Energy Conversion and Management 70 (2013) 83–89

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Energy Conversion and Management

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Model based rapid maximum power point tracking for photovoltaicsystems

0196-8904/$ - see front matter Crown Copyright � 2013 Published by Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.enconman.2013.02.018

⇑ Corresponding author. Tel.: +852 27666145; fax: +852 23301544.E-mail addresses: steve.tsang@polyu.edu.hk (K.M. Tsang), eewlchan@polyu.e-

du.hk (W.L. Chan).

K.M. Tsang, W.L. Chan ⇑Department of Electrical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

a r t i c l e i n f o

Article history:Received 16 October 2012Accepted 26 February 2013Available online xxxx

Keywords:Photovoltaic systemMaximum power point trackingPolynomial model

a b s t r a c t

This paper presents a novel approach for tracking the maximum power point of photovoltaic (PV) sys-tems so as to extract maximum available power from PV modules. Unlike conventional methods, a veryfast tracking response with virtually no steady state oscillations is able to obtain in tracking the maxi-mum power point. To apply the proposed method, firstly, output voltages, output currents under differ-ent conditions and temperatures of a PV module are collected for the fitting of environmental invariantnonlinear model for the PV system. Orthogonal least squares estimation algorithm coupled with the for-ward searching algorithm is applied to sort through all possible candidate terms resulted from the expan-sion of a polynomial model and to come up with a parsimonious model for the PV system. It is notnecessary to test all PV modules as the resultant model is valid for other modules. The power deliveredby the PV system can be derived from the fitted model and the maximum power point for the PV systemat any working conditions can be obtained from the fitted model. Consequently, rapid maximum powerpoint tracking could be achieved. Experimental results are included to demonstrate the effectiveness ofthe fitted model in maximum power point tracking.

Crown Copyright � 2013 Published by Elsevier Ltd. All rights reserved.

1. Introduction

Global market penetration of photovoltaic (PV) technology hasincreased tremendously over the last decade [1]. In fact, PV sourceshave been predicted to be the biggest contributors to electricitygeneration among all renewable energy candidates by 2040 [2].In PV systems, maximum power point tracking (MPPT) is essentialbecause it enables the extraction of maximum available energyfrom the system. It is well known that current flow from PV mod-ules change with the solar radiation and temperature conditions.PV modules present a nonlinear current–voltage (I–V) curve [3]and for each solar radiation as well as temperature conditionsthere exists an optimum working load which leads to extract themaximum power from the PV modules [3]. There are a numberof different approaches for MPPT [4]. They are the constant voltagemethod, open circuit voltage method, short-circuit method, per-turb and observe (P&O) method and the incremental conductancemethod. The constant voltage method is the simplest method but ithas been commented that the method could only collect about 80%of the available maximum power under varying irradiance. Animprovement on the constant voltage method uses the open circuitvoltage to estimate the maximum power output voltage while theshort circuit current method uses the short-circuit current to esti-

mate the maximum power output current. P&O method [3]searches for the maximum power point by changing the PV voltageor current and detecting the change in PV output power. The stepsize for the search affects the rate of convergence of the tracking.Also, the method may fail under rapidly changing atmosphericconditions [5]. The continuous oscillation around the optimal oper-ating point is another problem of the P&O tracking algorithm. Itmay cause power loss and system instability [6]. The incrementalconductance (IC) method [3,7] is developed to eliminate the oscil-lations around the maximum power point but experiments showthat there were still oscillations when digital control was used.Another drawback for both algorithms is that the perturbation stepis difficult to choose when dealing with the tradeoff betweensteady-state performance and fast dynamic response. Neural net-work (NN) based MPPT techniques have also been proposed [4].In comparing with the conventional methods, NN based controlalgorithm is capable of improving the tracking performance. Themethod requires no knowledge of the PV parameters, and thetrained NN can provide a sufficiently accurate MPPT. However,large amount of training data is needed and the training processcould be very slow. Recently [8] reported an implementation onFPGA of an efficient MPPT based on fuzzy heuristic rules that doesnot need any PV parameters. It consists on a stepwise adaptivesearch that leads to fast convergence and performs well with re-spect to change in climate conditions. However, knowledge ofVHDL and specific hardware are required for successful implemen-tation. Recursive least squares based MPPT techniques have also

84 K.M. Tsang, W.L. Chan / Energy Conversion and Management 70 (2013) 83–89

been proposed [9]. As the tracking model is time varying, the prob-lem of convergence arises.

