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A maximum power point tracking method with variable weather parameters based on input resistance for photovoltaic system
Energy Conversion and Management 106 (2015) 290–299
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A maximum power point tracking method with variable weather parameters based on input resistance for photovoltaic system Shaowu Li ⇑ Science and Technology College, and School of Information Engineering, Hubei Minzu University, Enshi, China
a r t i c l e
i n f o
Article history: Received 24 April 2015 Accepted 12 September 2015 Available online 1 October 2015 Keywords: PV system MPPT VWP Input resistance
a b s t r a c t In order to greatly improve the maximum power point tracking (MPPT) speed and adaptability to the varying weather conditions for photovoltaic (PV) system, in this paper, a MPPT method with variable weather parameters (VWP) considered specially from input resistance standpoint is proposed. As well as other VWP methods, it can also track the maximum power point (MPP) as quickly as possible. In this method, the approximate relationship between control signal and PV cell parameters (Vm and Im) is firstly built by studying the input resistance of PV system deeply, then the relationship between VWP and Vm, Im is found by the curve fitting technique. Through these relationships, the bridge between control signal and VWP is built successfully, which is the key work to implement the direct MPPT control. Finally, some simulation experiments show that the proposed method is feasible and available to track the MPP successfully and has better MPPT rapidity, accuracy, stability and adaptability than conventional perturbation and observation (P&O) method and fuzzy control method. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction In order to avoid the produced power losses, now almost all PV systems use the DC/DC converters as MPPT control units. There are many existing MPPT methods such as the constant voltage tracking [1], the P&O method [2,3], the incremental conductance (IncCond) method [3–5], the genetic algorithm [6,7], the fuzzy logic control method [6–8], the neural network method [9,10], the sliding mode control method [11] and the predictive control technique [12–14]. In them, the P&O method is one of most widely used MPPT techniques. Its advantages mainly include the low-cost hardware, the easy implementation and the good performance without solar irradiance and temperature varying quickly with time. However, there are also some shortcomings including its slow tracking speed and oscillation around the MPP. In this paper, in order to study the output performance of proposed MPPT method, the P&O method is selected as the main compared object. With respect to the issue how the changing weather influence on MPPT control of PV system, some works have been done in papers [15,16]. Remarkably, some VWP methods have been proposed in papers [16–18] to implement the real-time MPPT control for PV systems with different topology. In these methods, the key technique is to find out the relationship between control signal
⇑ Tel.: +86 13997799701. E-mail address: [email protected] http://dx.doi.org/10.1016/j.enconman.2015.09.055 0196-8904/Ó 2015 Elsevier Ltd. All rights reserved.
and VWP, and their main advantages are the fast MPPT speed and strong adaptability under varying weather conditions. In this paper, the VWP technique will be studied continuously and considered specially from input resistance of PV system standpoint, which is one of the main aims and innovations in this work. In existing MPPT methods, except for paper [19], there is hardly a technique specially considered from input resistance of PV system standpoint because of its variability and difficult measurement. In paper [19], a real-time identification scheme using the Lambert W-Function is proposed to estimate the model of PV module and its desired resistance at the MPP. By contrast, in this paper, the input resistance of PV system is only used as a bridge in connecting control signal and cell parameters (Vm and Im), which lays the foundation of the relationship between control signal and VWP. It is clear that the innovative and reasonable use of input resistance is the key work to study the principle of proposed MPPT method. On the other hand, there are some MPPT method using the cell parameters Vm and Im. In paper [20], a linear current control is proposed on the basis of the linear relationship between Im and the level of irradiance. In paper [21], a feedback MPPT control method is proposed by computing from equations involving temperature and irradiance. In addition, the fractional open-circuit voltage method [22], the ripple correlation control (RCC) method [23] and so on also use Vm or Im as the key parameter to implement the MPPT control of PV system. However, in all these methods,
S. Li / Energy Conversion and Management 106 (2015) 290–299
the direct relationship Vm or Im and weather parameters (irradiance and temperature) is not still found out. Therefore, another purpose of this paper is to build this relationship. This paper is divided into the following sections: the principle of proposed MPPT method is analyzed in Section 2. The relationship between control signal of PV system and VWP is found out, then the description and implementation of proposed MPPT method are given in Section 3. The feasibility and availability of proposed method are verified and the MPPT rapidity, accuracy, stability and adaptability are compared with P&O method and fuzzy method in Section 4. Some discussions are had in Section 5. Finally, some conclusions are drawn in Section 6.
2.1. Model and input resistance of PV system The configuration of common PV system can be shown in Fig. 1. Where V and I represent the output voltage and current of PV panel, respectively; Vo and Io represent the output voltage and current of DC/DC converter, respectively. In Fig. 1, the DC/DC converter between PV panel and load or grid-connected inverter is usually an indispensable unit to implement the MPPT function. In practical application, the simplified mathematical model of PV panel can be shown in Eq. (1) [24,15]. This model is usually called as ‘‘Four-Parameter Model” [15].
h V i I ¼ Isc 1 C 1 eC2 V oc 1
ð1Þ
where C1 = (1 Im/Isc) exp (Vm/C2Voc); C2 = (Vm/Voc 1)/ ln (1 Im/Isc); Isc, Voc, Im and Vm represent the short circuit current, the open circuit voltage, the MPP current and voltage at standard conditions (1000 W/m2 and 25 °C), respectively. All data above are given by the PV panel manufacturer. When the DC/DC converter shown in Fig. 1 is the buck circuit, the structure of PV system can be shown in Fig. 2. Where Ri represents the input resistance on the right of PV panel. Here, assume that the buck DC/DC converter, which is operating in the continuous conduction operational model, is the ideal circuit and its output load or inverter can be regarded as a pure resistance RL. According to Fig. 2, Eqs. (2)–(4) can be given by laws of circuit and conservation of energy.
V Ri ¼ I
ð2Þ V 2o RL
ð3Þ
Io
I
PV panel
V
DC/DC converter
Vo
I V
Ri
PWM
D
Ri ¼
RL
ð5Þ
D2
Moreover, according to Eqs. (1), (3) and (4), the mathematical model of PV system can be expressed as
Po ¼
RL I2sc D2
pffiffiffiffiffiffi 2 P o RL 1 C 1 eC2 DV oc 1
ð6Þ
2.2. Acquisition of control signal based on input resistance at MPP Firstly, in practical application, PV system usually operates at the MPP, now Eq. (5) can be replaced as
RiMPP ¼
RL RL ¼ D2 MPP D2max
ð7Þ
where RiMPP and Dmax represent the input resistance and duty cycle of PWM signal, of PV system at the MPP, respectively. Secondly, according to Fig. 2, the input resistance of PV system can also be expressed as
RiMPP ¼
V MPP IMPP
ð8Þ
where VMPP and IMPP represent the values of V and I corresponding to the MPP, respectively. According to paper [17], Eq. (9) can be given.
V MPP ¼ C
ð9Þ
where C is a variable parameter whose value can be calculated by Eq. (10).
1 þ C1 1 C ¼ C 2 V oc lambertw e C1
ð10Þ
Now, Eq. (11) can be expressed as
RiMPP ¼
C IMPP
ð11Þ
Thirdly, in order to study the input resistance at the MPP in depth, the cell parameters Vm and Im can be taken into account. Here, a virtual input resistance Rim can be introduced and expressed as
Vm Im
ð12Þ
According to Eqs. (11) and (12), it is obvious that the inequality Rim – RiMPP is usually satisfied because Vm – C under a given weather condition. Here, assume that there is the relationship between RiMPP and Rim as shown in Eq. (13). Where RiE represents the error between RiMPP and Rim.
RiE ¼ RiMPP Rim
Io
L
S
where Po and D represent the output power and duty cycle of PWM signal, respectively. According to Eqs. (2)–(4), the input resistance Ri can be expressed easily as Eq. (5) [25].
Rim ¼
Load or inverter
Fig. 1. Structure of common PV system.
PV panel
ð4Þ
It is obvious that the model shown in Eq. (6) is the mathematical relationship between control signal D and Po, which is one of the ideal mathematical models of PV system [17].
2. Principle of proposed MPPT method
Po ¼ VI ¼ V o Io ¼
V o ¼ DV
291
ð13Þ
Finally, according to Eqs. (7), (12), and (13), Eq. (14) can be given. C
Vo
Buck converter
Fig. 2. Structure of PV system with buck DC/DC converter.
RL
Dmax ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Im RL V m þ V iE
ð14Þ
where ViE = ImRiE. Eq. (14) shows the relationship between control signal Dmax and cell parameters (Vm and Im), which is the
292
S. Li / Energy Conversion and Management 106 (2015) 290–299
equation of studying the MPPT method based on input resistance. It is obvious that Dmax can be calculated by the values of parameters Vm, Im, RL and ViE. Therefore, the acquisition of these parameters is playing a key role in proposing and implementing this new MPPT method. 3. Proposition and implementation of new MPPT method 3.1. Proposition In order to further study the relationships among parameters shown in Eq. (14), some simulation experiments are conducted by MATLAB software under various weather conditions. In these experiments, PV system shown in Fig. 2 is selected as experimental circuit; the cell parameters Isc, Voc, Im and Vm are selected as 9.19 A, 22 V, 8.58 A and 17.5 V, respectively; the selected value of RL is 1 X. Meanwhile, assuming: (1) the inductance and capacitor are all ideal electronic components and their values are large enough to ensure the output current continuous; (2) the switch and diode are all ideal electronic components; and (3) the wires have no resistance and the output load is the pure resistance. The experimental results under given arbitrary weather conditions can be shown in Table 1. It can be seen from Table 1 that, firstly, VMPP is always greater than Vm under every given weather condition and the equation VMPP = Vm + 0.411 is approximately satisfied. Secondly, Im and RiMPP are always greater than IMPP and Rim, respectively, under every given weather condition. Thirdly, there is only a small change for ViE under various weather conditions. Therefore, in practical application, ViE in Eq. (14) can be considered as a constant which can be represented by VC. Now Eq. (11) can be simplified to
Dmax ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I m RL Vm þ VC
ð15Þ
According to Eq. (15), the MPPT strategy with variable weather parameters based on input resistance can be proposed and described as follows: PV system can operate around MPP by controlling the duty cycle of DC/DC converter to equal to Dmax calculated by Eq. (15) after the parameters Im, Vm, VC and RL have been acquired. 3.2. Implementation It is clearly seen from Eq. (15) that, in order to calculate the control signal Dmax, the parameters Im, Vm, VC and RL must be acquired in time under every weather condition, which is the key work in the implementation of proposed MPPT method. Firstly, some simulation experiments, whose parameters is the same as Section 3.1, can also be done to verify the accuracy of the assumption that ViE = VC, and to acquire the value of VC. The experiment results are shown in Figs. 3 and 4. Where Figs. 3 and 4 show the ViE S curves under various T and RL = 1 X conditions and under various RL and T = 20 °C conditions, respectively. In these experiments, the overall range of S is assumed from 300 W/m2 to 1200 W/m2 and its sampling points are selected as every 50 W/m2. With respect to the rationalization of ViE = VC, on the one hand, it is can be clearly seen from Fig. 3 that, under various T and constant RL conditions, the values of ViE only change within a small range (about [0.72, 0.85]), just as the Table 1 has been showing. On the other hand, according to Table 1, the relationship Vm ViE is always satisfied, so the effect of various ViE changing within interval [0.72, 0.85] to Dmax is very small and can be ignored according to Eq. (15). Finally, it is clear from Fig. 4 that ViE is hardly influenced by the various RL. Therefore, ViE can be regarded as a
constant VC whose value can be represented with the average value of ViE in Fig. 3, which is approximately 0.79. That is to say, VC 0.79. Secondly, according to Eqs. (14) and (15), the parameters Vm and Im are playing a key role in acquiring the control signal Dmax. Therefore, to find out the relationships between them and weather parameters (S and T), some simulation experiments are also conducted by MATLAB software, and experiment results can be shown in Figs. 5 and 6. Where Fig. 5(a) and (b) show the Vm S and Vm T curves, respectively; Fig. 6(a) and (b) show the Im S and Im T curves, respectively. It can be seen from Fig. 5(a) that the approximation function fitting Vm S curve can be given in Eq. (16) when T keeps at 25 °C.
