1
Single Stage Photovoltaic Inverter
Luis Miguel Pereira Nascimento IST Lisbon
Portugal, Lisbon 1049-001 Email: lumipena@hotmail.com
1 – Introduction
The present society social economical context, leads to the need to find new alternative technologies to generate electrical power.
In the year 2000, the global electricity production accounted for 39% of 𝐶𝑂2 released into the atmosphere, an amount that increases with the growth of the world’s population which has access to electrical power [1].
The use of fossil fuel for electrical energy generation causes environmental problems. Besides this problem, fossil fuel is scarce.
Solar photovoltaic energy production may be part of the solution to generate clean electrical energy, since the sun is a non-depletable energy source.
Electrical energy production by means of an solar power plant using photovoltaic panels, connected in series and/or in parallel, require the use of switched converters between the grid and the panels. The converter is needed because the panels voltage and current are continuous while the grid’s voltage and current are sinusoidal.
This paper is in the area of power electronics and it’s main objective is to connect a system made of several photovoltaic panels with the electrical grid using a single stage photovoltaic inverter. The necessary steps are the following.
1) Study the converter;
2) Implement a controller for the output of the photovoltaic panels and the grid’s electrical current;
3) Design a controller for the DC voltage of the converter;
4) Use a non-linear load and a linear load as the converter’s output;
Validate the operation of the global system implemented using the simulation tool of Matlab Simulink;
2 – Photovoltaic Panel
A photovoltaic cellule is mainly made of silicon, which is added boron to create a region with excess gaps (type p) and phosphor to create an area with excess electrons (n- type).
Contiguous union of these materials results in a pn junction, which are the conditions necessary to establish that the photoelectric effect, in the presence of solar radiation.
However a photovoltaic cellule requires it’s n junction to be sufficiently thin and coated by a anti-reflexion surface. Additionally the n and p zones must have output terminals.
Photovoltaic cellule typical electrical power production is 1,5 𝑊𝑃. A bigger amount of power (50 to 100 𝑊𝑃) can be obtained by connecting several cellules in series and or in parallel, which is called a module [2].
The photovoltaic panel used is the combination in series and in parallel of 21 photovoltaic modules BP5170. Table 1 presents the characteristics of the photovoltaic module BP5170 provided by the manufacturer, which are the operating characteristics of the module with standard working temperature 𝑇𝑟= 298,16°𝐾 and standard incident radiation 𝐺𝑟= 1000𝑊/𝑚2.
Assuming the parameters for the association of modules are the same as the parameters for a single module, Table 1 is the following.
Table 1 – Features of BP 5170 photovoltaic module provided by the manufacturer.
Based on the mathematical model, where a photovoltaic cell can be described using the equivalent electrical circuit of Figure 1 and by using the Matlab Simulink, the desired photovoltaic panel is implemented, which represents an association of 21 photovoltaic modules BP5170.
2 Figure 1 – Electrical circuit equivalent of a photovoltaic cell fed a
load Z.
Analyzing Figure 1 the output current of an photovoltaic cellule is [2]:
(1) Where 𝐼𝐶𝐶 is the short circuit current of a photovoltaic cellule, which varies linearly witch the incident radiation and is written as:
𝐼𝑐𝑐= 𝐼𝑐𝑐𝑟 𝐺
𝐺𝑟 (2)
The maximum reverse current of the diode saturation I0 is influenced directly by the operating temperature of photovoltaic cell and is given by:
𝐼0= 𝐼0𝑟(𝑇𝑇𝑟)3𝑒
𝜀 𝑚 ,(1
𝑉𝑇𝑟−1 𝑉𝑇)
(3)
The ideality factor of the diode, is calculated based on parameters that are part of the characteristic of the panel, which is calculated by:
𝑚 = 𝑉𝑚𝑎𝑥𝑟 −𝑉𝑐𝑎𝑟
𝑉𝑇𝑟 ln(1−𝐼𝑚𝑎𝑥𝑟
𝐼𝑐𝑐𝑟 ) (4)
While the ideality factor equivalent 𝑚, is given by:
𝑚,=𝑁𝑚
𝑆𝑀 (5)
To calculate the theoretical value of the voltage of a photovoltaic cell V is necessary to know the value of the gap of silicon, 𝜀 = 1,12𝑒𝑉 and set the value of heat potential 𝑉𝑇, which is given by:
𝑉𝑇 =𝐾𝑇𝑞 (6) Using the mathematical model described and with the help of simulations, the block diagram [x] which represents the operation of an photovoltaic panel with 𝑃𝑚𝑎𝑥 = 3568,32 𝑊𝑝 is validated.
Figure 2 shows the plot of theoretical curves for the photovoltaic panel, with the parameters of Table 1, versus the plot of the curves generated by the block diagram designed using the Matlab Simulink.
