A Sensorless Control of H-bridge Multilevel Converter for Maximum Power Point Tracking in Grid Connected Photovoltaic Systems G. Brando, A. Dannier, R. Rizzo Department of Electrical Engineering University of Naples, Via Claudio 21 – 80125 Naples (Italy)
N
Abstract—There is a high development of grid-connected photovoltaic systems that utilize static converter which influence the efficiency of the system. The energy conversion depends on the architecture of the converter, different solutions are available and H-bridge multilevel converters seem to be an optimal solution also for the power quality. It is important to utilize a proper architecture of converter but also to set up a control optimized to have the energy conversion at maximum efficiency. The persistence of the maximum efficiency condition is related, depending on the atmospheric conditions, to the tracking of the Maximum Power Point (MPP) by modifying the operating conditions of the system (Maximum Power Point Tracking – MPPT). In the paper is proposed a sensorless control set up to deliver the maximum power to the grid in presence of variations of incident irradiation on the photovoltaic arrays. The control technique is presented and validated by simulation implemented on a photovoltaic system with Hbridge 5-levels converter. The simulation results confirm that the control is able to effectively track the MPP and to stabilize immediately in the new steady-state condition.
is , vs
is , j , vs , j
instantaneous space-vector of line currents, voltages instantaneous space-vector of the converter voltages current, voltage of the j-th line-phase
vf ,j
converter voltage of the j-th phase
vr , j
voltage on the r-th dc-link capacitor of the j-th
sr , j
phase j-th converter phase switching funct. of vr , j
vf
tk
k-th sampling instant
Ck , j
k-th dc-link capacitance of the j-th converter phase
L, R Ts
line-phase inductance, resistance sampling interval
R. Rizzo is corresponding author. e-mail:
[email protected]
1-4244-0632-3/07/$20.00 ©2007 IEEE.
'h
h-th subinterval of the sampling interval
idc
dc-link current
si
current slope
sv
voltage slope
si , p
positive slope current
si , n
negative slope current
si ,0
start-up slope current
Photovoltaic systems make possible the energy conversion from solar to electrical, nowadays is diffused the connection to the grid of these systems that are based on photovoltaic cells which generate the energy. It is possible to individuate the operating point of a photovoltaic cell, corresponding to the maximum efficiency parameterized with the irradiation and the temperature, by means of the characteristic V-I of the cell. It is very important to guarantee constantly that the cells work at maximum efficiency because the energy production systems based on Photovoltaic (PV) have their main problem in a low efficiency. Taking into account that a PV system is connected to the grid through an inverter, the optimization of the performances of PV arrays can be realized by choosing a proper converter and set up a specific control strategy. In this paper is proposed a control algorithm that aim to maximize the cells efficiency for irradiation variations at constant temperature. The influence of the temperature is not taken into account because, when the thermal steady-state is reached, the variation is negligible due to the particularly slow dynamic. Once fixed the control objective, the main decision to be taken is to define the architecture of the converter. The PV system grid-connected and the nature of the energy source in this case (DC modules) suggest to adopt a multilevel H-bridge topology as an optimal solution that synthesizes a desired voltage output from several levels of dc voltages as inputs. Several topologies for multilevel inverters have been proposed in the literature, the cascade
I. NOMENCLATURE number of converter voltage levels number of capacitors N = 3 m 1 / 2
n. of dc-link capacitors per converter phase time constant of the supplying line duty-cycle of the h-th modulation vector
II. INTRODUCTION
Index Terms—H-bridge, Multi-level Converter, MPPT Maximum Power Point Tracking.
m N
W Gh
789
H-bridge inverter power circuit is simple and is ideally suited for systems such as photovoltaic where isolated input dc source is available. Photovoltaic cells generate electric energy from solar energy, the electrical energy from solar cells is dc form and it has to be processed to required form to suite the load requirements. This can be achieved by cascaded H-bridge multilevel inverter [1]. The architecture considered in this paper is a 5 level H-bridge that gives good response also to power quality problems that are more and more significant in the considered application. Currently power quality indexes of distribution networks are strongly influenced by the presence of a high number of loads where power electronics systems are widely used. Therefore a possible solution to overcome or to reduce this problem can be given by multilevel conversion topologies. III. CHARACTERISTIC V-I OF A PHOTOVOLTAIC ARRAY
Fig. 2. PV cell V-I characteristics, at different irradiation.
