Maximum-Power-Point Tracking Method of
Photovoltaic Power System Using Single Transducer
Toshihiko Noguchi, IEEE, Senior Member, and Hiroyuki Matsumoto
Nagaoka University of Technology
1603-1 Kamitomioka, Nagaoka 940-2188, Japan
Phone: +81-258-47-9510, Fax: +81-258-47-9500
e-mail: tnoguchi@vos.nagaokaut.ac.jp
URL: pelab.nagaokaut.ac.jp/

Index
terms:
photovoltaic;
maximum-power-point
tracking; power estimation; voltage-source DC-DC
converter; current-source DC-DC converter.

I. INTRODUCTIONRESEARCH and development on alternative energy resources
to fossil fuels have intensively been promoted due to growing
concern on an environmental issue since 1990’s. Above all,a photovoltaic power generation system is one of the mostpromising environment-friendly solutions and has extensively
been used by various residential houses, rural power networks,
huge commercial power plants and so forth. Photovoltaic
power systems are capable to produce electric power with noCO2 emission and their energy resource is actually infinite,which is the most attractive point. Also, they can easily be
installed without very strict restrictions. However, lowefficiency and higher cost per unit power are the fatal
drawbacks of the systems, which prevents them from being
more and widely used, and how to overcome these drawbacks
is an important technical issue so far and toward the future.
A technique to utilize the photovoltaic effectively is known
as a maximum-power-point tracking (MPPT) method, whichmakes it possible to acquire as much power as possible fromthe photovoltaic. Since an electric characteristic of the

0-7803-7906-3/03/$17.00 ©2003 IEEE.

output power has a convex property with respect to theoperating voltage or current as shown in Fig. 1, there exists
only one optimum operating point on the power vs. voltage (orcurrent) curve. The MPPT is a method to let the controller
operate at the optimum operating point. There have beenvarious kinds of MPPT methods reported and the mostcommon technique of them is a hill-climbing method, whichseeks the optimum operating point by changing the operating
point until the maximum power point is found [1]-[4].
Therefore, this method essentially requires power calculation
using both the voltage and the current transducers, whichshould be reduced from a viewpoint of system simplificationand cost reduction.
This paper focuses on reduction of the count of suchtransducers. In order to achieve this goal, either a current
ripple or a voltage ripple, which is inherently generated by aswitched power converter, is effectively utilized in the
proposed system and the output power of the converter isestimated with an average value and ripple amplitude of thecurrent or the voltage detected in the converter. Applying theconventional hill-climbing method to the system, themaximum-power-point can be sought with only a single
transducer. The paper discusses a theoretical aspect of the
proposed technique, assuming a current-source DC-DC
converter or a voltage-source DC-DC converter is employed
as an MPPT controller, and presents experimental results todemonstrate excellent MPPT operations of the system.
20 (℃) 4000 (lx)20 (℃) 22000 (lx)60 (℃) 22000 (lx)1.6
1.4
Output Power (W)Abstract — This paper proposes a maximum-power-point
tracking (MPPT) method of a photovoltaic power systemwith less transducer count. A unique feature of thismethod is capability to seek the maximum power point,using only a single transducer implemented in a switched
DC-DC power converter, i.e., a current transducer or a
voltage transducer. Output power of the converter can be
estimated with an average value and ripple amplitude of
the reactor current or the capacitor voltage detected in theconverter. The output power obtained from only thecurrent or the voltage information allows seeking the
maximum power point on the basis of a common
hill-climbing method. In this paper, not only a theoreticalaspect of the proposed method is described, but also
several experimental results are presented to provefeasibility of the method.

1.2
1.0
0.8
0.6
0.4
0.2
0.0
0

50

100
150
Output Current (mA)

200

Fig. 1. Example of power vs. current characteristics.

2350

250

II. MPPT ALGORITHM WITH SINGLE TRANSDUCER

L

DA. Power Estimation in Voltage-Source DC-DC Converter
Fig. 2 shows a configuration of the voltage-source DC-DC
converter. The system consists of a photovoltaic, a boost
chopper with a single current transducer and a load resistor.
This circuit operates in switched modes and is regarded as a
non-linear system; thus analysis of the circuit is rather
complicated. Therefore, a state space averaging method isapplied to analyze the circuit operation in order to treat thecircuit as a linear system. In the linearization process, thephotovoltaic is regarded as an equivalent circuit composedwith a DC voltage source VPV and a series connectedinternal resistance RPV , which varies with the operating

