eZdsp PV-MPPT Steepest Decent Control 2007-11 Consult

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 54, NO. 5, OCTOBER 2007 2539

Application of Centered Differentiation and SteepestDescent to Maximum Power Point Tracking

Weidong Xiao, Student Member, IEEE , William G. Dunford, Senior Member, IEEE ,Patrick R. Palmer, Member, IEEE , and Antoine Capel

Abstract—This paper concentrates on two critical aspects toimprove the performance of maximum power point tracking(MPPT). One improvement is to accurately locate the positionof the maximum power point (MPP) by using the centered dif-ferentiation. Another effort is to reduce the oscillation aroundthe MPP in steady state by controlling active perturbations. Thispaper also adopts the method of steepest descent for MPPT,which shows faster dynamic response and smoother steady statethan the method of hill climbing. Comprehensive experimentalevaluations have successfully illustrated the effectiveness of theproposed algorithm.

IndexTerms—Control systems, digital control, maximum powerpoint tracking (MPPT), photovoltaic power systems.

I. INTRODUCTION

IN PHOTOVOLTAIC power systems, a particular control al-

gorithm, which is maximum power point tracking (MPPT),

is utilized because it takes full advantage of available solar

energy. A variety of techniques has been developed in re-

cent years. Studies [1]–[6] indicate that the optimal operating

voltage of a photovoltaic module is always very close to a

fixed percentage of the open-circuit voltage. This implies that

MPPT could simply use the open-circuit voltage to predictthe optimal operating condition. This is called voltage-based

MPPT. Similarly, studies [2]–[4] and [7] illustrate a current-

based MPPT method, which approximates a linear relationship

between the maximum power point (MPP) and the short-

circuit current. Another technique was developed according to a

linear approximation between the maximum output power and

the optimal operating current [7], [8]. A fourth algorithm [9]

measures and compares the output of two modules to track the

MPP. Various other techniques [10]–[13] are also available to

estimate the characteristics of photovoltaic output and track the

MPP by way of some specific mechanisms such as a sliding-

mode observer and neural network.

Generally, most recent applications of MPPT are based onthe extremum value theorem [14]–[19]. According to the the-

orem, as described in [20], the extremum, which is maximum

or minimum, occurs at the critical point. If a function y = f (x)

Manuscript received October 9, 2006; revised February 28, 2007.W. Xiao and W. G. Dunford are with the University of British Columbia,

Vancouver, BC V6T 1Z4, Canada (e-mail: [email protected]).P. R. Palmer is with the Department of Engineering, University of

Cambridge, Cambridge CB2 1PZ, U.K.A. Capel is with the School of Engineering, Universitat Rovira i Virgili,

Tarragona 43007, Spain.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TIE.2007.899922

Fig. 1. Block diagram of P&O and IncCond MPPT topologies.

is continuous on a closed interval, it has the critical points at

all point x0, where f (x0) = 0. In photovoltaic power systems,

the local MPP can be continuously tracked and updated to

satisfy a mathematical equation:dp/dx = 0, where p represents

the photovoltaic power and x represents the control variable

chosen from the photovoltaic voltage, photovoltaic current, or

switching duty cycle of power interfaces.

In photovoltaic power applications, two popular algorithms

are the perturbation and observation (P&O) method [17] and

the incremental conductance (IncCond) method [19]. The op-

eration is regulating the voltage of photovoltaic array to follow

an optimal set point, which represents the voltage of the MPPV MPP, as shown in Fig. 1. The focus of the MPPT is determin-

ing when dp/dv = 0, where p represents the output power and

v represents the photovoltaic voltage. Without a mathematical

model, this is generally based on numerical differentiation (the

Euler method), which is a process of finding a numerical value

of a derivative of a given function at a given point. The Euler

method is simple in use, but shows local truncation errors [21],

[22]. The definition of local truncation errors is how well the

exact solution satisfies the numerical scheme.