V m ¼ 4:4 106 S2 5:6104 103 S þ 18:711
ð16Þ
Meanwhile, Fig. 5(b) shows that the approximation function fitting Vm T curve can be given in Eq. (17) when S keeps at 1000 W/m2.
V m ¼ 0:0504T þ 18:76
ð17Þ
According to Eqs. (16) and (17), the mathematical relationship between Vm and S, T, can be expressed
V m ¼ V m ðS; TÞ ¼ 4:4 106 ðS 638:25Þ2 þ 16:918 þ 0:0504 ð25 TÞ
ð18Þ
It can be seen from Fig. 6(a) that the approximation function fitting Im S curve can be given in Eq. (19) when T keeps at 25 °C.
Im ¼ 0:00858S þ 2:8782 1015
ð19Þ
Meanwhile, Fig. 6(b) shows that the approximation function fitting Im T curve can be given in Eq. (20) when S keeps at 1000 W/m2.
Im ¼ 0:02145T þ 8:0438
ð20Þ
According to Eqs. (19) and (20), the mathematical relationship between Im and S, T, can be expressed
Im ¼ Im ðS; TÞ ¼ 8:58 103 S þ ðT 25Þ 2:145 105 S ð21Þ Thirdly, after VC, Vm and Im have been determined, the structure of PV system can be designed and shown in Fig. 7. In Fig. 7, to ensure the PV system operating at the MPP, the real-time value of RL must be acquired by taking sample Vo and Io, then the realtime value of the duty cycle Dmax can be calculated by MPPT controller after S and T have been measured. Finally, according to Eqs. (15), (18), (21) and Fig. 7, the MPPT method with variable weather parameters based on input resistance can be described as follows: PV system can operate around MPP by controlling the duty cycle of DC/DC converter to equal to Dmax calculated by Eqs. (15), (18) and (21) after the parameters S, T and RL have measured. At this time, the proposed MPPT method can be expressed again as
Dmax
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Im ðS; TÞ RL ¼ V m ðS; TÞ þ 0:79
ð22Þ
4. Simulation experiments and results analysis To verify the feasibility and availability of proposed MPPT method and test its MPPT performance, many simulation experiments whose circuit structure is shown in Fig. 7 are conducted. Where PV cell model whose parameters are the same as Section 3.1 is built by Simulink according to Eq. (1); the inductance and capacitor whose selected values are 1 mH and 1000uF, respectively, are
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S. Li / Energy Conversion and Management 106 (2015) 290–299 Table 1 Experimental results of different parameters under given arbitrary weather conditions. (S, T) (W/m2, °C)
Vm (V)
Im (A)
Rim (X)
VMPP (V)
IMPP (A)
RiMPP (X)
RiE (X)
ViE (V)
(300, 0) (300, 10) (300, 20) (500, 0) (500, 10) (500, 20) (800, 10) (800, 20) (800, 30) (1000, 20) (1000, 30) (1000, 40) (1200, 20) (1200, 30) (1200, 40)
18.679 18.177 17.675 18.207 17.718 17.229 17.753 17.263 16.772 17.752 17.248 16.744 18.584 18.056 17.528
2.413 2.477 2.542 4.022 4.129 4.236 6.607 6.778 6.950 8.473 8.687 8.902 10.167 10.425 10.682
7.740 7.337 6.954 4.527 4.291 4.067 2.687 2.547 2.413 2.095 1.985 1.881 1.828 1.732 1.641
19.112 18.599 18.085 18.630 18.129 17.629 18.165 17.663 17.162 18.164 17.648 17.133 19.015 18.475 17.935
2.357 2.428 2.483 3.943 4.041 4.150 6.475 6.639 6.802 8.30 8.513 8.722 9.965 10.217 10.469
8.108 7.659 7.285 4.725 4.486 4.248 2.805 2.660 2.523 2.187 2.073 1.964 1.908 1.808 1.713
0.3674 0.3221 0.3311 0.1983 0.1952 0.1814 0.1181 0.1136 0.1098 0.0922 0.0878 0.0834 0.0804 0.0762 0.0722
0.886 0.798 0.842 0.798 0.806 0.769 0.780 0.770 0.763 0.781 0.762 0.742 0.818 0.794 0.771
0.95 T=0 T=10 T=20 T=30 T=40
0.9
iE
V (V)
0.85
0.8
0.75
0.7 300
400
500
600
700
800
900
1000
1100
1200
S (W/m2) Fig. 3. ViE S curves with various T and RL = 1 X.
all ideal components; IGBT with 1 mX internal resistance and 500 kX snubber resistance is chosen as the switch; the internal resistance, snubber resistance and forward voltage of the diode are 1 mX, 500 X and 0.8 V, respectively; the load resistance is 1 X in Sections 4.1.1, 4.2.1 and 4.2.2; the frequency of PWM control
0.9 R =0.3 L
0.88
L L
0.86
R =1 L
0.84 iE
V (V)
4.1. Feasibility and availability experiments of proposed MPPT method 4.1.1. Experiments under given arbitrary weather conditions With respect to the verification of the feasibility and availability of proposed MPPT method, lots of experiments whose results are shown in Table 2 are done by MATLAB simulation under given arbitrary weather conditions. In Table 2, Pomax and Dmax represent the ideal values of output power and duty cycle corresponding to the MPP; P omax and Dmax represent the experiment results of output power and duty cycle corresponding to the MPP when the proposed MPPT method is used; P omax& and Dmax& represent the experiment results of output power and duty cycle corresponding to the MPP when the P&O method is used. It can be seen from Table 2 that every Dmax is approximately equal to its corresponding Dmax and the error between them is always less than 0.1%, and that every Dmax is approximately equal to its corresponding Dmax& and the error between them are always less than 0.2%. Meanwhile, It can be also seen from Table 2 that, under a given weather condition, P omax is approximately equal to its corresponding Pomax& and the error between them are always less than 0.05 W, and that there is a small difference between Pomax and Pomax or P omax& because the switch and diode are all the nonideal elements. Moreover, the results of Dmax& and P omax& given in Table 2 are all the mean values because of output oscillation of P&O method. Meanwhile, in Table 2, every Pomax& is smaller than its corresponding P omax , which results from the step size 0.003.
R =0.5 R =0.8
0.82 0.8 0.78 0.76 0.74 300
signal is 20 kHz; the conventional P&O method whose step size is 0.003 is selected as the compared object.
400
500
600
700
800
900
1000
S (W/m 2) Fig. 4. ViE S curves with various RL and T = 20 °C.
1100
1200
4.1.2. Experiments for testing the error between Dmax and Dmax In Section 4.1.1, the feasibility and availability of proposed MPPT method have been directly verified under given arbitrary weather conditions. Here, some experiments will be conducted to verifies indirectly the feasibility and availability of proposed MPPT method by testing the error between Dmax and Dmax, and the experiment results are shown in Figs. 8–10. Where Figs. 8–10 show the Dmax S, Dmax T and Dmax RL curves under 20 °C and 1 X, 1000 W/m2 and 1 X, and 1000 W/m2 and 20 °C conditions, respectively. It can be seen from Fig. 8 that the error between Dmax and Dmax are always less than 0.1% under 20 °C, 1 X and various S conditions. Meanwhile, Fig. 9 shows that the error between Dmax and Dmax are always less than 0.25% under 1000 W/m2, 1 X and various T conditions. Finally, Fig. 10 shows that the error between Dmax and Dmax are always less than 0.07% under 1000 W/m2, 20 °C and various
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S. Li / Energy Conversion and Management 106 (2015) 290–299 20.5
19
20 18.5
19.5 18
Vm (V)
Vm (V)
19 18.5
17.5
18 17
17.5 16.5
17 16.5
0
500
1000
16
1500
S (W/m 2)
(a)
0
5
10
15
20
25
30
35
40
45
50
T
Vm S curve with T =25 C
(b)
Vm T curve with S =1000 W/m2
Fig. 5. Vm S and Vm T curves.
9.2
12 9
10 8.8
Im (A)
Im (A)
8
6
8.4
4
8.2
2
0
8.6
0
500
1000
1500
8
0
5
10
15
20
25
(a)
I m S curve with T =25 C
30
35
40
45
50
T
S (W/m2)
(b)
I m T curve with S =1000 W/m2
Fig. 6. Im S and Im T curves.
Dmax Sensors
DC load or inverter
Buck DC/DC converter
PV panel
S, T
MPPT controller
Vo Io
Fig. 7. Structure of PV system controlled by proposed MPPT method.
