Page 1

Power Loss Analysis of Grid Connection Photovoltaic Systems

T.-F. Wu, C.-H. Chang, Y.-D. Chang, and K.-Y. Lee

Elegant Power Application Research Center

(EPARC)

Department of Electrical Engineering

National Chung Cheng University

Ming-Hsiung, Chia-Yi, Taiwan, R.O.C.

E-mail: tfwu@ee.ccu.edu.tw

Abstract—This paper presents power loss analysis of

grid-connection PV systems, based on the loss factors of double

line-frequency voltage ripple, fast irradiance variation, fast dc

load variation, non-uniform solar cell characteristic, and limited

operating voltage range. These loss factors will result in power

deviation from the maximum power points (MPP). In the power

loss analysis, both single-stage and two-stage grid connection PV

systems are considered. The effects of these loss factors on

two-stage grid-connection PV systems are insignificant due to an

additional maximum power point tracker (MPPT), but it will

reduce the system efficiency typically about 3 %. The power loss

caused by these loss factors in single-stage grid-connection PV

systems is also around 3 %; that is, a single-stage grid-connection

PV system has the merits of saving components and reducing cost,

while does not scarify overall system efficiency. Simulation re-

sults with a MATLAB software package are presented to confirm

the analysis.

Index Terms -- power loss analysis, loss factor, single-stage

grid-connection PV system, and two-stage grid-connection PV

system.

I. INTRODUCTION

Photovoltaic (PV) grid-connection systems based on either

a single-stage or a two-stage configuration have been widely

studied. The grid-connection PV systems can draw maximum

power from PV modules and inject the power into utility grid

with unity power factor. However, the loss factors, such as

operational conditions, components, and grid voltage, will

deviate effective PV output power.

In a grid-connection PV system (GCPVS), PV power varies

with operational conditions, such as irradiance, temperature,

light incident angle, reduction of sunlight transmittance on

glass of module, and shading [1]-[6]. These factors have been

investigated in detail and the authors present diagnosis me-

thods to estimate the reduction of PV power. Moreover, a

special condition, snow coverage, has been also discussed [7],

which was compared with other coverage situations like

shading and dirt. Then, another loss factor, such as compo-

nents, solar cell serial resistance, and capacity loss in PV bat-

teries, has been reported [2], [8]-[10]. The above mentioned

loss factors are related to cells themselves, which do not cause

deviation from the maximum power points (MPPs) during

system operation. Even though a PV system is under a specific

operating condition, there are still loss factors like double

line-frequency voltage ripple from ac grid [11]-[16], and fast

irradiance variation [17], causing deviation from the MPPs

and also resulting in power loss. However, these factors have

not been taken into account in power loss analysis for both

single-stage and two-stage GCPVS.

This paper presents power loss analysis according to five

loss factors, which might deviate the operating points from the

MPPs. First, modeling of solar cells and numerical analysis

including I-V characteristics of PV modules to derive the ef-

fect of double line-frequency voltage ripple on both sin-

gle-stage and two-stage GCPVS are conducted. Then, power

loss analysis due to fast irradiance and dc load variation is

performed, which is based on the above derivation. Moreover,

non-uniform solar cell characteristics, which also cause power

loss, due to different MPPs in each PV module will be taken

into account. Finally, since the operating voltage range of PV

modules in a single-stage GCPVS might be directly applied to

dc electric appliances and grid-connection inverters, their

input dc voltage should be limited, such as 360 ~ 400 V. This

limitation will result in power loss, since the effect of tem-

perature and irradiance will drive the MPPs of the PV modules

out of the operating voltage range.

In a two-stage GCPVS, an additional boost converter can

alleviate the effect of the loss factors. However, the converter

will cause power loss typically about 3 % and increase com-

ponent count and cost. According to a preliminary analysis of

the above mentioned effects on a single-stage GCPVS, the

power loss is not significant and even smaller than 3 %.

Therefore, a single-stage GCPVS is feasible in dc distribution

and grid-connection applications, which will be analyzed in

detail in this paper.

