Analytical Method to Estimate the Maximum

Power for a Photovoltaic Inverter System

Eduardo I. Ortiz-Rivera

Dr. Fang Z. Peng

Michigan State

University Dept. of

Electrical and Computer

Engineering

What is PV Inverter System (PVIS)?

It is a series connection of solar panels or

photovoltaic modules with a dc-ac power
electronics inverter circuit, to generate ac
voltage from a solar source.

The inverter is controlled by the maximum

power point tracking (MPPT) control, that it
can compensate for the reduction in output
power caused by the shadow covering the
photovoltaic modules. Also, the MPPT
controls the inverter to produce the
maximum power and the ac power to be
connected to the load or utility grid. Figure
1 shows a PVIS with the three principal
stages of operation.

Fig. 1. PVIS with MPPT for utility application

PV Dynamic Model Equations

(

)













−

+

⋅

−

+

⋅

⋅

⋅

⋅

⋅

−

⋅

⋅

=

b

V

b

V

I

I

V

I

V

i

i

1

)

1

(

exp

)

(

max

max

max

τ

γ

α

γ

τ

α

τ

α

(

)

V

i

V

b

I

V

I

dV

V

dI

τ

γ

α

γ

τ

α

+

⋅

−

+

⋅

⋅

⋅

⋅

−

=

max

max

)

1

(

)

(

)

(

(1)

(2)

(

)

V

i

V

b

I

V

V

I

V

V

I

dV

V

dP

τ

γ

α

γ

τ

α

+

⋅

−

+

⋅

⋅

⋅

⋅

⋅

−

⋅

−

=

max

max

)

1

(

)

(

)

(

)

(

(3)

( )

b

I

I

sc

1

exp

1

max

−

−

=

V

V

V

τ

γ

+

−

=

max

min

1

max

)

1

(

1

V

V

b

b

⋅

−

+

⋅

−

=

γ

α

γ

(4)

(5)

(6)

Fig. 2. I-V Characteristics for any intensity of light

Proposed Method Characteristics

The proposed method is named Linear

Reoriented Coordinates Method (LRCM). It is a
novel simple method to estimate (symbolic and
analytically), the optimal voltage (Vop) and the
maximum power (P

max

) for a PVIS for solar

distributed generation (SDG) using the I-V
Characteristic Curves.

LRCM can find an approximate symbolic

solution for the P

max

calculated by the MPPT.

The main idea is to find the I-V curve knee
point, see figure 5. It is the optimal current (I

op

)

and voltage (V

op

) that produces P

max

.

Linear Reoriented Coordinates Method

Unfortunately, it is not possible to find a

symbolic

solution

for

P

max

using

the

differentiation on (3). Hence LRCM can provide
at least an estimate solution for P

max

using the

following method:
Linear current equation using the I-V Curve

(

)













−

+

⋅

−

⋅

⋅

⋅

=

γ

α

γ

τ

α

1

1

)

(

max

V

I

V

IL

i

I

V

P

⋅

=

(7)

(8)



























⋅

⋅

⋅

−

+

⋅

+

⋅

⋅

−

+

⋅

⋅

=

max

max

)

1

(

ln

1

)

1

(

I

I

b

b

V

b

V

sc

ap

α

γ

α

γ

γ

α

γ

(9)

Now lets substitute (9) into (1) to obtain I

ap

then

using (3), an estimate of P

max

can be obtained.

More important a symbolic and analytical
solution is given for an exponential function
without the homeomorphism property.

ap

ap

op

op

I

V

I

V

P

⋅

≅

⋅

=

max

V

opt

15 V

I

max

= 2 A & V

max

= 25 V

VI Curve

Knee

V

max

P

max

≅

i(V)

IL(V)

Fig. 5. Linear Reoriented Coordinates Method

Proof of efficiency for LRCM

The inequalities (11) and (12) can be obtain

using geometric analysis from the Fig. 4.
P

max

can be seen as the maximum

rectangular area inside of the I-V curve
produced by (1). Using the fact that the P-V
Characteristic Curve has a unique P

max

hence P

max

is more or equal than the

estimate P

max

. The last statement to estimate

P

max

is shown in the figure 6.

max

0

)

(

)

(

max

P

V

V

I

ds

s

I

op

op

V

=

⋅

>

∫

max

max

V

315

.

0

⋅

⋅

>

⋅

≥

sc

ap

ap

I

V

I

P

The slope of the I-V Curve at the knee point is

approximated by the slope of the linear current
equation. We will approximate V

op

with V

ap

.

The current equations (1) and (8) are
differentiated and set equal to each other to
solve for V then the solution is V

ap

.

Fig. 3. P-V Curves for different b

with estimated Pmax curve

(10)

Fig. 4. P-V Characteristics for

different intensities of light

(11)

(12)

Fig. 6. Simulation to prove (12)

Fig. 7. Error Curve for Pmax

and Pmax estimated

LRCM Results and Conclusions

The main advantage of the LRCM is that only

the characteristic constants of the VI curve
(i.e. b, α , γ, V

max

) are required. The maximum

error for the estimation of P

max

is near to 3%

using the LRCM.

Approximate symbolic solutions are given for

P

max

,V

op

and I

op

. Numerical and approximate

symbolic solutions are found for a PVIS.

If V

ap

is a solution when (4) is equal to 0 then

we found the exact solution for P

max

.

The

LRCM is a novel simple but powerful method,
to estimate P

max

,V

op

and I

op

.

INV

MPPT

G

L

Ii

Vi

Io

Vo

S

PV

DC Power

AC

Power

Max Power