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11THE DEVELOPMENT OF A NEW MAXIMUM POWER POINT TRACKER
FOR A PV PANEL
M. Salhi*, R. El-Bachtiri*, E. Matagne**
*REPEER Group, LESSI laboratory, Faculty of sciences dhar el-Mehrez, USMBA University
te
Higher school of technology, km5, R Imouzzer, BP 2427, Fez, Morocco. Telephone: +212 35 600584/Fax: +212 35 600588 E-mail: [email protected]; [email protected]
**
LEI laboratory, Université Catholique de Louvain Louvain-la-Neuve, Belgium E-mail: [email protected]
Received: 21 Sept 2007; accepted: 29 Oct 2007
The PV systems are rapidly expanding and have increasing roles in electric power technologies, providing more secure power sources and pollution free electric supplies. Since the PV electricity is expensive compared to the electricity from the utility grid, users want to use all the available output PV power. Therefore, the PV systems should be designed to operate at their maximum output power for any temperature and solar radiation level. In this paper, we consider a photovoltaic panel supplying a battery. For maximizing the output power of this panel, we have using a boost dc/dc converter controlled by a PI regulator. For synthesizing this regulator, we have replaced the converter with an equivalent continuous model when we have considered the mean values, over the chopping period, of the electric quantities. Then, we have developed the transfer function of the system by using the small signal modeling around an optimal operating point. A PI synthesis has been achieved by using Bode method. In the study, we have taken into account the converter losses. Coefficients Kp and K obtained of PI regulator lead to good simulations. The theoretical results confirm excellent tracking effectiveness response. The regulator operates correctly on a large range.
Keywords: solar powerplants, photovoltaic panel, maximum power point tracking, boost dc/dc converter, losses
Rachid El-Bachtiri
Organization: Assist master in Sidi Mohamed Ben Abdellah University (USMBA), Higher School of Technology (EST), in Fez, Morocco.
Education: - Diploma of thorough higher study (DESA) in automatic and systems analysis (20022004) from faculty of sciences dhar el-Mehrez at Sidi Mohamed Ben Abdellah University (USMBA). - Researcher student in solar energy photovoltaic since January 2005.
Experience: Member of "Team research in electrical engineering, power electronics and renewable energies" (REEPER at the higher school of technology in Fez) belonging to the "Laboratory of electronics, signal-systems and data processing" (LESSI at the Faculty of Science in Fez). Main range of scientific interests: renewable energy, optimal use of the photovoltaic power. Publications: participation at congress on renewable energy; automatic control and systems engineering.
Organization: Researcher teacher in Sidi Mohammed Ben Abdellah University (USMBA), Higher School of Technology (EST), in Fes, Morocco. Assisting Master (21/10/88). Ability Professor (22/01/97). Higher teaching Professor (22/01/01).
Education: Engineer (July 1988); Mohamed V University (Rabat), Mohammadia School of Engineers (EMI, 1983-1988), Electrical engineering, Electro technics and Industrial Electronics. Doctor of Sciences Applied (January 1997), Catholic University of Louvain (UCL, 1992-1997) at Louvain-La-Neuve (Belgium), Faculty of Science Applied (FSA), Department of electricity, Laboratory of electrotechnics and instrumentation (LEI).
Experience: Lectures and directed work: Power electronics and electrotechnics. - Person in charge for a "Team of research in electrical engineering, power electronics and renewable energies" (REEPER at the higher school of technology in Fez) belonging to the: "Laboratory of electronics, signal-systems and data processing" (LESSI at the Faculty of Science in Fez). Main range of scientific interests: electrical engineering and industrial electronics; resonance static conversion, effects of the harmonics, and their attenuation. Renewable energies; optimal use of the photovoltaic electrical power.
Publications: Papers in the power electronics and renewable energy field.
International Scientific Journal for Alternative Energy and Ecology № 6 (62) 2008
© Scientific Technical Centre «TATA», 2008
Organization: Associate Professor in Université catholique de Louvain (UCL), Belgium, Assistant (1974), Senior staff member (1979), Associate professor (1999). Education: Engineer (1971) in UCL, Doctor in Applied Sciences (1991). Experience: Lectures and directed work in physics, electrical machines, power electronics, (^s instrumentation and sensors, mechatronics, photovoltaic energy.
Main range of scientific interest: physics, analytical field computation, optimal design of electromechanical and electrical converters. ^ \ Publications: Physics, circuit theory, electrical machines, photovoltaic energy.