Typical cells V-I characteristics are shown in fig. 1, 2, the power delivered by a photovoltaic source depends on the operating point. The maximum efficiency is obtained when the load characteristic intersects the V-I characteristic of the source in the point corresponding to the maximum V·I product. This maximum point vary in the V-I plane as a function of the solar irradiation and cells temperature. The persistence of the maximum efficiency condition is related, depending on the atmospheric conditions, to the tracking of the Maximum Power Point (MPP) by modifying the operating conditions of the system (Maximum Power Point Tracking – MPPT). The objective of the control algorithm proposed in this paper is to drive opportunely the converters which interface the PV system and the grid, the aim is to track the working point at maximum efficiency. The control must identify the direction where the MPP is moving and make the necessary correction on the duty-cycle of the converter to drive the system on the new MPP. The V-I characteristic of a PV array is strongly nonlinear. It is evident that the temperature variations are extremely slow, the oscillations of the maximum efficiency operating point are quite exclusively due to the solar irradiation variations.
IV. H-BRIDGE M-LEVEL MODEL A mathematical model of multilevel PWM-rectifiers is pointed out and described in details in the following. The model formulation is generalized to “m” voltage levels. The mathematical model links currents and voltages of the supplying line to the currents and voltages in the dclink. The 2nd Kirchhoff law applied to the j-th phase gives: vs , j
Rs is , j Ls
d is , j v f , j voo ' dt
with j 1, 2,3 (1)
The meaning of the used symbols is explained in the list of symbols and in Fig. 3. In (1) the “internal converter phase-voltages” v f , j can be expressed by: N
vf ,j
¦ sr , j vr , j
(2)
r 1
where sr , j is the switching function of the j-th phase which modulates the vr , j voltage on the r-th dc-link capacitor of the considered phase, with r =1, 2, …, N. The expressions of sr , j is sr , j ^1, 0,1` . Equations (1) can be combined using the instantaneous space-vectors of the different quantities, defined as: vs
2 3 ¦ vs , j e 2S ( j 1) / 3 ; 3 j 1
vf
2 3 ¦ v f , j e 2S ( j 1) / 3 (3) 3 j 1
for line and converter voltage respectively, and in a similar way for the other quantities. The following complex equation can be assumed:
vs
Rs is Ls
d is vf dt
(4)
The mathematical model is completed adding to (4) the equations where the derivatives of the voltages vr , j in (2) are expressed as function of the currents
Fig. 1. PV cell V-I characteristics, at different temperatures.
790
i.e. v0, j is the total dc-link voltage of j-th phase and er , j
i0, j , i1, j ,..., i N 1 , j (see Fig. 3).
is the difference between r-th and (r-1)-th capacitor voltage of j-th phase.
st
1 Module H-Bridge
s11, j
s21, j
i0, j
s31, j
Besides, the i eq currents in (5) are combinations of currents defined as: eq °i0, j ° ® ° eq °ir , j ¯
V1, j
s41, j
i0, j 1 N
1 N
N 1
¦ ( N k ) ik , j
j 1, 2,3
k 1
N 1
§ · ¨ ( N r ) ¦ k ik , j r ¦ ( N k ) ik , j ¸ ¨ ¸ k 1 k r 1 © ¹ r
(7)
r 1,..., N 1
nd
2 Module H-Bridge
s12, j
s22, j
i0, j i1, j
s32, j
V. CONTROL TECHNIQUE FOR MPPT The proposed sensorless control is set up to deliver the maximum power to the grid in presence of variations of incident irradiation on the photovoltaic arrays. The control technique, synthesized by the scheme shown in fig. 4, carries out the following tasks: x to force the line currents to be sinusoidal and in phase with the fundamental harmonic of the positive linevoltage sequence; x to keep constant the total dc-link voltages vo i ; In some cases the first two tasks could not be satisfied simultaneously: this occurs when the link load is unbalanced and/or when the three-phase grid voltages are unsymmetrical. A priority is needed in the selection of the task to be satisfied. The input quantities of the block “predictive control and modulation” are: x and components in the stationary frame of the actual line voltages and currents symmetrical components; x actual dc-link voltages vo i ; x neutral voltages vni (reference value for these voltages is vn* 0 ); x the reference current i *s . The reference current i *s is obtained by processing the current outputs I si* of the MPPT modules, as described in details in the following. In the 1st step, the magnitudes I si* are transformed in a positive sequence with magnitude I sp* and a negative sequence with magnitude I sn* and phase n* by means of the equation I si* I sp* I sn* cos Mn* 2S / 3 i 1 with i=1,2,3. This equation is obtained by imposing that two currents absorb the same active power from the positive sequence of the line-grid voltages.
