A fractional-order model of a photovoltaic (PV) system is proposed in this paper. The system identification method is used to construct an accurate dynamical model for a PV system, a real PV module and boost converter are used to collect experimentally input-output data for the identification process. A black box modeling is considered for system identification to obtain a transfer function, without needing to make mathematical analysis. the input-output data which is measured is fitted by the least-square curve fitting method, and the parameters of the curve fitting are optimized by the Levenberg-Marquardt algorithm. the proposed fractional-order model is employed with the MATLAB and Simulink software to obtain a fractional order PID controller, which is used to extract maximum power from the PV system.
Research Article
System identification of photovoltaic system based on fractional-order model
https://doi.org/10.21203/rs.3.rs-1865340/v1
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fractional-order model
System identification
PV system
least-square curve fitting
and PV system.
The generation of electrical energy has been dependent on fossil fuels for many years, such as coal, natural gas, and oil 1, which cause environmental issues, and fossil fuel reserves decrease. These problems have made electricity generation go towards renewable energy sources such as gravitational energy, geothermal energy, wind energy, tides, and solar energy 2,3. renewable energy sources have been taking a significant position in the replacement or complement of conventional power generation, but solar energy source among all renewable energy sources is used widely because of advantages such as having no noise due to the absence of mechanical parts, low maintenance cost, long working life, the great amount of solar irradiation that covers the most of the earth’s surface 4. The electric energy is generated from solar energy by a small unit called a photovoltaic (PV) cell, which produces a little amount of energy, several PV cells are connected in parallel and series to form a PV module to have more power, and PV panel is a group of PV modules which are joined together in parallel or series form. In a PV system, PV panels are connected in series and parallel to form PV arrays for getting high current and high voltage, to increase the output power of PV arrays 5.
The productivity of PV modules to convert solar energy to electric energy is dependent on the solar irradiance level, the surrounding temperature, and others 6,7, so the variable environmental conditions cause variation in the voltage and the current of the PV module. therefore, the output power which is generated by the PV module changes also. Hence In a PV module, voltage and current have a nonlinear relation 8, and in the electrical characteristic of the PV module, there is only one operating point, at which PV output power is maximum, this operating point is called Maximum Power Point (MPP), which makes PV modules are working efficiently. As this point fluctuates with variable environmental conditions, it should design a maximum power point tracking (MPPT)algorithm 9, MPPT algorithms increase PV system efficiency by moving the PV voltage or current to make PV operate at the maximum power point (MPP).
Another problem that has a great effect on the PV efficiency is Partial shading (PS), PS decreases the amount of sunlight that is allocated to the PV array resulting in lower output power from the PV array, under PS multiple local maximum power points (LMPP) emerge on PV curve 10, but there is one MPP of LMPP is called global maximum power point.
Therefore, to overcome the above issues, many MPPT algorithms are proposed to get maximum power from PV array under variable environmental conditions, in11 the MPPT algorithms are classified into conventional techniques (CTs), soft computing techniques (SCTs), linear and nonlinear control techniques, all these algorithms are used in a simulation environment before real systems, PV model has many important roles in design optimization, fault diagnosis 12, and prediction performance in the future of any PV system, therefore it should develop an accurate PV model to design MPPT.
The mathematical models of real systems can be divided into static and dynamic models 13, the dynamical model for the PV system is preferred due to the loads always are not constant, and the PWM which is used for switching converters or inverters, the static model for PV module is introduced in many pieces of literature such as in 14–18, a single diode model is considered as a static PV model and is the most one which is used, the static model can be derived from parameter identification, parameter identification method tray to make approximation and solve analytically the equation of the PV as a polynomial of the first order [15], this means the static model is simplified and has low accuracy.
Alternatively, a second-order dynamic model for PV module is proposed in 19, the technique which is proposed increases the model accuracy, but it doesn’t consider the power electronics system, another dynamic model is introduced for a PV system based on system identification and considered buck converter, MPPT, and load in 20, this technique used experimental input-output data and nonlinear Hammerstein-Wiener model to extract dynamic model. This paper proposes a dynamic fractional order model for a PV system, boost converter, MPPT, and load.