In this paper, a model based maximum power point trackingalgorithm is proposed. Nonlinear models are fitted to a PV systemand maximum power points under different working conditionsare derived based on the fitted models. A very fast trackingresponse with less steady state oscillations is able to obtain intracking the maximum power point using the fitted nonlinearmodel. Apart from supplying DC loads, the proposed method couldbe used with multi-level inverter [10] for delivering power to ACloads. Experimental results are included to demonstrate the effec-tiveness of the new maximum power point tracking technique.

2. Nonlinear model for PV systems

The traditional equivalent circuit of a solar cell is represented bya current source in parallel with one or two diodes. A single-diodemodel [11] includes a photocurrent source, a diode parallel to thesource, a series resistor and a shunt resistor. There are fiveunknown parameters in a single-diode model and the mathemati-cal description of the current versus voltage (I–V) characteristicsfor the equivalent circuit is generally difficult to solve using analyt-ical methods in real-time operation. To handle the nonlinear I–Vcharacteristics and the temperature effect on the characteristics,polynomial models of the form

isc ¼ F1ðT; ip;vpÞvp ¼ F2ðT; isc; ipÞip ¼ F3ðT; isc;vpÞ

ð1Þ

where F1(�), F2(�) and F3(�) are polynomial type nonlinear functions,vp is the PV output voltage, ip is the PV output current, isc is the shortcircuit current and T is the temperature of the PV system away from25 �C are proposed for the approximation of the PV system undernormal working environments. For example, a second order nonlin-earity model for vp will give

vp ¼ h1 þ h2ip þ h3T þ h4isc þ h5i2p þ h6ipT þ h7ipisc þ h8T2 þ h9Tisc

þ h10i2sc

where hi, i = 1,2, . . . ,10 are the model parameters.

3. Nonlinear model identification

A normal expansion of (1) may involve hundreds of linear andnonlinear terms if the order of nonlinearity is high. A lot of the lin-ear and nonlinear terms may be redundant. Hence an estimationalgorithm which involves procedures for the selection of signifi-cant candidate terms and the estimation of the corresponding coef-ficients is therefore required. The orthogonal least squaresestimation algorithm [12] has been found to be efficient tools forthe estimation of nonlinear systems. The error reduction ratio[12] which is a by-product of the estimation algorithm providesinformation regarding the significance of individual linear andnonlinear terms. As an example, F1(�) of (1) can be representedby the regression equation

iscðkÞ ¼XM

i¼1

piðkÞhi þ eðkÞ ð2Þ

where M is the number of the unknown parameters, hi is the un-known parameters, isc(k) is the short circuit current for sample k,e (k) is the estimation error for sample k, pi(k) represents a termin the expansion of F1(�) for sample k and no two pi(k)’s are identical.The objective of the orthogonal least squares estimation algorithmis to minimize the cost function

J ¼ 1N

XN

k¼1

e2ðkÞ ð3Þ

where N is the number of collected samples by transforming (2)into an equivalent orthogonal equation

iscðkÞ ¼XM

i¼1

giwiðkÞ þ eðkÞ ð4Þ

Using the procedures [12]

w1ðkÞ ¼ p1ðkÞ

wiðkÞ ¼ piðkÞ �Xi�1

j¼1

ajiwkðkÞ; i ¼ 1; . . . ;M

aji ¼1N

PNk¼1wjðkÞpiðkÞ

1N

PNk¼1w2

j ðkÞ;

j ¼ 1; . . . ; i� 1i ¼ 1; . . . ;M

�

gi ¼1N

PNk¼1iscðkÞwiðkÞ

1N

PNk¼1w2

i ðkÞ; i ¼ 1; . . . ;M

ð5Þ

The original system parameters hi can then be recovered as [12]

hM ¼ gM

hj ¼ gj �XM

i¼jþ1

ajihi; j ¼ M � 1; . . . ;1ð6Þ

The relative contribution of each individual orthogonal data setwi(k) can be determined using the error reduction ratio defined as[12]

eRRi ¼1N

PNk¼1g2

i w2i ðkÞ

1N

PNk¼1i2

scðkÞ; i ¼ 1; . . . ;M ð7Þ

A large value of e RRi indicates the significance or importance ofa particular term to the output whereas a small value indicates theterm is insignificant. Hence incorporating the error reduction ratiotest with the orthogonal estimation algorithm to sort through allthe linear and nonlinear candidate terms will result with a parsi-monious model and all redundant terms are eliminated.

