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International Review on Modelling and Simulations (I.RE.MO.S.), Vol. 9, N. 2 ISSN 1974-9821 April 2016
The Partial Linearization of Power-Voltage Curve for Grid-Connected Photovoltaic System Faicel EL Aamri1, Hattab Maker2, Azeddine Mouhsen1, Mohammed Harmouchi1 Abstract – This paper aims to present a new method to track the maximum power point for single-stage single-phase grid connected PV inverter, which is based on the partial linearization of the power-voltage PV panel curve coupled to the MPPT algorithm. The proposed approach generates the reference current in order to regulate the power fed into the grid, and it needn’tany accuracy of the PV panel model which may change over time. The current controller approach is based on the stability of the internal dynamic due to the nonlinearity of the PV system and it uses the Lyapunov function that gives globally asymptotically stable trajectories of the closed-loop system. The effectiveness of the control algorithm has been verified by simulating it in Matlab/Simulink software. The proposed control is compared with the conventional approach using PI controller. Copyright © 2016 Praise Worthy Prize S.r.l. -All rights reserved.
Keywords: Single-Phase Grid-Connected PV Inverter, MPPT, Partial Linearization, Lyapunov’s Stability
Nomenclature ( )
m MPPT
PV – ( ) µ
( )
( )* () () ()
Input DC link capacitor in µF Given function Inverter switching frequency in Hz Grid current in A Inductor current in A Current reference in A Solar panel current in A Short-circuit Current Integral gain Proportional gain Output filter inductor in mH Dimensionless value Maximum power point tracking Average power of the grid in W Power reference in W Photovoltaic panel Full-bridge inverter switch Control input, dimensionless Steady state control input, dimensionless Nonlinear control input, dimensionless Votlage at maximum power point in V Open-circuit voltage in V Solar panel voltage in V Lyapunov function Grid voltage in V Data point on a curve Variable state of panel output voltage Variable state of panel output current Switch statue Dimensionless value
Denotes dynamic value Denotes reference value Denote actual value Denote predicted value Denote former value
I.
Introduction
In the last decade, several researchers have investigated the integration of renewable energy systems on the grid. In this field, photovoltaic generation system has been considered as one of the most rapid growing new source of energy and it has become widely used due to the development of power electronics and the reduction in PV panel price. The Photovoltaic system is based on absorbing and converting a sunlight into electricity, and it falls into two main categories: off-grid and grid-connected. Generally, there are two typical configurations to connect the PV system to the grid, by using a single stage or two-stage converter. In the two stages converter configuration, the first stage converter is generally a Boost converter [1], [2]. Its aim is to track the maximum power and to increase the PV panel voltage. The second stage converter, a DC/AC converter, ensures the conversion of the DC power obtained from the PV module to AC power to be injected into the grid. Although, the efficiency of the whole converter decrease and it contains many energy dissipative components [3]-[9]. However, the single stage PV inverter is able to ensure the high efficiency and reliability [10], furthermore, the inverter in the single stage is used for both functions, it injects the maximum
Copyright © 2016 Praise Worthy Prize S.r.l. - All rights reserved
DOI: 10.15866/iremos.v9i2.8134
75
Faicel El Aamri, Hattab Maker, AzeddineMouhsen, Mohammed Harmouchi
available power and transfer it from the PV array to the electric grid simultaneously. Nevertheless, the design of an appropriate control strategy of this system is the most interesting and valuable factor of study since it is considered as a strongly nonlinear system [11]-[12]. A new control scheme design will be introduced in this paper for a single-stage single-phase Grid-Connected PV inverter that is still more widely used for residential application [1].The main components of the topology, shown in Fig. 1, are a PV string which is the combination of series and parallel connections of PV module [2], a capacitor (C) as an energy stored element, full bridge inverter, an inductor (L) as a filter, and the grid.
