Methods for Grid Connected Photovoltaic Systems. G. M. S. Azevedo, M. C. Cavalcanti, K. C. Oliveira, F. A. S. Neves and Z. D. Lins. Universidade Federal de ...
Evaluation of Maximum Power Point Tracking Methods for Grid Connected Photovoltaic Systems G. M. S. Azevedo, M. C. Cavalcanti, K. C. Oliveira, F. A. S. Neves and Z. D. Lins Universidade Federal de Pernambuco - Departamento de Engenharia Eletrica e Sistemas de Potencia, Recife - PE, Brazil
Abstract ! " " # #
I. I NTRODUCTION The photovoltaic (PV) energy is on the way to become a well-established energy source in the coming decades. The PV energy is particularly attractive as a renewable source for distributed generation systems due to their relative small size, noiseless operation, simple installation and the possibility to put it close to the user. In this kind of application all available PV power is delivered to the electric grid and the system should operate in order to improve its energy conversion. Therefore it is necessary a control system that detect variations in the PV array conditions and lead the system to a new operation point where the maximum power can be extracted. The voltage-power characteristic of a PV array is nonlinear and time-varying because of the changes caused by solar irradiance and temperature. Thus the linear control theory cannot be easily used to obtain the maximum power point (MPP) of the PV array [1]. To overcome this problem several methods have been developed to continuously tracking the MPP [1][11]. These methods are used for MPP tracking (MPPT) and they are an essential part of a PV system. The most popular MPPT methods in PV systems are perturbation and observation (P&O) and incremental conductance (IncCond). In most applications two conversion stages are used, the first stage is a DC-DC converter to increase the PV voltage and, in some cases, to provide galvanic isolation. Moreover, this converter decouples the energy change between the PV array and the DC link capacitor. The second stage is an inverter used to deliver the energy to the electric grid. The inner control loop of the grid currents and the DC link voltage is done by the inverter while the MPPT control is done by the DC-DC converter.
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iPV
iDC iC
PV array
vDC
Fig. 1.
C
3 phase inverter
to electric grid
PV system with one conversion stage.
However, in some applications it is possible to use only one stage (one inverter), as show in Fig. 1, in order to increase the system efficiency. In this case, the control has three inner loops, the current loop is the most inner, the voltage loop is the intermediate and the last is the MPPT control. Thus, the MPPT control method, as well as the value of DC link capacitor have great influence on the power quality in the grid side. In most cases the MPPT method perturbs the DC link voltage and if a large value of capacitance is used, much more energy must be exchange with the grid causing a pulsating power injected in the grid. On the other hand, if a small value of capacitance is used, variations in the available power of the PV array are transmitted to the grid. Improvements of the MPPT techniques can be obtained with the best adjustment of the sampling rate and the perturbation size [7], both in accordance with the dynamics of the converter. In this paper, a procedure to determine the parameters of the P&O and IncCond techniques is explained and it helps to define which technique is better to the grid connected PV system with one stage. The techniques have been verified on a PV system modeled in Matlab and experimental results corresponding to the operation of a grid connected PV converter controlled by a digital signal processor are also presented. II. S YSTEM M ODEL To evaluate the MPPT techniques it is necessary to have the models of the PV array and the inverter. The PV array model is based on the single-diode equivalent circuit. The inverter model is simplified to take into account only the dynamic behavior of the DC link voltage control to achieve fast simulation time. A. PV Array Model A mathematical description of the PV cells output characteristic can be modeled by a double-exponential, that is derived from the p-n junction physics and it is generally accepted as a good model for the PV cells, especially those made of
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polycrystalline silicon. The cells made of amorphous silicon, do not exhibit a sharp “knee” in the curve characteristic as the crystalline types do, and therefore the single-exponential model provides a better fit to such cells [2]. From these models, an equivalent circuit can be determined. In Fig. 2, it is shown the equivalent circuit for the single-exponential model (or single-diode model) [3], that is used in this work. The output characteristic, at constants temperature and irradiance, is given by the following equation [2]-[4]: IP V = Ig − Io
V q P V − IP V RS , (VP V − IP V RS ) − 1 − exp AkT RP (1)
where IP V and VP V are the PV cell output current and voltage, respectively, Ig is the photo-generated current, Io is the reverse saturation current, q is the charge of an electron, A is the ideality factor of the p-n junction, k is the Boltzmann’s constant, T is the temperature in Kelvin, RS and RP are the PV cell series and shunt resistances, respectively. The equivalent circuit used in this study is further simplified by neglecting the shunt resistor. Considering the effects of irradiance and temperature on the current, the photo-generated current can be approximated as [2], [3]: S ISCST C Ig = 1 + ki T − TST C , (2) SST C where ki is the temperature coefficient of the short-circuit current, S is the irradiance and TST C , ISCST C and SST C are the temperature, the short-circuit current and irradiance in the Standard Test Condition (STC) provided by the manufacturer’s data sheet. The saturation current does not depend on the irradiance conditions, but it shows a strong dependence with the temperature, that can be approximated as:
A3 1 T q Ego 1 Io = IoST C exp − , (3) TST C Ak TST C T
was neglected):
IP V = np Ig − np Io exp
RS Ig
Fig. 2.