II. MODELING OF PV ARRAY

This section models a PV array, consisting of 8 in series and

2 in parallel PV modules (sun power 315) for a single-stage

GCPVS or 4 in series and 4 in parallel for a two-stage GCPVS.

The power loss analysis based on the loss factors will be then

presented in the next section.

Output current ipv of the solar cell can be expressed as

⎡

−−=

satp scp pv

eInIni,1⎥

⎥

⎦

⎤

⎢

⎣

⎢

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

s

pv

n

v

kTA

q

(1)

where

,

1

T

1

T

exp

3

⎥

⎦

⎤

⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

⎥⎦

⎤

⎢⎣

⎡

=

kA

qE

T

T

II

r

gap

r

rrsat

current source Isc is the light induced current, np is the number

of solar cells in parallel, ns is the number of solar cells in series,

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q is the electric charge, k is the Boltzmann constant, T is the

temperature of solar cells (in degrees Kelvin), A is the diode

ideal factor (A = 1~5), Tr is the reference temperature of the

solar cell, Irr is the reverse saturation current at Tr, and Egap is

the band gap of the solar cell (in electro-volts) [18]. Table 1

lists parameters of a sun power 315 PV module [19], which are

used in (1). By tuning ideal diode factor A = 1.43 and energy

band gap Egap = 1.11 eV, a PV module can be then simulated

and compared with the commercially available results, as

shown in Fig. 1. It can be observed that the simulated results

are relatively close to the commercially available ones, which

can insure accuracy of the following analysis.

Table 1. Specifications of the 315 W solar module (sun power 315)

Parameter

Peak Power (+/-5%) Pmax

Rated Voltage Vmpp

Rated Current Impp

Open Circuit Voltage Voc

Short Circuit Current Isc

Temperature Coefficient of Voltage

Temperature Coefficient of Current

Value

315 W

54.7 V

5.76 A

64.6 V

6.14 A

-176.6 mV/°C

3.5 mA/°C

(a) (b)

Fig. 1. PV module I-V characteristics of (a) the simulated results and (b) the

commercially available results (sun power 315)

Fig. 2. Circuit configuration of a single-stage GCPVS

≈

Fig. 3. Conceptual current waveforms of isd, iin, and iL

III. ANALYSIS OF POWER LOSS

This section presents power loss analysis of both sin-

gle-stage and two-stage GCPVS, which includes the effects of

double line-frequency voltage ripple, fast irradiance variation,

fast dc load change, non-uniform solar cell characteristic, and

limited operating voltage range, 360 ~ 400 V. The analysis is

based on the simulated PV module model.

A. Double Line-Frequency Voltage Ripple

(A) Single-stage GCPVS

Circuit configuration of a single-stage GCPVS is shown in

Fig. 2, in which PV array output power is injected into ac grid

through an inverter. We assume that the system reaches a

maximum power point Pmpp under a specific irradiance. That is,

its PV voltage vpv and current ipv are constant, and equal Vmpp

and Impp, respectively. However, the effect of double

line-frequency voltage ripple on deviation of PV voltage away

from the MPP will cause power loss, which can be expressed as

(

1

s losssloss

ΔpT k-p kTp

+=

where

(

pvs pvmpps loss

i kTvP kTΔp

⋅−=

First, expressions for vpv and ipv are derived, and then the power

loss can be determined. Voltage vpv can be expressed as

(

1

pvs pvs pv

kTΔvT k-v) (kTv

+=

where

(

1

s pv

pv

C

( )

,0

mpp pv

Vv

=

and current iinv stands for the inverter input average current. PV

voltage vpv will vary with input capacitance Cpv, the difference

between current ipv and iinv, and switching period Ts.