To clearly demonstrate this subject, we define the derivative

in a general form of dp/dv, as shown in (1). Consequently,

the numerical differentiations of forward Euler and backwardEuler [21] can be demonstrated in (2) and (3), correspondingly,

where pk and pk−1 represent the sequence of the photovoltaic

power, vk and vk−1 symbolize the sequence of the photovoltaic

voltage, and ∆V is the incremental step of photovoltaic voltage,

which is equal to vk − vk−1. As shown in (2) and (3), the local

truncation error for the Euler methods is equal to O(∆V 2),

which stands for the order of ∆V 2 and indicates a first-order

accuracy. In mathematics, the big O notation is usually used

to characterize the residual term of a truncated infinite series.

The Euler methods also demonstrate both magnitude errors

and phase errors in the frequency domain [23], [24]. The local

truncation error is the reason why both P&O and IncCond

0278-0046/$25.00 © 2007 IEEE

Authorized licensed use limited to: Weidong Xiao. Downloaded on October 22, 2008 at 19:33 from IEEE Xplore. Restrictions apply.



XAIO et al.: APPLICATION OF CENTERED DIFFERENTIATION AND STEEPEST DESCENT TO MPPT 2541

Fig. 3. Normalized power–voltage curve of photovoltaic module.

requires Lipschitz classes [26]. According to the definition,

the numerical differentiation shall always satisfy the Lipschitz

condition shown in (6), where C LIP is a constant independent

of ∆V . In MPPT, the value of C LIP can be selected accord-

ing to the maxima of f (v, p), because this value is known

through offline analysis of power–voltage curves. For a specific

photovoltaic module, i.e., BP350, the absolute value of the

steepest slope is 37 under the standard test condition. Therefore,

the value of C LIP is chosen as 37. The BP350 is a product

of BP Solar International Inc. To promise numerical stability,

the controller will evaluate this following condition after each

numerical differentiation:

| pk+1 − pk−1| ≤ C LIP| 2∆V |. (6)

Disturbances and measurement noise are two major reasons

that cause a malfunction of numerical differentiation.

C. Selection of Tracking Methods

Two mathematical methods are relevant to applications of

MPPT. One optimization method is the steepest descent, which

is also called gradient descent method [27], which is originally

an optimization method in applied mathematics. Another is theGauss–Newton method, which is also called Newton–Raphson

method, which is a root-finding algorithm [27], [28].

The method of steepest descent can be applied to find the

nearest local MPP when the gradient of the function can be

computed. Based on the method of steepest descent, the algo-

rithm of MPPT can be demonstrated by (7), where K ε is the

step-size corrector, and dp/dv is the derivation. The value of

K ε decides how steep each step takes in the gradient direction.

Based on a specific application, the tuning of K ε will be further

discussed in Section II-D. The Newton–Raphson method uses

a first and second derivative of the change with parameter value

to estimate the direction and distance the program should to

go to reach a better point. When it is used to track MPPs, thecomputation of operating point can be illustrated in (8). This

Fig. 4. Comparison of different algorithms for MPPT.

Fig. 5. Gauss–Newton method is theoretically efficient in MPPT.

algorithm needs to numerically perform both single and double

differentiations, as shown in the following:

vk+1 = vk +

dp

dv

v=vk

K ε(7)

vk+1 = vk −

dp

dv

v=vk

d2

pdv2v=vk

. (8)

In theory, the Gauss–Newton method is the fastest algorithm

in comparison to the steepest descent and the hill climbing. The

plots of simulation results shown in Figs. 4 and 5 demonstrate

their efficiency in seeking the voltage of MPP. The normaliza-

tion of the photovoltaic voltage in Fig. 4 was based on the open-

circuit voltage. As shown in Fig. 5, the algorithm can find the

MPP after four steps of movement. Nevertheless, this procedure

can be unstable regarding the initial condition [28]. To avoid

this problem, the proposed algorithm of maximum power point

tracking (PAMPPT) chooses the method of steepest descent,

which shows faster dynamic response and smoother steady statethan the method of hill climbing, as illustrated in Fig. 4.




2542 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 54, NO. 5, OCTOBER 2007

D. Oscillation Reduction

Continuous tracking operations cause unnecessary oscil-

lations around the MPP. This can be reduced by stopping

continuous perturbations when a local MPP is accurately lo-

cated. The controller shall be able to achieve these operations:

1) to evaluate if the true MPP is found; 2) to stop the pertur-

bation operation to make the operating point stay at MPP; and3) to estimate if the MPP has drifted to a new location.