RP
VPV
Equivalent circuit of the PV cell.
−1 .
(4)
The PV module manufacturer does not provide all parameters of the model, but the unknown parameters can be extract from the data sheet values through the procedure described in [3]. B. Inverter Model The operation point of the PV array can be set through the control of its output voltage, that is the same voltage of the inverter DC link. Therefore the DC link voltage control must be used by the MPPT technique to set the system operation point. Considering the electric diagram in Fig. 1, the DC link voltage can be controlled by the current (iDC ) in the inverter DC side. Some considerations are assumed for the simulation: the dynamics of the converter current control is much faster than the dynamics of the control loop that is being designed, in such a way that the current control can be neglected; the sample rate of the measured signals is sufficiently high in such a way that the system can be considered continuous; the deviations in the current caused by the non linearity of the AC loads do not disturb the DC link voltage. In Fig. 3, it is shown the block diagram of the DC link voltage control. kp is the proportional gain of the control loop. The PV array current appears as a perturbation in the diagram, but it can be compensated in the control since it is measured to implement the MPPT algorithm. By the block diagram, the transfer function of the voltage control can be obtained: T (s) = where τvdc =
C kp
1 VDC (s) , ∗ (s) = 1 + s τ VDC vdc
Plant
* v DC
kP
+ -
Fig. 3.
(5)
is the time constant of the voltage control. iPV
Control
where some simplifications, discussed in [4], are done. IoST C is the saturation current in the STC and Ego is the band-gap energy of the semiconductor used in the cell. The same model presented for the PV cell can be used for the PV array, being necessary some modifications in the output current equation. Considering np and ns as the number of parallel and series connections of the PV array, respectively, the following modification in (1) must be done (Where RP IPV
q VP V − IP V RS AkT ns
i *C
+ -
* i DC
1
iDC -
+
iC
1 sC
vDC
Block diagram of the DC link voltage control.
Using the PV array model, the MPPT algorithm and the inverter model, as shown in Fig. 4, a simple and fast tool for evaluation of the system behavior is achieved. It has been used to define criteria to choose the MPPT parameters. III. P ERTURBATION AND O BSERVATION The P&O method compares the power of the previous step with the power of the new step in such a way that it can increase or decrease the array voltage [6]-[8]. This method changes the reference value by a constant factor of voltage. It moves the operating point toward the MPP by periodically
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MPPT Algorithm
* v DC
1 1 + stvcc
vDC
iPV
Fig. 4.
Simulation model for MPPT evaluation.
increasing or decreasing the array voltage. From Fig. 5, it can be seen that incrementing (decrementing) the voltage increases (decreases) the power when operating on the left of the MPP and decreases (increases) the power when on the right of the MPP. Therefore, if there is an increase in power, the subsequent perturbation should be kept the same to reach the MPP and if there is a decrease in power, the perturbation should be reversed.
Fig. 5. Characteristic of the output power of the PV module for S = 1000 W/m2 and T = 50o C .