Ernest Matagne
Introduction
Photovoltaic energy is a promising alternative energy source for the future, due to the world's limited conventional energy sources. Its disadvantages are that the initial cost is very high and the energy conversion efficiency is relatively low. Therefore, it is desirable to extract the highest possible power at any moment from the solar array source. The amount of power obtained from a photovoltaic array depends on its operating voltage. From its typical V-I and V-P characteristics (Fig. 1, a and b respectively), a unique operating point (v = Vmpp), known as the maximum power point (MPP), delivers the maximum available power Pmax. When operated at the MPP, the array is best utilized. The MPP of a photovoltaic array varies with irradiation, temperature and other effects. Up to now, a large variety of MPP seeking algorithms exists: look-up table [1, 2], perturbation and observation (P&O) [3], incremental conductance [4, 5] etc. Another maximum power point tracking (MPPT) is proposed in [6, 7], where a dc/dc converter is controlled so that dIou/dV and dP/dV equal zero, where Iout and P are the dc/dc converter output current and input power respectively. However, these methods cannot take into account the losses in the dc/dc converter, particularly, the switch losses in the MOSFET transistor.
mpp
b
Fig. 1. Typical PV module: a - current-voltage and b -powervoltage characteristic
In this paper, we pick up the work exposed in [7], and we consider that the dc/dc converter is not ideal. This converter is used between the PV and the battery to track the maximum power point of the PV module (Fig. 2, a).
The battery is considered as a constant voltage E in series with a constant resistance Rb [8, 9]. The MPP (Vmpp, Pmax) is reached when dP/dV = 0, P = VI being the PV power. Then, the control circuit must keep (dP/dV) equals zero. That is possible with action on duty cycle a (0 < a < 1) according to the solar irradiation X and the temperature T. Duty cycle is a signal produced by a PI regulator. For synthesizing this regulator, we have developed a transfer function for the system using the small signal model. The coefficients Kp and K of the PI regulator are obtained by frequency synthesis.
PV module
F
V
dc-dc converter
Qui Д-
-EL
Battery
Controller
a
R
V
R
sh
b
Fig. 2. System bloc diagram (a) and - equivalent circuit(b) of a PV module
Theoretical analysis
The power electronic converter is a boost converter inserted between the PV generator and the battery. It is characterised by its duty cycle a (0 < a < 1) that gives the ratio input between the input and the output voltage when the conduction is continuous (Fig. 3). The transistor is ON during aT and OFF during the rest of the period i.e. (1 - a)T. The diode state, in continuous conduction, is complementary of the transistor one. The inductance is charged by the input through the transistor, and it discharges at the output through the diode.
a
a
V
Ut) i (t)
-tmxr
-r-c MOSFETI 'r
T1
aTt
battery
T
(L) .
battery
Fig. 3. Boost converter (a); converter "mean" (b) equivalent circuit
If the chopping frequency is sufficiently higher than the system characteristic frequencies, we can replace the converter with an equivalent continuous model. We will consider, for that, the mean values, over the chopping period, of the electric quantities (Fig. 3, b). The transistor can be replaced by a voltage source whose value equals its mean voltage. At the same, the diode can replaced by a current source.
Optimal operating point of PV module
(= 1,740), Tr - reference temperature (= 298,18 K), Ior -cell saturation current at Tr, Rsh - shunt resistance, Rs -series resistance.
The output power of PV panel is P = VI, at optimal point, we have:
dP dI dI -1
dP = I+VdF =0 = - ■ (4)
Hence:
I = ( - RSI ){IosA expU(iV + RSI )] + }, (5)
where: A = q/(jkTNceli) and NceIl is the number of series cells in the module.
The PV module considered in this paper is the SM55. It has 36 series connected mono-crystalline cells. The manufacturer ratings of this PV photovoltaic under standard conditions (irradiation 1 = 1000 W/m2, A.M. 1.5, solar spectrum and cell temperature T = 25 °C) is shown in Table 1. The values of the rest parameters are as follow: Rs = 0,1124 Q, Rsh = 6500 Q and Ior = 4,842 pA.