C2

C1

PV
V PV , R PVSI L , ∆I L

Fig. 2. Photovoltaic and voltage-source DC-DC converter.
RPV

VPV

rL

C1

L

rS C2

vc

points, depending on irradiance as well as temperature.
Figs. 3 (a) and 3 (b) show equivalent circuits in an on-mode
and an off-mode of the switching device. In these circuits,rL , rS and rD represent an equivalent resistance that

(a)

1

 − RPV C1
 1
AON = 
 L

0


 1
bON =  RPV C1

1−
C1
rL + rS
−
L
0

RPV

VPV

vc

rD

C2

RL vo

iL

dx
= Ax + bVPV ,
dt

(3)

where
A = AON D + AOFF D ′T1− RPV C1
 1
=
 L

0



In a similar manner, when the switching device is turned off,the state variable equation of the converter is expressed asdx(2)= AOFF x + bOFFVPV ,dt

1
C1
rL + rS D + rD D ′
−
L
D′
−
C2
−


0 
D′  , and
−
L 
1 
RL C 2 T 1
0 0 .
b = bON D + bOFF D ′ =  RPV C1
In (3), D and D′ ( = 1 − D ) are duties of the on-mode and

where

 1
bOFF =  RPV C1

C1

L

(b)
Fig. 3. Equivalent circuits of voltage-source DC-DC converter.
(a) On-mode circuit. (b) Off-mode circuit.

0 
0  , and
1 
RL C 2 
0 0 .
1

− RPV C1
 1
AOFF = 
 L

0



rL

(1)

where

RL vo

iL

corresponds to losses dissipated in the reactor, an on-mode
resistance of the switching device and an equivalent resistanceof the diode that corresponds to its forward drop, respectively.
The following state variable expression is derived in terms ofa capacitor voltage vC1 , a reactor current iL and an output
voltage vo from the equivalent circuit illustrated in Fig. 3 (a)when the switching device is turned on:
dx= AON x + bONVPV ,
dt

RL

1
C1
rL + rD
−
L
1
−
C2
−


0 
1 
−  , andL 
1 
RL C 2 

the off-mode of the switching device, respectively.
Since dx / dt = 0 in the steady state, the average values of
the capacitor voltage VC , the reactor current I L and theoutput voltage Vo can be represented by the followingT
0 0 .
In the above equations, the state variable vector isT
x = [vC i L vo ] . Combining (1) and (2) by using the state
space averaging method, an averaged state variable expression[

can be obtained as follows, where x = vC

iL

vo

]

T

expression:
VC 
VPV
 
 I L  = rL + rS D + rD D′ + RPV + RL D′ 2 ∗Vo 
rL + rS D + rD D′ + RL D′ 2 


1
∗



R
L D′



is an

average state variable vector:

2351

.

(4)

VC1 ,∆VC1

On the other hand, steady sate ripple amplitude of thereactor current ∆I L is expressed as the following equation,
which is obtained by making product between (1) and an
on-mode time duration:
DTS
VPV
*
∆I L =
L rL + rS D + rD D ′ + RPV + RL D ′ 2,(5)* rS D + rD D′ − rS + RL D ′ 2(

L1

because they are manipulating quantities specified by thecontroller and that L , rS and rD are circuit parameters of

C1

PV
V PV , R PV)

where TS is a switching period.
The load power Wo can be estimated by (4) and (5) as
follows, paying attention to that VPV , RPV , VC , Vo and
RL are all unknown variables and constants:
Vo2
L
Wo =
=
∆I L I L + (rS − rD )D′I L2 .
(6)
RL DTSIt should be noted that D , D′ and TS are known variables

L2

D

S

RL

C2

Fig. 4. Photovoltaic and current-source DC-DC converter.

vC1
rL1

RPV

L1

C1

rL 2

iL1

L2

iL 2
rS

VPV

which rough values, e.g., nominal values, are preliminarily
available in the circuit design process. Therefore, (6) showsthat the output power can be estimated with information of
I L and ∆I L from only the current transducer.

C2

RL vo

(a)
vC1

B. Power Estimation in Current-Source DC-DC Converter
Fig. 4 shows a schematic diagram of a current-source
DC-DC converter, which is known as a Cük converter. The
system is composed with a photovoltaic, the Cük converterwith a single voltage transducer, i.e., an isolation amplifier,and a load resistor. In the same manner as the voltage-source
converter, state variable equations of the current-source
converter are represented by the following on-mode andoff-mode expressions; in the on-mode as illustrated in Fig. 5
(a),
dx
(7)
= AON x + bONVPV ,dt
where RPV + rL1 + rS
−
L1


0
AON = 

rS
−

L
2


0

1
bON = 
 L1

rS
L1
1
−
C1
rL 2 + rS
−
L2
1
C2
−

0
0
1
L2
0




0 
 , and
1 
−
L2 
1 
−
C 2 RL 0T
0 0 0 .