There are two parameters to be defined in the P&O technique. The first parameter is the sample rate (TM P P T ), that is the time interval in which the voltage and the current of the PV array is sampled and the reference voltage is determined. The second parameter is the perturbation size (ΔV ) that the reference voltage is changed. Considering the perturbation size, there is a practical limitation that do not allows the choice of very small perturbations in relation to the DC link voltage. For instance, in a DC link voltage of 400V , it is very difficult to modify its voltage with precision of ±1.0V (0.25%). Actually, the DC link voltage oscillates around these values due to the non linearities system, measurements noises and oscillation in the instantaneous active power in the converter output. In the prototype developed during this work, good results were obtained with ΔV > 1.0% of the DC link voltage. A choice of high ΔV provides a fast tracking for the MPP voltage. If ΔV has low value, the MPPT is slower, but it has small oscillations around the MPP. The choice of a low sample rate seems viable because it
would allow the algorithm to rapidly detect ambient changes and track the new MPP. However, there is a limitation for the minimal value of TM P P T imposed by the dynamics of the voltage control. The TM P P T should be chosen as being higher than the stabilization time of the DC link voltage to avoid instability of the MPPT algorithm [7]. Therefore, in a first order system, TM P P T ≥ 4τvdc . To evaluate the algorithm behavior in two different irradiance conditions (low irradiance and high irradiance) and its tracking speed, in a given instant, the irradiance changes from 500W/m2 to 1000W/m2 . The temperature is considered constant during the simulation since the PV module has high thermic inertia [5]. Figure 6(a) shows the result of the MPPT when a TM P P T lower than τvdc is chosen. In this case, TM P P T = 0.5τvcc . It can be seen that the reference voltage has more oscillations than the case TM P P T = 4τvdc , shown in Fig. 6(b). These oscillations increase if TM P P T decreases in relation to τvdc and this makes the converter current control difficult. When TM P P T ≥ 4τvdc is chosen the PV array voltage can track the reference voltage imposed by the algorithm. In this case, the voltage oscillation around the MPP is determined, being twice the value for the perturbation (ΔV ) as shown in Fig. 6(b). Figure 6(c) shows the simulation result with perturbation size of ΔV = 5V . It can be seen that the tracking time for the MPP decreases for the same TM P P T . However, the PV array voltage has more oscillations, wasting more energy. It is difficult to quantify how much energy is wasted in function of the perturbation size because it depends of the operation point, irradiance and temperature. In the P&O technique, the reference voltage usually does not coincide with the MPP voltage in steady state due to discrete process of the reference voltage. The reference voltage ∗ ∗ ∗ assumes three values: (VDC −ΔV ), VDC and (VDC +ΔV ), ∗ where VDC is the reference average voltage that does not coincide with VM P P . IV. I NCREMENTAL C ONDUCTANCE In the IncCond method [9]-[11], the slope of power versus voltage characteristic (Fig. 5) is used to define the direction of the perturbation. The slope (dPP V /dVP V ) is zero at the MPP, positive on the left of the MPP, and negative on the right. Since dPP V dIP V d(IP V VP V ) = = IP V + VP V , (6) dVP V dVP V dVP V it can be rewritten as 1 dPP V IP V dIP V = + . (7) VP V dVP V VP V dVP V Defining g as the sum of the incremental and the instantaneous dIP V array conductance g = VIPPVV + dV , the signal of the slope PV is the same as g, because the voltage VP V is always positive. The flowchart of IncCond in [9]-[11] was modified to take into account some issues. In the IncCond method, as well as in the P&O method, it is necessary to define the parameters TM P P T and ΔV . The
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shown in Fig. 7. Now, there are other three parameters to be defined: the tolerable voltage VT ; the tolerable current IT and the tolerable conductance variation ΔG.
365
VMPP 360
* vDC
vDC
355
Voltages (V)
350 345 340 335 330 325 320
4
3
2
1
0
5
Time (s)
(a) ΔV = 2.5V , TM P P T = 0.5τvcc 360
V
MPP *
vDC
355
vDC
Voltages (V)
350
345
340
335
330
325
10
5
0
15
Time (s)
Fig. 7.