The PV system in this paper is off-grid, and consists of a 60-watt PV module, boost converter, MPPT controller, and load. the circuit diagram of the PV system is shown in Fig. 1
The PV current measurement is used as input data for the system identification and output power of the boost converter as output data. The identification method to minimize the error of model output is based on Nonlinear least-squares estimation, Levenberg-Marquardt algorithm is used to solve nonlinear least-squares problems21, and for determining the parameters of the model.
The proposed PV model is based on the fractional order model, fractional calculus is a generalization of ordinary differentiation and integration to arbitrary (non-integer) order, it is supposed that fractional order models will generally provide a more accurate description of the system dynamics than those based on conventional differential equations because it is given more degree of freedom for determining parameters of the model, the general definition for an integral differential operator is22 :
Where \(\alpha \in \mathbb{ }\mathbb{R}\) and \(\alpha\) can equal a complex number, there are many multiple definitions of the fractional operator, we consider the definition of Grünwald-Letnikov, which is used throughout this paper to drive numerical solutions for fractional-order differential equations.
Fractional order systems modeling and controllers are implemented through the FOMCON toolbox for MATLAB/Simulink, the toolbox Fractional Order Modeling and Control “FOMCON” is developed in 23, this toolbox will be used for system identification to extract the fractional order model of the PV system.
A single diode model can describe the static behavior of a PV module and is proposed in many kinds of literature such as in24,25, the electric circuit for the single diode model is shown in Fig. 2.
From Fig. 2, the mathematical equation which describes the static PV model can be driven as follow:
Where \({I}_{ph}\) is the photocurrent, \({I}_{sat}\) is the reverse saturation current of the diode, q is the electron charge, A is the diode ideality factor, K is the Boltzmann constant, T is the PV module temperature, \({R}_{s}\) is the series resistor, \({R}_{sh}\) is the shunt resistor, and \({N}_{s}\) is the number of connected PV cells in series, the photocurrent (\({I}_{ph}\)) which is produced by the PV module is influenced by the solar irradiance falling on the PV module plane and its influenced by the ambient temperature of the PV module as shown below26,27:
Where \({I}_{phs}\) is the photocurrent at standard conditions (STC), \({I}_{rrs}\) is the irradiation at STC and is 1000 \(\frac{W}{{m}^{2}}\),\({ K}_{i}\) is the temperature coefficient, \(\varDelta T=T-{T}_{s}\) is the difference between the PV temperature (\(T\)) and STD temperature (\({T}_{s}=25℃\)), from equations (4) and (5) the output current (\({I}_{PV}\) ) depends on the amount of sunlight and the PV temperature, the characteristics curves(P-V), and(I-V) for different irradiance and temperature are shown in Fig. 2,
PV System Modeling
The modeling process for dynamical systems is used to drive mathematical representation for the system, it depends on two techniques: mathematical representation is derived from physics or measured input-output data, in general, the modeling process can be organized into three techniques: white box, grey box, and black box modeling28, in white box modeling the physics of the system, are completely known; and the mathematical model can be constructed entirely from prior knowledge and physical insight29, in grey box modeling, some of the physical parameters are known, the other parameters are determined from measured input-output data, but the black box modeling is used when the physical properties of the model are unknown, and the model is constructed from measured input-output data, but the chosen model structure belongs to the candidate models that are known to have good flexibility and have been successful in the past 30. The back box modeling is chosen for the system identification methodology in this paper, in the back box modeling method, many models are compared with the real data, and the best is determined by an optimization criterion. Four basic stages characterize the system identification technique (I) The measured input-output data, (II) the proposed model, (III) a criterion of fit and parameter estimation, and (IV) Validate the model which is obtained.
4.1. The measured input-output data. The data used in system identification is obtained from the PV system, shown in figure .1, where the input data is the PV current and output data is the PV system output power.
