IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 54, NO. 5, OCTOBER 2007
2539
Application of Centered Differentiation and Steepest
Descent 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 to
improve the performance of maximum power point tracking
(MPPT). One improvement is to accurately locate the position
of the maximum power point (MPP) by using the centered differentiation. Another effort is to reduce the oscillation around
the MPP in steady state by controlling active perturbations. This
paper also adopts the method of steepest descent for MPPT,
which shows faster dynamic response and smoother steady state
than the method of hill climbing. Comprehensive experimental
evaluations have successfully illustrated the effectiveness of the
proposed algorithm.
Index Terms—Control systems, digital control, maximum power
point tracking (MPPT), photovoltaic power systems.
I. I NTRODUCTION
I
N PHOTOVOLTAIC power systems, a particular control algorithm, 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 recent 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 predict
the optimal operating condition. This is called voltage-based
MPPT. Similarly, studies [2]–[4] and [7] illustrate a currentbased MPPT method, which approximates a linear relationship
between the maximum power point (MPP) and the shortcircuit 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 slidingmode observer and neural network.
Generally, most recent applications of MPPT are based on
the extremum value theorem [14]–[19]. According to the theorem, 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: weidongx@ece.ubc.ca).
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 operation is regulating the voltage of photovoltaic array to follow
an optimal set point, which represents the voltage of the MPP
VMPP , as shown in Fig. 1. The focus of the MPPT is determining 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 backward
Euler [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
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 54, NO. 5, OCTOBER 2007
numerical differentiation. However, the lower bound of ∆V
is limited by the achievable resolution of digital controllers
and the accuracy of measurement. Further, the signal-to-noise
ratio of measurement can cause instability problems of numerical differentiation when the ∆V value is not significant
enough [25].
For MPPT, deviations from MPPs result in power losses. In
this paper, there are two key parts to improve tracking performance. One improvement is to accurately locate the position
of MPP by using the centered differentiation. Another effort is
to reduce the oscillation around the MPP in steady state. This
paper also presents the method of steepest descent to improve
the tracking performance. The following sections will discuss
these issues and provide relative solutions.
II. P ROPOSED A LGORITHM OF MPPT
Fig. 2. Measured signals of photovoltaic module with the operation of P&O.
algorithms can never accurately locate the optimal operating
point by finding the root of f (v, p) = 0, as shown in the
following:
dp
= f (v, p)
dv
pk − pk−1
f (vk−1 , pk−1 ) =
+ O(∆V 2 )
∆V
pk − pk−1
f (vk , pk ) =
+ O(∆V 2 ).
∆V
(1)
(2)
(3)
Continuous oscillation around the optimal operating point is
an intrinsic problem of the P&O algorithms, as shown in Fig. 2.
The plots illustrate the measured signals of the photovoltaic
voltage VPV , the photovoltaic current IPV , and the output
power PPV . The normalization was based on the MPP. In the
steady state, the continuous oscillation of the operating point
around the voltage of MPP VMPP makes the averaged power
level biased from the MPP. The IncCond method was developed to eliminate the oscillations around the MPP and avoid
the deviation problem. According to [19], the performance of
IncCond is 8.4% better than the P&O method. However, as
explained in [18], the P&O parameters in [19] might not be
properly optimized. As a result, it is difficult to justify which
algorithm is better because there is not a standard method to
evaluate and compare the performance of MPPT.
In theory, both algorithms control the system in steady state
when the approximation of dp/dv is equal to zero. However,
the experiments [19] showed that there were still oscillations
under stable environmental conditions because the digitalized
approximation of maximum power condition of di/dv = −i/v,
which is equivalent to dp/dv = 0, only rarely occurred. As
shown in (2) and (3), the problem is actually caused by the
local truncation error of the numerical differentiation, i.e., the
Euler method, of which the approximation always deviates
from the true MPP. Another drawback for both algorithms
is the difficulty in choosing the proper perturbation step of
photovoltaic voltage [18]. Generally, according to (2) and (3),
a smaller perturbation size ∆V implies a better accuracy of
The improvements of MPPT are illustrated by the following
facts: the reduction of local truncation error; the evaluation of
numerical stability; the selection of tracking methods; and the
oscillation reduction.
A. Reduction of Local Truncation Error
The centered differentiation is symmetric, as expressed in
(4) and (5). This method requires three-point measurements:
(vk−1 , pk−1 ); (vk , pk ); and (vk+1 , pk+1 ) to approximate the
derivative value at the center point (vk , pk ). As shown in (5),
the local truncation error for the centered differentiation is equal
to O(∆V 3 ), which indicates a second-order accuracy. This
means that this method is more accurate than the Euler methods
for numerical differentiation. Further, it was proved that the
centered differentiation method, as shown in the following,
does not have a phase error in the frequency domain [23]:
dp
= f (v, p)
dv
pk+1 − pk−1
+ O(∆V 3 ).
f (vk , pk ) =
2∆V
(4)
(5)
For MPPT, a controller needs to find the point where
f (v, p) = 0. The symmetric characteristics allow the extremum
value to be located with a better precision, as demonstrated by
the normalized power–voltage curve of a photovoltaic module
shown in Fig. 3. The normalization is based on the MPP. When
the calculation shows pk+1 ≈ pk−1 or pk+1 − pk−1 ≈ 0, the
differentiation approximation of f (v, p) is very close to zero
and the MPP is located at the center point (vk , pk ) instead of
either (vk+1 , pk+1 ) or (vk−1 , pk−1 ). However, Euler methods
can never make the operating point remain stably at the actual
MPP (vk , pk ) because their approximation is always biased due
to the phase error.