Table 1
PV module specifications under standard test conditions (STC)
Cell temperature, °C 25
Open-cicuit voltage, V 21,7
Short-circuit current, A 3.45
Maximum power current, A 3.15
Maximum power voltage, V 17.4
Maximum power, W 55
The equivalent circuit of the PV module considered in this paper is shown in Fig. 2, b. The relationship between Vand I is given by [10, 11]:
1 - Isoi - Ios 1 exp
— (v + R1 )
- U-
V + RSI
R„
where:
1os 1 or
T_
T
exp
QEgo f 1 1 A
ßk {Tr T
Isa =hsc + K (T - 298.18^
1000
(1)
(2) (3)
Choice of L and C
The inductor value, L, required such the converter operates in the continuous conduction mode (Fig. 4, a) is calculated such that the peak inductor current at maximum input power does not exceed the power switch current rating [12]. Hence, L is calculated as:
L >
Vom (1-am )a„
(6)
and I and V - cell output current and voltage, Ior - cell reverse saturation current, T - cell temperature in degree Kelvin, k - Boltzmann's constant (1.381e-23 J/K), q -electronic charge (1,602e-19 C), KI - short-circuit current temperature coefficient at Isc (= 0,0004 A/K), Isc - short-circuit current at 25 °C and 1000 W/m2, 1 - solar irradiation in W/m2, Isoi - light-generated current, Eqo -band gap for silicon (= 1.12 eV), y (= ß) - ideality factor
f\AILm\
where fs (= 1/Ts) - switching frequency, am - duty cycle at maximum converter input power, AILm - peak-to-peak ripple of the inductor current, Vom - maximum of dc component of the output voltage, Iom - dc component of the output current at maximum output power. Taking into account that the ripple of the PV output current must be less than 2 % of its mean value [12], the input capacitor value is calculated to be:
C >■
I a2
om m
(7)
0.02(1 -a m )VmnL
where Vinm - PV input voltage at the maximum power point.
a
b
3
When the boost converter is used in PV applications, the input power, voltage and current change continuously with atmospheric conditions. Thus, the converter conduction mode could change since it depends on them. Also, the duty cycle a is changed continuously in order to track the maximum power point of the PV array. The choice of the converter switching frequency and the inductor value is a compromise between the converter efficiency, the cost, the power capability and the weight. For example, higher is the switching frequency, lower is the inductor core size, but the power switch losses increase. Also, by using a large L value, the peak-to-peak current ripple AIL is smaller; requiring lower current rating power switches. But the converter size is increased substantially because a larger inductor core is required.
*Lpeak
and Pon.oJf - switching losses, VDS - drain-to-source voltage, Idso„ - drain-to-source current, tri - rise time of transistor current, tfv - fall time of the voltage, trv - rise time of the voltage, tft - fall time of current at the state OFF. The switching power losses in the diode are neglected. And
r2 „ , T7- T , .. j 2
Pd ~ RDSonIDSona+ VsIout + rLIL -
(10)
where Pd - average power dissipation in dc/dc converter, RDSon - static drain-to-source on-resistance, ampp - duty cycle at MPP, Vs - threshold voltage of diode, ID -average current of diode, rL - resistance of inductor, IL -average current of inductor.
So, the output power of dc/dc converter at optimal operating point (Pout) is:
P = p - ( p + p )
out mpp V on-off d '
(11)
where Pmpp - maximum power point.
The efficiency (n) of dc/dc converter is defined as:
Pou
n = "
p
(12)
ILpeak 1Lmean
b
Comnrand signal
OFF
OFF
ON
OFF
ON
OFF
Fig. 4. Boost converter waveforms: (a) continuous conduction mode and (b) discontinuous conduction mode
Losses in the MOSFET transistor
In this study, we have taken into account the losses in the MOSFET transistor for determining the power transmitted to the battery at the optimal operating point. At ON state (Fig. 5), the transistor is equivalent to the RDSon resistance. This resistance is considered as a constant. The average switching power dissipation in dc/dc converter is given as follow [13]:
1
Pon—off = 2 DSon '
= л VDS1 DSon fs у cON + tcOFF ) 7
(8)
where:
VDS = E + Vs + RJout, VDSon ~ Il, tcON — tri + tfv tcOFF — trV + tft (9)
b
Fig. 5. Switching time: a -ideal waveforms and b - real waveforms
Frequencial synthesis of PI regulator
We deduce from the continuous model equations (Fig. 3, b) the following equations:
dV
C-= I — IL
dt L
(13)
a
a
s
т.