RL conditions. According to these results, it is clear that the calculated values Dmax by MPPT controller using the proposed MPPT strategy are always approximately equal to their corresponding ideal values Dmax regardless of various S, T and RL, which means that PV system can always operate around the MPP. According to these experiment results and analysis in this Section, a conclusion can be drawn that the proposed MPPT method is
Table 2 Experimental results under various weather conditions. (S, T) (W/m2, °C)
Dmax
Dmax
Dmax&
Pomax (W)
P omax (W)
P omax& (W)
(300, 0) (300, 10) (300, 20) (500, 0) (500, 10) (500, 20) (800, 10) (800, 20) (800, 30) (1000, 20) (1000, 30) (1000, 40) (1200, 20) (1200, 30) (1200, 40)
0.3512 0.3613 0.3705 0.4600 0.4721 0.4852 0.5971 0.6134 0.6307 0.6761 0.6945 0.7135 0.7239 0.7437 0.7640
0.3520 0.3614 0.3710 0.4594 0.4718 0.4845 0.5963 0.6123 0.6289 0.6761 0.6941 0.7126 0.7249 0.7437 0.7631
0.351 0.361 0.370 0.460 0.472 0.485 0.598 0.613 0.631 0.676 0.694 0.7135 0.724 0.744 0.763
45.05 45.165 44.90 73.45 73.26 73.15 117.62 117.275 116.75 150.83 150.23 149.43 189.48 188.77 187.77
41.51 41.47 41.36 69.75 69.68 69.51 113.94 113.66 113.215 147.16 146.59 145.80 185.815 185.10 184.105
41.48 41.44 41.33 69.72 69.65 69.46 113.90 113.63 113.20 147.14 146.57 145.79 185.78 185.08 184.07
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S. Li / Energy Conversion and Management 106 (2015) 290–299 0.75
0.8 D*
D*
max
0.7
max
D
max
D
0.7
max
0.65
0.6 max
0.622
0.55
0.62
D
D
max
0.6
0.618
0.5
0.5 0.552
0.616
0.55
0.4
0.614
0.45
780
785
790
0.548
795
0.3
0.4 0.35 300
400
500
600
700
800
900
1000
1100
1200
S (W/m 2)
0.2 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.64
0.645
0.65
0.8
0.9
1
1.1
R ( ) L
Fig. 8. Dmax S curves under 20 °C and 1 X conditions.
Fig. 10. Dmax RL curves under 1000 W/m2 and 20 °C conditions.
Table 3 Values of experimental parameters under T = 25 °C and RL = 1 X conditions.
0.72 D*
max
0.71
Range of time (s)
D
max
2
S (W/m ) Dmax Pomax (W)
0.7
From 0 to 0.4
From 0.4 to 0.7
From 0.7 to 1
1000 0.6852 150.55
500 0.4915 72.96
800 0.6216 117.16
D
max
0.69 0.68 0.716
0.67
0.714
0.66
0.712 0.71
0.65 0.64
39
0
5
10
15
20
25
39.5
30
40
35
40
T ( C) Fig. 9. Dmax T curves under 1000 W/m2 and 1 X conditions.
feasible and available in controlling successfully PV system to operate around the MPP. 4.2. Analysis of MPPT performance To analyze the MPPT performance of proposed control method, some experiments are also conducted by MATLAB. In these experiments, PV system is assumed to operate under invariable T and RL conditions, invariable S and RL conditions and invariable S and T conditions, respectively. Here, to make a MPPT performance comparison between proposed MPPT method and conventional P&O method, their output power curves or duty cycle curves are shown together in all following figures. 4.2.1. Experiments under invariable T and RL conditions When T and RL are selected as 25 °C and 1 X, respectively, an experiment is done. In this experiment, the changing values of S and their corresponding ranges of time are shown in Table 3. Meanwhile, in order to compare and analyze the experiment results easily, two key ideal values (Dmax and Pomax) corresponding to every weather condition are also shown in Table 3. The experiment results are shown in Figs. 11 and 12. They represent the compared output power curves and duty cycle curves, respectively,
between proposed method and conventional P&O method when S changes according to Table 3. Fig. 11 shows that, firstly, when the proposed MPPT method is used, the settling times of output powers corresponding to three time intervals are about 17 ms, 14 ms and 13 ms, respectively. By contrast, the settling times of P&O method are about 230 ms, 87 ms and 59 ms, respectively. That is to say, the rapidity of proposed MPPT method is far better than that of P&O method. Secondly, the output powers of PV system using proposed MPPT method corresponding to three time intervals are 146.91 W, 69.38 W and 113.30 W, respectively. By contrast, the average output powers of P&O method are about 146.88 W, 69.34 W and 113.27 W, respectively. Comparing these results with corresponding ideal values Pomax shown in Table 3, it is obvious that the output powers using proposed MPPT method are always closer to Pomax than P&O method. That is to say, the accuracy of proposed MPPT method is better than that of P&O method. Thirdly, the proposed MPPT method can make the output power stabilize at the MPP until the changes of solar irradiance have taken place, just as the black line in Fig. 11 has been showing. By contrast, the output power is oscillating around the MPP because of the inherent shortcomings of P&O method. That is to say, the stability of proposed MPPT method is better than that of P&O method. It can be seen from Fig. 12 that, firstly, in every time interval, the duty cycle of proposed method always reaches its value corresponding to the MPP more quickly than that of P&O method. Secondly, the duty cycle of proposed method keeps stable at the MPP while oscillating around the MPP for P&O method. Thirdly, the duty cycles of proposed method corresponding to three time intervals are 0.685, 0.491 and 0.620, respectively. By contrast, the corresponding values of P&O method are 0.685, 0.493 and 0.622. It is clear that the errors between these results and corresponding ideal Dmax are very small. Therefore, according to these results and analysis, a conclusion can be drawn that, under variable S and invariable T, RL conditions,
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S. Li / Energy Conversion and Management 106 (2015) 290–299 200
200 146.9
180
146.85
160
140 0.31 0.32 0.33 0.34 0.35
120
Power (W)
Power (W)
160
146.8
140
100 80 60
147.5
100
147
80
146.5 146
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Proposed met P&O method
0.7 0.6 0.5
0.495 0.49 0.485 0.48 0.475 0.64 0.66 0.68
0.1
0
0.1
0.2
0.3
0.4
0.5
0.74
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 13. Compared power curves under variable T conditions.
0.8
0.2
0.72
Time (s)
Fig. 11. Compared power curves under variable S conditions.
0.3
0.7
20
Time (s)
0.4
145.5 0.68
40
Proposed method P&O method
20
D
120
60
40
0
Proposed method P&O method
180
0.6
0.7
0.7
0.8
0.72
0.9
1
Time (s) Fig. 12. Compared D curves under variable S conditions.
Table 4 Values of experimental parameters under S = 1000 W/m2 and RL = 1 X conditions. Range of time (s)
From 0 to 0.4
From 0.4 to 0.7
From 0.7 to 1
T (°C) Dmax Pomax (W)
40 0.7135 149.43
0 0.6409 151.36
40 0.7135 149.43
the rapidity, accuracy and stability of proposed MPPT method are better than P&O method. 4.2.2. Experiments under invariable S and RL conditions When S and RL are selected as 1000 W/m2 and 1 X, respectively, an experiment is also done. In this experiment, the changing values of T and their corresponding ranges of time are shown in Table 4. Meanwhile, in order to compare and analyze the experiment results easily, two key ideal values (Dmax and Pomax) corresponding to every weather condition are also shown in Table 4. The experiment results are shown in Figs. 13 and 14. They represent the compared output power curves and duty cycle curves, respectively, between proposed method and conventional P&O method when T changes according to Table 4.
Fig. 13 shows that, firstly, when the proposed MPPT method is used, the settling times of output powers corresponding to three time intervals are about 17 ms, 6 ms and 9 ms, respectively. By contrast, the settling times of P&O method are about 238 ms, 30 ms and 34 ms, respectively. That is to say, the rapidity of proposed MPPT method is far better than that of P&O method. Secondly, the output powers of PV system using proposed MPPT method corresponding to three time intervals are 145.80 W, 147.66 W and 145.80 W, respectively. By contrast, the average output powers of P&O method are about 145.785 W, 147.61 W and 145.785 W, respectively. Comparing these results with corresponding ideal values Pomax shown in Table 4, it is obvious that the output powers using proposed MPPT method are always closer to Pomax than P&O method. That is to say, the accuracy of proposed MPPT method is better than that of P&O method. Thirdly, the proposed MPPT method can make the output power stabilize at the MPP until the changes of temperature have taken place, just as the black line in Fig. 13 has been showing. By contrast, the output power is oscillating around the MPP because of the inherent shortcomings of P&O method. That is to say, the stability of proposed MPPT method is better than that of P&O method. It can be seen from Fig. 14 that, firstly, in every time interval, the duty cycle of proposed method always reaches the value corresponding to the MPP more quickly than that of P&O method. Secondly, the duty cycle of proposed method keeps stable at the MPP while oscillating around the MPP for P&O method. Thirdly, the duty cycles of proposed method corresponding to three time intervals are 0.7126, 0.6415 and 0.7126, respectively. By contrast, the corresponding values of P&O method are 0.7135, 0.64 and 0.7135. It is clear that the errors between these results and corresponding ideal Dmax are also very small. Therefore, according to these results and analysis, a conclusion can be drawn that, under variable T and invariable S, RL conditions, the rapidity, accuracy and stability of proposed MPPT method are better than P&O method.
4.2.3. Experiments under invariable S and T conditions When S and T are selected as 1000 W/m2 and 25 °C, respectively, an experiment is also done. In this experiment, the changing values of RL and their corresponding ranges of time are shown in Table 5. Meanwhile, in order to compare and analyze the experiment results easily, two key ideal values (Dmax and Pomax) corresponding to every weather condition are also shown in Table 5. The experiment results are shown in Figs. 15 and 16. They represent
297
S. Li / Energy Conversion and Management 106 (2015) 290–299 0.8
0.8
0.7
0.7
0.6
0.6 0.645
0.5
0.5
D
D
0.64
0.4
0.4 0.49
0.635
0.3
0.3
0.485
0.2
0.475
0.54 0.56 0.58 0.6 0.62
0.48
0.2
0.47
Proposed method P&O method
0.1 0
0
0.1
0.2
0.3
Proposed method P&O method
0.1
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0
0.1
0.2
0.3
0.4
Time (s)
0.5
0.6
0.7
0.6
0.62
0.8
0.9
1
Time (s)
Fig. 14. Compared D curves under variable T conditions.
Fig. 16. Compared D curves under variable RL conditions.
Table 5 Values of experimental parameters under S = 1000 W/m2 and T = 25 °C conditions.