According to the extremum value theorem, any MPP shall

satisfy a condition, either (9) or (10), where v is the photovoltaic

voltage, i is the photovoltaic current, p symbolizes photovoltaic

power, vmpp stands for the voltage of MPP, and impp represents

the current of MPP. Section II-C has shown an increasing

accuracy of numerical differentiation by the introduction of the

centered differentiation. However, the centered differentiation

cannot eventually eliminate the local truncation error and the

tracking iteration continues until the condition illustrated in

(11) is continuously satisfied for several cycles. Then, the local

extremum has been determined within a chosen accuracy εmpp.The choice of εmpp, as shown in the following, depends on a re-

quired sensitivity and a signal-to-noise ratio of measurements:

dp

dv

vmpp

= 0 (9)

dp

di

impp

= 0 (10)

dp

dv

≤ εmpp. (11)

A flowchart in Fig. 6 demonstrates the operation. When the

location of MPP is located, the controller records the value of

impp and vmpp for further estimation described in Section II-E.

As shown in Fig. 6, the variable MPP_index records how

many times condition (11) is continuously satisfied. When the

number of MPP_index is accumulated to a certain threshold

MPP_th, the controller presumes a local MPP is temporarily

found under current conditions. The controller needs to clear

the index variable MPP_index and records the current location

of MPP. Otherwise, the MPPT will continue until the MPP is

successfully located.Both insolation and temperature are time-variant parameters

of a photovoltaic power system in a daily period. A changing

environment can make the MPP drift to a new location. From

(9) and (10), we can derive (12) and (13), respectively. Each

MPP corresponds to a specific value of resistance Rmpp or

conductance Gmpp. Consequently, the controller can estimate

a shift of MPP by monitoring the variation of either resistance

Rmpp or conductance Gmpp. The absolute resistance error eRis characterized in (14), which illustrates the difference of the

present measurement and the recorded Rmpp. Likewise, the

absolute conductance error eG of the present measurement and

the recordedGmpp is characterized in (15). The averaged values

of these absolute errors are symbolized in (16) and (17) forresistance and conductance, correspondingly. N th, as shown

Fig. 6. Flowchart to evaluate if the MPP was located.

in the following, is the number of samples for each averaging

window:

di

dv

vmpp

+ Gmpp = 0 (12)

dv

di

impp

+ Rmpp = 0 (13)

eR =v

i−Rmpp (14)

eG =i

v−Gmpp (15)

E R =

N thi=1

eR(i)

N th(16)

E G =

N th

i=1

eG(i)

N th. (17)

The averaged values of these absolute errors E R or E G are

updated in each tracking cycle. An MPP that has drifted can be

detected by monitoring the change of either E R or E G. If there

is no variation of MPP, the values of E R or E G are generally

within a certain range. When the averaged error is larger than

a specified threshold, the controller supposes that the MPP has

drifted to a new level and initializes a procedure of MPPT. The

controller stops the calculation of the averaged errors when

it is tracking the new MPP. E R or E G will be reset to zero

after the existing MPP is successfully located in steady state,

whereas Rmpp and Gmpp should be updated accordingly. Then,the detection restarts for a new shift of MPP.





Fig. 7. Flowchart illustrating the main loop of the proposed MPPT.

With a regulation of photovoltaic voltage, the controller

needs to monitor the error of the photovoltaic current change

since the voltage is principally constant due to the regulation

process. The absolute error of the photovoltaic current and the

average error are respectively symbolized as follows:

eI = i− impp (18)

E I =

N th

i=1 eI (i)

N th(19)

where eI represents the absolute error of the photovoltaic

current, impp is the recorded current of MPP, E I symbolizes

the averaged error of the photovoltaic current, and N th is the

number of samples for each averaging window. When E I is

larger than a specified threshold E I _THRED, the controller can

determine that the MPP has drifted to a new level and initializes

a procedure of MPPT. E I _THRED is chosen by analyzing the

nominal current of the photovoltaic array and the dynamics of

weather condition. The controller stops the calculation of the

averaged errors when it is tracking the new MPP. E I will bereset to zero after the existing MPP is successfully located in

steady state. Then, the detection restarts to search for a new

shift of MPP. This is applicable for systems with a photovoltaic

regulation function. For the general case, E R or E G shall be

used as the criterion to determine a shifted MPP.