3.1. Forward searching algorithm

A candidate term selected at each interaction by the errorreduction ratio is optimal but there is no guarantee that the finalmodel fitted is optimal. A combination of candidate terms withsmaller error reduction ratios at different iterations may producea model with smaller cost function. To overcome this, the forwardsearching algorithm is proposed and it can be summarized asfollows:

(a) Start the estimated model with no term and define the orderof nonlinearity l and number of candidate terms v to betested at each iteration stage.

(b) Employ the orthogonal estimator (5) and the error reductionratio (7) to search through all possible candidate variables.

(c) Apply the error reduction ratio on each variable in turn andselect the top v variables which have the largest error reduc-tion ratios as plausible candidates to enter into the finalmodel.

(d) Try each of the v variables in turn and pass the rest of thevariables to the next stage of the estimation.

(e) Repeat steps (b–e) until the sum of error reduction ratiosreaches a certain threshold or the variables have beenexhausted.

(f) Reconstruct the actual system parameters using (6).

sc

p

p

Fig. 1. A PV simulator.

K.M. Tsang, W.L. Chan / Energy Conversion and Management 70 (2013) 83–89 85

(g) Obtain the model predicted output and compare it with theactual system output to produce the estimation erroreðkÞ ¼ iscðkÞ � iscðkÞ where iscðkÞ is the model predicted out-put. Calculate the cost function J ¼ 1

N

PNk¼1e2ðkÞ:

(h) Repeat steps (b–g) until all possible combination of termshave been tested.

(i) Select the model which produces the least cost function J.

4. Short-circuit current determination and maximum powerpoint identification

Even though the short-circuit current isc can easily be deter-mined from a PV system by performing a short-circuit test onthe system, this will seldom carry out as it highly affect the effi-ciency of the energy conversion process. It would be useful if theshort-circuit current can be determined based on ip, vp and T as thiswill not affect the normal operation of the PV system. Assuming aset of collected samples isc, ip, vp and T are available, combining theforward searching algorithm and the orthogonal least squares esti-mation algorithm can provide estimates for F1(�), F2(�) and F3(�) as

isc ¼ bF 1ðT; ip;vpÞvp ¼ bF 2ðT; isc; ipÞip ¼ bF 3ðT; isc;vpÞ

ð8Þ

The power delivered from the PV panel can be approximated asP ¼ vpip and the change of power with respect to change of ip forgiven T and isc is given by

f ðipÞ ¼dPdip¼ bF 2ðT; isc; ipÞ þ

dbF 2ðT; isc; ipÞdip

ip ð9Þ

For given T, vp and ip, an estimate of the short circuit current canbe obtained from bF 1ðT; ip;vpÞ. If the temperature and the short-circuit current remained unchanged, the optimal power can beobtained when

f ðipÞ ¼ 0 ð10Þ

As (9) is a high order polynomial in ip, the solution to (10) can beobtained using the Newton–Raphson method. The gradient andHessian of f(ip) are given by

f 0ðipÞ ¼ 2dbF 2ðT; isc; ipÞ

dipþ d2bF 2ðT; isc; ipÞ

di2p

ip

f 00ðipÞ ¼ 3d2bF 2ðT; isc; ipÞ

di2p

þ d3bF 2ðT; isc; ipÞdi3

p

ip

ð11Þ

and the solution to (10) can be obtained iteratively by

iðnþ1Þop ¼ iðnÞop �

f 0ðiðnÞop Þf 00ðiðnÞop Þ

ð12Þ

where iop is the required PV output current at the maximum powerpoint and n denotes the iteration number. A good starting point forthe iteration process is to set to

ið0Þop ¼ 0:9isc ð13Þ

where isc ¼ bF 1ðT; ip; vpÞ.

