IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 24, NO. 5, OCTOBER 2014 0600806

Novel Linear Iteration Maximum Power PointTracking Algorithm for Photovoltaic

Power GenerationWei Xu, Chaoxu Mu, and Jianxun Jin

Abstract—A novel maximum power point tracking (MPPT)algorithm is proposed in this paper based on the linear iterationmethod for photovoltaic (PV) power generation for improvingsteady-state performance and fast dynamic response simultane-ously. In this MPPT algorithm, linear prediction estimates theMPP of PV with a very high accuracy due to the constant-voltageand constant-current characters of PV. Then, further iterations forerror correction are executed to get a better approximation to theMPP. Experiments verify theoretical analysis and demonstrate itsfast dynamic response and high working efficiency.

Index Terms—Convergence, dynamic response, linear iteration,maximum power point tracking (MPPT) algorithm, photovoltaic(PV) power generation.

I. INTRODUCTION

R ECENTLY, there has been an increasing interest in usingPV systems to supply electricity for various consumers

since it is clean, noiseless, inexhaustible, and little maintenance[1]. However, the PV power generation performs a nonlinearvoltage–current (U−I) curve, and the output power of a PVsystem largely depends on the array temperature and the solarinsulation, so it is necessary to constantly track the maximumpower point (MPP) of the solar array. Several methods, such asconstant-voltage tracing method, perturbation and observation(P&O) method, incremental conductance method (INC), etc.,have been used to draw the maximum power of the solar arrayin the past couple of years. Generally, each method mainlyincludes three criteria, i.e. tracking velocity, tracking accuracy,and stability. The constant-voltage method can track the MPPrapidly, but its accuracy is poor. The P&O method has beenwidely used in practice due to its simplicity and easy implemen-tation, but the tracking speed is slow and always has oscillationsduring the MPP tracking period [2], [3]. The INC method couldbe regarded as an improved P&O method. It can track the

Manuscript received May 22, 2014; accepted June 18, 2014. Date of pub-lication June 30, 2014; date of current version July 31, 2014. This work wassupported by theNational Natural Science Foundation under Grant 51377065,Grant 61301035, and Grant 61304018. (Corresponding author: Chaoxu Mu.)

W. Xu is with the School of Electrical and Electronic Engineering,Huazhong University of Science and Technology, Wuhan 430074, China(e-mail: weixu@hust.edu.cn).

C. Mu is with the School of Electrical and Automatic Engineering, TianjinUniversity, Wuhan 300072, China (e-mail: cxmu@tju.edu.cn).

J. Jin is with the School of Automation Engineering, University of Elec-tronic Science and Technology of China, Chengdu 610054, China (e-mail:jxjin@uestc.edu.cn).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TASC.2014.2333534

MPP more accurately, but the fixed step size is determined bythe requirements of steady-state calculation accuracy and theMPPT dynamic response. Hence, its tracking speed is also veryslow [4], [5]. The variable step size algorithms can track MPPfast and accurately, but some essential parameters, such as themaximal step size and scaling factor, are difficult to estimateaccurately [6]. In general, fast tracking can be achieved withbigger increments, but the system might not operate exactly atthe MPP, resulting in oscillations around it [3]. This situationcould turn to the contrary when the MPPT operates with asmaller increment [6].

To overcome the limitations of the traditional MPPT meth-ods, a novel MPPT algorithm based on linear iteration isproposed in this paper to achieve much faster tracking speedand higher efficiency. In the proposed algorithm, the step sizecan modify its response speed intelligently without step sizereference.

II. BASIC CHARACTERISTICS AND MODEL OF THE PV

The PV power generation system has many parameters in-fluenced by irradiance and temperature, which are difficult todetermine accurately. In engineering application, a simple PVcharacteristic equation based on PV cell physical character hasbeen widely used as follows [7]:

i = ISC

{1− C1

[exp

(u

C2UOC

)− 1

]}(1)

where ISC is the PV generation short circuit current relative tothe solar radiation and temperature, UOC is the PV open circuitvoltage, i expresses the solar cell output current and u is thesolar cell output voltage. Two coefficients C1 and C2 can bedescribed as

C1 =

(1− Im

ISC

)exp

(− Um

C2UOC

)(2)

C2 =

(Um

UOC− 1

)/ln

(1− Im

ISC

)(3)

where Um is the voltage of the PV model when the output poweris maximal, and Im is the maximum of the PV model outputcurrent. Then the PV array output power is

P (u) = u · ISC{1− C1

[exp

(u

C2UOC

)− 1

]}. (4)

1051-8223 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

0600806 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 24, NO. 5, OCTOBER 2014

Fig. 1. The DC module structure of PV system.