(b) ΔV = 2.5V , TM P P T = 4τvcc 360
V
MPP * v DC
355
vDC
Voltages (V)
350 345 340 335 330 325 320
10
5
0
15
Time (s)
(c) ΔV = 5V , TM P P T = 4τvcc Fig. 6.
Simulation results of the P&O method.
same considerations discussed before are valid here. However, there are other parameters to be analyzed. Due to the discrete process of the reference voltage, the condition g = 0 usually will not be achieved. A practical aspect is considering that the condition is true when it is between a ΔG tolerable range (ΔG) around zero − ΔG 2 < g < 2 . Similar problems occur for the other comparisons in the technique, dVP V = 0 and dIP V = 0, shown in the flowchart in [9]-[11]. The solution is the same, substituting zero by tolerable ranges. Therefore, the technique is modified to have the flowchart
The modified flowchart of the IncCond technique.
An estimation for VT can be obtained by simulation of the complete system. For implementation of the prototype, this value must be reevaluated in such a way to consider the phenomena non modeled in simulation, as noises. Suppose that the PV array voltage oscillates around VM P P with amplitude lower than VT . Therefore the algorithm is lead to the stage where the current is compared to IT , as can be seen in the flowchart in Fig. 7. If the PV array voltage oscillates the PV array current also oscillates in according to (4). The method must guarantee that this oscillation in the current does not seem a climatic change. Therefore, IT should be chosen as being the maximum variation in the PV array current when the PV array voltage is deviated by ±VT of VM P P . Using the approximation based on the Taylor’s series of the function given by (4) around the MPP, yields vˆ2 ∂ 2 I ˆi = ∂I + ··· , (8) v ˆ + 2 ∂V M P P ∂V M P P 2! where the symbol ( ˆ ) denotes small variation around the MPP. In (8) the partial derivatives of the current in relation to the irradiance and temperature are not considered since the analysis is for steady-state and these values are constant. Substituting (4) into (8) and considering only the first derivative of the Taylor’s series:
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q (VM P P − IM P P RS ) ˆi = − np q VM P P I0 exp vˆ. ns AkT AkT ns
(9)
Defining
q (VM P P − IM P P RS ) ˆ ≡ np q VM P P I0 exp , G ns AkT AkT ns
it is obtained
ˆ vˆ. ˆi = −G
(10)
(11)
ˆ determines how much the PV array The coefficient G current changes when its voltage has a small variation around VM P P . This coefficient is a function of the irradiance and temperature and it can be seen from (11) that it is similar to a conductance. The influence of irradiance appears in VM P P ˆ and IM P P . The function G(S, T ) is shown in Fig. 8. The maximum point of this function occurs for the condition of maximum irradiance and temperature. Therefore, knowing the extreme operational conditions of the PV modules, maximum irradiance (Smax ) and maximum temperature (Tmax ), IT can be determined as follows: ˆ max , Tmax ) VT . IT = G(S
(12)
When the method is tracking a new operation point the reference voltage is modified in discrete steps ΔV and consequently the quantity g will also presents discrete variations and usually will not be zero. Suppose a minimal positive value (g + ) and a minimal negative value (g − ) when the PV array voltage changes of ΔV around VM P P as shown in Fig. 9. Therefore ΔG should be higher than the maximum g = g + + g − that can occurs to guarantee that the method will lead the voltage to the nearest point of the MPP. If ΔG is high the reference voltage will be far of VM P P (Fig. 10(a)). On the other hand, if ΔG is low the reference voltage will oscillate around the VM P P (Fig. 10(b)). In both cases the efficiency decreases. Figure 10 shows the simulation results for ΔV = 5.0V and TM P P T = 4τvdc . A analytical criteria for the choice of ΔG is very complex and the value for ΔG was obtained by using simulation. V. C OMPARISON : P&O AND I NC C OND M ETHODS To compare the two methods, the same simulation conditions are used. Comparing figures 6(c) and 10 it can be
Variation of g due to the voltage perturbation around VM P P .