5. Experimental results

To illustrate the effectiveness of the polynomial models in cap-turing the I–V characteristics, a 300 V PV simulator with a maxi-mum short circuit current of 5 A was built in connecting 200 V6 A DII6A2 diodes in series and a constant current source wasconnected to the diodes as shown in Fig. 1. Different amplitudes

of current ranging from 1 A to 5 A were injected into the simulatorat various temperatures ranging from 20 �C to 45 �C and the corre-sponding output currents and voltages at various temperatureswere recorded. Fig. 2 shows the 1950 collected data records at var-ious temperatures and short-circuit currents.

Initially, the order of nonlinearity l was set to eight, v was set tothree and the termination threshold for the sum of error reductionratios was set to 0.9999. There were 165 terms in the initial spec-ified model. After applying the orthogonal least squares estimationalgorithm and the forward searching algorithm, parsimoniousmodels with only 10 terms was resulted for the approximation ofthe short circuit current and it was given by

isc ¼ 5:481� 10�4vp þ 0:9186ip þ 2:297� 10�6v2pip

� 1:869� 10�16Tv6p � 1:940� 10�17v7

p þ 1:081

� 10�19v8p þ 1:079� 10�18Tv7

p þ 2:228� 10�18T2v6p

� 9:275� 10�17v6pi2

p þ 3:161� 10�15v5pi3

p ð14Þ

Looking at (14), the approximation could easily run into numer-ical problem because of the very small coefficients associated withsome of the nonlinear terms. One simple way to get around theproblem was to normalize the current and voltage by the rated val-ues of the PV systems to give

iscN ¼isc

ir

ipN¼ ip

ir

vpN¼ vp

v r

ð15Þ

where vr and ir are the rated voltage and current for the PV system.The normalized data records were passed to the orthogonal leastsquares estimation algorithm and the fitted model became

iscN ¼ 0:03288vpNþ 0:9186ipN

þ 0:2067v2pN

ipN

� 0:02724Tv6pN� 0:8485v7

pNþ 1:419v8

pN

þ 0:04721Tv7pNþ 0:0003248T2v6

pN� 0:3381v6

pNi2pN

þ 0:1920v5pN

i3pN

ð16Þ

The sum of error reduction ratios was equal to 0.9999 and thecost function was 7.563 � 10�5. Fig. 3a shows the estimation errorfor (16).

For the identification of the output voltage, 12 terms wereresulted and it was given by

0 200 400 600 800 1000 1200 1400 1600 18000

10

20

30

40

50

60

Sample number

Tem

pera

ture

(Cen

tigra

de)

(a) Temperature

0 200 400 600 800 1000 1200 1400 1600 18000

1

2

3

4

5

6

Sample number

Cur

rent

(A)

(b) Short-circuit current

0 200 400 600 800 1000 1200 1400 1600 18000

1

2

3

4

5

6

Sample number

Cur

rent

(A)

(c) Output current

0 200 400 600 800 1000 1200 1400 1600 1800150

200

250

300

350

Sample number

Volta

ge (V

)

(d) Output voltage

Fig. 2. Collected output currents and voltages at different temperatures and short-circuit currents.

0 200 400 600 800 1000 1200 1400 1600 1800-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

Sample number

Nor

mal

ized

erro

r

(a) Normalized error for fitted short-circuit current

0 200 400 600 800 1000 1200 1400 1600 1800-0.02

-0.01

0

0.01

0.02

Sample number

Nor

mal

ized

erro

r

(b) Normalized error for the fitted output voltage

Fig. 3. Approximation errors for the fitted nonlinear models.

0 1 2 3 4 5150

200

250

300

350

Current (A)

Volta

ge (V

)

I =5AI =4AI =3AI =2AI =1A

sc

scsc

scsc

Fig. 4. Approximated I–V characteristics for different short-circuit currents at 25 �C.

86 K.M. Tsang, W.L. Chan / Energy Conversion and Management 70 (2013) 83–89

vpN¼ 0:7855� 0:003510T � 0:3157ipN

þ 0:4857iscN

� 0:2162i2scNþ 0:1716iscN ipN

� 918:7i4pNþ 5735:0iscN i4

pN

� 13000:0i2scN

i4pNþ 12670:0i3

scNi4pN� 4492:0i4

scN

� 0:2040i8pN

ð17Þ

The sum of error reduction ratios was to 0.9999 and the costfunctions was 3.055 � 10�5. Fig. 3b shows the estimation errorfor (17). Clearly very good approximations had been obtained forthe fitted short circuit current and PV output voltage. Fig. 4 showsthe denormalized I–V characteristics of (17) for different short cir-cuit currents at 25 �C.