There are various linear control techniques such as the hysteresis controller [15], [16], the predictive controller [17], the PI and the PR controller [18] and so on. The hysteresis controller has a low Total harmonic distortion output and a fast response time, but it has an irregular switching frequency. This issue is overcome by adding an adaptive hysteresis band. Another concept is the predictive control, which has been receiving recently a particular attention due to the remarkable improvement in the industry. Its implementation is simple, but it is a very fast accurate controller only at a nominal operating conditions. All the previously described procedures control the variation of the duty cycle of the PWM signal of the inverter without taking into account the internal stability of the whole system, which provides a good performance only in the linear vicinity of the operating point. However, the system may become unstable if affected by an external disturbance or even a sudden irradiation change, as the single phase grid-connected PV inverter is less stable than three phase Grid connected PV inverter. To overcome the limitation of the linear controller, a Lyapunov-Based control scheme is presented as an innerloop control for the current field into the grid. The obtained Lyapunov candidate function was useful for designing the globally stabilizing controls in order to eliminate the perturbations allocated to the system [19]. The Lyapunov function is formed by the error energy stored in the inductor and the capacitor due to the fact that the system state converges to the equilibrium point if the total stored energy is continuously dissipated. The rest of this paper is organized as follows. In the next section, the change of output power of PV string is presented and then the MPPT algorithm is elaborated. In Section III, the full mathematical development for linearization of Power-Voltage PV string curve is designed. In section IV, the current controller is designed together with the conditions that guarantee the stability of the overall system. Section V is dedicated to the presentation of simulation results obtained with the implemented controller based on Matlab/Simulink and discussion. Finally, Section VI concludes this paper.
Fig. 1. Typical configuration of single stage grid -connected PV system
As the grid voltage is out of the control, the power transferred to the grid is controlled by the output current of the inverter. The control structure used for the system is based on multi-loop in cascade: respectively an outerloop and an inner-loop, the outer-loop consists into settling the DC-Link voltage for achieving the maximum power point, while the inner-loop is intended to regulate the inverter output current. The aim of this paper is to design a controller for the outer-loop, based on the linearization step-by-step of the nonlinear characteristic of (Power-Voltage) PV string curve. It is a new idea, used for the first time used. In the literature, there have been several control strategies processed, the most widely used, namely the voltage controller is based on PI controller [13], [14], it depends on the measured of the PV voltage as an input of the PI controller and it is compared to the reference voltage given by the (P&O) MPPT algorithm, the output signal after the process is multiplied through the phase of the grid and it will be used to control the output current field into the grid. Nevertheless, the use of PI controller suffers from many drawbacks, as the necessity of tuning the gain values in each changes in atmospheric conditions (temperature or irradiation) [9], as they depend on an accurate irradiation and they require knowledge of the PV panel model. Consequently, an adaptive algorithm that might work for all irradiation levels and not needing any accurate of the PV panel model is a paramount challenge, achieved through the partial linearization stepby-step approach. As for the inner closed loop, called PWM control, is intended to regulate the PV inverter output by controlling the current that is injected into the grid.
II.
Output Power of PV String
The PV string curve is shown in Fig. 2 for different radiation levels. In this figure it can be observed that the output power is maximum only if the voltage equals to , which correspondsto the maximum power point. As the irradiance changes, the P-V curve will change accordingly. The maximum power pointwillalso shift as the irradiance changes. Therefore the system needs to be controlled by an MPPT system to extract the maximum power from it [25], [26]. Many MPP tracking methods have been developed in the literature [20]-[24]:among them the perturbation and observation (P&O), the incremental conductance and the hill-climbing.
Copyright © 2016 Praise Worthy Prize S.r.l. - All rights reserved
International Review on Modelling and Simulations, Vol. 9, N. 2
76
Faicel El Aamri, Hattab Maker, AzeddineMouhsen, Mohammed Harmouchi
given point. It should be considered a function ( ) of a single variable , and is a given point. The Taylor Series expansion of ( )around the point is given by: ( )= ( )+ + 3
+ Fig. 2. Influence of the solar irradiation
In this paper a P&O algorithm is chosen because of the simplicity of its control structure as shown in Fig. 3. It requires the measurement of the PV voltage and the PV current as an input values: the ratio of the derivative of the power on the derivative of the voltage determines the direction of the operating point, if the ratio is positive, then the reference voltage is increased by some constant magnitude. If it is not, then the reference voltage is decreased with every MPPT cycle. The choice of the constant magnitude value is very important: in fact a poor choice of the constant value causes the system to oscillate, thereby reducing the power generation.