E. Main Loop

Fig. 7 illustrates a main loop of the proposed MPPT. The

tracking procedure can be activated or deactivated according

to the estimation if an MPP has been changed or located,

correspondingly. For the evaluation of drifting or locked MPP,refer to the former sections.

Fig. 8. Flowchart of the proposed MPPT.

F. Flowchart of MPPT

According to [2], for silicon photovoltaic modules, the volt-

age of MPP V MPP is about 71% of the open-circuit voltage.

For the PAMPPT, the startup time can be shortened noticeably

because the controller knows the range of MPP by measuring

the open-circuit voltage in the initial condition. However, the

optimal duty cycle for MPP is generally unknown at the be-

ginning because it depends on the specific system and the load

condition.The flowchart in Fig. 8 demonstrates an implementation of

the proposed MPPT. The numerical differentiation is based on

the centered differentiation according to the analysis in the

former sections. To avoid numerical instabilities, the Lipschitz

function is evaluated in each control cycle according to the

criterion, as shown in (6). The tracking goes back to P&O

method when the stability condition is not satisfied. The value

of K ε determines how big the step takes in the gradient di-

rection. In this paper, the system starts with a chosen value

for K ε. Then, a further tuning proceeds to make sure that

the tracking converges to the local extremum value. When a

fixed-point digital signal processor (DSP) or microcontroller is

used, it is important to choose binary numbers for both ∆V and K ε for efficient computation. The variable of “Sched” is





TABLE IIMPORTANT PARAMETERS USED IN FIG S. 6 AN D 8

Fig. 9. Measured maximum photovoltaic power of the BP350 acquired inVancouver from 10:00 A.M. to 11:00 A.M. in June 15, 2006.

used for scheduling the computation load of MPPT. As shown

in the flowcharts in Figs. 6 and 8, several important parame-

ters of the proposed algorithm are listed in Table I. Theseare used for the system evaluation, which will be illustrated

in Section III.

III. EVALUATION

The efficiency of MPPT is defined as a ratio of the practi-

cal power output divided by the true maximum power value.

However, a fair comparison is never easy because a true MPP

is unknown at a specific moment when the system is oper-

ated by MPPT. One recommendation [29] is to periodically

interrupt the operation of MPP and switch the photovoltaic

output to an I –V curve tracer, which is able to acquire theI –V curve and show the MPP quickly. However, this method

is only suitable for evaluating the steady-state performance

since the assumption is that environmental conditions do not

significantly change in a short time period. Furthermore, it is

important to assess the dynamic performance of MPPT caused

by rapid and unpredictable change of environmental conditions.

Fig. 9 shows the weather effect on the photovoltaic power of a

specific solar module, i.e., the BP350, which is a product of BP

Solar International Inc. In 1-h period, a specific photovoltaic

power system is operated by the P&O MPPT algorithm to

deliver the maximum power of the photovoltaic module. The

photovoltaic power variation is sampled by a data acquisition

system recording the photovoltaic voltage and current. Thismeasurement demonstrates a dramatic change of MPP in 1-h

TABLE IISYSTEM SPECIFICATIONS

TABLE IIIVARIABLES AND RELATIVE RESOLUTION

period. The sampling rate is 10 kHz. Under this condition,

periodical interruptions will eventually affect the dynamic eval-

uations of MPPT in response of the fast variation of solar

insolation.

To properly evaluate the MPPT, a DSP-controlled photo-

voltaic power system is specially designed and constructed

for evaluation purposes, as illustrated in Fig. 10. This consists

of two identical photovoltaic modules, two equal dc–dc boost

converters as the power interfaces, and a 24-V battery bank.

The system specifications are summarized in Table II. The

eZDSP LF2407 acts as a digital controller, in which the control

algorithms are implemented. The DSP has 16 channels of 10-bit

analog–digital converter. The measured variables and relative

resolutions are listed in Table III. To design a switching modeconverter, there is always a tradeoff between switching fre-

quency, size of the system, and power losses in all components.