V2, j
s42, j i0, j ... iN 1, j m-Module H-Bridge
VN , j
Fig. 3. H-bridge topology for multilevel converter.
Starting from these well-known equations and introducing some suitable auxiliary variables, very compact and generalized expressions can be yield for a m-level converter structure. By properly handling these equations finally results: d ° vs Rs is Ls dt is vf ° N °d 1 eq N 1 § 1 1 · eq i0, j ¦ ¨ ¸i j 1, 2,3 ° v0, j ¦ ¨ Ckj ¸¹ k , j °° dt k 1 Ck , j k 1 © Ck 1, j (5) ® § · § · 1 1 1 1 °d e eq ¸ i0, ¸ ireq, j j ¨¨ ° dt r , j ¨¨ C C ¸ ¸ C C r 1, j ¹ r 1, j ¹ © r, j © r, j ° ° 1 eq 1 ir 1, j ireq1, j r 1,..., N 1 ° C C °¯ r, j r 1, j
The total number of equations of the set (5) is 3N 2 and number of capacitors is 3 m 1 / 2 .
In the 2nd step, the reference current i *s is computed as I sp* e jM * I sn* e j (M *Mn *) where * is the phase of the
In (5) the auxiliary voltages are:
i*s
v0, j v1, j v2, j ... vN , j ° °e1, j v1, j v2, j ° ®e2, j v2, j v3, j ° ° °eN 1, j vN 1, j vN , j ¯
fundamental positive sequence of the line-grid voltages v s , shifted by 2Ts to take into account the delay introduced by the digital predictive control. The MPP, corresponding to the maximum value of the irradiation, is determined by means of an hysteresis control which modulates the variation (slope of variation) of the current that the dc-link absorbs from the cells in function of the variation of the dc-link voltage.
j 1, 2,3
(6)
791
vsi
L
R
1, 2,3 o D , E
voi
Active V.S.R.
isi 1, 2,3 o D , E
vni
sai
is
Cai Cbi
vni
PREDICTIVE CONTROL AND MODULATION vs
PLL
e
jM
e e
i*s
jM *
REFERENCE CURRENT COMPUTATION
j 2ZTs
Is*i
voi
MPPT
Fig. 4. Control scheme.
track the MPP against variations of irradiation. The oscillation of the current will decrease when decreasing si , p , while the time response of the control - when the
The control technique can be described referring to the simple scheme that is shown in Fig. 5. idc
L
irradiation varies - will decrease when increases si , n .
R
+
This control technique can be realized by means of an hysteresis controller which imposes the value of si as
PV
function of sv and of the hysteresis band sv* , sv* . On
-
the other side the optimal value si , p , that guarantees at Fig. 5. Equivalent circuit with current generator.
steady-state a reduced oscillation of the current idc , determine a too long assessment time when the system start operating at no load condition; therefore during the start-up of the system, the current is increased with a slope si ,0 with si ,0 !! si , p .
The current generator represents in this scheme the converter. The PV arrays current idc is assumed to be: idc
i0 si t
(8)
where si is the current slope. Increasing the current idc, if si ! 0 , vdc decreases with a slope sv assumed to be sv 0 . The slope sv depends on si and on the working point of the cells. In particular, assuming si constant, sv increases strongly, in absolute value, when the cells are working in the stable part of the characteristic. This condition is taken in consideration to determine the MPP. If a positive slope si,p is considered for the current, for each value of si , p is possible to determine a specific
VI. SIMULATION RESULTS The control technique has been validated by means of simulations implemented on the system shown in Fig. 6. It was considered an H-bridge, 5-level, 7.5 kVA. Each Hbridge module is fed by an array of 8 series-connected photovoltaic modules. Each module has a maximum power of 175 W at irradiation of 1000 W/m2 and temperature of 25°C. The converter is connected to a three-phase grid 400 V-50 Hz. The control sampling frequency is fp= 10 kHz and determines an average switching frequency of the controlled components fm=1.25 kHz. The data assumed for the proposed MPPT are: sv* 10 V/s, si , p 0.1 A/s, si , n 2.5 A/s, si ,0 2.5 A/s.
value sv* , where sv* ! 0 , that identify the operation in the unstable part of the characteristic. Obviously imposing in (8) si si , p when sv sv* the current idc is higher than the maximum current idc , M which correspond to the optimal operation. To guarantee that the system works closed to the MPP it is sufficient to invert the slope of the current assuming in (8) si si , n with si , n 0 .