Also, in the off-mode shown in Fig. 5 (b),dx
= AOFF x + bOFFVPV ,
dt

RPV

rL1

L1

C1

iL1

rL 2

L2

iL 2

VPV

rD

C2

RL vo

(b)
Fig. 5. Equivalent circuits of current-source DC-DC converter.
(a) On-mode circuit. (b) Off-mode circuit.
1
bOFF =  L1T
0 0 0 .


In the above equations, the state variable vector is defined asT
x = [i L1 vC1 i L 2 vo ] where iL1 , vC1 , iL 2 and vo area current of the reactor L1 , a voltage across the capacitor C1 ,a current of the reactor L2 and the output voltage,
respectively. Therefore, the following expression can bederived on the basis of the state space averaging method,
where

[

x = i L1

vC 1

iL 2

vo

]

T

denotes an average statevariable vector of the converter:dx= Ax + bVPV ,
dt

(9)

where
A = AON D + AOFF D ′(8)rS D + rD D ′D′ R PV + rL1 + rS D + rD D ′
0 
−
−
−
L1
L1
L1


D′
D

0
0 ,
−
C1
C1

=

rS D + rD D ′rL 2 + rS D + rD D ′ D
1 
−
−
−

L2
L2
L2
L2 

1
1 

0
0 −
C2
C 2 RL 


where
1
rD
 RPV + rL1 + rD

−
−
0 
−
L1
L1
L1


1

0
0
0 
C1
 , and
AOFF = 1 rDrL 2 + rD
0
−
−
−

L2
L2
L2 

1
1 

0
0 −

C2
C 2 RL 

1and b = bON D + bOFF D′ = 
 L1

2352

T


0 0 0 .


2.5
Maximum
peak holder+-

Minimum peakholderAveragePower
Estimatorand MPPTController

2.0
Gate
SignalCurrent (A)iL or vC1

∆IL or ∆VC1

IL or VC1

Maximum peak

1.5
1.0

Current ripple

0.5
0.0

Digital Signal Processor

-0.5

(a)

-1.0
0.000

Minimum peak
0.001

Maximum peak holder

−
＋

＋

0.004

2.5
Maximum peak

2.0
Current (A)iL or vC1

＋

0.003

(a)

−

∆iL or ∆vC1

−

0.002
Time (s)

−

1.5
1.0

Current ripple

0.5
0.0

＋

-0.5
-1.0
0.000

Minimum peak holder

Minimum peak
0.004

0.008

0.012

0.016

0.020

Time (s)(b)
Fig. 6. Block diagram of proposed MPPT controller. (a)Configuration of whole controller. (b) Detailed schematic ofripple amplitude detector.

(b)Fig. 7. Responses of peak holders. (a) Response in case ofincreasing ripple. (b) Response in case of decreasing ripple.

Average values of the state variables in the steady state are
obtained by substituting dx / dt = 0 into (9) as described inIII. EXPERIMENTAL SETUP AND RESULTS
A. Configuration of Single Transducer Based MPPT System(10): I L1  
VPVVC1  =* I L 2  ( RPV + rL1 ) D 2 + rS D + ( RL + rL 2 ) D′2 + rD D′  Vo 

.

(10)



D2


′
r
D
r
r
R
D
+
+
+
(
)
S
D
L2
L

*


DD′


RL DD′


Also, steady state ripple amplitude of the capacitor voltage∆VC1 is given by the following equation:
D 2 D ′TS
∆VC1 =∗C1. (11)V PV∗( R PV + rL1 ) D 2 + rS D + ( R L + rL 2 ) D ′ 2 + rD D ′Canceling out the unknown variables and constants, i.e.,
V PV , R PV , I L1 , I L 2 and R L , from (10) and (11), theoutput power Wo can be estimated only with VC1 and∆VC1 from the voltage transducer as follows:

Wo =

Vo2
RL

rS D + (rD + rL 2 ) D ′  ∆VC1 ∆V C 1
VC 1 −
= C1
 C1

TS
D′
DTS 


2

. (12)