By considering the tradeoff, the switching frequency of the

converter is 40 kHz in this system design. According to the

40-MHz clock of the DSP used for this system, the resolution

of duty cycle is only 0.1%. The capacity of the battery bank

is sufficient to handle the maximum power outputs from both

photovoltaic modules. To avoid any overcharge of batteries, the

system also includes a charge protection unit and a discharge

circuit, which are not directly related to this paper and not

illustrated in Fig. 10.

The structure allows testing two photovoltaic modules inde-pendently and simultaneously under the same environment. The

experimental results will be used to illustrate the effectiveness

of the proposed control method and MPPT. With this structure,

one of two identical modules is used as the benchmark shar-

ing the same load and control bandwidth, whereas another is

implemented with a developed algorithm.

The benchmark algorithm of MPPT is the well-established

method of P&O. For the parameterization and implementation,

refer to the study [18] published in 2005, which gives detailed

analysis and optimization for the P&O method. According to

the analysis [18], the perturbation time interval for this bench

system shall be longer than 0.0072 s. Therefore, we choose the

tracking frequency as 100 Hz, which satisfies this requirement.The perturbation level of switching duty cycle is alternatively





Fig. 10. Block diagram of the bench system used to evaluate MPPT under the same condition.

Fig. 11. Plots of startup procedures of PAMPPT and P&O algorithms underlow radiation. The available power is about 8 W for 50-W solar module.(a) Comparison of normalized power waveforms controlled by PAMPPT andP&O, (b) normalized voltage waveform controlled by PAMPPT, (c) normalizedvoltage waveform controlled by P&O, (d) normalized current waveform con-trolled by PAMPPT, and (e) normalized current waveform controlled by P&O.

chosen either 0.01 or 0.005. The following evaluations will

show the dynamic response, steady-state performance, and

daily 8-h tests. As shown in Figs. 11–16, the power and voltage

waveforms were normalized by the MPP and the voltage of

MPP, respectively. The PAMPPT is the abbreviation of the

proposed algorithm of MPP, which was developed and illus-trated in Section II.

Fig. 12. Plots of startup procedures of PAMPPT and P&O algorithms underweak radiation. Theavailable poweris about 3.53 W forthe 50-W solar module.(a) Comparison of normalized power waveforms controlled by PAMPPT andP&O, (b) normalized voltage waveform controlled by PAMPPT, (c) normalizedvoltage waveform controlled by P&O, ∆d was equal to 0.005, (d) normalizedcurrent waveform controlled by PAMPPT, and (e) normalized current waveformcontrolled by P&O,∆d was equal to 0.005.

A. Startup Performance

A routine operation of photovoltaic power systems is that the

controller starts the MPPT in the morning whenever sunlight

is available. However, the radiation is generally low at this

moment. Fig. 11 demonstrates that PAMPPT can locate theMPP much faster than the P&O method at low power condition,





Fig. 13. Normalized power waveforms in steady state that the 50-W photo-voltaic module outputs about 23-W power. (a) Normalized power waveformsin steady state when the system was operated by PAMPPT. (b) Normalizedpower waveforms in steady state when the system was operated by P&O andthe perturbation step∆d was equal to 0.005.

Fig. 14. Normalized voltage waveforms in steady state that the 50-W photo-voltaic module outputs about 23-W power. (a) Normalized voltage waveformsin steady state when the system was operated by PAMPPT. (b) Normalizedvoltage waveforms in steady state when the system was operated by P&O andthe perturbation step∆d was equal to 0.005.

where the available power is about 8 W for a 50-W photovoltaic

module. Compared to 0.23 s for P&O method, the PAMPPT

used 0.04 s to stabilize the operating point at the MPP.Ideally, photovoltaic power systems should start even earlier

to harvest more solar energy in the early morning. This requires

that the MPPT can start and work well under weak radiation.

Fig. 12 shows that the PAMPPT can initialize the tracking op-

eration and stabilize the operation at MPP when the achievable

photovoltaic power is only 3.53 W for a 50-W module due to

the weak insolation. However, when the perturbation step ∆d is

0.005, the P&O method was not able to find the MPP because

it was trapped due to the low signal-to-noise ratio and large

local truncation errors. Experiments show that the P&O method

can operate normally when the perturbation step is increased

to 0.03 under the weak insolation. However, the large ripples

of photovoltaic voltage and power result in significant loss insteady state.