In Figs. 7,8,9 can be easily seen that the control tracks and holds the MPP of the system at constant irradiation equal to 800 W/m2. The equilibrium condition is reached within 10 seconds. At steady-state, as evidenced in fig. 9, the current oscillation is reduced, even if the voltage slope changes. Thanks to the hysteresis control the PV cells operate in proximity of the MPP crossing the unstable part of their characteristic. The current oscillation is, finally, reduced
After inverting the slope, the system will work again at the optimal condition. When sv ! sv* , it will be sufficient to invert again the current slope assuming si
si , p .
It is possible to contain the oscillation of the current idc just around the maximum point and contemporarily
792
under 1%. In figs. 7,8 is shown the control response against a variation of the irradiation; it was considered the variation from 800 W/m2 to 600 W/m2 in 10 seconds. Looking at the power absorbed by the cells it can be SA1,1 PV System
SC1,1
SB2,1
SB1,1
SA2,1
deduced that the control is able to effectively track the MPP and to stabilize immediately in the new steady-state condition.
PV System
SA3,1
SA4,1
SA1,2
PV System SB4,1
SB3,1
SA2,2
SC3,1
SB2,2
SB1,2
SA3,2
is1 R L vs1 is 2 vs 2 vs 3 is 3
O
SC2,2
PV System
SB3,2
SA4,2
SC4,1
SC1,2
PV System
PV System
SC2,1
SB4,2
SC3,2
SC4,2
O'
0
200 Wm-2/div
800 Wm-2 0
5
10
15
20
25
25 V/div
Dc-link voltage (V) Voltage slope (V/s)
1,5 kW/div
30
35
20 Vs-1/div
300 V 0
0
5
10
Time (s)
15
20
25
30
35
Time (s)
Fig. 7. Behaviour of the power output and power irradiation.
Line current is (A)and Voltage slope (V/s)
-2 -2 Irradiation (Wm ) ) Power radiaton (Wm Dc Power (kW)
Fig. 6. Photovoltaic system, grid-connected.
Fig. 8. Behaviour of the dc-link voltage and voltage slope.
5 A/div
20 Vs-1/div
0,5 s/div
Fig. 9. Line current and Voltage slope in steady-state condition.
793
Line currents is (A)
5 A/div
30 ms/div Time (s)
Fig. 10. Behaviour of the line currents around the maximum power output.
VII. CONCLUSIONS
REFERENCES
It has been proposed a sensorless control of H-bridge multilevel converter for MPPT in grid-connected photovoltaic systems. The control was set up to deliver the maximum power to the grid in presence of variations of incident irradiation on the photovoltaic arrays. The control technique has been validated by simulation implemented on a photovoltaic system with H-bridge 5levels converter. The simulation results have been reported and discussed in the paper, they essentially confirm that the control is able to effectively track the MPP and to stabilize immediately in the new steady-state condition. It has been verified the satisfactory control response against variation of the irradiation. The current oscillation is reduced, at steady-state, even if the voltage slope changes. The results of simulation demonstrate that the current oscillation is, finally, reduced under 1%.
Naik, R.L.; Udaya Kumar, R. Y.: A novel technique for control of cascaded multilevel inverter for photovoltaic power supplies. Power Electronics and Applications European Conference 2005, Sept. 2005, pp. 9. [2] Calais, M.; Vassilios, G. A.: Multilevel converters for single-phase grid connected photovoltaic systems-an overview. Proc. of ISIE '98 IEEE International Symposium Industrial Electronics, July 1998, vol.1 pp. 224–229. [3] Brando, G.; Dannier, A.; Del Pizzo, A: An Optimized Control Technique of Cascaded H-bridge Multilevel Active Front-ends. Proc. 12th International Power Electronics and Motion Control Conference , Aug. 2006, pp. 793-799. [4] Alonso, O.; Sanchis, P.; Gubia, E.; Marroyo, L.: Control Cascaded H-bridge multilevel converter for grid connected photovoltaic generators with independent maximum power point tracking of each solar array. Power Electronics Specialist Conference, 2003 (PESC '03), June 2003, vol. 2 pp. 731–735. [1]
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