For determining inverter input average current iinv, we as-

sume that this system has reached the MPP and is operated by a

Bang-Bang control and with a fixed frequency of 1/Ts. The

uperbound current command isd of the control sets a current

limit for inverter inductor current iL to insure a sinusoidal cur-

rent waveform injected to ac grid, as shown in Fig. 3. With an

approximation, inverter input average current iinv is close to a

multiplication of output current is and duty ratio d in every

switching cycle Ts, which can be expressed as:

(

ssss inv

kTd kTi kTi

≅≅

)()()(),

s loss

kT

(2)

)()().

s

kT

)()(),

s

(3)

)()()()()

[],1

s inv

i

s

s pv

TkTki

T

kTΔv

−−−=

)( ) ()

()()( )(

d

),

2

s

sLs sd

kT

kTi kTi

+

(4)

where

i

≈

i

sd

(kTi

, 1 . 1

kT

sds

i

≅

=

⋅

()(

k

), t

Δ

)

s

T

(

pv

kT

kT

−

i

s sd

(

L

s

)

+

)

(()

)

(),11

sLsL

T

) (

d

⋅

kΔii

−+

)(

(

)()()()

[],1

sssssss

ac

s

sL

kTd kTv kT kTvv

L

T

)(kTΔi

−⋅−−=

From the above equations, we can determine the turn-on time

interval as follows:

(

spv

vv

−

where Lac is the inverter inductor and vs is the ac grid voltage.

From (1) to (5), power loss ploss deviating from the MPP during

one line period (1/60Hz) can be calculated.

)

()()

[],

sLs sd ac

ss

kTi kTiL

T kTd

−

≅

(5)

(B) Two-stage GCPVS

Circuit configuration, with a boost converter functioning as

an MPPT, of a two-stage GCPVS is shown in Fig. 4. The effect

of double line-frequency voltage ripple on the PV modules

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could become insignificant due to its filtering inductor Lb, PV

terminal capacitor Cpv, and dc-link capacitor Clink. Assuming

that boost switching cycle is the same as inverter switching

cycle Ts, the variation of PV voltage vpv is the same as (3)

except inductor current iLb. Determination of boost inductor

current iLb is illustrated in Fig. 5. Before the next perturbation

of maximum power point tracking, it is necessary to keep the

average value of iLb equaling MPP current Impp. Therefore, we

can adopt an average method to determine iLb where the av-

erage value of iLb should equal Impp during turn-on interval dbTs.

It can be expressed as

(

−Δ+−=

LbsLbs Lb

kiTki kTi1

(

1

links pv

L

where

(

1

b

L

( )

2

and

( )

,00

sb

b

L

From (6), we can have

(

(

ss pv

T kTv

⋅

Power loss analysis according to the loss factor of double

line-frequency voltage ripple can be attained from (1) to (8).

)(

(

)

)

()

)

(

k

)

T

(

(

)−

[

⋅

s

)

T1

)

(

()()

] ,T11

1

ssb

b

s

Tkd

vTkv

−−

−−−

(6)

)()

()()

()()

,1

1

ssb

s pv

s Lb

TTkd

Tkv

TkΔi

⋅−⋅

−

=−

In both single-stage and two-stage GCPVS, when the dc

load starts to absorb power, dc-link capacitor current ic will

drop and PV voltage vpv will drop correspondingly. PV voltage

vpv then will deviate from the MPP and result in power loss.

We assume that capacitors Cpv and Clink can hold enough

energy to supply the bi-directional inverter, and the required

power Pdc for dc load and rising time Trise from no load to the

full load are given. Thus, the expression for dc load power

pdc_load is shown as follows:

(

1

__ loaddcs load dc

Tkp kTp

−=

where

P

p

⋅=Δ

(7)

( ),

0

0

Lb

mppLb

i

Ii

Δ

−=

( )

mpp

Lb

Td

V

i

⋅⋅=Δ

)

()

)

.

2

bs Lb mpp

sb

L) (kTiI

kTd

⋅−

=

(8)

Fig. 4. Circuit configuration of a two-stage GCPVS with a boost converter

functioning as an MPPT

Fig. 5. Illustration of inductor current iLb varying with time

B. Fast Irradiance Variation

The output power of a PV array is strongly dependent on

irradiance and temperature, where irradiance would change

rapidly in a cloudy day. Under certain cloudy days, the varia-

tion can be dramatic and fast. A cloud passing over partial PV

modules will cause operating point deviating from the MPP

and resulting in power loss. Then, by setting the change of

irradiance ΔSi over one line period (1/60Hz) and combining (1)

with the analysis of double line-frequency voltage ripple, as

well as the effect of fast irradiance variation on PV voltage vpv,

power loss Ploss for the two GCPVS can be determined as

,/

lT loss

TEP

=

(9)

where

(),

0∑

=

k

lTT

(

kT

(

loss

(

s

kT

=

N

s lossT

kTEE

,

=

)