141
dIL
V - L~dL~ + rLIL + Vout
Iout -R - a)lL ,
(14)
(15)
where:
dAV
C — =A ~AIl
dAIL
L~dL = AV- K +amppRDSon)) -
- Rb AIout + (e + Vs - RDSonIlmpp )a
a
AIout = (l - ampp )AIL - ILmpp Aa
At steady state, we have:
Imp = I
V
Rb ^ outmpp
mpp Lmpp
mpp — (rL +amppRDSon )ILmpp + + Rb AIoutmpp +(E + Vs )(l-a mpp )
T =1 - a )- T
outmpp mpp Lmpp
(20)
(21) (22)
For a small variation around an optimal operating point, the system can be shown by a functional diagram like that used in [14] for synthesizing the regulator. Thus, the system can be presented as (Fig. 6):
Fig. 6. Functional diagram of the system
According to [14], the transfer function Ga1(s) and Ga2(s) must have the following form:
Rai(s) Ra 2(s)
Ga1(s) = KaX sa^ and Ga2 (s) = Ka2 -sa^ , (23)
where Ra1(s) and Ra2(s) - rational fractions with Ra1(0) = Ra2(0) = 1, Ka1 and Ka2 - static gains of Ga1(s) and Ga2(s), a1 and a2 - integration numbers of Ga1(s) and Ga2(s) respectively. In this paper, we have obtained:
^i =
RG - K2 )(l + RsGm + Vs + (R - RDSon )J (24)
(25)
Ka 2 -
Ki + K2 K33 Kl + K 2 K 3
(1 + RsGm)(1 + GK 3)
Vout _ Rdso„vIl + Rb (1 - *)h + (E + Vs) (1 - a) (16)
and rL - inductor resistance, Rb - battery resistance, Vs -threshold voltage of the diode.
For expanding in series equations (11-14) around an optimal operating point, we write: q = qmpp+ Aq, for each quantity q in the set (V, a, IL, I, Iout} defining the operating point. So, the equations system become:
Ra1 (s) = K LC 2 KK3C + K2L (26)
-s 2 + 13 J^2 s + 1
K1 + K2 K3 K1 + K2 K3
K1 LC 2 K1K3C + K2 L
~s +——---;— s +1
K1 + K 2 K 3 K1 + K 2 K 3
Ra2 (S) = -LC
2 GL + GK 3
s2 + -—-—3s +1
-, (27)
1 + GK 3 1 + GK 3
(17) where:
(18) (19)
ARs I RsImpp - Vmpp
Rdm I1 + RsGm
K -1 +
RsImpp Vtnpp y
K 2 --7-— - G
Rdm I1 + RsGm j
K 3 = rL +amppRDSon + Rb 1 -ampp ) > G = 1 + ^^ >
Gm
Gm and Rdm =
1
R.
dm
IosA eXP{ A(Vmpp + RsImpp )}
Ka1 and Ka2 are constants for a given temperature T and solar irradiation X.
The transfer function of the open loop used to synthesizing a PI regulator is:
Go(s) = Ga1(s)Ga2 (s) =
_((G - K2 ){E + Vs + (Rb - RDSon kmpp I
1 + GK 3 1
LC 2 GL + CK 3
-s +——-—3 s +1
(28)
1 + GK 3 1 + GK 3 Simulation procedure
The bloc used for simulations is given by Fig. 7, a. In PV module block, equations (1-3) are used; and in block (converter + battery) equations (11-14) are used. The switching losses are cutting off the output converter power according to equations (8-11). The proposed controller circuit that forces the system to operate at its optimal operating point under variable temperature and insolation conditions, is shown in Fig. 7, b. On one hand, we multiply the PV output current I by the PV output voltage V. Then, we obtain the PV output power P who is derived in order to obtain the (dP/dt) signal. On the other hand, we drive the signal voltage and invert it. Thus, the signal 1/(dV/dt) is obtained.
x
International Scientific Journal for Alternative Energy and Ecology № 6 (62) 2008
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The product of 1/(dV/dt) by (dP/dt) signals gives the (dP/dV) signal, who is compared to zero. The resulting difference signal (error signal) is the input signal of the PI regulator. This PI regulator is used to regulate the duty cycle signal of the dc/dc converter until that the condition: dP/dV = 0 is satisfied.
T ¥
dP/dt
dV/dt
I
MATLAB Function
Sati ration
г
l/< dV/dt)
PI
regulatoi
dP/9v
Ground
Fig. 7. a - bloc diagram for system and b -MPPT tracker circuit
Results and discussion
Theoretical results The battery voltage, the threshold voltage of the diode, the resistance of battery and the series resistance of the inductor, used in this paper, are respectively E = 24 V, Vs = 0.7 V, Rb = 0.65 Q and rL = 0.05 Q. The MOSFET transistor utilized here, is an IRFP250. Their characteristics used in this paper are: RDSon = 0.085 Q, ty = 86 ns, try = 62 ns, tri = 16 ns and tft = 70 ns. Fifty kilohertz switching frequency is used. The PI controller gain and the integral time constant obtained by frequency synthesis using Bode method are respectively Kp = 0.01 and Tt ( = 1/Ki) = 1.8 ms. Using equation (6), the boost inductance choice is L = 1 mH. With equation (7), the choice of input capacitance is C = 4.7 ^F.
