Bond graph modelling of a 4-parameter photovoltaic array

Figures & data

Table 1. Generalized variables.

Domain	Effort	Flow
Translational	Force	Velocity
Rotational	Torque	Angular speed
Electric	Voltage	Current
Hydraulic	Pressure	Flow rate

Figure 1. Fields that form a causal bond graph model.

Figure 2. Distributed current trajectories in a PV cell.

Figure 3. (a) Two-diode equivalent. (b) One-diode equivalent with a series resistor.

Figure 4. Bond graph of the one diode equivalent circuit.

Figure 5. Bond graph model of a PV system with assigned causality.

Table 2. Calculated values for the four parameters.

	$G=1000W/m^2$		$G=100W/m^2$
	Implicit	Explicit	Implicit	Explicit
$I_{ph}$	$9.0$	$9.0$	$0.9$	$0.9$
$I_0$	$2.8743\times {10}^{-8}$	$2.8744\times {10}^{-8}$	$2.8738\times {10}^{-9}$	$2.8744\times {10}^{-9}$
$R_s$	$0.2283$	$0.2283$	$2.283$	$2.2829$
$n$	$74.4129$	$74.4130$	$74.4122$	$74.4130$
Error	$1.415\times {10}^{-6}$		$1.085\times {10}^{-6}$

Figure 6. Flow diagram to get the four parameters.

Figure 7. 20-sim submodel of a PV source.

Figure 8. 20-sim use of the PV source.

Figure 9. MOSFET-based $V-I$ tracer.

Table 3. Comparison between the parameters calculated by 4-parameter methods and 5-parameter method.

Module	Method	$a\left(V\right)$	$I_{ph}\left(A\right)$	$I_0\left(A\right)$	$R_s\left(\mathrm{\Omega }\right)$
Solartec	4PE	1.86	8.74	1.859$\times {10}^{-8}$	0.19
S60MC-250	4PI	1.8679	8.74	1.858$\times {10}^{-8}$	0.19
	5P	1.628	8.745	9.672$\times {10}^{-10}$	0.2488
Canadian Solar	4PE	1.91	9	2.87$\times {10}^{-8}$	0.228
CS6P-255P	4PI	1.92	9	2.874$\times {10}^{-8}$	0.2283
	5P	1.62	9	8.36$\times {10}^{-10}$	0.2963
Kyocera	4PE	3.06	8.79	7.869$\times {10}^{-7}$	0.164
KD33GX-LFE	4PI	3.074	8.79	7.87$\times {10}^{-7}$	0.165
	5P	2.223	8.8	1.53$\times {10}^{-9}$	0.3625

Figure 10. $V-P$ curve comparative between 5-parameter and 4-parameter models.

Figure 11. $V-P$ Curves.

Figure 12. Photovoltaic module with partial shading.

Figure 13. Two instances of the PV_Source model to simulate two different sections of a module.

Figure 14. Shading effect on one section of the module.

Figure 15. Model involving a KD33GX-LFE PV module and a D-C machine.

Figure 16. Angular speed response of the D-C motor.

Figure 17. Proposed model interconnected with an MPPT controller.

Figure 18. Power in the PV module under the control of the MPPT.

Table

$a$	Five-parameter model modified ideality factor
$D_{in}$	Input data set
$D_{in,STC}$	Input data set under STC
$G$	Irradiance
$G_{STC}$	Irradiance under STC
$I$	PV current
$I_0$	Reverse saturation current
$I_D$	Diode current
$I_{MP}$	Maximum power current
$I_{MP,STC}$	Maximum power current under STC
$I_{ph}$	Photocurrent
$I_{SC}$	Short circuit current
$I_{SC,STC}$	Short circuit current under STC
$k$	Boltzmann constant
$n$	Ideality factor
$n_s$	Number of series cells in an array
$n_{se}$	Number of series cells in a section
$NOCT$	Nominal operating cell temperature
$P_{MP}$	Maximum power
$P_{MP,STC}$	Maximum power under STC
$q$	Electron charge
$R_p$	Parallel resistance
$R_s$	Series resistance
$T_a$	Ambient temperature
$T_C$	PV temperature
$T_{C,STC}$	PV temperature under STC
$V$	PV voltage
$V_D$	Diode voltage
$V_{MP}$	Maximum power voltage
$V_{MP,STC}$	Maximum power voltage under STC
$V_{OC}$	Open circuit voltage
$V_{OC,STC}$	Open circuit voltage under STC
${\alpha }_I$	Temperature coefficient of short circuit current
${\alpha }_P$	Temperature coefficient of power
${\alpha }_V$	Temperature coefficient of open circuit voltage
$\mathrm{\Lambda }$	4-parameter model exponential factor