180 Proposed method P&O method
160 140 120
Power (W)
0.54 0.56 0.58
Range of time (s)
From 0 to 0.4
From 0.4 to 0.7
From 0.7 to 1
RL (X) Dmax Pomax (W)
1 0.6852 150.55
0.5 0.4845 150.52
1 0.6852 150.55
100 80
146.9
60
146.85 146.8
40 146.75
20 0
0.4
0
0.1
0.2
0.3
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
1
Time (s) Fig. 15. Compared power curves under variable RL conditions.
the compared output power curves and duty cycle curves, respectively, between proposed method and conventional P&O method when RL changes according to Table 5. Fig. 15 shows that the output powers of using two MPPT methods are influenced by the change of RL because of those nonideal circuit elements. Meanwhile, the transient performance of output power without drastic adjustment of using proposed MPPT method is better than P&O method. On the other hand, the stead-state values of output powers using proposed MPPT method corresponding to three time intervals are 146.91 W, 146.83 W and 146.91 W, respectively. By contrast, these values of P&O method are about 146.87 W, 146.83 W and 146.87 W, respectively. It is clear that from these results that the changes of output powers using two MPPT methods always keep in 0.1 W range, which illustrates that the effect of varying RL to the stead-state value of output power is very small, and that, comparing with corresponding ideal values of Pomax shown in Table 5, the output powers using proposed MPPT method are generally closer to Pomax than P&O method, which illustrates that the accuracy of proposed MPPT method is better than that of P&O method. It can be also seen from Fig. 15 that, on the one hand, when the proposed MPPT method is used, the settling times of output powers corresponding to three time intervals are
about 18 ms, 3 ms and 6 ms, respectively. By contrast, the settling times of P&O method are about 230 ms, 82 ms and 79 ms, respectively. That is to say, the rapidity of proposed MPPT method is far better than that of P&O method. On the other hand, the proposed MPPT method can make the output power stabilize at the MPP until the changes of load resistance have taken place, just as the black line in Fig. 15 has been showing. By contrast, the output power is oscillating around the MPP because of the inherent shortcomings of P&O method. That is to say, the stability of proposed MPPT method is better than that of P&O method. It can be seen from Fig. 16 that, firstly, in every time interval, the duty cycle of proposed method always reaches its value corresponding to the MPP more quickly than that of P&O method. Secondly, the duty cycle of proposed method keeps stable at the MPP while oscillating around the MPP for P&O method. Thirdly, the duty cycles of proposed method corresponding to three time intervals are 0.6850, 0.4844 and 0.6850, respectively. By contrast, the corresponding values of P&O method are 0.685, 0.484 and 0.685. It is clear that the errors between these results and corresponding ideal Dmax are very small. Therefore, according to these results and analysis, a conclusion can be drawn that, under variable RL and invariable S, T conditions, the rapidity, accuracy and stability of proposed MPPT method are better than P&O method. According to three experiment results and analysis in this Section, a conclusion can be drawn that the rapidity, accuracy and stability of proposed MPPT method are always better than P&O method. 4.3. Comparison with other MPPT methods To make a comparison between proposed MPPT method and other MPPT methods, a simulation experiment is done under fast changing weather conditions. Here, the fuzzy method in paper [26] and P&O method are selected as the compared objects, and the step size of P&O method is selected as 0.003. Meanwhile,
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influence on the accuracy of the calculated control signal of MPPT controller, just as Figs. 8–10 has been showing. If it must be reduced, some measures can be taken as follows. On the one hand, a changing constant VC can be used in substitution for ViE to match the different temperature condition and represented as
1400
S (W/m 2)
1200 1000
V C ¼ 0:79 þ ð20 TÞ 0:0025:
800
On the other hand, for a higher degree of accuracy, VC can be also represented as a function of S and T by curve-fitting technique, and expressed as
600 400
ð23Þ
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
V C ¼ 2:5856 107 S2 þ 3:7347 104 S þ 0:9 þ ð20 TÞ 0:0025
1
Time (s)
ð24Þ
Fig. 17. Changing curve of S in simulation experiment.
assume that the solar irradiance changes according to Fig. 17 when the temperature and load resistance keep at 25 °C and 0.8 X, respectively. Finally, the experiment result is shown in Fig. 18. Moreover, to judge whether PV system is operating at the MPP, the ideal MPP values corresponding to the varying irradiance (represented by ideal MPP) are also shown in Fig. 18. It can be clearly seen from Fig. 18 that, firstly, all three MPPT methods can track the MPP successfully while there is a MPPT failure of P&O method within 0.2 s because of its slow tracking speed. Secondly, the settling time of proposed method is always better than fuzzy method and P&O method, which also illustrates its stronger adaptability to the varying weather conditions. Thirdly, the output power of proposed method can be stabilized at the MPP while there is some oscillation around the MPP for P&O method or fuzzy method. Therefore, it is obvious that the proposed method has better MPPT performance than other two methods under fast changing weather conditions.
5. Discussions Firstly, according to Fig. 3, it is clear that a error arises from using 0.79 in substitution for ViE. However, this error has a small
Secondly, in Section 3.2, VC = 0.79 (or VC 0.79) has been determined by simulation experiments for PV system with buck DC/DC converter when the cell parameters Isc, Voc, Im and Vm are selected as 9.19 A, 22 V, 8.58 A and 17.5 V, respectively. In practical application, on the one hand, the four cell parameters are usually different from these values, which causes VC – 0.79. Aiming at this question, we can find out easily the different VC to match different PV cell four parameters by some simulation experiments, and build a list for user. On the other hand, for PV system with boost DC/DC converter, it can be easily proved that VC = 0.79 is still approximately satisfied when the cell parameters Isc, Voc, Im and Vm are selected as 9.19 A, 22 V, 8.58 A and 17.5 V, respectively. Thirdly, the accuracy and rapidity of conventional P&O method are usually influenced by its MPPT step size. The bigger its step size is, the better its MPPT rapidity is while the poorer its MPPT accuracy is. In this paper, the rapidity and accuracy are all considered comprehensively, so the P&O method with step size 0.003 has been selected as compared object in all experiments. Finally, in this work, the measured parameters (S, T and RL) have been assumed that there is no delay and measurement error. It is because of these real-time measured data that the control signal Dmax can be calculated out quickly to implement the real-time MPP tracking. It is because of these accurate measured data that the control signal Dmax can be calculated out accurately to make
Proposed method P&O method Fuzzy method Ideal MPP
200 124.2 124 123.8
Power (W)
150
0.52
0.53
0.54
0.55
0.56
0.57
100
50
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Time (s) Fig. 18. Compared output power curves of three MPPT methods.
0.8
0.9
1
S. Li / Energy Conversion and Management 106 (2015) 290–299
PV system operating around the MPP. It is because of these realtime and accurate measured data that the MPPT control of output power should be almost not influenced by varying RL according to Fig. 4 and Eq. (22). However, in practical application, there are some measurement inaccuracy, uncertainty and delay, but the proposed MPPT strategy can still work well because, on the one hand, with the high development of modern electronic technology, these shortcomings of measurement circuit (or sensor) will be overcome to a great extent. For example, the high precision sensors have been invented and the high-speed processor have been applied successfully and widely. On the other hand, the little inaccuracy and uncertainty of measured data have a small influence on the MPPT accuracy. For example, it can be easily known from simulation experiments that when T keeps constant and S is varying within ±10 W/m2, the output power only changes within 2 W, and that when S keeps constant and T is varying within ±1 °C, the output power only changes within 0.1 W. 6. Conclusions In this paper, a MPPT control method with VWP based on input resistance, which has the advantages of faster MPPT speed and stronger MPPT adaptability to the varying weather conditions, has been proposed. In this study, through analyzing the input resistance of PV system deeply and using curve fitting technology and MATLAB simulation experiment, finding out the direct relationship between VWP and control signal is the key work. Finally, simulation experiments have verified that this proposed method can track the MPP successfully, and the MPPT rapidity, accuracy and stability are better than conventional fuzzy control method and P&O method. Acknowledgment This work was supported by guidance project of Hubei Provincial Department of Education (No. B2015111). References [1] Salas V, Barrado A. Review of the maximum power point tracking algorithms for stand-alone photovoltaic systems. Sol Energy Mater Sol Cells 2006;90 (10):1555–78. [2] Abdelsalam Ahmed K, Massoud Ahmed M, Ahmed Shehab, Enjeti Prasad N. High-performance adaptive perturb and observe MPPT technique for photovoltaic-based microgrids. IEEE Trans Power Electron 2011;26 (4):1010–21. [3] Sera Dezso, Mathe Laszlo, Kerekes Tamas, Spataru Sergiu Viore, et al. On the perturb-and-observe and incremental conductance MPPT methods for PV systems. IEEE J Photovoltaics 2013;3(3):1070–8. [4] Liu F. A variable step size INC MPPT method for PV systems. IEEE Trans Industr Electron 2008;55(7):2622–8.