For different values of irradiation X and temperature T, the computation of the theoretical optimum quantities Vmpp, Pmpp, Pout and n are assembled in Table 2.
Table 2
Theoretical quantities Vmpp, Pmpp, Pout and n for different values of X and T
Values of X (W/m2) and T (K) Optimum voltage V (V) Optimum power P (W) Optimum power Pout (W) Efficiency n (%)
X = 100 and T = 298.18 14.23 4.393 4.224 9б.1б
X = 1000 and T = 298.18 17.39 34.8Q 32.Q7 93.Q1
X = 1000 and T = 320.18 13.б3 48.б1 46.Q7 94.7б
X = 100 and T = 320.18 12.31 3.713 3.37Q 9б.11
Simulation results
The simulation study was made to illustrate the response of the proposed method to rapid temperature and solar irradiance change. For this purpose, the irradiance X and the temperature T, which are initially 100 W/m2, and 298.18 K, are switched, at 0.02 s and 0.05 s, to 1000 W/m2 and 320.18 K respectively (Fig. 8, a and b). and vice versa (Fig. 9 a and b), i.e., the solar irradiance changes from 1000 W/m2 to 100 W/m2 at 0.02 s and the temperature changes from 320.18 K to 298.18 K at 0.05 s.
6Q
)Q Í4Q Pd 3Q
3 2Q 1Q Q -1Q -2Q
-3Q
1
— PV output power P "" dc/dc converter output power "
Q.Q2 Q.Q4 Q.Q6 Q.Q8 Time, s
a
Q.1
b
Fig. 8. Variation of: a -PV output power and dc/dc converter output power, and b -PV output voltage for a step change on irradiation and temperature from 100 W/m2 to 1000 W/m2 and 29818 K to 320.18 K respectively
Международный научный журнал «Альтернативная энергетика и экология» № 6 (62) 2008 © Научно-технический центр «TATA», 2008
a
X
b
•o
§
a.
— PV output power P dc/dc converter output power
0.02 0.04 0.06 0.08 0.1 Time, s
>
-10
-20
PV output voltage V(V)
0.02
0.04
0.06 0.08 Time, s
0.1
b
Fig. 9. Variation of: a -PV output power and dc/dc converter
output power; b -PV output voltage for a step change on irradiation and temperature from 1000 W/m2 to 100 W/m2 and 320.18 K to 298.18 K respectively
In Fig. 8, a and 9, a the variation of instantaneous PV power (P) and dc/dc converter output power for a step change of temperature and solar irradiance are shown. And the Fig. 8, b and 9, b give the variation of PV output voltage for a step change of temperature and solar irradiance.
The optimum values of PV output voltage, instantaneous PV power and dc/dc converter output power obtained by simulations are assembled in Table 3.
Table 3
Simulated quantities Vmpp, Pmpp, Pout and n for different values of X and T
Values of 1 (W/m2) and T (K) Optimum voltage V (V) Optimum power P (W) Optimum power Po«t (W) Efficiency n (%)
1 = 100 and T = 298.18 14.25 4.396 4.205 96.66
1 = 1000 and T = 298.18 17.41 54.83 51.99 94.82
1 = 1000 and T = 320.18 15.66 48.64 45.95 94.47
1 = 100 and T = 320.18 12.31 3.717 3.549 95.48
To watch Tables 2 and 3, then Fig. 8 and 9, it is clear, on one hand, that the average voltage (V), instantaneous PV power (P) and dc/dc converter output power (Pout) are very close to their optimal values Vmpp, Pmpp and Poumpp. And on the other hand, the values obtained by simulation coincide with their obtained by programming. The losses in the dc/dc converter varies from 0.168 W to 3.04 W for 1 =100 W/m2, T = 320.18 K, and 1 = 1000 W/m2, T = 298.18 K respectively. They can be minimized for adequate choices of the component of dc/dc converter.
The simulations of the MPPT show that the system is stable. The oscillations about the computed optimal operating point are due to the switching action of the dc/dc converter. The transients between operating points are natural for a dynamic system which is controlled by a PI type controller.
Conclusion
In this paper, a method that forces a photovoltaic panel to operate at its maximum power point under variable temperature and irradiation conditions is developed. This method is tested by simulations in Matlab software. It has been concluded that the method was able to track the irradiance and the temperature level change rapidly. The PI regulator used in this work for controlling the boost dc/dc converter in order to get the system operating at the PV maximum power is synthesized by frequencial synthesis using Bode method. So, we have developed a transfer function of global model using a small signal method by taking into account the losses in the dc/dc converter. Simulations show that the regulation is robust against disturbances.
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