B. Evaluation of Numerical Stability
Numerical stability is an essential property for any numerical
algorithm. In general, numerical differentiation is more difficult
than numerical integration because numerical differentiation
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XAIO et al.: APPLICATION OF CENTERED DIFFERENTIATION AND STEEPEST DESCENT TO MPPT
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Fig. 4. Comparison of different algorithms for MPPT.
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 CLIP is a constant independent
of ∆V . In MPPT, the value of CLIP can be selected according 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 CLIP 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 | ≤ CLIP | 2∆V |.
(6)
Disturbances and measurement noise are two major reasons
that cause a malfunction of numerical differentiation.
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:
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 the
Gauss–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 algorithm of MPPT can be demonstrated by (7), where Kε is the
step-size corrector, and dp/dv is the derivation. The value of
Kε decides how steep each step takes in the gradient direction.
Based on a specific application, the tuning of Kε 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, the
computation of operating point can be illustrated in (8). This
vk+1 = vk +
vk+1
dp dv v=v
k
Kε
dp dv = vk − 2 v=vk .
d p
dv 2 (7)
(8)
v=vk
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 normalization of the photovoltaic voltage in Fig. 4 was based on the opencircuit 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 state
than the method of hill climbing, as illustrated in Fig. 4.
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D. Oscillation Reduction
Continuous tracking operations cause unnecessary oscillations around the MPP. This can be reduced by stopping
continuous perturbations when a local MPP is accurately located. The controller shall be able to achieve these operations:
1) to evaluate if the true MPP is found; 2) to stop the perturbation operation to make the operating point stay at MPP; and
3) 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 required sensitivity and a signal-to-noise ratio of measurements:
dp =0
dv vmpp
dp =0
di (9)
(10)
impp
dp ≤ εmpp .
dv (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 eR
is 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 recorded Gmpp is characterized in (15). The averaged values
of these absolute errors are symbolized in (16) and (17) for
resistance and conductance, correspondingly. Nth , 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 + Gmpp = 0
(12)
dv vmpp
dv + Rmpp = 0
(13)
di impp
v
eR = − Rmpp
(14)
i
i
eG = − Gmpp
(15)
v
N
th
eR (i)
i=1
ER =
(16)
Nth
N
th
eG (i)
.
(17)
EG = i=1
Nth
The averaged values of these absolute errors ER or EG are
updated in each tracking cycle. An MPP that has drifted can be
detected by monitoring the change of either ER or EG . If there
is no variation of MPP, the values of ER or EG 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. ER or EG 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.
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Fig. 7.
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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
N
th
EI =
(18)
eI (i)
i=1
Nth
(19)
where eI represents the absolute error of the photovoltaic
current, impp is the recorded current of MPP, EI symbolizes
the averaged error of the photovoltaic current, and Nth is the
number of samples for each averaging window. When EI is
larger than a specified threshold EI _THRED , the controller can
determine that the MPP has drifted to a new level and initializes
a procedure of MPPT. EI _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. EI will be
reset 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, ER or EG 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 voltage of MPP VMPP 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 beginning 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 Kε determines how big the step takes in the gradient direction. In this paper, the system starts with a chosen value
for Kε . 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 Kε for efficient computation. The variable of “Sched” is
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TABLE I
IMPORTANT PARAMETERS USED IN FIGS . 6 AND 8
TABLE II
SYSTEM SPECIFICATIONS
TABLE III
VARIABLES AND RELATIVE RESOLUTION
Fig. 9. Measured maximum photovoltaic power of the BP350 acquired in
Vancouver 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 parameters of the proposed algorithm are listed in Table I. These
are used for the system evaluation, which will be illustrated
in Section III.
III. E VALUATION
The efficiency of MPPT is defined as a ratio of the practical 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 operated 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 the
I–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. This
measurement demonstrates a dramatic change of MPP in 1-h
period. The sampling rate is 10 kHz. Under this condition,
periodical interruptions will eventually affect the dynamic evaluations of MPPT in response of the fast variation of solar
insolation.
To properly evaluate the MPPT, a DSP-controlled photovoltaic 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 mode
converter, there is always a tradeoff between switching frequency, 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 independently 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 sharing 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
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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 under
low radiation. The available power is about 8 W for 50-W solar module.
(a) Comparison of normalized power waveforms controlled by PAMPPT and
P&O, (b) normalized voltage waveform controlled by PAMPPT, (c) normalized
voltage waveform controlled by P&O, (d) normalized current waveform controlled 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 illustrated in Section II.