4.2. The proposed model structure. a fractional order model is considered in this paper, A generalized fractional-order dynamic system can be represented by the next Eq. 31:
The dynamic system in (6) can be written in the next form:
$$\sum _{k=0}^{n}{a}_{k}{\mathcal{D}}^{{\alpha }_{k}}\mathcal{Y}\left(t\right)= \sum _{k=0}^{m}{b}_{k}{\mathcal{D}}^{{\beta }_{k}}\mathfrak{u}\left(t\right) \left(7\right)$$
Where( \({a}_{k}\),\({ b}_{k})\in \mathcal{ }\mathcal{R}\), and (\({\alpha }_{k},{\beta }_{k}\)) \(\in {\mathcal{R}}_{+}^{2}\). in (6) if all the derivation orders are integer multiples of a base order q such that \({\alpha }_{k}\), \({\beta }_{k}=qk\), \(q\in {\mathcal{R}}_{ }^{+}\), the system is said to be of commensurate order. If \(q=\frac{1}{\mathcal{g}}\), \(\mathcal{g}\in {\mathbb{Z}}_{+}\), the system is said to be of rational order.
The Laplace transform of (7) can be written in the form:
$$\sum _{k=0}^{n}{a}_{k}{s}^{{\alpha }_{k}}Y\left(s\right)= \sum _{k=0}^{m}{b}_{k}{s}^{{\beta }_{k}}U\left(s\right), \left(8\right)$$
Then the fractional order transfer function can be written in the form:
$$G\left(s\right)=\frac{Y\left(s\right)}{U\left(s\right)}=\sum _{k=0}^{m}{b}_{k}{s}^{{\beta }_{k}}/\sum _{k=0}^{n}{a}_{k}{s}^{{\alpha }_{k}}, \left(9\right)$$
By considering the system is commensurate order q, We can write \(\text{ɦ}={s}^{q}\), then the continuous time rational transfer function is:
$$H\left(\lambda \right)= \frac{\sum _{k=0}^{m}{b}_{k}{ɦ}^{k}}{\sum _{k=0}^{n}{a}_{k}{ɦ}^{k}} . \left(10\right)$$
In fractional calculus the differential operator has many definitions, The three most frequently used definitions for the general fractional differ integral are the Grunwald-Letnikov (GL) definition, the Riemann-Liouville (RL), and the Caputo definitions32, Grünwald-Letnikov definition is considered in this work for the drive numerical solutions to fractional-order differential equations, which is given by the next equation:
$${}_{\text{a}}{}^{\text{ }}{\text{D}}_{\text{t}}^{ \alpha }=\underset{h\to 0}{\text{lim}}\frac{1}{{h}^{\alpha }} \sum _{j}^{\left[\frac{t-a}{h}\right]}{\left(-1\right)}^{k}\left(\begin{array}{c}\alpha \\ k\end{array}\right)f\left(t-kh\right), \left(11\right)$$
Where \(h\) is the step size, and the binomial coefficient can be expressed with Gamma function \(\varGamma \left(.\right)\)in the following expression:
$$\left(\begin{array}{c}\alpha \\ k\end{array}\right)=\frac{\alpha !}{k!\left(\propto -k\right)!}=\frac{\varGamma \left(\alpha +1\right)}{\varGamma \left(k+1\right)\varGamma \left(\alpha -k+1\right)} ,$$
12
Laplace transform of the Grünwald-Letnikov fractional operator is:
$$\mathcal{L}\left[{\mathcal{D}}^{\alpha }f\left(t\right)\right]={S}^{\alpha }F\left(S\right)$$
13
.