Fig. 6 shows system configuration of the proposed MPPT
controller. The controller requires current detection with aHall-effect CT or voltage detection with an isolation amplifier
in order to estimate the output power by (6) or (12).
According to (6) and (12), both of the maximum and theminimum peak values of the reactor current or the capacitor
voltage must be detected to calculate the ripple amplitude of
the current or the voltage. Also, an average value of thereactor current or the capacitor voltage can be calculated fromthe maximum and the minimum peak values detected.
It is requisite to sample the reactor current or the capacitor
voltage synchronously with a switching pattern of theconverter because the maximum and the minimum peakvalues are detected at rising edges and falling edges of thepattern. However, this approach demands sophisticated A/Dconverters with high-sampling rate and high-resolution, which
is a disadvantage from a viewpoint of circuit implementationand cost. In order to overcome this difficulty, a simple
OP-amp based analog circuit is employed in the proposedsystem as shown in Fig. 6 (b), which is an external front-endof the A/D converters embedded in a digital signal processor.
The ripple amplitude of the reactor current ∆I L or the ripple

2353

3.5
3.0

Measured
Estimated

3.0

2.5

Output Power (W)Output Power (W)

3.5

Measured
Estimated

2.0
1.5
1.0
0.5
0.0

2.5
2.0
1.5
1.0
0.5
0.0

0.0

0.2

0.4
0.6
Duty Ratio

0.8

1.0

0.0

0.2

1.0

Measured
Estimated

3.0

2.5

Output Power (W)Output Power (W)

3.5

Measured
Estimated

3.0

0.8

(a)

(a)
3.5

0.4
0.6
Duty Ratio

2.0
1.5
1.0
0.5
0.0

2.5
2.0
1.5
1.0
0.5
0.0

0.0

0.2

0.4
0.6
Duty Ratio

0.8

1.0

0.0

0.2

0.4
0.6
Duty Ratio

0.8

1.0

(b)
Fig. 8. Comparison between measured and estimated power ofvoltage-source DC-DC converter with single current
transducer. (a) Irradiance: 2.43 (kW/m2). (b) Irradiance:
1.06 (kW/m2).

(b)Fig. 9. Comparison between measured and estimated power ofcurrent-source DC-DC converter with single voltage
transducer. (a) Irradiance: 2.43 (kW/m2). (b) Irradiance:
1.06 (kW/m2).

amplitude of the capacitor voltage ∆VC1 is calculated fromcondition listed in TABLE I. Fig. 8 shows experimental
results of the voltage-source DC-DC converter andcomparison between the measured output power and theestimated one by (6) from current transducer information.
Small error can be observed in a low-duty range but theestimated output power conforms to the measured power very
well both in a higher irradiance condition and in a lower
irradiance condition. Fig. 8 (a) indicates that the maximum
power points of the estimated and the measured results areobserved commonly at D = 0.70 , while the both maximum
power points can be seen at D = 0.40 in Fig. 8 (b).

difference between the maximum and the minimum peakvalues, while the average reactor current I L or the averagecapacitor voltage VC1 is obtained from the sum of the bothpeak values.
B. Operation of Ripple Amplitude Detector
Figs. 7 (a) and 7 (b) show tracking characteristics of theripple amplitude detector when the ripple amplitude of a test
signal is changed stepwise. Fig. 7 (a) corresponds to the caseof increased amplitude and Fig. 7 (b) is the case of decreasing.
As can be seen in Fig. 7 (a), the ripple amplitude detector
properly tracks the step changes of the ripple without large
delays.
However, Fig. 7 (b) indicates that it takes
approximately 8 (ms) and 14 (ms) for the ripple amplitudedetector to track new maximum and minimum values,
respectively. These delays are caused by discharge time
constant of the capacitors in the OP-amp circuit. However,these delays hardly affect MPPT performance of the systembecause solar irradiation does not vary in a few milliseconds.
C. Evaluation of Estimated Power from Single Transducer
Several experimental tests were conducted to examine
estimation performance of the proposed method under a test

Fig. 9 shows similar characteristics of the current-source
DC-DC converter and comparison between experimental
results of the measured and the estimated output power by(12). It is confirmed that the estimated values are slightly
larger than measured values in a whole range of the duty ratio.
However, the estimated power reaches the maximum power
point at D = 0.42 similarly with the measured one as shownin Fig. 9 (a), while Fig.9 (b) shows that the maximum power
points of both data are observed at D = 0.26 .
Since the optimum operating points are sought by acommon hill-climbing method as described before, absoluteaccuracy in estimating the output power is not important toachieve accurate MPPT operation. In other words, it is2354