Fig. 12. Plots of startup procedures of PAMPPT and P&O algorithms under
weak radiation. The available power is about 3.53 W for the 50-W solar module.
(a) Comparison of normalized power waveforms controlled by PAMPPT and
P&O, (b) normalized voltage waveform controlled by PAMPPT, (c) normalized
voltage waveform controlled by P&O, ∆d was equal to 0.005, (d) normalized
current waveform controlled by PAMPPT, and (e) normalized current waveform
controlled 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 the
MPP much faster than the P&O method at low power condition,
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Fig. 13. Normalized power waveforms in steady state that the 50-W photovoltaic module outputs about 23-W power. (a) Normalized power waveforms
in steady state when the system was operated by PAMPPT. (b) Normalized
power waveforms in steady state when the system was operated by P&O and
the perturbation step ∆d was equal to 0.005.
Fig. 15. Normalized power waveforms in steady state that the 50-W photovoltaic module outputs about 23-W power. (a) Normalized power waveforms
in steady state when the system was operated by PAMPPT. (b) Normalized
power waveforms in steady state when the system was operated by P&O and
the perturbation step ∆d was equal to 0.01.
Fig. 14. Normalized voltage waveforms in steady state that the 50-W photovoltaic module outputs about 23-W power. (a) Normalized voltage waveforms
in steady state when the system was operated by PAMPPT. (b) Normalized
voltage waveforms in steady state when the system was operated by P&O and
the perturbation step ∆d was equal to 0.005.
Fig. 16. Normalized voltage waveforms in steady state that the 50-W photovoltaic module outputs about 23-W power. (a) Normalized voltage waveforms
in steady state when the system was operated by PAMPPT. (b) Normalized
voltage waveforms in steady state when the system was operated by P&O and
the perturbation step ∆d was equal to 0.01.
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 operation 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 in
steady state.
B. Steady-State Performance
The benefit of PAMPPT is that the controller stops the tracking 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 performance when P&O-type algorithms are used. Figs. 13 and 14
illustrate the photovoltaic power and voltage waveforms, respectively, when the step size is 0.005 for the P&O method.
Figs. 15 and 16 show normalized power waveforms and normalized 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 deviation summarized in Table IV, where ∆d stands for the pertur-
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XAIO et al.: APPLICATION OF CENTERED DIFFERENTIATION AND STEEPEST DESCENT TO MPPT
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TABLE IV
PERFORMANCE 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 commanded 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 voltages and currents every second for 8-h periods. The arithmetic
mean of a set of power values P is expressed as
P =
N
1 V (k)I(k).
N
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 controlled
by PAMPPT and P&O, (b) voltage waveform controlled by PAMPPT, and
(c) voltage waveform controlled by P&O.
(20)
k=1
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, the
system 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. Broken 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 different
weather conditions.
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 controlled
by 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 tracking 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 computational 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 to
40% computational power.
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2548
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 54, NO. 5, OCTOBER 2007
IV. C ONCLUSION
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 secondorder accuracy. The algorithm also occasionally stops tracking
operations 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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Weidong Xiao (S’03) received the M.A.Sc.
degree in electrical engineering from the University
of British Columbia, Vancouver, BC, Canada, in
2003, where he is currently working toward the
Ph.D. degree.
His research interests include power electronics
and applications of renewable energy sources.
William G. Dunford (S’78–M’81–SM’92) received
the B.S. degree in engineering from the Imperial
College London, London, U.K., and the Ph.D. degree from the University of Toronto, Toronto, ON,
Canada.
He is with the Clean Energy Research Centre,
University of British Columbia, Vancouver, BC,
Canada. He was also part of a team modeling satellite
batteries at Alcatel, Toulouse, France. He has worked
on 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 was the General Chair of the IEEE Power Electronics Specialists
Conference in 1986 and 2001.
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XAIO et al.: APPLICATION OF CENTERED DIFFERENTIATION AND STEEPEST DESCENT TO MPPT
Patrick R. Palmer (M’87) received the B.S. and
Ph.D. degrees in electrical engineering from the
Imperial College of Science and Technology, University of London, London, U.K., in 1982 and 1985,
respectively.
He joined the faculty of the Department of Engineering, 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 Computer Engineering, University of British Columbia,
Vancouver, BC, Canada, in 2004, returning to Cambridge as a Reader in
electrical engineering in 2005. He has extensive publications in his areas of
interest. He is the holder of two patents. His research is mainly concerned
with the characterization and application of high-power semiconductor devices,
computer analysis, simulation, and design of power devices and circuits, and he
has further interests in fuel cell hybrid vehicles.
Dr. Palmer is a Chartered Engineer in the U.K.
2549
Antoine Capel received the Ph.D. degree from the
University of Toulouse, Toulouse, France.
After teaching at Toulouse and at the University of Pernambouc, Récife, Brazil, he joined the
European Space Agency as a Power System
Engineer. In 1983, he joined Alcatel Espace,
Toulouse, where he first managed the Power Supply
Laboratory and was Head of the Power System Simulation 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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