The efficiency of PV cell depends on the internal shuntresistance, sunlight intensity, array temperature, and load. SetISC_ref , UOC_ref , Um_ref , Im_ref as the PV reference pa-rameters under standard conditions, and solar irradiation isgiven by Sref = 1000 W/m2 and circumstance temperature isTref = 25 centi degree(◦C), then ISC, UOC, Um, and Im can beexpressed as

ISC = ISC_ref ·S

Sref(1 + a ·ΔT ) (5)

UOC =UOC_ref · ln(e+ b ·ΔS)(1− c ·ΔT ) (6)

Im = Im_ref ·S

Sref(1 + a ·ΔT ) (7)

Um =Um_ref · ln(e+ b ·ΔS)(1− c ·ΔT ) (8)

where ΔS = (S/Sref − 1), ΔT = (T − Tref), and typicallya = 0.025/◦C, b = 0.5/(W/m2), c = 0.00288/◦C, respec-tively, e is constants. It is obvious that the PV output changesdepending upon circumstance. Especially, when the system isunder partially shade condition, the output characteristics ofPV array will become more complex. In order to overcomethis shortcoming, a MPPT PV system with DC module isproposed shown in Fig. 1, the PV arrays are connected inparallel through DC/DC converter and the maximum outputpower is controlled by some independent micro programmedcontrol unit (MCU) controllers. A constant-voltage strategy canbe adapted to DC/AC converter to maintain the voltage of DCbus. Therefore, every unit of PV array, converter and controllercan be regard as separated with each other.

III. LINEAR PREDICTION METHOD

FOR MPPT ALGORITHM

A. Parameters Choice for Linear Iteration Method

Normally, the U−I curve of PV includes two regions, i.e. theconstant-current region when u ∈ [0, Um − ξ] and the constant-voltage region when u ∈ [Um + ρ, UOC], ξ, ρ > 0, as shownin Fig. 2. The derivative of the power to voltage (dP/dU)depicted by Curve (1) in Fig. 2, can be described by

P ′(u) = ISC

(1 + C1 − C1 exp

(u

C2UOC

))

− C1ISCu

C2UOCexp

(u

C2UOC

). (9)

Fig. 2. The curves of P (u), 1/[1 + P ′(u)2], P ′(u) and angle θ.

H(ux) is set as the tangent line through point (ux, P (ux)) ofthe U−P curve, which can be calculated by

H(ux) = P ′(ux)× (u− ux) + P (ux) (10)

θ is supposed as a angle between the tangent line H(ux) andx-axis depicted by Curve (2). Its expression is

θ=arctandP

du=arctan

[i−C1ISCu

C2UOCexp

(u

C2UOC

)]. (11)

The derivative of arctan(P ′(u)) can be estimated by

d arctan (P ′(u))

du=

1

(1 + P ′(u)2). (12)

When u∈ [0, Um−ξ] or u∈ [Um+ρ, UOC], 1/[1+P ′(u)2]shown in Fig. 2 is less than 0.1. It illustrates that i and unearly keep constants in the constant-current region and theconstant-voltage region, respectively. Hence, the estimation ofMPP saves time consumption and achieves high accuracy byusing two first-order Taylor series expansions. From the viewof geometry, θ is the most intuitive representation of P ′(u), andit changes very slowly when PV system works on the regionsof constant current or constant voltage.

B. Basic Analytic Steps of Linear Prediction Algorithm

The new algorithm includes linear prediction and error cor-rection. The procedures are summarized as follows:

S1: Take any two points uL and uR from the PV curve. LetuL ∈ [0, Um − ξ] and uR ∈ [Um + ρ, UOC]).

S2: Define HL(u) is the left tangent line through point(uL, P (uL)), HR(u) is the right tangent line throughpoint (uR, P (uR)). The corresponding slopes are

XU et al.: NOVEL LINEAR ITERATION MPPT ALGORITHM FOR PV POWER GENERATION 0600806

Fig. 3. The procedure of linear prediction algorithm (a) Linear prediction.(b) Error correction.