seen that both methods can have errors during the intervals with changes in the atmospherical conditions. During these intervals, the operation point can move in a wrong direction instead of keeping near to the MPP [7]. To confirm the capacity of the methods extract the maximum available power in the PV array, two steady state conditions are shown in Table I. These results were obtained by simulations in the same conditions of figures 6(b), 6(c) and 10(c). The transient voltage interval was not considered. The efficiency of the MPPT techniques are obtained by: Tvdc Tvdc 1 P dt P dt Tvdc 0 0 = , (13) ηM P P T = Tvdc 1 V I M P P M P P Tvdc PM P P dt Tvdc
0
where Tvdc is the reference voltage period. The comparison of efficiency for P&O and IncCond techniques in Table I shows that these techniques present similar efficiencies with the IncCond being a little better. Even in steady-state, the reference voltage in the P&O technique oscillates around VM P P . These oscillations also appear in iC (Fig. 1 and Fig. 3) as shown in the transfer function: IC (s) sC . (14) ∗ (s) = s τ VDC vcc − 1 The reference voltage is a step of amplitude ΔV and iC will have a peak of amplitude ΔIC = ΔV kp . The currents that the PV system injects in the grid are function of iDC = iP V − iC , meaning that the currents injected in the electric grid will be distorted. Figure 11 shows the reference voltages and the grid currents in phase a obtained by the P&O and IncCond
0.015
0.01 ˆ (S ) G 0.005
0 1000 800
60
2
S (W/m )
TABLE I E FFICIENCIES OF THE METHODS WITH T = 500 C
AND
TM P P T = 4τvcc
50
600
Fig. 8.
Fig. 9.
40
400
30 200
T (ºC)
Irradiance conditions S = 500W/m2 S = 1, 000W/m2
20
ˆ in function of irradiance and temperature. Coefficient G
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ΔV = 2.5V 99.9870 % 99.9885 %
P&O ΔV = 5.0V 99.9559 % 99.9518 %
IncCond ΔV = 5.0V 99.9975 % 99.9875 %
360
V
Reference voltages
MPP *
vDC
355
vDC
356
352 2.1
345
340
335
∗ VDC IncCond
354
Grid current PO Method
Voltages (V)
350
∗ VDC PO
358
2.15
2.2
2.25
2.3
2.15
2.2
2.25
2.3
2.15
2.2
2.25
2.3
5 0 −5
330
10
5
0
Grid current IncCond Method
2.1 325
15
Time (s)
(a) ΔV = 5V , TM P P T = 4τvcc ,ΔG = 4 m0
5 0 −5 2.1
360
Time (s)
V
MPP v*DC
355
v
DC
Fig. 11.
Voltages (V)
350
345
VI. E XPERIMENTAL R ESULTS
340
The experimental results are obtained by using power measurements in the real ambient conditions. The PV array is composed by 24 modules M SX77 connected in series. Each PV module has a rated power of 77W and a rated voltage of 16.9V. The DC link capacitance is 2.35 mF and the proportional gain of voltage loop, kp , is set to obtain a time constant τvdc = 100 ms. Figures 12(a) and 12(b) show the reference voltage and the DC link voltage during the tracking of power point for the P&O and IncCond methods, respectively. In both results, the MPPT parameters are TM P P T = 4τvdc = 400ms and ΔV = 5V . It can be seen a very good agreement between simulation (figures 6(c) and 10(c)) and experiment. Figure 13(a) shows the instant that the voltage reference is changed by the P&O method, when it operates in steadystate. This change causes a small dip in the grid current as shown in Fig. 13(a). When IncCond is used, the reference voltage becomes constant resulting in a constant grid current peak as shown in Fig. 13(b). These figures are obtained with identical conditions of irradiance and temperature. It can be seen again a very good agreement between simulation (Fig. 11) and experiment.
335
330
325
10
5
0
15
Time (s)
(b) ΔV = 5V , TM P P T = 4τvcc ,ΔG = 0.5 m0 360
V
MPP *
vDC
355
v
DC
350
Voltages (V)
Grid currents using the P&O and IncCond methods
345
340
335
330
325
0
10
5
15
Time (s)
(c) ΔV = 5V , TM P P T = 4τvcc ,ΔG = 1.5 m0 Fig. 10.