( −
1 2
( −
1 6
)+
) +
( −
(1)
) +⋯
This can be written as: ( )= ( )+ +
( −
)+
(2)
ℎ
For sufficiently close to , these higher order terms will be very close to zero, and so they can be dropped to obtain the approximation: ( )= ( )+
( −
)
(3)
The main aim of this method is to estimate an area of points belonging to a nonlinear curve in an approximately linear modality. Applying these definitions to the output power of the PV panel leads to: (
)= (
)+
(
−
)
(4)
where ( ) is the actual average power measured from the PV panel at the operating point, and is the actual measurement of DC-Link capacitor voltage. ( − ) represents the variable step size adjusted in the Power-Voltage curve of the PV panel where: =
(
)− ( −
)
(5)
where: is the predictive voltage given by the P&O MPPT algorithm. The linearization control objective is to regulate the reference predictive output power of the inverter using adaptive and variable step as shown in Fig. 4. In the tracking process, the Power-voltage curve of the PV panel is almost linear if the reference point is far from the MPP, thus the presented controller adjusts progressively the reference power using a too large step size to exhibit a fast dynamic response as shown in Fig. 4 and will decrease the step size to be small in order to maintain a stable output power as the operating point approaches near the MPP.
Fig. 3. Flow chart of the P&O Method
In the next section, a new method, determined by theconsideration that the inverter can be only controlled byits reference output current, will be presented.
III. Partial Linearization Using Taylor Series The Taylor Series expansion of a function is widely used to find the linear approximation to a function at a
Copyright © 2016 Praise Worthy Prize S.r.l. - All rights reserved
International Review on Modelling and Simulations, Vol. 9, N. 2
77
Faicel El Aamri, Hattab Maker, AzeddineMouhsen, Mohammed Harmouchi
Finally, the reference current that will be injected into the grid can be written as: =
Furthermore, when the reference point is at the maximum power point, the slope of the curve is zero, and therefore the step size is approximately near to zero, then the MPP is reached and the voltage and the current will be as small as possible in order to minimize the power oscillation extracted from the PV panel. The power fed to the grid is equal to: =
2
(1 −
2
)
IV.
1 2
(1 −
2
)=
(6)
a unipolar PWM technique. The switch status can be represented by the input denoted , defined as follows: =
2
Stability Study of the System
Considering the single phase single-stage grid connected photovoltaic system as shown in Fig. 1, C is the DC-Link Capacitor which is parallel to the PV string in order to reduce the ripple caused by the PWM switching. Moreover it is a storage element, L that indicates as a filter interfacing the inverter and the grid with zero resistance value, utility grid , and four power switches (S1-S4). The inverter switching frequency is denoted by and the switching period is = . The inverter is driven by
where is the amplitude of the injected current and is the amplitude of the grid voltage; it should be remembered that is constant and it is out of the control. The average power ( ) called also active power fed into the grid can be written as: =
(7)
0→
+1 → , : on, , ∶ off −1 → , : on, , ∶ off and : , , and ∶ on
=
+
(8)
(14) =
The reference output power generated by the controller is: (
)=
·
(13)
∈ {−1, 0, 1} the state equation
when the switch input could be written as:
It should be noted that the single-phase inverter has no losses, and thus the power at its input equals to the power delivered at its output, which means that: =
(12)
where can be obtained using a phase locked loop (PLL). After having determined the outer-loop, there remains only the inner-loop, but it must be mentioned that to control the current it is necessary to consider the stability of the whole system.