Table 1Maximum power point at 25 �C with different short circuit currents.

Short circuit current Actual maximum power point Estimated maximum power point

isc (A) iop (A) vop (V) P (W) iop (A) vp (V) P (W)

5 4.42 244.6 1081 4.35 248.3 10804 3.55 236.1 838 3.67 225.4 8273 2.67 228.1 609 2.76 218.4 6032 1.77 217.1 384 1.84 204.5 3761 0.87 196.8 171 .84 203.5 171

Fig. 5. Boost circuit for battery charging.0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

5

Time (s)

Cur

rent

(A)

model based trackingextremum seeking control

start tracking

(a) PV output current

0 0.2 0.4 0.6 0.8 1150

200

250

300

350

Time (s)

Volta

ge (V

)

model based trackingextremum seeking control

(b) PV output voltage

0 0.2 0.4 0.6 0.8 10

200

400

600

800

1000

Time (s)

Pow

er (W

)

model based trackingextremum seeking control

(c) PV output power

Fig. 6. Comparison between model based maximum power point tracking andextremum seeking control.

K.M. Tsang, W.L. Chan / Energy Conversion and Management 70 (2013) 83–89 87

From (17),

f ðipNÞ ¼ �1:836i8

pNþ ð28680:0iscN � 64990:0i2

scN

þ 63370:0i3scN� 22460i4

scN� 4593:0Þi4

pN

þ ð0:3432iscN � 0:6313ÞipNþ ð0:7855� 0:003510T

þ 0:4857iscN � 0:2162i2scNÞ ð18Þ

for specific T and isc. Table 1 shows the actual optimal power pointobtained from the test and the estimated maximum power pointobtained from (18) at 25 �C. The power shown in Table 1 was ob-tained in drawing the required output current from the simulator.For the estimated maximum power points, it could achieve morethan 97.9% of actual maximum power. This clearly demonstratedthe effectiveness of using the fitted model in maximum power pointtracking.

5.1. Maximum power point tracking for battery charging

As a demonstration, a boost converter is connected to the PVsimulator for battery charging as shown in Fig. 5. If the switch Sb

is driven by pulse width modulation signal, the state averagingdynamics of the converter can be described as

L_ipðtÞ ¼ vpðtÞ � ð1� dðtÞÞvBðtÞ ð19Þ

where L is the inductor connected to the PV system, d(t) is the dutyratio of the switching operation for Sb, vB(t) is the terminal voltageof the charging battery, ip(t) is the PV output current and vp(t) is thePV output voltage, respectively. A proportional plus integral (PI)controller [13] of the form

GPIðsÞ ¼KPsþ KI

sð20Þ

where Kp and KI are constants can be applied for the control of thePV output current. As the battery voltage can be approximated toclamp at its rated voltage Ur, the closed-loop characteristic equationfor the current control loop can be approximated to

DðsÞ � Ls2 þ KPUrsþ KIUr ð21Þ

The bandwidth of the current control loop can be approximated to

xb �ffiffiffiffiffiffiffiffiffiffiKIUr

L

rð22Þ

and the damping ratio of the current control loop can be approxi-mated to

0 10 20 30 40 500

1

2

3

4

5

6

Time (s)

Cur

rent

(A)

actual short circuit currentpredicted short circuit current

(a) PV predicted short circuit current superimposed on actual short-circuit current

0 10 20 30 40 500

1

2

3

4

5

Time (s)

Cur

rent

(A)

(b) PV output current

0 10 20 30 40 50150

200

250

300

350

Time (s)

Volta

ge (V

)

(c) PV output voltage

0 10 20 30 40 500

200

400

600

800

1000

Time (s)

Pow

er (W

)

(d) PV output power

0 10 20 30 40 5020

25

30

35

40

45

Time (s)

Tem

pera

ture

(Cen

tigra

de)

(e) PV temperature

Fig. 7. Performance of model based maximum power point tracking for varying short-circuit current.

88 K.M. Tsang, W.L. Chan / Energy Conversion and Management 70 (2013) 83–89

fb �KPUr

2xbð23Þ

For given bandwidth, damping ratio and rated battery voltage,the required controller setting can easily be obtained from (22)and (23).