Figure 2 – Curves of the photovoltaic panel, obtained based on the theoretical model and the block diagram representative of
operation of the panel to the terms of reference.
3 – Single Stage Photovoltaic Inverter For the interconnection of photovoltaic panels with the power grid, the converter chosen to perform this function is a single stage photovoltaic inverter, which can be seen in Figure 3.
Figure 3 – Schematic of Single Stage Photovoltaic Inverter.
In the analysis of converter operation, it is considered that the semiconductors 𝑆1, 𝑆2, 𝑆3 e 𝑆4 behave as ideal switches, when (on) they act as a short circuit, when (off) they act as an open circuit. The diodes 𝐷1 and 𝐷2 when their terminal voltage 𝑉𝐴𝐾 are positive they act as a short circuit, which results in 𝑉𝐴𝐾 = 0 e 𝐼𝐷> 0 and when the voltage 𝑉𝐴𝐾 is negative, then act as an open circuit.
Defining 𝛾𝑖 for each of the arms of the converter:
(7) Also defining 𝑉12 as the voltage between the two arms of the converter, 𝑉𝐷 as the voltage of the diodes anode and 𝑉𝑟 as the grid’s voltage, which are given by:
𝑉12= 𝑈𝑐(𝛾1− 𝛾2) (8) 𝑉𝐷 = 𝑈𝑐 𝛾1 𝛾2 (9) 𝑉𝑟= 2 𝑉𝑟𝑒𝑓 cos(𝜔𝑡) (10) Where 𝑈𝐶 is the voltage of the capacitor C, which should have a constant value of 800 V and 𝑉𝑟𝑒𝑓 is the RMS voltage of the electricity grid This value is equal to 230 V.
Table 2 shows how 𝐼𝐿𝑓, current output of photovoltaic panels, 𝐼𝐿𝑟, grid’s current and 𝑈𝐶, converter’s DC voltage, for different operating states [3].
3 Table 2 – Evolution 𝐼𝐿𝑟, 𝐼𝐿𝑓 and 𝑈𝐶 for the diferent states of
operation of the converter.
For the dimensioning of the converter reactive elements, coils and condensers, it is assumed that under the continuous operation, the voltages and currents that are present in almost continuous, with only a small variation due to the switching semiconductors.
The dimensioning of the coil𝐿𝑓 takes into account that the single stage photovoltaic inverter operation is similar in certain situations to that of a boost converter [4], where the coil 𝐿𝑓 is given by:
𝐿𝑓=Δ𝐼𝛿𝑒𝑇
𝐿𝑓𝑈𝐶(1 − 𝛿𝑒) (11) Where 𝛿𝑒 is the duty cycle of the boost converter, 𝑇 is the average value of the switching period and ∆𝐼𝐿𝑓 is the variation around the average value of the photovoltaic panels output current 𝐼𝐿𝑓.
𝛿𝑒 =13 𝑇 = 2 × 10−5 𝑠 ∆𝐼𝐿𝑓 = 0,15 𝐼𝐿𝑓𝑎𝑣 Which leads to 𝐿𝑓= 3,5 𝑚𝐻.
For the dimensioning of the inductor Lr it is necessary to resort to an analogy between the single stage photovoltaic inverter and a four-quadrant converter [4], in which the coil Lr is given by:
𝐿𝑟 =2𝑈𝐶(1−𝛿∆𝐼 𝑞)𝛿𝑞𝑇
𝐿𝑟 (12)
Where 𝛿𝑞 is the duty cycle of a four-quadrant converter and ∆𝐼𝐿𝑟 the variation around the average value of grids current 𝐼𝐿𝑟.
𝛿𝑞=1 2 ∆𝐼𝐿𝑟 = 0,15𝐼𝐿𝑟𝑎𝑣 Which leads to 𝐿𝑟= 12 𝑚𝐻.
The capacitor C is scaled based on a rectifier power factor unitátrio [4], in which the capacitor C is given by:
𝐶 =𝜔 Δ𝑈𝑃0
𝐶 𝑈𝐶 (13)
Where 𝑃0 is the power output of the converter and ∆𝑈𝐶
is the tremor of the voltage across the capacitor C around its mean value, which is equal to:
𝑃0= 𝑉𝑚𝑎𝑥𝑟 𝐼𝑚𝑎𝑥𝑟 ∆𝑈𝐶= 0,1𝑈𝐶
Calculating the value of the capacitor 𝐶, one obtains 𝐶 = 0,177 𝑚𝐹.
It should be noted that the maximum voltage to be borne by semiconductor 𝑆1, 𝑆2, 𝑆3 e 𝑆4 is de 840 𝑉 and maximum current that runs is 53,22 𝐴, while the maximum voltage to the terminals of the diode 𝐷1 e 𝐷2 is −840 𝑉 e and the maximum current that runs is 33,04 𝐴 [3].