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[5] Elgendy Mohammed A, Zahawi Bashar, Atkinson David J. Assessment of the incremental conductance maximum power power point tracking algorithm. IEEE Transactions on sustainable energy 2013;4(1):108–17. [6] Shaiek Yousra, Smida Mouna Ben, Sakly Anis, Mimouni Mohamed Faouzi. Comparison between conventional methods and GA approach for maximum power point tracking of shaded solar PV generators. Sol Energy 2013;90:107–22. [7] Messai A, Mellit A, Guessoum A, Kalogirou SA. Maximum power point tracking using a GA optimized fuzzy logic controller and its FPGA implementation. Sol Energy 2011;85:265–77. [8] Altin Necmi, Ozdemir Saban. Three-phase three-level grid interactive inverter with fuzzy logic based maximum power point tracking controller. Energy Convers Manage 2013;69:17–26. [9] Liu Yihua, Liu Chunliang, Huang Jiawei, Chen Jinghsiau. Neural-network-based maximum power point tracking methods for photovoltaic systems operating under fast changing environments. Sol Energy 2013;89:42–53. [10] Syafaruddin, Engin Karatepe, Takashi Hiyama. Performance enhancement of photovoltaic array through string and central based MPPT system under nonuniform irradiance conditions. Energy Convers Manage 2012;62:131–40. [11] Chiu Chian-Song, Ouyang Ya-Lun, Chan-Yu Ku. Terminal sliding mode control for maximum power point tracking of photovoltaic power generation systems. Sol Energy 2012;86:2986–95. [12] Kakosimos Panagiotis E, Kladas Antonios G. Implementation of photovoltaic array MPPT through fixed step predictive control technique. Renewable Energy 2011;36:2508–14. [13] Tsang KM, Chan WL. Model based rapid maximum power point tracking for photovoltaic systems. Energy Convers Manage 2013;70:83–9. [14] Jain Sachin, Agarwal Vivek. New current control based MPPT technique for single stage grid connected PV systems. Energy Convers Manage 2007;48:625–44. [15] Mutoh Nobuyoshi, Ohno Masahiro, Inoue Takayoshi. A method for MPPT control while searching for parammeters corresponding to weather conditions for PV generation systems. IEEE Trans Ind Electron 2006;53(4):1055–65. [16] Gao Xianwen, Li Shaowu, Gong Rongfen. Maximum power point tracking control strategies with variable weather parameters for photovoltaic generation systems. Sol Energy 2013;93:357–67. [17] Li Shaowu, Gao Xianwen, Wang Lina, Liu Sanjun. A novel maximum power point tracking control method with variable weather parameters for photovoltaic systems. Sol Energy 2013;97:529–36. [18] Li Shaowu, Attou Amine, Yang Yongchao, Geng Dongshan. A maximum power point tracking control strategy with variable weather parameters for photovoltaic systems with DC bus. Renewable Energy 2015;74:478–88. [19] Roshan Yaser M, Moallem M. Maximum power point estimation and tracking using power converter input resistance control. Sol Energy 2013;96:177–86. [20] Ching-Tsai Pan, Jeng-Yue Chen, Chin-Peng Chu, Yi-Shuo Huang. A fast maximum power point tracker for photovoltaic power systems. In: Proc 25th Annu Conf IEEE Ind Electron Soc, 1999. p. 390–3. [21] Takashima T, Tanaka T, Amano M, Ando Y. Maximum output control of photovoltaic (PV) array. In: Proc 35th Intersociety Energy Convers Eng Conf Exhib 2000. p. 380–3. [22] Masoum MAS, Dehbonei H, Fuchs EF. Theoretical and experimental analyses of photovoltaic systems with voltage and current-based maximum power point tracking. IEEE Trans Energy Convers 2002;17(4):514–22. [23] Esram Trishan, Chapman Patrick L. Comparison of photovoltaic array maximum power point tracking techniques. IEEE Trans Energy Convers 2007;22(2):439–48. [24] Moura Scott. A switched extremum seeking approach to maximum power point tracking in photovoltaic systems. Grid Integrat Alternative Energy Sources 2009:1–17. [25] Enrique JM, Durán E, Sidrach-de-Cardona M, Andújar JM. Theoretical assessment of the maximum power point tracking efficiency of photovoltaic facilities with different converter topologies. Sol Energy 2007;81:31–8. [26] Bouchafaa F, Hamzaoui I, Hadjammar A. Fuzzy logic control for the tracking of maximum power point of a PV system. Energy Procedia 2011;6:633–42.
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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
A maximum power point tracking method with variable weather parameters based on input resistance for photovoltaic system Shaowu Li ⇑ Science and Technology College, and School of Information Engineering, Hubei Minzu University, Enshi, China
a r t i c l e
i n f o
Article history: Received 24 April 2015 Accepted 12 September 2015 Available online 1 October 2015 Keywords: PV system MPPT VWP Input resistance
a b s t r a c t In order to greatly improve the maximum power point tracking (MPPT) speed and adaptability to the varying weather conditions for photovoltaic (PV) system, in this paper, a MPPT method with variable weather parameters (VWP) considered specially from input resistance standpoint is proposed. As well as other VWP methods, it can also track the maximum power point (MPP) as quickly as possible. In this method, the approximate relationship between control signal and PV cell parameters (Vm and Im) is firstly built by studying the input resistance of PV system deeply, then the relationship between VWP and Vm, Im is found by the curve fitting technique. Through these relationships, the bridge between control signal and VWP is built successfully, which is the key work to implement the direct MPPT control. Finally, some simulation experiments show that the proposed method is feasible and available to track the MPP successfully and has better MPPT rapidity, accuracy, stability and adaptability than conventional perturbation and observation (P&O) method and fuzzy control method. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction In order to avoid the produced power losses, now almost all PV systems use the DC/DC converters as MPPT control units. There are many existing MPPT methods such as the constant voltage tracking [1], the P&O method [2,3], the incremental conductance (IncCond) method [3–5], the genetic algorithm [6,7], the fuzzy logic control method [6–8], the neural network method [9,10], the sliding mode control method [11] and the predictive control technique [12–14]. In them, the P&O method is one of most widely used MPPT techniques. Its advantages mainly include the low-cost hardware, the easy implementation and the good performance without solar irradiance and temperature varying quickly with time. However, there are also some shortcomings including its slow tracking speed and oscillation around the MPP. In this paper, in order to study the output performance of proposed MPPT method, the P&O method is selected as the main compared object. With respect to the issue how the changing weather influence on MPPT control of PV system, some works have been done in papers [15,16]. Remarkably, some VWP methods have been proposed in papers [16–18] to implement the real-time MPPT control for PV systems with different topology. In these methods, the key technique is to find out the relationship between control signal
⇑ Tel.: +86 13997799701. E-mail address: [email protected] http://dx.doi.org/10.1016/j.enconman.2015.09.055 0196-8904/Ó 2015 Elsevier Ltd. All rights reserved.
and VWP, and their main advantages are the fast MPPT speed and strong adaptability under varying weather conditions. In this paper, the VWP technique will be studied continuously and considered specially from input resistance of PV system standpoint, which is one of the main aims and innovations in this work. In existing MPPT methods, except for paper [19], there is hardly a technique specially considered from input resistance of PV system standpoint because of its variability and difficult measurement. In paper [19], a real-time identification scheme using the Lambert W-Function is proposed to estimate the model of PV module and its desired resistance at the MPP. By contrast, in this paper, the input resistance of PV system is only used as a bridge in connecting control signal and cell parameters (Vm and Im), which lays the foundation of the relationship between control signal and VWP. It is clear that the innovative and reasonable use of input resistance is the key work to study the principle of proposed MPPT method. On the other hand, there are some MPPT method using the cell parameters Vm and Im. In paper [20], a linear current control is proposed on the basis of the linear relationship between Im and the level of irradiance. In paper [21], a feedback MPPT control method is proposed by computing from equations involving temperature and irradiance. In addition, the fractional open-circuit voltage method [22], the ripple correlation control (RCC) method [23] and so on also use Vm or Im as the key parameter to implement the MPPT control of PV system. However, in all these methods,
S. Li / Energy Conversion and Management 106 (2015) 290–299
the direct relationship Vm or Im and weather parameters (irradiance and temperature) is not still found out. Therefore, another purpose of this paper is to build this relationship. This paper is divided into the following sections: the principle of proposed MPPT method is analyzed in Section 2. The relationship between control signal of PV system and VWP is found out, then the description and implementation of proposed MPPT method are given in Section 3. The feasibility and availability of proposed method are verified and the MPPT rapidity, accuracy, stability and adaptability are compared with P&O method and fuzzy method in Section 4. Some discussions are had in Section 5. Finally, some conclusions are drawn in Section 6.
2.1. Model and input resistance of PV system The configuration of common PV system can be shown in Fig. 1. Where V and I represent the output voltage and current of PV panel, respectively; Vo and Io represent the output voltage and current of DC/DC converter, respectively. In Fig. 1, the DC/DC converter between PV panel and load or grid-connected inverter is usually an indispensable unit to implement the MPPT function. In practical application, the simplified mathematical model of PV panel can be shown in Eq. (1) [24,15]. This model is usually called as ‘‘Four-Parameter Model” [15].
h V i I ¼ Isc 1 C 1 eC2 V oc 1
ð1Þ
where C1 = (1 Im/Isc) exp (Vm/C2Voc); C2 = (Vm/Voc 1)/ ln (1 Im/Isc); Isc, Voc, Im and Vm represent the short circuit current, the open circuit voltage, the MPP current and voltage at standard conditions (1000 W/m2 and 25 °C), respectively. All data above are given by the PV panel manufacturer. When the DC/DC converter shown in Fig. 1 is the buck circuit, the structure of PV system can be shown in Fig. 2. Where Ri represents the input resistance on the right of PV panel. Here, assume that the buck DC/DC converter, which is operating in the continuous conduction operational model, is the ideal circuit and its output load or inverter can be regarded as a pure resistance RL. According to Fig. 2, Eqs. (2)–(4) can be given by laws of circuit and conservation of energy.
V Ri ¼ I
ð2Þ V 2o RL
ð3Þ
Io
I
PV panel
V
DC/DC converter
Vo
I V
Ri
PWM
D
Ri ¼
RL
ð5Þ
D2
Moreover, according to Eqs. (1), (3) and (4), the mathematical model of PV system can be expressed as
Po ¼
RL I2sc D2
pffiffiffiffiffiffi 2 P o RL 1 C 1 eC2 DV oc 1
ð6Þ
2.2. Acquisition of control signal based on input resistance at MPP Firstly, in practical application, PV system usually operates at the MPP, now Eq. (5) can be replaced as
RiMPP ¼
RL RL ¼ D2 MPP D2max
ð7Þ
where RiMPP and Dmax represent the input resistance and duty cycle of PWM signal, of PV system at the MPP, respectively. Secondly, according to Fig. 2, the input resistance of PV system can also be expressed as
RiMPP ¼
V MPP IMPP
ð8Þ
where VMPP and IMPP represent the values of V and I corresponding to the MPP, respectively. According to paper [17], Eq. (9) can be given.
V MPP ¼ C
ð9Þ
where C is a variable parameter whose value can be calculated by Eq. (10).
1 þ C1 1 C ¼ C 2 V oc lambertw e C1
ð10Þ
Now, Eq. (11) can be expressed as
RiMPP ¼
C IMPP
ð11Þ
Thirdly, in order to study the input resistance at the MPP in depth, the cell parameters Vm and Im can be taken into account. Here, a virtual input resistance Rim can be introduced and expressed as
Vm Im
ð12Þ
According to Eqs. (11) and (12), it is obvious that the inequality Rim – RiMPP is usually satisfied because Vm – C under a given weather condition. Here, assume that there is the relationship between RiMPP and Rim as shown in Eq. (13). Where RiE represents the error between RiMPP and Rim.
RiE ¼ RiMPP Rim
Io
L
S
where Po and D represent the output power and duty cycle of PWM signal, respectively. According to Eqs. (2)–(4), the input resistance Ri can be expressed easily as Eq. (5) [25].
Rim ¼
Load or inverter
Fig. 1. Structure of common PV system.
PV panel
ð4Þ
It is obvious that the model shown in Eq. (6) is the mathematical relationship between control signal D and Po, which is one of the ideal mathematical models of PV system [17].
2. Principle of proposed MPPT method
Po ¼ VI ¼ V o Io ¼
V o ¼ DV
291
ð13Þ
Finally, according to Eqs. (7), (12), and (13), Eq. (14) can be given. C
Vo
Buck converter
Fig. 2. Structure of PV system with buck DC/DC converter.