Fig. 15. Normalized power waveforms in steady state that the 50-W photo-voltaic module outputs about 23-W power. (a) Normalized power waveformsin steady state when the system was operated by PAMPPT. (b) Normalizedpower waveforms in steady state when the system was operated by P&O and

the perturbation step∆d

was equal to 0.01.

Fig. 16. Normalized voltage waveforms in steady state that the 50-W photo-voltaic module outputs about 23-W power. (a) Normalized voltage waveformsin steady state when the system was operated by PAMPPT. (b) Normalizedvoltage waveforms in steady state when the system was operated by P&O andthe perturbation step∆d was equal to 0.01.

B. Steady-State Performance

The benefit of PAMPPT is that the controller stops the track-ing process and runs only the voltage regulation when the MPP

is located. In steady state, this dramatically reduces ripples

in photovoltaic power and voltage compared to continuous

perturbations caused by P&O-type algorithms.

The value of perturbation step affects the steady-state perfor-

mance when P&O-type algorithms are used. Figs. 13 and 14

illustrate the photovoltaic power and voltage waveforms, re-

spectively, when the step size is 0.005 for the P&O method.

Figs. 15 and 16 show normalized power waveforms and normal-

ized voltage waveforms, correspondingly, when the step size

is 0.01 for the P&O method. Comparing to the P&O methods

with two difference step sizes, the improvement of PAMPPT in

steady state can be demonstrated by the values of standard devi-ation summarized in Table IV, where ∆d stands for the pertur-





TABLE IVPERFORMANCE COMPARISON IN STEADY STATE

bation step of duty cycle. Table IV also demonstrates the mean

values of power outputs. The standard deviation of photovoltaic

voltage controlled by PAMPPT is much smaller than that com-

manded by P&O algorithm. As a result, the proposed control

system delivers more solar power under the same condition.

C. Results of Long-Term Test

To sufficiently evaluate the efficiency of photovoltaic power

systems, daily long-term tests with natural sunlight have been

performed for 14 days in Vancouver, BC, Canada, from

June 14, 2006 to July 3, 2006. Most of them are conducted for

8 h a day. Both PAMPPT and P&O were being tested under

the same weather conditions because two power interfaces are

available for operations at the same time. A multichannel data

acquisition system is available to record the photovoltaic volt-

ages and currents every second for 8-h periods. The arithmetic

mean of a set of power values P is expressed as

P =1

N

N

k=1

V (k)I (k). (20)

Calibrations have shown that two photovoltaic modules

used for tests give slightly different output characteristics.

Module #2 demonstrates more power output than module #1

under same conditions. For better comparison, two modules

were alternatively connected to the power interfaces in a daily

basis. When module #2 is operated by PAMPPT and module #1

is controlled by P&O, module #2 outputs average 5.3% more

energy than module #1 in an 8-h test period. When module #1

is operated by PAMPPT and module #2 is controlled by P&O,

module #2 harvests an average of 3.3% more energy than

module #1. After 14 days 8-h tests with natural sunlight, thesystem controlled by PAMPPT harvested total 3.67-kWh solar

energy, and the system controlled by P&O controlled collected

3.64 kWh, which is 1.0% less. Fig. 17 demonstrates waveforms

of power and photovoltaic voltage, in one 8-h test performed in

June 24, 2006, which is a perfect sunny day. Evaluations show

that the PAMPPT performs even better in rapidly changing

atmospheric conditions. One example is illustrated in Fig. 18,

where the PAMPPT made module #2 harvest 5.8% more solar

energy than P&O algorithm that controlled module #1. Bro-

ken clouds happened very often in the 8-h testing period of

July 9, 2006. As a result, the 14 daily tests demonstrate that

the PAMPPT is an efficient algorithm that always harvests

about 1.0% more energy than the P&O method under differentweather conditions.

Fig. 17. Waveforms of power and voltage acquired by the 8-h test on June 24,

2006, which was a sunny day. (a) Comparison of power waveforms controlledby PAMPPT and P&O, (b) voltage waveform controlled by PAMPPT, and(c) voltage waveform controlled by P&O.