S

/

s

N =

E

Δp

)(

S

)

T

)()

),

()

[]

, 2/1

ss loss

p

s

kT

)

+

losss loss

TTk

(

Δp

)

)

ΔS

kTΔp

P

=

(

i

⋅−+

(

(

s pv

,

s

si mpp

−

s

kT kT

)

=

−

(

/1

isi

TkS

ET is the total energy loss in one line period, Tl is the line

period, and Eloss is the energy loss over one switching period Ts.

Moreover, for a boost MPPT converter in a two-stage GCPVS,

sampling period and current command perturbation are 1 ms

and 1 %, respectively.

C. Fast DC Load Variation

)()()

,

_load dcs

p

Δ+

(10)

,

_s

rise

dc

loaddc

T

T

and

p

Analysis of power loss due to fast dc load variation can be

accomplished from (10), and based on the derivation of double

line-frequency voltage ripple from (1) to (8).

( )

0 . 0

_

=

loaddc

D. Non-uniform Solar Cell Characteristic

In a PV array, the characteristics of solar cells might vary

from cells to cells, which will result in different maximum

power point voltage Vmpp and thus power loss. By tuning the

diode ideal factor A in (1) slightly, we can simulate a voltage

difference of about

5

±

V in a PV array. The power loss

caused by the characteristic mismatch can be expressed by

(

t mppo mpp loss

APAP CMp

−=

where Ao is the original diode ideal factor, and At is the tuned

diode ideal factor. With equations (1) and (11), comparison of

the power loss due to this loss factor in a single-stage and a

two-stage GCPVS can be then simulated.

)() ( ),

(11)

E. Limited operating voltage range

In a single-stage GCPVS, operating voltage range of PV

arrays must be limited due to finite ac grid voltage. For in-

stance, the peak value of 220 Vac grid voltage is close to 311 V,

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so that the MPP voltage of PV arrays should be higher than

311V for possible grid connection. Therefore, we set the MPP

voltage range, from 360 to 400 V for a single-stage GCPVS. In

general, MPP voltage changes with irradiance and temperature.

Therefore, from I-V characteristic equations of the cells, the

power loss caused by the limited operating voltage range can

be determined as

(

,,

,L pvi mppLloss

PTSPP

−=

where Pmpp(Si,T) is the MPP under certain irradiance and tem-

perature, and Ppv,L is the PV output power under a limited op-

erating voltage range.

)

(12)

IV. SIMULATION RESULTS AND DISCUSSION

This section shows simulation results of the power loss

analysis due to the loss factors, and the plots of power loss

versus different operating conditions such as irradiance,

dc-link capacitance, and PV-side capacitance. In addition,

effects of the loss factors on both single-stage and two-stage

GCPVS are discussed.

A. Simulation of the Power Loss Analysis

(A). Double Line-Frequency Voltage Ripple

For the PV module model shown in section II, the operating

conditions, irradiance Si = 1000 W/m2 and temperature 60 °C,

are selected. In both a single-stage and a two-stage GCPVS,

maximum power Pmpp of PV arrays is about 4.5 kW. The range

of PV capacitor Cpv has six levels from 470 μF to 4700 μF in a

single-stage GCPVS. According to the above power loss

analysis of double line-frequency voltage ripple with MAT-

LAB, the waveforms of vpv and ppv can be simulated, as shown

in Fig. 6, in which the system is operated with Cpv = 2000 μF.

Then, by calculating the voltage ripple of vpv and the power

loss, plots of Cpv versus voltage ripple and power loss are

illustrated in Fig. 7. When PV capacitor Cpv is larger than 1000

μF, voltage ripple is lower than 9 % and power loss is below

0.6 %; that is, for a 4.5 kW PV system, capacitance above

1000 μF is high enough to alleviate the effect of double

line-frequency voltage ripple.