RL
Dmax ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Im RL V m þ V iE
ð14Þ
where ViE = ImRiE. Eq. (14) shows the relationship between control signal Dmax and cell parameters (Vm and Im), which is the
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equation of studying the MPPT method based on input resistance. It is obvious that Dmax can be calculated by the values of parameters Vm, Im, RL and ViE. Therefore, the acquisition of these parameters is playing a key role in proposing and implementing this new MPPT method. 3. Proposition and implementation of new MPPT method 3.1. Proposition In order to further study the relationships among parameters shown in Eq. (14), some simulation experiments are conducted by MATLAB software under various weather conditions. In these experiments, PV system shown in Fig. 2 is selected as experimental circuit; the cell parameters Isc, Voc, Im and Vm are selected as 9.19 A, 22 V, 8.58 A and 17.5 V, respectively; the selected value of RL is 1 X. Meanwhile, assuming: (1) the inductance and capacitor are all ideal electronic components and their values are large enough to ensure the output current continuous; (2) the switch and diode are all ideal electronic components; and (3) the wires have no resistance and the output load is the pure resistance. The experimental results under given arbitrary weather conditions can be shown in Table 1. It can be seen from Table 1 that, firstly, VMPP is always greater than Vm under every given weather condition and the equation VMPP = Vm + 0.411 is approximately satisfied. Secondly, Im and RiMPP are always greater than IMPP and Rim, respectively, under every given weather condition. Thirdly, there is only a small change for ViE under various weather conditions. Therefore, in practical application, ViE in Eq. (14) can be considered as a constant which can be represented by VC. Now Eq. (11) can be simplified to
Dmax ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I m RL Vm þ VC
ð15Þ
According to Eq. (15), the MPPT strategy with variable weather parameters based on input resistance can be proposed and described as follows: PV system can operate around MPP by controlling the duty cycle of DC/DC converter to equal to Dmax calculated by Eq. (15) after the parameters Im, Vm, VC and RL have been acquired. 3.2. Implementation It is clearly seen from Eq. (15) that, in order to calculate the control signal Dmax, the parameters Im, Vm, VC and RL must be acquired in time under every weather condition, which is the key work in the implementation of proposed MPPT method. Firstly, some simulation experiments, whose parameters is the same as Section 3.1, can also be done to verify the accuracy of the assumption that ViE = VC, and to acquire the value of VC. The experiment results are shown in Figs. 3 and 4. Where Figs. 3 and 4 show the ViE S curves under various T and RL = 1 X conditions and under various RL and T = 20 °C conditions, respectively. In these experiments, the overall range of S is assumed from 300 W/m2 to 1200 W/m2 and its sampling points are selected as every 50 W/m2. With respect to the rationalization of ViE = VC, on the one hand, it is can be clearly seen from Fig. 3 that, under various T and constant RL conditions, the values of ViE only change within a small range (about [0.72, 0.85]), just as the Table 1 has been showing. On the other hand, according to Table 1, the relationship Vm ViE is always satisfied, so the effect of various ViE changing within interval [0.72, 0.85] to Dmax is very small and can be ignored according to Eq. (15). Finally, it is clear from Fig. 4 that ViE is hardly influenced by the various RL. Therefore, ViE can be regarded as a
constant VC whose value can be represented with the average value of ViE in Fig. 3, which is approximately 0.79. That is to say, VC 0.79. Secondly, according to Eqs. (14) and (15), the parameters Vm and Im are playing a key role in acquiring the control signal Dmax. Therefore, to find out the relationships between them and weather parameters (S and T), some simulation experiments are also conducted by MATLAB software, and experiment results can be shown in Figs. 5 and 6. Where Fig. 5(a) and (b) show the Vm S and Vm T curves, respectively; Fig. 6(a) and (b) show the Im S and Im T curves, respectively. It can be seen from Fig. 5(a) that the approximation function fitting Vm S curve can be given in Eq. (16) when T keeps at 25 °C.
V m ¼ 4:4 106 S2 5:6104 103 S þ 18:711
ð16Þ
Meanwhile, Fig. 5(b) shows that the approximation function fitting Vm T curve can be given in Eq. (17) when S keeps at 1000 W/m2.
V m ¼ 0:0504T þ 18:76
ð17Þ
According to Eqs. (16) and (17), the mathematical relationship between Vm and S, T, can be expressed
V m ¼ V m ðS; TÞ ¼ 4:4 106 ðS 638:25Þ2 þ 16:918 þ 0:0504 ð25 TÞ
ð18Þ
It can be seen from Fig. 6(a) that the approximation function fitting Im S curve can be given in Eq. (19) when T keeps at 25 °C.
Im ¼ 0:00858S þ 2:8782 1015
ð19Þ
Meanwhile, Fig. 6(b) shows that the approximation function fitting Im T curve can be given in Eq. (20) when S keeps at 1000 W/m2.
Im ¼ 0:02145T þ 8:0438
ð20Þ
According to Eqs. (19) and (20), the mathematical relationship between Im and S, T, can be expressed
Im ¼ Im ðS; TÞ ¼ 8:58 103 S þ ðT 25Þ 2:145 105 S ð21Þ Thirdly, after VC, Vm and Im have been determined, the structure of PV system can be designed and shown in Fig. 7. In Fig. 7, to ensure the PV system operating at the MPP, the real-time value of RL must be acquired by taking sample Vo and Io, then the realtime value of the duty cycle Dmax can be calculated by MPPT controller after S and T have been measured. Finally, according to Eqs. (15), (18), (21) and Fig. 7, the MPPT method with variable weather parameters based on input resistance can be described as follows: PV system can operate around MPP by controlling the duty cycle of DC/DC converter to equal to Dmax calculated by Eqs. (15), (18) and (21) after the parameters S, T and RL have measured. At this time, the proposed MPPT method can be expressed again as
Dmax
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Im ðS; TÞ RL ¼ V m ðS; TÞ þ 0:79
ð22Þ
4. Simulation experiments and results analysis To verify the feasibility and availability of proposed MPPT method and test its MPPT performance, many simulation experiments whose circuit structure is shown in Fig. 7 are conducted. Where PV cell model whose parameters are the same as Section 3.1 is built by Simulink according to Eq. (1); the inductance and capacitor whose selected values are 1 mH and 1000uF, respectively, are
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S. Li / Energy Conversion and Management 106 (2015) 290–299 Table 1 Experimental results of different parameters under given arbitrary weather conditions. (S, T) (W/m2, °C)
Vm (V)
Im (A)
Rim (X)
VMPP (V)
IMPP (A)
RiMPP (X)
RiE (X)
ViE (V)
(300, 0) (300, 10) (300, 20) (500, 0) (500, 10) (500, 20) (800, 10) (800, 20) (800, 30) (1000, 20) (1000, 30) (1000, 40) (1200, 20) (1200, 30) (1200, 40)
18.679 18.177 17.675 18.207 17.718 17.229 17.753 17.263 16.772 17.752 17.248 16.744 18.584 18.056 17.528
2.413 2.477 2.542 4.022 4.129 4.236 6.607 6.778 6.950 8.473 8.687 8.902 10.167 10.425 10.682
7.740 7.337 6.954 4.527 4.291 4.067 2.687 2.547 2.413 2.095 1.985 1.881 1.828 1.732 1.641
19.112 18.599 18.085 18.630 18.129 17.629 18.165 17.663 17.162 18.164 17.648 17.133 19.015 18.475 17.935
2.357 2.428 2.483 3.943 4.041 4.150 6.475 6.639 6.802 8.30 8.513 8.722 9.965 10.217 10.469
8.108 7.659 7.285 4.725 4.486 4.248 2.805 2.660 2.523 2.187 2.073 1.964 1.908 1.808 1.713
0.3674 0.3221 0.3311 0.1983 0.1952 0.1814 0.1181 0.1136 0.1098 0.0922 0.0878 0.0834 0.0804 0.0762 0.0722
0.886 0.798 0.842 0.798 0.806 0.769 0.780 0.770 0.763 0.781 0.762 0.742 0.818 0.794 0.771
0.95 T=0 T=10 T=20 T=30 T=40
0.9
iE
V (V)
0.85
0.8
0.75
0.7 300
400
500
600
700
800
900
1000
1100
1200
S (W/m2) Fig. 3. ViE S curves with various T and RL = 1 X.
all ideal components; IGBT with 1 mX internal resistance and 500 kX snubber resistance is chosen as the switch; the internal resistance, snubber resistance and forward voltage of the diode are 1 mX, 500 X and 0.8 V, respectively; the load resistance is 1 X in Sections 4.1.1, 4.2.1 and 4.2.2; the frequency of PWM control
0.9 R =0.3 L
0.88
L L
0.86
R =1 L
0.84 iE
V (V)
4.1. Feasibility and availability experiments of proposed MPPT method 4.1.1. Experiments under given arbitrary weather conditions With respect to the verification of the feasibility and availability of proposed MPPT method, lots of experiments whose results are shown in Table 2 are done by MATLAB simulation under given arbitrary weather conditions. In Table 2, Pomax and Dmax represent the ideal values of output power and duty cycle corresponding to the MPP; P omax and Dmax represent the experiment results of output power and duty cycle corresponding to the MPP when the proposed MPPT method is used; P omax& and Dmax& represent the experiment results of output power and duty cycle corresponding to the MPP when the P&O method is used. It can be seen from Table 2 that every Dmax is approximately equal to its corresponding Dmax and the error between them is always less than 0.1%, and that every Dmax is approximately equal to its corresponding Dmax& and the error between them are always less than 0.2%. Meanwhile, It can be also seen from Table 2 that, under a given weather condition, P omax is approximately equal to its corresponding Pomax& and the error between them are always less than 0.05 W, and that there is a small difference between Pomax and Pomax or P omax& because the switch and diode are all the nonideal elements. Moreover, the results of Dmax& and P omax& given in Table 2 are all the mean values because of output oscillation of P&O method. Meanwhile, in Table 2, every Pomax& is smaller than its corresponding P omax , which results from the step size 0.003.
R =0.5 R =0.8
0.82 0.8 0.78 0.76 0.74 300
signal is 20 kHz; the conventional P&O method whose step size is 0.003 is selected as the compared object.
400
500
600
700
800
900
1000
S (W/m 2) Fig. 4. ViE S curves with various RL and T = 20 °C.