P ′(uL) and P ′(uR), where P ′(uL) > 0 and P ′(uR) <0. HL(u) and HR(u) can be expressed by{

HL(u) = P ′(uL)× (u− uL) + P (uL)HR(u) = P ′(uR)× (u− uR) + P (uR)

. (13)

S3: As P ′(uL) > 0 and P ′(uR) < 0 are always true, thetwo tangent lines will intersect at O1, and set the corre-sponding point on the PV curve as P1. Then it considerspoint P1 as the prediction MPP of the PV curve, sothe segment O1P1 is the truncation error referring toFig. 4(a). And the abscissa of point P1 is

uP1 =P (uR)− P (uL) + P ′(uL)× uL − P ′(uR)× uR

P ′(uL)− P ′(uR).

(14)S4: The tangent line through point P1 is H(uP1) and its

slope is P ′(uP1). If P ′(uP1) > 0, set uL = uP1; other-wise, set uR = uP1. Referring to (13) and (14), updateP (uL), P (uR), P ′(uL), P ′(uR), HL(u) and HR(u).

S5: Update the intersection point Oi of HL(u) and HR(u)and the corresponding Pi. Calculate uPi referring to(14). If |H(uOi)− P (uPi)| ≥ ε, return S3 until theterminal condition is satisfied. If the truncation error|H(uOi)− P (uPi)| < ε, ε > 0, the iteration stops, andmaintain the operating voltage uPi until the systemrestarts or the irradiance changes.

Fig. 3(a) and (b) describe the processes of linear predictionand error correction. When |H(uOi)− P (uPi)| < ε, the algo-rithm stops and the operating voltage is maintained until thenext duty cycle.

Fig. 4. The maximum truncation error of linear prediction.

C. Estimation of Truncation Error

The truncation error is defined as R(u) = |H(uoi)−P (uPi)|, where H(u) is the tangent line of U−P curve, Oi theintersection point of the ith iteration cycle. Due to that H(ui)goes through point (ui, P (ui)), R(ui) = H(ui)− P (ui) = 0and R′(ui) = H ′(ui)− P ′(ui) = 0, it can calculate the errorR(u) by

R(u) = λ(u)× (u− ui)2 (15)

where λ(u) is the scale factor. In order to get the specificexpression of λ(u), it should calculate P ′′(u) according toRolle theorem expressed by

P ′′(u) = −2C1ISCC2UOC

· eu

C2UOC − C1ISCu

(C2UOC)2· e

uC2UOC . (16)

It is obvious that P ′′(u) < 0 is always true. When u ∈[0, UOC], there is at least one ε that satisfies

P (ε)−H(ε)− λ(ε)× (ε− ui)2 = 0. (17)

By further investigation, it can get the relationship betweenλ(u) and ε, as illustrated by

λ(u) = P ′′(ε)/2 (18)

where P ′′(ε) is a boundary function and ε is a variable duringthe range of [0, UOC]. Then, the estimation maximum error is

R(u)=P ′′(ε)

2×[P (uR)−P (uL)+P ′(uL)×(uL−uR)

P ′(uL)−P ′(uR)

]2.

(19)

As shown in Fig. 4, the maximal absolute value of truncationerror is less than 30, and it could become smaller gradually withthe increasing times of error correction.

D. Proof of New Algorithm Convergence

Due to P ′′(u) < 0, the P (u) can be regarded as one con-tinuous convex function. According to the PV curve, uO ∈[uL, uR], the interval [uL, uR] will become [ui

L, uiR] after i times

error correction, and the equations are obtained as follows:∣∣uiO − ui

R

∣∣ =Li ·∣∣ui

L − uiR

∣∣ (20)

Li ·∣∣ui

L − uiR

∣∣ = ∣∣ui+1L − ui+1

R

∣∣ (21)

0600806 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 24, NO. 5, OCTOBER 2014

Fig. 5. Tracking trajectory in mathematical theory.

