5).

       1.2.2 Effect of Irradiance and Temperature

      The I_LG and I_sc are directly proportional to solar irradiance and temperature which can be expressed as below [15],

      (1.14)

      (1.15)

      where, G_stc and G are the standard and actual irradiance in W/m². Similarly, the open-circuit voltage is also varying as function of irradiance which can be expressed as,

      (1.16)

      The V_oc equation is a non-linear transcendental equation and also need to be solved using numerical methods for determining the parameters under dynamic condition. The procedure for obtaining this value using GS or NR technique is similar as that of five parameter estimation of SDM of SPV. For a given operating temperature, the I_sc and V_oc can be evaluated using the following expressions:

      (1.17)

      (1.18)

      Where T_stc and T denotes the standard and actual operating cell temperature in K. On the other hand, the light generated current as a function of temperature can be determined using (1.4) and is rewritten as,

      (1.19)

For any operating temperature and irradiance, the value of I_LG, I_SC and V_oc can be estimated using the equations (1.20) to (1.22) as below,

(1.20)

      (1.21)

(1.22)

      The thermal voltage (V_t) as function of cell temperature can be represented as,

(1.23)

      The diode dark current also called reverse saturation current as a function of cell operating temperature and irradiance can be represented using (1.6) and (1.23) as,

      (1.24)

       1.2.3 Estimation of Maximum Power Point

      For accurate evaluation of maximum power point of SPV panel, the initial values of V_mpp and I_mpp should be selected properly using the known values of V_oc and I_sc under STC. The variation of parameters like V_t(GT), R_se(GT), and R_sh(GT) with respect to operating temperature and irradiance can be deduced using equations shown below [15],

      (1.25)

(1.26)

      (1.27)

      To determine the voltage at MPP, (1.6) has been modified as a function of irradiance and temperature as represented below,

(1.28)

      Where,

      Also, the current at MPP can be deduced by rearranging (1.7) as a function of irradiance and temperature as,

(1.29)

      Where,

      Equation (1.28) and (1.29) are transcendental equation and the solution can be obtained through numerical techniques.

1.3 Numerical Techniques for Parameter Estimation

This section presents the numerical methods of Gauss-Seidel and Newton-Raphson to estimate the five parameters (A, R_se, R_sh, I_sat and I_LG) by solving the non-linear transcendental equations written for SDM of PV panel under STC and dynamic environmental conditions.

       1.3.1 Gauss-Seidel Technique

      The GS is an iterative technique for solving the non-linear transcendental equations. The generalized form of equation to be solved using GS can be represented as [15, 20]:

      (1.30)

      Where ‘x’ is the variable to be determined and ‘k’ denotes the number of iterations, x^k+1 is the new value obtained and x^k represents the old value. The algorithm converges if the absolute error of new and old values is less than the tolerance of 10^-6. In this work, the GS method is employed to determine the unknown parameters by solving the equations of (1.10), (1.11), and (1.13) to get the values of V_t, R_se, and R_sh with the input values of V_mpp, I_mpp, V_oc,