For a two-stage GCPVS, dc-link capacitor Clink is fixed at

2000 μF, and the range of Cpv varies from 47 to 470 μF. Fig. 8

shows the ripple waveform of dc-link voltage vlink, PV voltage

vpv, and PV output power ppv with Cpv = 470 μF. The voltage

ripple of vpv and the power ripple are approximated to be zero,

hence the ripple effect can be even ignored. Finally, the plots

of Cpv versus the percentage of PV voltage ripple and power

loss are shown in Fig. 9. The component of the power ripple is

almost due to switching-frequency. Therefore, we can select

PV capacitor Cpv = 100 μF, which is sufficient enough to filter

out the effect of double line-frequency voltage ripple.

(a) (b)

Fig. 6. Ripple waveforms of (a) PV voltage vpv and (b) PV output power ppv in

a single-stage GCPVS when PV capacitor Cpv is fixed at 2000 μF

0.00%

5.00%

10.00%

15.00%

20.00%

470u 630u1000u2000u 3300u 4700u

PV Capacitor Cpv(F)

Voltage Ripple (%)

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

470u 630u 1000u2000u 3300u4700u

PV Capacitor Cpv(F)

Power Loss (%)

(a) (b)

Fig. 7. Plots of PV capacitor Cpv versus (a) PV voltage-ripple percentage and

(b) power-loss percentage in a single-stage GCPVS

(a) (b)

Fig. 8. Ripple waveforms of (a) PV voltage vpv and (b) PV output power ppv in a

two-stage GCPVS when PV capacitor Clink is fixed at 2000 μF and Cpv is 470μF

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

47u100u 220u330u470u

PV capacitor Cpv(F)

Voltage Ripple (%)

0.00%

0.01%

0.01%

0.02%

0.02%

0.03%

0.03%

0.04%

47u100u 220u 330u470u

PV capacitor Cpv(F)

Power Loss (%)

(a) (b)

Fig. 9. Plots of PV capacitor Cpv versus (a) PV voltage-ripple percentage and

(b) power-loss percentage in a two-stage GCPVS

(B). Fast Irradiance Variation

Assuming that fast variation of irradiance ΔSi during one

line cycle (1/60 Hz) is 100 W/m2 and the PV array temperature

does not change. Moreover, the operation conditions, Cpv =

2000 μF for a single-stage GCPVS, and Clink = 2000 μF and Cpv

= 330 μF for a two-stage GCPVS, are selected. According to

the above power loss analysis of fast irradiance variation, the

ripple waveforms of PV output power ppv can be simulated as

shown in Fig. 10. The power loss in a single-stage and a

two-stage GCPVS are 0.16 % and 0.11 %, respectively. It can

be seen that the operating power do not deviate from the MPP

significantly; that is, this effect is not significant.

(C). Fast DC Load Variation

For common

air-conditioners consume high power above 500 W per set and

others such as lamps and computers just consume low power

below 300 W. Assuming that a set of dc equipment includes

two air-conditioners (maximum power is 2000 W), ten lamps

(about 400 W), two computers (about 500 W) and the rising

times of their power are 3 min, 1 ms, and 0.5 ms, respectively.

That is, the fastest variation is 500 W/0.5ms. Moreover, the

dc home appliances, some like

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operating conditions, Cpv = 2000 μF for a single-stage GCPVS,

and Clink = 2000 μF and Cpv = 330 μF for a two-stage GCPVS,

are selected. The simulated results can be obtained from (10)

with parameters Pdc = 500 W and Trise = 0.5 ms, as shown in

Fig. 11 and Fig. 12. The power loss caused by a deviation from

the MPP under fast dc load variation is 0.34 % in a single-stage

GCPVS and 0.005 % in a two-stage GCPVS.

(D). Non-uniform Solar Cell Characteristic

With the parameters shown in (1) and setting an ideal diode

factor A = 1.43, one can simulate a PV module to fit a real one.