1100
1200
4.1.2. Experiments for testing the error between Dmax and Dmax In Section 4.1.1, the feasibility and availability of proposed MPPT method have been directly verified under given arbitrary weather conditions. Here, some experiments will be conducted to verifies indirectly the feasibility and availability of proposed MPPT method by testing the error between Dmax and Dmax, and the experiment results are shown in Figs. 8–10. Where Figs. 8–10 show the Dmax S, Dmax T and Dmax RL curves under 20 °C and 1 X, 1000 W/m2 and 1 X, and 1000 W/m2 and 20 °C conditions, respectively. It can be seen from Fig. 8 that the error between Dmax and Dmax are always less than 0.1% under 20 °C, 1 X and various S conditions. Meanwhile, Fig. 9 shows that the error between Dmax and Dmax are always less than 0.25% under 1000 W/m2, 1 X and various T conditions. Finally, Fig. 10 shows that the error between Dmax and Dmax are always less than 0.07% under 1000 W/m2, 20 °C and various
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S. Li / Energy Conversion and Management 106 (2015) 290–299 20.5
19
20 18.5
19.5 18
Vm (V)
Vm (V)
19 18.5
17.5
18 17
17.5 16.5
17 16.5
0
500
1000
16
1500
S (W/m 2)
(a)
0
5
10
15
20
25
30
35
40
45
50
T
Vm S curve with T =25 C
(b)
Vm T curve with S =1000 W/m2
Fig. 5. Vm S and Vm T curves.
9.2
12 9
10 8.8
Im (A)
Im (A)
8
6
8.4
4
8.2
2
0
8.6
0
500
1000
1500
8
0
5
10
15
20
25
(a)
I m S curve with T =25 C
30
35
40
45
50
T
S (W/m2)
(b)
I m T curve with S =1000 W/m2
Fig. 6. Im S and Im T curves.
Dmax Sensors
DC load or inverter
Buck DC/DC converter
PV panel
S, T
MPPT controller
Vo Io
Fig. 7. Structure of PV system controlled by proposed MPPT method.
RL conditions. According to these results, it is clear that the calculated values Dmax by MPPT controller using the proposed MPPT strategy are always approximately equal to their corresponding ideal values Dmax regardless of various S, T and RL, which means that PV system can always operate around the MPP. According to these experiment results and analysis in this Section, a conclusion can be drawn that the proposed MPPT method is
Table 2 Experimental results under various weather conditions. (S, T) (W/m2, °C)
Dmax
Dmax
Dmax&
Pomax (W)
P omax (W)
P omax& (W)
(300, 0) (300, 10) (300, 20) (500, 0) (500, 10) (500, 20) (800, 10) (800, 20) (800, 30) (1000, 20) (1000, 30) (1000, 40) (1200, 20) (1200, 30) (1200, 40)
0.3512 0.3613 0.3705 0.4600 0.4721 0.4852 0.5971 0.6134 0.6307 0.6761 0.6945 0.7135 0.7239 0.7437 0.7640
0.3520 0.3614 0.3710 0.4594 0.4718 0.4845 0.5963 0.6123 0.6289 0.6761 0.6941 0.7126 0.7249 0.7437 0.7631
0.351 0.361 0.370 0.460 0.472 0.485 0.598 0.613 0.631 0.676 0.694 0.7135 0.724 0.744 0.763
45.05 45.165 44.90 73.45 73.26 73.15 117.62 117.275 116.75 150.83 150.23 149.43 189.48 188.77 187.77
41.51 41.47 41.36 69.75 69.68 69.51 113.94 113.66 113.215 147.16 146.59 145.80 185.815 185.10 184.105
41.48 41.44 41.33 69.72 69.65 69.46 113.90 113.63 113.20 147.14 146.57 145.79 185.78 185.08 184.07
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0.8 D*
D*
max
0.7
max
D
max
D
0.7
max
0.65
0.6 max
0.622
0.55
0.62
D
D
max
0.6
0.618
0.5
0.5 0.552
0.616
0.55
0.4
0.614
0.45
780
785
790
0.548
795
0.3
0.4 0.35 300
400
500
600
700
800
900
1000
1100
1200
S (W/m 2)
0.2 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.64
0.645
0.65
0.8
0.9
1
1.1
R ( ) L
Fig. 8. Dmax S curves under 20 °C and 1 X conditions.
Fig. 10. Dmax RL curves under 1000 W/m2 and 20 °C conditions.
Table 3 Values of experimental parameters under T = 25 °C and RL = 1 X conditions.
0.72 D*
max
0.71
Range of time (s)
D
max
2
S (W/m ) Dmax Pomax (W)
0.7
From 0 to 0.4
From 0.4 to 0.7
From 0.7 to 1
1000 0.6852 150.55
500 0.4915 72.96
800 0.6216 117.16
D
max
0.69 0.68 0.716
0.67
0.714
0.66
0.712 0.71
0.65 0.64
39
0
5
10
15
20
25
39.5
30
40
35
40
T ( C) Fig. 9. Dmax T curves under 1000 W/m2 and 1 X conditions.
feasible and available in controlling successfully PV system to operate around the MPP. 4.2. Analysis of MPPT performance To analyze the MPPT performance of proposed control method, some experiments are also conducted by MATLAB. In these experiments, PV system is assumed to operate under invariable T and RL conditions, invariable S and RL conditions and invariable S and T conditions, respectively. Here, to make a MPPT performance comparison between proposed MPPT method and conventional P&O method, their output power curves or duty cycle curves are shown together in all following figures. 4.2.1. Experiments under invariable T and RL conditions When T and RL are selected as 25 °C and 1 X, respectively, an experiment is done. In this experiment, the changing values of S and their corresponding ranges of time are shown in Table 3. Meanwhile, in order to compare and analyze the experiment results easily, two key ideal values (Dmax and Pomax) corresponding to every weather condition are also shown in Table 3. The experiment results are shown in Figs. 11 and 12. They represent the compared output power curves and duty cycle curves, respectively,
between proposed method and conventional P&O method when S changes according to Table 3. Fig. 11 shows that, firstly, when the proposed MPPT method is used, the settling times of output powers corresponding to three time intervals are about 17 ms, 14 ms and 13 ms, respectively. By contrast, the settling times of P&O method are about 230 ms, 87 ms and 59 ms, respectively. That is to say, the rapidity of proposed MPPT method is far better than that of P&O method. Secondly, the output powers of PV system using proposed MPPT method corresponding to three time intervals are 146.91 W, 69.38 W and 113.30 W, respectively. By contrast, the average output powers of P&O method are about 146.88 W, 69.34 W and 113.27 W, respectively. Comparing these results with corresponding ideal values Pomax shown in Table 3, it is obvious that the output powers using proposed MPPT method are always closer to Pomax than P&O method. That is to say, the accuracy of proposed MPPT method is better than that of P&O method. Thirdly, the proposed MPPT method can make the output power stabilize at the MPP until the changes of solar irradiance have taken place, just as the black line in Fig. 11 has been showing. By contrast, the output power is oscillating around the MPP because of the inherent shortcomings of P&O method. That is to say, the stability of proposed MPPT method is better than that of P&O method. It can be seen from Fig. 12 that, firstly, in every time interval, the duty cycle of proposed method always reaches its value corresponding to the MPP more quickly than that of P&O method. Secondly, the duty cycle of proposed method keeps stable at the MPP while oscillating around the MPP for P&O method. Thirdly, the duty cycles of proposed method corresponding to three time intervals are 0.685, 0.491 and 0.620, respectively. By contrast, the corresponding values of P&O method are 0.685, 0.493 and 0.622. It is clear that the errors between these results and corresponding ideal Dmax are very small. Therefore, according to these results and analysis, a conclusion can be drawn that, under variable S and invariable T, RL conditions,
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S. Li / Energy Conversion and Management 106 (2015) 290–299 200
200 146.9
180
146.85
160
140 0.31 0.32 0.33 0.34 0.35
120
Power (W)
Power (W)
160
146.8
140
100 80 60
147.5
100
147
80
146.5 146
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Proposed met P&O method
0.7 0.6 0.5
0.495 0.49 0.485 0.48 0.475 0.64 0.66 0.68
0.1
0
0.1
0.2
0.3
0.4
0.5
0.74
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 13. Compared power curves under variable T conditions.
0.8
0.2
0.72
Time (s)
Fig. 11. Compared power curves under variable S conditions.
0.3
0.7
20
Time (s)
0.4
145.5 0.68
40
Proposed method P&O method
20
D
120
60
40
0
Proposed method P&O method
180
0.6
0.7
0.7
0.8
0.72
0.9
1
Time (s) Fig. 12. Compared D curves under variable S conditions.
Table 4 Values of experimental parameters under S = 1000 W/m2 and RL = 1 X conditions. Range of time (s)
From 0 to 0.4
From 0.4 to 0.7
From 0.7 to 1
T (°C) Dmax Pomax (W)
40 0.7135 149.43
0 0.6409 151.36
40 0.7135 149.43
the rapidity, accuracy and stability of proposed MPPT method are better than P&O method. 4.2.2. Experiments under invariable S and RL conditions When S and RL are selected as 1000 W/m2 and 1 X, respectively, an experiment is also done. In this experiment, the changing values of T and their corresponding ranges of time are shown in Table 4. Meanwhile, in order to compare and analyze the experiment results easily, two key ideal values (Dmax and Pomax) corresponding to every weather condition are also shown in Table 4. The experiment results are shown in Figs. 13 and 14. They represent the compared output power curves and duty cycle curves, respectively, between proposed method and conventional P&O method when T changes according to Table 4.
Fig. 13 shows that, firstly, when the proposed MPPT method is used, the settling times of output powers corresponding to three time intervals are about 17 ms, 6 ms and 9 ms, respectively. By contrast, the settling times of P&O method are about 238 ms, 30 ms and 34 ms, respectively. That is to say, the rapidity of proposed MPPT method is far better than that of P&O method. Secondly, the output powers of PV system using proposed MPPT method corresponding to three time intervals are 145.80 W, 147.66 W and 145.80 W, respectively. By contrast, the average output powers of P&O method are about 145.785 W, 147.61 W and 145.785 W, respectively. Comparing these results with corresponding ideal values Pomax shown in Table 4, it is obvious that the output powers using proposed MPPT method are always closer to Pomax than P&O method. That is to say, the accuracy of proposed MPPT method is better than that of P&O method. Thirdly, the proposed MPPT method can make the output power stabilize at the MPP until the changes of temperature have taken place, just as the black line in Fig. 13 has been showing. By contrast, the output power is oscillating around the MPP because of the inherent shortcomings of P&O method. That is to say, the stability of proposed MPPT method is better than that of P&O method. It can be seen from Fig. 14 that, firstly, in every time interval, the duty cycle of proposed method always reaches the value corresponding to the MPP more quickly than that of P&O method. Secondly, the duty cycle of proposed method keeps stable at the MPP while oscillating around the MPP for P&O method. Thirdly, the duty cycles of proposed method corresponding to three time intervals are 0.7126, 0.6415 and 0.7126, respectively. By contrast, the corresponding values of P&O method are 0.7135, 0.64 and 0.7135. It is clear that the errors between these results and corresponding ideal Dmax are also very small. Therefore, according to these results and analysis, a conclusion can be drawn that, under variable T and invariable S, RL conditions, the rapidity, accuracy and stability of proposed MPPT method are better than P&O method.