Fig. 18. Waveforms of power and voltage acquired by the 8-h test on July 9,2006, which was a cloudy day. (a) Comparison of power waveforms controlledby PAMPPT and P&O, (b) voltage waveform controlled by PAMPPT, and

(c) voltage waveform controlled by P&O.

D. Computation Requirement

Based on this specific system, as described in Fig. 10 and

Tables II and III, the PAMPPT requires more computation

power than the simple P&O algorithm. According to the track-

ing frequency of 100 Hz, the P&O can finish the MPPT in 1 µs

in each tracking cycle, which is equal to 0.01% of total compu-

tational power. In the application of the PAMPPT system, the

sampling frequency of the photovoltaic voltage is 40 kHz. The

PAMPPT algorithm requires 10 µs in one 25-µs period to fulfill

the regulation control of photovoltaic voltage, which is equal to40% computational power.





IV. CONCLUSION

This paper proposed an algorithm of MPPT, which can

accurately locate the position of MPP and reduce the oscillation

around the MPP in steady state. Instead of the Euler method

of numerical differentiation, this paper proposes the centered

differentiation, which improves the approximation to a second-

order accuracy. The algorithm also occasionally stops trackingoperations to avoid unnecessary oscillations around the MPP.

Long-term evaluations show that it is an efficient algorithm that

always harvests about 1% more energy than the P&O method

under different weather conditions. However, a much simpler

controller could be used for the P&O algorithm because its

computational requirement is much less than the PAMPPT. This

paper also illustrates the effectiveness of the test bench system

and the evaluation method with natural sunlight.

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point tracking performance,” Energy Res. Centre Netherlands, Petten,The Netherlands, 1997. RX97040.

Weidong Xiao (S’03) received the M.A.Sc.degree in electrical engineering from the Universityof British Columbia, Vancouver, BC, Canada, in2003, where he is currently working toward thePh.D. degree.

His research interests include power electronicsand applications of renewable energy sources.

William G. Dunford (S’78–M’81–SM’92) receivedthe B.S. degree in engineering from the ImperialCollege London, London, U.K., and the Ph.D. de-gree from the University of Toronto, Toronto, ON,Canada.

He is with the Clean Energy Research Centre,University of British Columbia, Vancouver, BC,Canada. He wasalso part of a team modeling satellitebatteries at Alcatel, Toulouse, France. He has workedon photovoltaic applications for a number of years.His other interests include distributed systems in

general with particular emphasis on efficiency, power quality, and automotive

applications.Dr. Dunford wasthe General Chair of theIEEE Power Electronics SpecialistsConference in 1986 and 2001.





Patrick R. Palmer (M’87) received the B.S. andPh.D. degrees in electrical engineering from theImperial College of Science and Technology, Uni-versity of London, London, U.K., in 1982 and 1985,respectively.

He joined the faculty of the Department of Engi-neering, University of Cambridge, Cambridge, U.K.,in 1985. He is an Engineering Fellow (elected 1987)

at St. Catharine’s College, University of Cambridge.He joined the Department of Electrical and Com-puter Engineering, University of British Columbia,

Vancouver, BC, Canada, in 2004, returning to Cambridge as a Reader inelectrical engineering in 2005. He has extensive publications in his areas of interest. He is the holder of two patents. His research is mainly concernedwith the characterization and application of high-power semiconductor devices,computer analysis, simulation, and design of power devices and circuits, and hehas further interests in fuel cell hybrid vehicles.

Dr. Palmer is a Chartered Engineer in the U.K.

Antoine Capel received the Ph.D. degree from theUniversity of Toulouse, Toulouse, France.

After teaching at Toulouse and at the Univer-sity of Pernambouc, Récife, Brazil, he joined theEuropean Space Agency as a Power SystemEngineer. In 1983, he joined Alcatel Espace,Toulouse, where he first managed the Power SupplyLaboratory and was Head of the Power System Sim-

ulation Division. He is currently with the School of Engineering, Universitat Rovira i Virgili, Tarragona,Spain. He is the holder of five U.S. patents.

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