Additionally, we assume that the voltage difference of about

± 5 V at Si = 1000 W/m2 and 60 °C for three PV arrays, A, B,

and C, where array B is the reference model which is kept with

the original PV modules, while the operating voltage of arrays

A and C are Vmpp,B ± 5 V. By tuning the factors, AA = 1.56 for

array A and AC = 1.31 for array C, which can achieve the

voltage difference of about ± 5 V, the simulation results for

both single-stage and two-stage GCPVS can be obtained and

are shown in Table 2 and Table 3, respectively. The power

deviation in arrays A and C from array B is around ± 2.5 %,

and the difference between single-stage and two-stage GCPVS

is insignificant.

(E). Limited Operating Voltage Range

As mentioned above, operating voltage Vmpp deviating out

of the limited range, 360 ~ 400 V, will cause power loss. In the

following, the power loss versus irradiance and temperature

under the limited voltage range is investigated and plotted.

Notice that seldom are occurring operating conditions re-

moved from the plots, since some temperature levels could not

happen under certain irradiance, such as 25 ~ 30 °C under Si =

700 ~ 1000 W/m2, 50 °C under Si = 100 ~ 200 W/m2, and 60

°C ~ 70 °C under Si = 100 ~ 300 W/m2. With simulation, plots

of irradiance versus MPP voltage and plots of irradiance ver-

sus power loss are illustrated in Fig. 13. The power loss due to

the limited operating voltage range is about 0 ~ 2 %, and the

maximum one under 600 W/m2 and 25 °C is 2.05 %.

B. Efficiency Comparison

In a two-stage GCPVS, the loss factor, limited operating

voltage range, does not result in power loss, since a PV array is

not directly connected to a grid-connection inverter. As a

result, the power loss due to the factor is set to 0 %. Table 4

shows the operating conditions for the two systems. A boost

converter loss typically about 3 % is considered. Then, ac-

cording to the above simulation, the power loss due to the loss

factors under the discussed operating conditions are shown in

Table 5. It can be seen that the total loss in a single-stage

GCPVS is smaller than that in a two-stage GCPVS.

(a) (b)

Fig. 10. PV power ripple in (a) single-stage GCPVS and (b) two-stage

GCPVS under fast irradiance variation of 100 W/m2/16.6ms

(a) (b)

Fig. 11. Ripple waveforms of (a) PV voltage and (b) PV output power under

fast dc load variation in a single-stage GCPVS

(a) (b)

Fig. 12. Ripple waveforms of (a) PV voltage and (b) PV output power under

fast dc load variation in a two-stage GCPVS

Sun Power 315 (mono)

340

350

360

370

380

390

400

410

420

430

440

100 200300 400500 600700800 9001000

Insolation (W/m^2)

MPP Voltage (V)

25℃

30℃

40℃

50℃

60℃

70℃

Sun Power 315 (mono)

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

100 200300 400500 600 700800 900 1000

Insolation (W/m^2)

Power loss (% )

25℃

30℃

40℃

50℃

60℃

70℃

Fig. 13. Plots of irradiance versus the (a) MPP voltage and (b) power loss

under the limited operating voltage range of 360 ~ 400 V

V. CONCLUSIONS

This paper has presented numerical analysis with MAT-

LAB to simulate the power loss caused by deviation from the

MPPs for both a single-stage and a two-stage GCPVS. The

deviation is primarily due to the loss factors of double

line-frequency voltage ripple, fast irradiance variation, fast dc

load variation, non-uniform solar cell characteristic, and li-

mited operating voltage range. These loss factors are almost

independent so that the overall power loss can be summarized

from each individual one. Analysis and simulation procedure

of the power loss has been described, which provides engi-

neers a guide for building a PV array model and performing

the power loss analysis. According to the loss analysis, the

total power loss in a single-stage GCPVS is close to a

two-stage GCPVS, while the single-stage one can save a stage

of maximum power point tracker. That is, from the viewpoint

of efficiency, cost and system size, a single-stage GCPVS is

feasible in dc distribution and grid-connection applications.

Table 2. Power deviation due to non-uniform PV array in a singe-stage GCPVS at 60 °C

Vmpp,C

(V) (W) (W) (W)

Irradiance

(W/m2)

Vmpp,A

(V)

Vmpp,B

(V)

Pmpp,A

Pmpp,B

Pmpp,C

Power Deviation

(Array A)

Power Deviation (Array

B: reference model)

Power Deviation

(Array C)

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