Fig. 4. Operating point movement for the power reference change
=
2
−
Combining the two equations below, the state-space average model of the whole system can be obtained by the dynamic equation described as follows:
(9)
where and are the average values of the voltage and the current PV panel. Therefore:
=− · ( )+ (15)
· 2
=
=
(10)
·
where ( ) is the average value of the switch status in a period . The capacitor voltage and the inductor current ] =[ ] , the define the state vector = [ state equation becomes:
As mentioned previously, since the grid voltage is out of the control, the only way to regulate the output power of the inverter is by controlling the current that is fed into the grid; at this stage, the magnitude of the reference current will be . Therefore, the above equation can be written as: =
2 (
)
· ( )−
̇ =− · ( )+ ̇ = · ( )−
(11)
(16)
where is the PV panel current, depending on the PV panel voltage, thereby causing a nonlinearity of the
Copyright © 2016 Praise Worthy Prize S.r.l. - All rights reserved
International Review on Modelling and Simulations, Vol. 9, N. 2
78
Faicel El Aamri, Hattab Maker, AzeddineMouhsen, Mohammed Harmouchi
system, and as a result, the system can be expressed in the general form as: ̇ = ( , )+ ( )· ( )
During energy transfer from the PV string to the grid, there is a part of energy exchanged between the capacitor and the inductor causing the unbalance of the system, consequently in order to make the system converging to a steady state, all the total energy stored in these dual elements must be continuously dissipated. The Lyapunov function can be constructed by considering the error energy stored in the DC-Link capacitor and the inductor [19]. In this case:
(17)
where ( , ) and ( ) are given as follows: ( , )=
( ) −
, ( )=
−
(18)
For analyzing an appropriate controller, the chosen control input has the following structure, made up of two terms, steady state and perturbed terms, hence: ( )=
+
( )=
If the system operates in the steady state, which means the reference point, the state-space average model could be rewritten as follows:
̇∗ +
(20)
̇( ) = +
∗ ∗
+ ) + ) ( ∗) +
= −( ∗ + ) · ( + ( ∗) +
̇∗ + ̇
=(
∗
+
)·(
(26)
∗
− ( + (
∗
∗
)+ )
+ +
+
(27)
∗
∗ )+
+
(28)
Therefore, to check the previous condition ( ̇ ( ) < 0), both the terms of the previous equation should be negative. Thus, the second term is replaced by its expression, to get:
(22)
̇ ( ) = (− +(
−
∗
)
∗
∗)
+ ( )−
+
(29)
( ∗)
Therefore, let consider as the actual voltage of the PV panel and ∗ as the future estimate voltage; it ( )− involves that if ( − ∗ ) < 0, therefore ( ∗ ) > 0. In the second state if ( − ∗ ) > 0, it will ( ) − ( ∗ ) < 0. As a result, it involves that be
+ )+ (23) + )−
( ) − ( ∗ ) < 0. − ∗) Thereby the former term becomes negative, that is why the controller signal μ will be written in the dynamic error of the system in the following form: (
Substituting Eq. (22) in Eq. (25), it can be obtained: ̇ =− ∗ − ( ∗+ ) ̇ = ∗ + ( ∗+ )
∗
−
̇ ( ) = (−
where , , represent the error between the actual value and its reference. Then the dynamic system can be rewritten as: ̇∗ + ̇
̇
Finally, by arranging the terms it can be obtained:
To add the second element to control signal, which eliminates the disturbance allocated to the system and renders the error dynamics stable, some state variables should be defined, as: =( =( ( )=
(25)
Substituting Eq. (24) in Eq. (26), it can be obtained:
(21)
∗
1 2
̇ +
̇( ) =
Accordingly, the control signal is equal to: =
+
The chosen Lyapunov function is positive for all ≠ 0, to ensure the global stability of the dynamic system, the derivative of the Eq. (25) will have to ascertain for all ≠ 0 that ̇ ( ) < 0, hence:
(19)
̇∗ = − ∗ · + ( ) ∗ ∗ ̇ = · −
1 2
(24)
∗
= (−
Therefore, to generate the global control signal for output current control, it is required to study the overall system stability using the Lyapunov function, since in this paper, it is interested to use the unipolar PWM technique, accordingly it exists three of switching states; direct connection between the DC-Link capacitor and the inductance, Crusader connection, and the third state is a state of separation [27].
∗)
+
(30)
where is a negative constant. Finally, the Lyapunov function verifies: ( ) · ̇( ) < 0
(31)
which means: ̇ ( ) = (−
Copyright © 2016 Praise Worthy Prize S.r.l. - All rights reserved
∗
+
∗)
+