4. Control Current Output Photovoltaic Panel and Electrical Network
In order to ensure that for any value of temperature and incident radiation, the power generated by the photovoltaic panels is maximum, and that the current injected into the grid is sinusoidal, it is necessary to control these currents.
For this reason 𝑒𝐼𝑟 is the grids current error, which results from comparing the grids current and its reference.
𝑒𝐼𝑟 = 1 𝑠𝑒 𝐶𝑜𝑚𝑝𝑎𝑟𝑎𝑑𝑜𝑟 1=1 𝑒 𝐶𝑜𝑚𝑝𝑎𝑟𝑎𝑑𝑜𝑟 2=1 −1 𝑠𝑒 𝐶𝑜𝑚𝑝𝑎𝑟𝑎𝑑𝑜𝑟 1=0 𝑒 𝐶𝑜𝑚𝑝𝑎𝑟𝑎𝑑𝑜𝑟 2=0 0 𝑠𝑒 𝐶𝑜𝑚𝑝𝑎𝑟𝑎𝑑𝑜 𝑟 1 ≠ 𝐶𝑜𝑚𝑝𝑎𝑟𝑎𝑑𝑜𝑟 2
(14)
Considering two hysteretic comparator, the comparator 1 with the limits [1, -1] and the comparator 2 with the limits [0.1, -0.1].
If 𝐼𝐿𝑟𝑒𝑓 − 𝐼𝐿𝑟 > 1 the comparators output is 1 ⇒ 𝐼𝐿𝑟
needs to ↑.
If 𝐼𝐿𝑟𝑒𝑓 − 𝐼𝐿𝑟 < −1 the comparators output is 0 ⇒ 𝐼𝐿𝑟 needs to ↓.
When the output of both comparators is the same, it is given priority to the grids current, making this current equal to the reference value.
When the comparators 1 and 2 output values are different, −1 < 𝐼𝐿𝑟𝑒𝑓 − 𝐼𝐿𝑟 < −0,1 ou 0,1 < 𝐼𝐿𝑟𝑒𝑓 − 𝐼𝐿𝑟 < 1, priority is given to the photovoltaic panels output current.
The photovoltaic panels output current 𝐼𝐿𝑓 value should be one that leads the panels to operate on maximum power operation point, therefore:
(15)
Using (15) and considering that the variations in current and voltage are small so that they can be considered almost linear, one makes the approximation:
(16)
In order to be able to relate the value of (16) with the value of 𝐼𝐿𝑓 the curves 𝑉𝑓(𝐼𝐿𝑓), 𝑃𝑖(𝐼𝐿𝑓) e 𝑑𝑃𝑖 𝐼𝐿𝑓 are plotted in Figure 4, for different operating conditions of the photovoltaic panels.
4 Figure 4 – Curvas Curves of photovoltaic panels, depending on
their current output.
By analizing Figure 4:
𝑑𝑃𝑖
𝑑𝐼𝐿𝑓 > 0 ⟹ 𝐼𝐿𝑓 < 𝐼𝑚𝑎𝑥 ⟹ 𝐼𝐿𝑓 ↑ 𝑑𝐼𝑑𝑃𝑖
𝐿𝑓 < 0 ⟹ 𝐼𝐿𝑓 > 𝐼𝑚𝑎𝑥 ⟹ 𝐼𝐿𝑓 ↓ (17)
𝑑𝑃𝑖
𝑑𝐼𝐿𝑓= 0 ⟹ 𝐼𝐿𝑓 ≈ 𝐼𝑚𝑎𝑥 ⟹ 𝐼𝐿𝑓 ≈
To analyze the first order derivate of panels power in order to the current output a hysteretic comparator (comparator 3 with limits [1, -1]) is added to the current controller. Also, 𝑒𝐼𝑓 is defined as the error in the output current of the panels.
𝑒𝐼𝑓=
1 𝑠𝑒 𝑑𝐼𝑑𝑃𝑖
𝐿𝑓 > 1 0 𝑠𝑒 𝑑𝐼𝑑𝑃𝑖
𝐿𝑓 < −1 (18) Using (14) and (18) it is possible to complete Table 3, where the command signal for the semiconductors 𝑆1, 𝑆2, 𝑆3 e 𝑆4 is shown [3].
Table 3 – Control signals of the semiconductor 𝑆1, 𝑆2, 𝑆3 and 𝑆4, and evolution of current 𝑰𝑳𝒓 e 𝑰𝑳𝒇.
From Table 3 it is possible to write logic functions to implement the photovoltaic panels currents controller.