4.2.3. Experiments under invariable S and T conditions When S and T are selected as 1000 W/m2 and 25 °C, respectively, an experiment is also done. In this experiment, the changing values of RL and their corresponding ranges of time are shown in Table 5. Meanwhile, in order to compare and analyze the experiment results easily, two key ideal values (Dmax and Pomax) corresponding to every weather condition are also shown in Table 5. The experiment results are shown in Figs. 15 and 16. They represent
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S. Li / Energy Conversion and Management 106 (2015) 290–299 0.8
0.8
0.7
0.7
0.6
0.6 0.645
0.5
0.5
D
D
0.64
0.4
0.4 0.49
0.635
0.3
0.3
0.485
0.2
0.475
0.54 0.56 0.58 0.6 0.62
0.48
0.2
0.47
Proposed method P&O method
0.1 0
0
0.1
0.2
0.3
Proposed method P&O method
0.1
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0
0.1
0.2
0.3
0.4
Time (s)
0.5
0.6
0.7
0.6
0.62
0.8
0.9
1
Time (s)
Fig. 14. Compared D curves under variable T conditions.
Fig. 16. Compared D curves under variable RL conditions.
Table 5 Values of experimental parameters under S = 1000 W/m2 and T = 25 °C conditions.
180 Proposed method P&O method
160 140 120
Power (W)
0.54 0.56 0.58
Range of time (s)
From 0 to 0.4
From 0.4 to 0.7
From 0.7 to 1
RL (X) Dmax Pomax (W)
1 0.6852 150.55
0.5 0.4845 150.52
1 0.6852 150.55
100 80
146.9
60
146.85 146.8
40 146.75
20 0
0.4
0
0.1
0.2
0.3
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
1
Time (s) Fig. 15. Compared power curves under variable RL conditions.
the compared output power curves and duty cycle curves, respectively, between proposed method and conventional P&O method when RL changes according to Table 5. Fig. 15 shows that the output powers of using two MPPT methods are influenced by the change of RL because of those nonideal circuit elements. Meanwhile, the transient performance of output power without drastic adjustment of using proposed MPPT method is better than P&O method. On the other hand, the stead-state values of output powers using proposed MPPT method corresponding to three time intervals are 146.91 W, 146.83 W and 146.91 W, respectively. By contrast, these values of P&O method are about 146.87 W, 146.83 W and 146.87 W, respectively. It is clear that from these results that the changes of output powers using two MPPT methods always keep in 0.1 W range, which illustrates that the effect of varying RL to the stead-state value of output power is very small, and that, comparing with corresponding ideal values of Pomax shown in Table 5, the output powers using proposed MPPT method are generally closer to Pomax than P&O method, which illustrates that the accuracy of proposed MPPT method is better than that of P&O method. It can be also seen from Fig. 15 that, on the one hand, when the proposed MPPT method is used, the settling times of output powers corresponding to three time intervals are
about 18 ms, 3 ms and 6 ms, respectively. By contrast, the settling times of P&O method are about 230 ms, 82 ms and 79 ms, respectively. That is to say, the rapidity of proposed MPPT method is far better than that of P&O method. On the other hand, the proposed MPPT method can make the output power stabilize at the MPP until the changes of load resistance have taken place, just as the black line in Fig. 15 has been showing. By contrast, the output power is oscillating around the MPP because of the inherent shortcomings of P&O method. That is to say, the stability of proposed MPPT method is better than that of P&O method. It can be seen from Fig. 16 that, firstly, in every time interval, the duty cycle of proposed method always reaches its value corresponding to the MPP more quickly than that of P&O method. Secondly, the duty cycle of proposed method keeps stable at the MPP while oscillating around the MPP for P&O method. Thirdly, the duty cycles of proposed method corresponding to three time intervals are 0.6850, 0.4844 and 0.6850, respectively. By contrast, the corresponding values of P&O method are 0.685, 0.484 and 0.685. It is clear that the errors between these results and corresponding ideal Dmax are very small. Therefore, according to these results and analysis, a conclusion can be drawn that, under variable RL and invariable S, T conditions, the rapidity, accuracy and stability of proposed MPPT method are better than P&O method. According to three experiment results and analysis in this Section, a conclusion can be drawn that the rapidity, accuracy and stability of proposed MPPT method are always better than P&O method. 4.3. Comparison with other MPPT methods To make a comparison between proposed MPPT method and other MPPT methods, a simulation experiment is done under fast changing weather conditions. Here, the fuzzy method in paper [26] and P&O method are selected as the compared objects, and the step size of P&O method is selected as 0.003. Meanwhile,
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influence on the accuracy of the calculated control signal of MPPT controller, just as Figs. 8–10 has been showing. If it must be reduced, some measures can be taken as follows. On the one hand, a changing constant VC can be used in substitution for ViE to match the different temperature condition and represented as
1400
S (W/m 2)
1200 1000
V C ¼ 0:79 þ ð20 TÞ 0:0025:
800
On the other hand, for a higher degree of accuracy, VC can be also represented as a function of S and T by curve-fitting technique, and expressed as
600 400
ð23Þ
0
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V C ¼ 2:5856 107 S2 þ 3:7347 104 S þ 0:9 þ ð20 TÞ 0:0025
1
Time (s)
ð24Þ
Fig. 17. Changing curve of S in simulation experiment.
assume that the solar irradiance changes according to Fig. 17 when the temperature and load resistance keep at 25 °C and 0.8 X, respectively. Finally, the experiment result is shown in Fig. 18. Moreover, to judge whether PV system is operating at the MPP, the ideal MPP values corresponding to the varying irradiance (represented by ideal MPP) are also shown in Fig. 18. It can be clearly seen from Fig. 18 that, firstly, all three MPPT methods can track the MPP successfully while there is a MPPT failure of P&O method within 0.2 s because of its slow tracking speed. Secondly, the settling time of proposed method is always better than fuzzy method and P&O method, which also illustrates its stronger adaptability to the varying weather conditions. Thirdly, the output power of proposed method can be stabilized at the MPP while there is some oscillation around the MPP for P&O method or fuzzy method. Therefore, it is obvious that the proposed method has better MPPT performance than other two methods under fast changing weather conditions.
5. Discussions Firstly, according to Fig. 3, it is clear that a error arises from using 0.79 in substitution for ViE. However, this error has a small
Secondly, in Section 3.2, VC = 0.79 (or VC 0.79) has been determined by simulation experiments for PV system with buck DC/DC converter when the cell parameters Isc, Voc, Im and Vm are selected as 9.19 A, 22 V, 8.58 A and 17.5 V, respectively. In practical application, on the one hand, the four cell parameters are usually different from these values, which causes VC – 0.79. Aiming at this question, we can find out easily the different VC to match different PV cell four parameters by some simulation experiments, and build a list for user. On the other hand, for PV system with boost DC/DC converter, it can be easily proved that VC = 0.79 is still approximately satisfied when the cell parameters Isc, Voc, Im and Vm are selected as 9.19 A, 22 V, 8.58 A and 17.5 V, respectively. Thirdly, the accuracy and rapidity of conventional P&O method are usually influenced by its MPPT step size. The bigger its step size is, the better its MPPT rapidity is while the poorer its MPPT accuracy is. In this paper, the rapidity and accuracy are all considered comprehensively, so the P&O method with step size 0.003 has been selected as compared object in all experiments. Finally, in this work, the measured parameters (S, T and RL) have been assumed that there is no delay and measurement error. It is because of these real-time measured data that the control signal Dmax can be calculated out quickly to implement the real-time MPP tracking. It is because of these accurate measured data that the control signal Dmax can be calculated out accurately to make
Proposed method P&O method Fuzzy method Ideal MPP
200 124.2 124 123.8
Power (W)
150
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0
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Time (s) Fig. 18. Compared output power curves of three MPPT methods.
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0.9
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PV system operating around the MPP. It is because of these realtime and accurate measured data that the MPPT control of output power should be almost not influenced by varying RL according to Fig. 4 and Eq. (22). However, in practical application, there are some measurement inaccuracy, uncertainty and delay, but the proposed MPPT strategy can still work well because, on the one hand, with the high development of modern electronic technology, these shortcomings of measurement circuit (or sensor) will be overcome to a great extent. For example, the high precision sensors have been invented and the high-speed processor have been applied successfully and widely. On the other hand, the little inaccuracy and uncertainty of measured data have a small influence on the MPPT accuracy. For example, it can be easily known from simulation experiments that when T keeps constant and S is varying within ±10 W/m2, the output power only changes within 2 W, and that when S keeps constant and T is varying within ±1 °C, the output power only changes within 0.1 W. 6. Conclusions In this paper, a MPPT control method with VWP based on input resistance, which has the advantages of faster MPPT speed and stronger MPPT adaptability to the varying weather conditions, has been proposed. In this study, through analyzing the input resistance of PV system deeply and using curve fitting technology and MATLAB simulation experiment, finding out the direct relationship between VWP and control signal is the key work. Finally, simulation experiments have verified that this proposed method can track the MPP successfully, and the MPPT rapidity, accuracy and stability are better than conventional fuzzy control method and P&O method. Acknowledgment This work was supported by guidance project of Hubei Provincial Department of Education (No. B2015111). References [1] Salas V, Barrado A. Review of the maximum power point tracking algorithms for stand-alone photovoltaic systems. Sol Energy Mater Sol Cells 2006;90 (10):1555–78. [2] Abdelsalam Ahmed K, Massoud Ahmed M, Ahmed Shehab, Enjeti Prasad N. High-performance adaptive perturb and observe MPPT technique for photovoltaic-based microgrids. IEEE Trans Power Electron 2011;26 (4):1010–21. [3] Sera Dezso, Mathe Laszlo, Kerekes Tamas, Spataru Sergiu Viore, et al. On the perturb-and-observe and incremental conductance MPPT methods for PV systems. IEEE J Photovoltaics 2013;3(3):1070–8. [4] Liu F. A variable step size INC MPPT method for PV systems. IEEE Trans Industr Electron 2008;55(7):2622–8.
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