Fig. 6. The response speed of Newton iteration and proposed methods.(a) Newton iteration method. (b) The proposed method.

where Li is a scale factor less than 1. Thus it can get

Li+k−1 ·∣∣ui+k−1

L − ui+k−1R

∣∣ = ∣∣uiL − ui

R

∣∣ ·i+k−1∏n=i

(Ln). (22)

As Li < 1, when k → ∞, |ui+kL − ui+k

R | will get close to0. Due to umax ∈ [ui+k

L , ui+kR ], the result will finally converge

to the MPP (ui+kP → umax), as shown in Fig. 5. Due to the

constant area of PV, it can be seen that the first step (linearprediction) can drive the working point to the vicinity of MPPrapidly, and then others will improve the tracking accuracythough repeated iterations.

Fig. 6 is the tracking speed comparison between Newtoniteration and the proposed methods. The formula of Newton’siterative algorithm is

uk+1=P (uk−1)−P (uk)−[P ′(uk−1)×uk−P ′(uk)×uk−1]

P ′(uk)−P ′(uk−1).

(23)

In Fig. 6(a), Newton iteration method has a character ofquadratic convergence, and it just takes eight calculation cyclesto obtain the optimal solution under standard conditions. But itis less stable for great dependence on its initial iteration value,which can be solved effectively by the new method in this paper.As shown in Fig. 6(b), when the initial point u0 ∈ [0, Um], the

Fig. 7. The accuracy analysis of in one step size. (a) The minimal error.(b) The maximal error.

Fig. 8. Simulation on the calculation accuracy of the two methods.

iteration result of Newton method will diverge. However, theproposed method has little requirement of the initial value, andit has almost similar tracking response speed (13 calculationcycles) for MPP.

E. Accuracy Analysis of New Method and P&O Method

P ′(u) is always calculated by ΔP/Δu, where Δu is a tinyvoltage increment and ΔP is the corresponding power incre-ment. Compared with other traditional methods with a fixedstep size (ΔuS), the new method has much higher accuracyeven when Δu = ΔuS.

In Fig. 7, the minimum error after line iteration is 0, whilethe maximum error is λΔu, λ < 1 is a scale factor. After thelast iteration, if the voltage uOi of intersection Oi is just on theMPP, the linear prediction algorithm has the minimum error 0,as depicted in Fig. 7(a). Meanwhile, the minimum error of fixstep size method is ρ∗Δu, ρ < 1 is also a scale factor. If theslop of H(uL) is zero, as shown in Fig. 7(b), the maximumerror of new method is λ∗Δu, which is different with those oftraditional methods with (1 + ρ∗Δu). It can be concluded thatthe new method has higher accuracy even when Δu = ΔuS.

Fig. 8 and Table I give the simulation result of the accuracy.It sets the increment of new algorithm Δu = 1 V which useto calculate the derivative of P ′(u) and P ′(u) can be expressby ΔP/Δu approximately. Set ΔuS = 1 V in P&O method.From Fig. 8, the voltage can converge to 36.59 by the proposedmethod, which is very close to the theoretical UMPP = 36.52even with a big increment, while the voltage oscillates between36 and 38 by P&O method. It is obvious that the proposedmethod is more accurate than the P&O method even with a bigperturbation step. Δu is much smaller than ΔuS in practice,and hence the new method could have much higher accuracy inthis condition.

F. Analysis of Stability Under ComplexCircumstance Conditions

The tracking speed and steady performance of the newmethod are affected by the value of P ′(ui) and the position

XU et al.: NOVEL LINEAR ITERATION MPPT ALGORITHM FOR PV POWER GENERATION 0600806

TABLE ISIMULATION RESULT OF THE ACCURACY

Fig. 9. The stability analysis under the solar irradiation step change. (a) Solarirradiation steps up. (b) Solar irradiation steps down.

TABLE IISIMULATION DATA OF THE RESPONSE SPEED AND GROSS ENERGY

of tangent line H(ui). When the position of H(u) has changedin steady state as shown in Fig. 9(a), the output power of PVgeneration will change in step from P (u

(n)R ) to P (u

(n)′R ), and

the slope of H(uR) will become greater than zero (P ′(uR) >0), then it needs to take some measures to ensure P ′(uR) < 0.Generally speaking, when the PV system keeps in trackingthe MPP, each u(i) should be stored in the memory. WhenP ′(u

(n)R ) > 0, it could go back from u

(n)R to u

(n−1)R and then

from u(n−1)R to u

(n−2)R until P ′(u

(i)R ) < 0. The same principle

can be applied to other side of the U−P curve, i.e. whenP ′(uL) < 0, it could also go back from u

(n)L to u

(n−1)L as shown

in Fig. 9(b). The small points in Fig. 9 are the simulationtracking trajectories, which demonstrates the operating voltageand power in the ith cycle. In order to make the system morestable, the uL and uR could be sampled in every calculationcycle.

