As shown, the SDM has five parametersIph, I0,Rs, A and Rsh which need to be
estimated from the data provided in the manufacturer’s datasheet or from exper- imental measurements. A variety of approaches are proposed in the literature to achieve this goal [37, 45, 56, 58, 59]. These techniques rely on analytical equations, iterative techniques, experimental measurements, least-squares or single-variable
optimisation approaches and artificial intelligence techniques [37, 45, 56, 58, 59].
The key parameters can be estimated at STC by considering the data available in the manufacturer’s datasheet or data from experimental testing. After determining these values, the temperature and irradiance dependencies, further discussed in Section 2.7, can be applied to the STC quantities.
In this section two key analytical methods for obtaining the parameters of the SDM will be described. Following this, a simple analytical technique to obtain the parameters of the ISDM will be discussed and then finally, a few other methods for parameter estimation described. These techniques rely on analytical equations and iterative techniques to converge to an appropriate set of parameters to model PV cell characteristics.
2.6.1
Analytical method one
In [45], a system of five equations is formed by considering the three remarkable points provided on the datasheet and two other key properties of the I-V and P-V characteristics. The three equations based on the remarkable points listed on the datasheet are given by (2.3) to (2.5).
Isc =Iph−I0 h expIscRs Vt −1i− IscRs Rsh (2.3) 0 =Iph−I0 h expVoc Vt −1i− Voc Rsh (2.4) Impp=Iph−I0 h expVmpp+ImppRs Vt −1i− Vmpp+ImppRs Rsh (2.5) where, Vt= ANsqKT is the junction thermal voltage.
The additional conditions include the MPP condition and the current slope con- dition at short-circuit and are given in (2.6) and (2.7), respectively.
dP
dV atM P P
dI
dV atIsc
≈ − 1
Rsh
(2.7)
To reduce the complexity of the algebraic manipulations required by this tech- nique, several simplifications are made in the analysis. These simplifications in- clude neglecting the ’−1’ term from the original SDM characteristic in (2.1), pro- ducing (2.8) and various other small simplifications throughout the process. The simplifications are generally justified by identifying that the component neglected is much smaller in value than what is retained in the equation and subsequently has a smaller effect on the results.
I =Iph−I0exp q(V +IRs) AKT − V +IRs Rsh (2.8)
A simplification process is outlined which reduces the system of equations to three implicit and two explicit equations. The Gauss-Seidel iterative technique can be applied to the implicit equations to solve within a certain tolerance and then the explicit equations can be solved to obtain the remaining parameters.
2.6.2
Analytical method two
A second analytical method [28] firstly assumes a diode ideality factor of 1 and defines equations for the light-generated and diode saturation currents. A re- lationship is then developed linking the resistances to consideration of the key properties of the MPP. In general, there should only be one unique pair of re- sistances (Rs, Rsh) which will result in the maximum calculated power matching
the maximum power at the actual MPP location. That is, a pair of resistances which leads to the condition
Pmax,m =Pmax,e =VmppImpp (2.9)
where, Pmax,m is the measured power calculated from the characteristic and
Pmax,e is the experimentally measured maximum power or that indicated on the
datasheet.
A relationship is defined which expresses the shunt resistance in terms of the series resistance and is given in (2.10). The resistances are solved by iteratively
incrementing the series resistance (as it should be small) until an appropriate shunt resistance is calculated.
Rsh = Vmpp(Vmpp+ImppRs) IphVmpp−I0Vmpp " exp q kT(Vmpp+ImppRs ANs −1 # −Pmax,e (2.10)
The light-generated current is approximated by the short-circuit current and the diode saturation current is given by (2.11) in this technique.
I0 = Isc exp qVoc ANskT −1 (2.11)
The basic parameters, that is Iph, I0 and A are determined in the algorithm
using the approximations and equations provided and then an iterative process is commenced to determine the most suitable resistance values. This process works by starting with a small series resistance (usually Rs = 0), evaluating what the
shunt resistance should be, and then determining the resulting characteristic. The characteristic provides a MPP voltage, current and power which can be compared with the anticipated values. If the maximum power does not match with the expected value, or the maximum power does not occur at the expected MPP location, the process is repeated for a small increment of Rs. The iterative
process continues until a suitable (Rs, Rsh) pair is determined which enables
the MPP to match the experimental MPP (or that from the datasheet) at the appropriate MPP location.
2.6.3
Analytical method three
The analytical method for the ISDM contains fewer parameters which need esti- mating [50]. The I-V characteristic of the ISDM is given in (2.12).
I =Iph−I0exp
V AKT
(2.12)
The approximation Iph = Isc is used in this method to reduce the number of
to determine the parameters by substituting the remarkable points in (2.12). As the ISDM characteristic is only a function of voltage, iterative techniques are not required to determine the parameters.
2.6.4
Other methods of parameter estimation
Other methods to enable parameter estimation of the SDM or the other analyt- ical circuit-based models include least-squares, and single-variable optimisation approaches. These methods are described below.
2.6.4.1 Least-squares
A non-linear least-squares optimisation approach based on a trust-region algorithm to determine the five parameters of the SDM is described in [60]. The process involves writing five functions similar to the approach in [45] based on the datasheet information, and then combining these functions to form a single function which can be minimised to determine the parameters simultaneously. Upper and lower bounds are specified for each parameter in this implementation. The process requires fewer iterations and computational time compared with other parameter estimation techniques.
2.6.4.2 Single-Variable Optimisation
A Single-Variable optimisation approach is described in [58] which reduces the optimisation process to only require the optimisation of a single parameter. This single parameter is the series resistanceRs. The process involves maximising the
coefficient of determination R2 between the experimental and model data. This
method has been tested on monocrystalline, polycrystalline and amorphous PV modules and demonstrates good robustness against noise.