Operating Points

2.0
22.0(kW/m )

1.5
1.0

2

1.5(kW/m )0.5

Operating Points

2.5
Output Power (W)Output Power (W)2.5

21.0(kW/m )2.0
22.0(kW/m )

1.5

2

1.0

1.5(kW/m )0.5

21.0(W/m )

0.0

0.0
0.0

0.2

0.4
0.6
Duty Ratio

0.8

0.0

1.0

0.2

0.4
0.6
Duty Ratio

0.8

1.0

Fig. 10. Output power vs. duty ratio characteristics andoperating points of voltage-source DC-DC converter withsingle current transducer.

Fig. 11. Output power vs. duty ratio characteristics andoperating points of current-source DC-DC converter withsingle voltage transducer.

TABLE I EXPERIMENTAL CONDITIONS

points even though irradiance widely changed. A tracking
error, which was a deviation of the operating point from thereal maximum power point, was within 4 % with respect to theduty ratio.
Therefore, the proposed single transducer
approach is quite effective to make an accurate MPPT
possible.

Test photovoltaic
Rated maximum power
Rated output voltage

GL418-TF
6.5 (W)
6 (V)Panel surface temperature
Switching frequency

50 (°C)
10 (kHz)

IV. CONCLUSION
possible to seek the maximum power point accurately as far asboth the estimated and the measured curves have their peaks at
the same duty ratio.
D. Discussion on Estimation Error
As can be seen in Fig. 8 (in the case of the voltage-source
DC-DC converter), the estimation error slightly increases in a
low-duty range. It is inferred that calculation error in (6) isdominant cause of the estimation error because thedenominator of the first term in (6) is detrimentally affectedby the duty ratio in the low-duty range. Therefore, theestimation error increases as the duty ratio is reduced if thecurrent detection error is always constant.
On the other hand, in the case of the current-source DC-DC
converter, almost constant estimation error, which looks
independent of the duty ratio, can be seen in Fig. 9. In thiscase, the estimation algorithm is expressed by (12) and its
principal term, i.e. the first term, does not include the duty.
This is the reason why the estimation error is hardly affectedby the duty ratio. It is a detection error of the voltage or aparameter mismatch of the capacitor rather than the duty ratiothat degrades the estimation characteristic.
E. Experimental Results of MPPT Operation
The MPPT operation was carried out by adopting acommon hill-climbing method to the proposed system. Figs.
10 and 11 show the experimental results of the voltage-source
DC-DC converter and the current-source DC-DC converter,
respectively. As can be seen in these figures, the MPPT
operation was properly performed in both converters and theiroperating points were placed close to the maximum power

This paper has discussed a novel strategy of an MPPT
operation of a photovoltaic power system with less transducer
count in power converters. The proposed strategy allows
output power estimation with only a single current transducer
or a single voltage transducer. Theoretical analysis of thisapproach has been developed on two types of converters, i.e.,the voltage-source DC-DC converter and the current-source
DC-DC converter.
Also, a practical implementation
technique to estimate the output power has been presented,
which allows a simple circuit configuration without sacrificing
accuracy of the power estimation.
Two types ofexperimental systems have been setup and proper estimation
characteristics have been confirmed through several
experimental tests. Consequently, an excellent tracking
performance to the maximum-power-points has been
confirmed with sufficient accuracy, using only a single
transducer.

REFERENCES[1] O. Wasynczuk, “Dynamic Behavior of a Class of Photovoltaic
Power Systems,” IEEE Trans. on Power Appl. Sys., vol. 102, no.
9, pp. 3031-3037, 1983.
[2] Katsumi Ohniwa, and Tsuyoshi Sato, ”A Simplified Maximum
Power Tracking Method for Photovoltaic Solar System,”
IEE-Japan Trans. on Power Sys., vol. 106-B, no. 7, pp. 72-78,
1984.
[3] C. Hua, J. Lin, and C. Shen, “Implementation of a
DSP-Controlled Photovoltaic System with Peak Power
Tracking,” IEEE Trans. on Ind. Elec., vol. 45, no. 1, pp. 99-107,
1998.
[4] K. Takahara, and T. Matsuda, “An Adaptive Control Method for
Maximum Power Tracking of Photovoltaic Power Generator,”
IEE-Japan Trans. on Ind. Appl., vol. 118-D, no. 6, pp. 810-811,
1998.

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