IV. EXPERIMENT ANALYSIS

The quantitative comparisons are made between the pro-posed method and the P&O method, as summarized in Table II.

Due to the response speed of new method depends less ontiny increment Δu, the linear prediction method in Fig. 10(b)can search MPP more rapidly compared with the P&O trackingmethod in Fig. 10(a). In this process, the voltage of proposedmethod could converge to MPP successfully through only7 times error correction. Nevertheless, the P&O method isclosely related to its step size.

Fig. 10. The initial PV output voltages waveform. (a) P&O method.(b) Proposed method.

Fig. 11. Experimental setup.

TABLE IIIPARAMETER VALUES FOR EXPERIMENTS

Different PV MPPT strategies have been presented to fur-ther verify theoretical analysis. The experimental platform isdepicted in Fig. 11, which is made up of the PV array, MCUcontroller and relative measurement equipment. The parametervalues are summarized in Table III. PV arrays are connectedwith buck circuit through filter capacitor C1 and the outputvoltage will be controlled by VT. A well-designed filter isintroduced to avoid the sampling noise the MPPT input voltageand current. The MCU is used as the main controller and thebuck circuit is responsible to control the DC voltage. Boththe PV voltage and PV current are fed back into analog todigital converters with 10-bit accuracy by sensors. Then it willbe transmitted to MCU directly. After the control effort iscalculated, the PWM generator in MCU will generate a PWMsignal to control the switching VT.

The experiments are certainly carried out outdoor and theirradiance cannot be very high. The experimental results of

0600806 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 24, NO. 5, OCTOBER 2014

Fig. 12. Experiment comparisons among different methods. (a) P&O method.(b) Proposed method.

tracking voltage, current and power by the P&O method andthe linear prediction method are given in Fig. 12. Fig. 12(a)shows the 0 voltage starting-wave of P&O method with stepsize of 1 V. The scanning process is so obvious that it couldbe easily caught from this figure. This method must measurethe voltage and current step by step, which needs up to 15 sto finish the scanning process. The proposed method with Δu =0.05 V is demonstrated in Fig. 12(b) where the point B andpoint C is uL and uR respectively. The process between uL

and uR is made by linear prediction, and after 3 times errorcorrections as shown clearly in Fig. 12(b), the new method willconverge to MPP finally. The convergence speed of the newmethod is faster than P&O method. But it takes 10 s to finishthe whole tracking process, 6 s for linear perdition and 4 s forerror correction, which do not exhibits a very good dynamicperformance like simulation results. It just because the systemshould reach the steady state in each MPPT cycle that is alwayscontrolled by the PI model before another MPPT cycle begins,when there are lots of outside interferences. And the trackingspeed performance is not only limited by the PI hardware circuitbut the UOC. Because the duty ratio working on Buck circuit isnot linear, the rising process of output experimental voltage isnot straight.

By covering the PV batteries with films which has differentdegrees of light transmission, it can get different solar irradi-ation conditions. Fig. 13 shows the experiment result of PVarray under changed solar irradiation conditions by the help ofthe proposed method. In Fig. 13(a), when the solar irradiationincreases, the voltage uR will become greater than 0. Then thecontroller will make the operate voltage go back from u

(n)R

to u(i−1)R where P (u

(i−1)R ) < 0. In Fig. 13(b), it is seen that

Fig. 13. Experiment results with changed irradiation conditions. (a) Irradia-tion up. (b) Irradiation down.

the solar irradiation has decreased under the similar workingprinciple. It is evident that the proposed algorithm has a gooddynamic performance.

V. CONCLUSION

This paper mainly focuses on a new linear prediction MPPTalgorithm. It can modify the response speed intelligently with-out step-size reference during the whole operation, and it isalso robust against external disturbances, such as solar irradi-ation change and so on. Compared with the P&O method andconstant-voltage tracing method, the new algorithm can find theMPP rapidly and accurately.

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