VII. C ONCLUSION
Simulation results of the IncCond method.
methods for a steady-state condition. In the P&O technique, the grid currents are distorted, having low power quality. When iP V is much higher than ΔIC , this effect is neglected. To decrease the distortion, a low value for kp can be used, but the converter dynamics will be slower because τvdc = kCp . This dynamics implies in higher tracking time. Due to these aspects, the IncCond method is chosen as the best technique to be implemented in the PV systems with one converter stage.
In this paper, the model of the photovoltaic array is simple and its parameters can be determined by the data supplied by the manufacturer, being very important for the definition of the parameters of the maximum power point tracking techniques. The implementation of the perturbation and observation algorithm in software is very simple, having only one multiplication and few sums and comparisons. The incremental conductance algorithm is a little more complex. The grid connected photovoltaic system presented in this work demands high computational capacity of the control system due to the current compensations and a digital signal processor
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Fig. 12.
(a) P&O
(a) P&O
(b) IncCond
(b) IncCond
Reference and DC Link voltages during tracking.
Fig. 13.
is used. Using digital signal processors the differences about the computational effort of the maximum power point tracking techniques can be neglected. Therefore, the best results of the incremental conductance method in relation to the current injected to the grid indicate that this method is more suitable for the grid connected photovoltaic system with only one conversion stage. ACKNOWLEDGMENT The authors would like to thank CNPq and CAPES for its financial support. R EFERENCES [1] C. Hua, J. Lin, C. Shen, ”Robust control for maximum power point tracking in photovoltaic power system”, Power Conversion Conference, 827-832, 2002. [2] J. A. Gow, C. D. Manning, ”Development of a photovoltaic array model for use in power-electronics simulation studies”, IEE Proc. Electric Power Applications, Vol. 146, No.2, pp. 193-200, March 1999. [3] Weidong Xiao, W. G. Dunford, A. Capel, ”A novel modeling method for photovoltaic cells”, IEEE Power Electronics Specialists Conference, Vol. 3, pp. 1950-1956, 2004. [4] E. Matagne, R. Chenni, R. El Bachtiri, ”A photovoltaic cell model based on nominal data only”, International Conference on Power Engineering, Energy and Electrical Drives, pp. 562-565, April 2007.
Reference voltage and grid current in phase a.
[5] Shengyi Liu, Roger A. Dougal, ”Dynamic Multiphysics Model for Solar Array”, IEEE Transactions on Energy Conversion, pp. 285-294, Vol. 17, No. 2, June 2002. [6] X. Liu, L. Lopes, ”An improved perturbation and observation maximum power point tracking algorithm for PV arrays”, IEEE Power Electronics Specialists Conference, pp. 2005-2010, 2004. [7] N. Femia, G. Petrone, G. Spagnuolo, M. Vitelli, ”Optimization of perturb and observe maximum power point tracking method”, IEEE Transactions on Power Electronics, 20(4), 963-973, 2005. [8] C. Hua, J. Lin, C. Shen, ”Implementation of a DSP-controlled photovoltaic system with peak power tracking”, IEEE Trans. on Industrial Electronics, 45(1), 99-107, 1998. [9] K. Hussein, I. Muta, T. Hoshino, M. Osakada, ”Maximum photovoltaic power tracking: an algorithm for rapidly changing atmospheric conditions”, IEE Generation, Transmission and Distribution, 142, 59-64, 1995. [10] Y. Kuo, T. Liang, J. Chen, ”Novel maximum power-point-tracking controller for photovoltaic energy conversion system”, IEEE Trans. on Industrial Electronics, 48(3), 594-601, 2001. [11] T. Kim, H. Ahn, S. Park, Y. Lee, ”A novel maximum power point tracking control for photovoltaic power system under rapidly changing solar radiation”, IEEE International Symposium on Industrial Electronics, pp. 1011-1014, 2001.
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