Whereas the output of the comparator 1 is equal to A, the comparator 2 equals B and the comparator 3 equal to C, we obtain:
𝑔𝑆1= 𝐵 + 𝐴𝐶 𝑔𝑆2= 𝑔 𝑆1 𝑔𝑆3= 𝐵 + 𝐴𝐶 𝑔𝑆4= 𝑔 𝑆3
5. Control of Supply Chain at Tier 𝑫𝑪 Converter To ensure that the power injected into the power grid is equal to the power generated by photovoltaic panels (neglecting losses in the converter), control is carried out on the continuous side of the converter.
This control requires maintaining the voltage of the capacitor C approximately constant and varying the power to injected into the grid at the expense of the grids current peak value. By analysis of Figure 3:
𝐼𝐶 = 𝐼𝑆1 𝑠𝑒 𝛾1 = 1
𝐼𝑆3 𝑠𝑒 𝛾2= 1 ⇒ 𝐼𝐶 = 𝐼𝑆1𝛾1+ 𝐼𝑆3𝛾2 (19) Where 𝐼𝑆1 is the semiconductor 𝑆1 current and 𝐼𝑆3 is the semiconductor 𝑆3 current, which are equal to:
(20) This leads to:
(21) After obtaining 𝐼𝐶 as a function of 𝐼𝐿𝑟 e 𝐼𝐿𝑓, the voltage control block diagram [3] is obtained, which is shown in Figure 5.
Figure 5 – Detailed block diagram of the control voltage of the converter.
Considering the closed loop transfer function with 𝐼𝐿𝑓 = 0 and a proportional controller 𝐶 𝑠 = 𝐾𝑃:
(22)
Based on (22) and considering 𝑈𝐶𝑟𝑒𝑓=𝑈𝛾𝐶
𝑘 the static error is guaranteed to be zero, with a proportional controller which has a gain equal to:
5 In the presence of disturbances, which means that, 𝐼𝐿𝑓 ≠ 0 it is necessary to add a integral gain 𝐶 𝑠 = 𝐾𝑝+
𝐾𝑖
𝑠 [5] to the previous controller. Analysing the effect of the disturbances:
(23)
Using (23) it can be ensured that the effect of disturbances is minimized with a proportional integral controller.
The integral gain is calculated based on the ITAE criterion-Integral of Time Weighted Absolute Error, and is equal to:
However the models shown up until now are valid under small disturbances working conditions. For large disturbances large current surges appear, which can compromise the integrity of the switched converter. To avoid this, a PI compensator with limiting anti-windup is used [4].
𝐾𝑖= −555.036 𝐾𝑝= −2,008 𝐾𝑤 = −0,5
6. Simulation System Harnessing Solar Energy
After the analysis made in the previous chapters the entire photovoltaic system is simulated using an
appropriate block diagram [x] and using MATLAB 7.5.0 together with SIMULINK.
In order to test the entire implemented system, a non linear load, bridge diode rectifier, is used in parallel with the electrical grid and a linear load which consists of a inductor in series with a resistor. The load is shown in Figure 6.
Figure 6 – Non linear load, the rectifier diodes and linear loads, resistance in series with coil placed in parallel with the
mains, the inverter.
Simulations are conducted on the working conditions of photovoltaic cells in Figure 7 and are shown in Figures 8, 9 and 10.
Figure 7 – Working conditions of photovoltaic cells.
Figure 8 – Voltage and current to the terminals of photovoltaic panels.
Figure 9 – Power generated by photovoltaic panels and power output of the converter.
Figure 10 – Current to inject into the electrical network and the voltage of the capacitor C.
6 7. Conclusion
This work arises from the need to develop new technologies for the production of electricity, due to the current society’s socio economic context. Scarcity of resources, high costs of electricity generation and environmental problems.
The use of a single stage photovoltaic inverter to interconnect the electrical grid with the photovoltaic panels ensures high efficiency. In this case the efficiency is about 92%
By controlling the currents of the converter it can be guaranteed that for any working conditions the photovoltaic cells generate maximum power.
To ensure that the electrical power injected in the grid is equal to the power generated by the panels (neglecting losses) the converters continuous voltage is controlled.
This way the overall efficiency of converting solar radiation into electrical power can be improved and electricity can be produced form a "clean" easy, and inexhaustible source.
References
[1] EDP - Energias de Portugal
http://www.edp.pt/EDPI/Internet/PT/Group/AboutEDP/Bus inessUnits/ElectricityDistribution/DistElectPT.htm
[2] Rui M. G. Castro, Introdução à Energia Fotovoltaica, Instituto Superior Técnico, 2009
[3] Dissertação de Mestrado, Inversor Fotovoltaico de Estágio Único, Luis Miguel Pereira Nascimento, 2009
[4] J. Fernando Alves da Silva, Projecto de Conversores Comutados (ERC_06_07)
[5] Ricardo Santos e Dinis Honrado - Conversor Elevador Quadrático para Aproveitamento de Energia Renovável, Trabalho Final de Curso 149/2005