Power MOSFET, SiHP22N60S, was used to realize the switch ofthe converter and the inductor was chosen as L = 12.6 mH.Although 360 V 10 Ah lead acid battery [14] was chosen as thestorage unit because the cost is lower, Lithium ion batteries couldalso be used if better performance is needed [15]. A temperaturesensor was also fitted to the PV simulator. The switching frequencyfor the PWM was set to 100 kHz. An industrial PC was used to sam-ple all required variables and to implement the current control forthe charging system. The sampling frequency for the industrial PC

was to set 10 kHz. The bandwidth of the current control loop wasset to 100 Hz and the damping ratio was set to 1, from (22) and(23) the required PI controller became

dðtÞ ¼ 0:044eðtÞ þ 13:82Z

eðtÞdt ð24Þ

where eðtÞ ¼ iopðtÞ � ipðtÞ and iopðtÞ is the required output currentfrom the PV system. Eq. (24) could be discretized to give

dðkÞ ¼ 0:044eðkÞ þ 0:001382X

eðkÞ ð25Þ

The search for the maximum power point and charging of thebattery can be summarized as follows:

(a) Sample T, vp and ip.

K.M. Tsang, W.L. Chan / Energy Conversion and Management 70 (2013) 83–89 89

(b) Calculate isc using (16).(c) For the measured T and estimated isc , initially set iop ¼ 0:9isc

and apply (12) to obtain the final iop.(d) Pass iop to (25) to obtain the required duty ratio d.(e) Pass the duty ratio d to the PWM driver for the switching of

the boost converter.(f) Repeat steps (a–e).

Notice that if iop starts with 0.9 isc the solution will converge intwo to three iterations using the Newton–Raphson method.

Fig. 6 showed the tracking performance of the proposed schemewhen the short-circuit current was fixed at isc = 4.2 A and the tem-perature was fixed at 25 �C. The maximum power point wastracked in about 0.02 s and there was no oscillation at the steadystate. A comparison had been made with a fast maximum powerpoint tracking algorithm based on the extremum seeking controlmethod [16] which is an improved version of [17]. Fig. 6 showedthe tracking performance using the extremum seeking controlmethod when the short circuit current was fixed at 4.2 A and thetemperature was fixed at 25 �C. The maximum power point wastracked in about 1 s and there was oscillation at the steady state.Even though faster tracking response could be obtained if the stepsize of the disturbance was increased, the amplitude of oscillationwould also be increased. This would further decrease the averagedpower extracted from the PV module. The amplitude of oscillationfor the tracked current was about 0.5 A. The steady state oscillationbecame more significant if the irradiance level was low becausethe corresponding short-circuit current would become less. Thepercentage of power that could be extracted using the extremumseeking control method would become even less. However, search-ing mechanism was not required for the proposed scheme and themaximum power point could easily be obtained based on the out-put voltage, output current and temperature of the PV system andthe fitted model. The steady state of the tracked current had virtu-ally no oscillation and the average power extracted would be closerto the ideal maximum power point.

Fig. 7 showed the tracking performance of the proposed schemefor varying short-circuit current. Fig. 7a showed the predictedshort circuit current obtained from (16) superimposed on theactual short-circuit current. A very close prediction had beenobtained. Fig. 7b showed that the PV output current closely fol-lowed the varying short circuit current. Clearly a very fast trackingresponse with less steady state oscillations was obtained in track-ing the maximum power point using the fitted nonlinear model.

6. Conclusions

A model based maximum power point tracking has been pro-posed for PV systems. Utilizing the orthogonal least squares esti-mation algorithm and the forward searching method can identifyenvironmental invariant polynomial type of nonlinear models forPV systems. Short-circuit current and maximum power point canbe obtained from the fitted models based on the measured temper-

ature, output voltage and output current. Maximum power pointcan easily be derived from the fitted model. Experimental resultsshow that the tracking performance of the proposed scheme is fastand there is no steady state oscillation. One limitation of the pro-posed method is complex mathematical operations are requiredduring the model fitting process. However, user friendly softwarewill be developed in future if people are interested in using themethod.

Acknowledgment

The authors gratefully acknowledge the support of the HongKong Polytechnic University.

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