Oustaloup’s recursive filter is an approximation for fractional differentiator and integrator, in a specified frequency range (\({\omega }_{b},{\omega }_{h}\)) and is capable of producing a very good fitting to the fractional-order elements 33,34, the algorithm for Oustaloup’s recursive filter is shown in the following form:
$${S}^{\alpha }\approx \prod _{K=1}^{N}\frac{S+{\omega }_{k}^{\text{'}}}{S+{\omega }_{k}} \left(14\right)$$
Where
\({\omega }_{k}^{\text{'}}= {\omega }_{b}* {\omega }_{u}^{\frac{2k-1-\alpha }{N}}\) , \({\omega }_{k}= {\omega }_{b}* {\omega }_{u}^{\frac{2k-1+\alpha }{N}}\), \(K={\omega }_{h}^{\alpha }\), \({\omega }_{u}= \sqrt{\frac{{\omega }_{h}}{{\omega }_{b}}}\)
And N is the approximation order in the frequency range (\({\omega }_{b},{\omega }_{h}\))
4.3. Criterion of fitting and parameters estimation
The system identification method depends on the error minimization between model output and system output, the least-square method is used in this work, and the problem is expressed as follows:
$$\mathcal{F}\left(x\right)=\sum _{p=1}^{n}{℮}_{p}$$
15
Where \({℮}_{p}={y}_{p}-{\widehat{y}}_{p}\) and \({\widehat{y}}_{p}=\widehat{\mathcal{H}}\left({\mu }_{p},\theta \right)\) is the output of the estimated model\(\widehat{\mathcal{H}}\) for the input signal (\({\mu }_{p}\)), \(p\) is the iteration, and \(\theta\) is the estimated parameters of the model \(\widehat{\mathcal{H}}\), from Eq. (6) the model parameter\(\theta\) is formed by:\({a}_{r}=\left[{a}_{0} {a}_{1}\dots { a}_{m}\right]\), \({\alpha }_{r}=\left[{\alpha }_{0} {\alpha }_{1}\dots { \alpha }_{m}\right]{b}_{\mathcal{F}}=\left[{b}_{0} {b}_{1} \dots {b}_{n}\right]\), \({\beta }_{\mathcal{F}}=\left[{\beta }_{0} {\beta }_{1} \dots { \beta }_{n}\right]\).
In this work the FOMCON toolbox for MATLAB/Simulink is used, the toolbox is based on fractional calculus for system modeling and control design. The FOMCON toolbox uses the Levenberg-Marquardt algorithm for optimization of the model parameters θ under the nonlinear least squares technique. The Levenberg-Marquardt algorithm is described by the Eq. (16) According to the references21,35,36 :
$${-J}_{p}^{T}{℮}_{p}=\varDelta {\theta }_{p} \left({J}_{p}^{T}{J}_{p}+\lambda I\right)$$
16
Where \(\lambda\) is a positive scaler, called a damping factor, \(I\) is the identity matrix, and \(J\) is the Jacobian matrix of the function \(\mathcal{F}\) and is defined as
$$J=\left[\begin{array}{ccc}\frac{\partial {℮}_{\text{1,1}}}{\partial {\theta }_{1}}& \frac{\partial {℮}_{\text{1,1}}}{\partial {\theta }_{2}} \dots & \frac{\partial {℮}_{\text{1,1}}}{\partial {\theta }_{N}}\\ \begin{array}{c}\frac{\partial {℮}_{\text{1,2}}}{\partial {\theta }_{1}}\\ \dots \end{array}& \begin{array}{c}\frac{\partial {℮}_{\text{1,2}}}{\partial {\theta }_{1}}\\ \dots \end{array} \dots & \begin{array}{c}\frac{\partial {℮}_{\text{1,2}}}{\partial {\theta }_{N}}\\ \dots \end{array}\\ \frac{\partial {℮}_{1,m}}{\partial {\theta }_{1}}& \frac{\partial {℮}_{1,m}}{\partial {\theta }_{1}} \dots & \frac{\partial {℮}_{1,m}}{\partial {\theta }_{N}}\end{array}\right] \left(17\right)$$
the update parameters are defined in Eq. (18).
$${\theta }_{p+1}={\theta }_{p}+\varDelta {\theta }_{p}$$
18
From equations (16) and (18) the update parameters can be determined by Eq. (19):
$${\theta }_{p+1}={\theta }_{p} - \frac{{-J}_{p}^{T}{℮}_{p}}{\left({J}_{p}^{T}{J}_{p}+\lambda I\right)}. \left(19\right)$$
4.4. The model quality
The quality of the identified model will be discussed in this section. let \({y}_{s}\) denote The output data of the system, and \({y}_{d}\) denote The output data of the identified model, for single input single output (SISO)system, both \({y}_{s}\) and \({y}_{d}\) are vectors of dimension N X 1, and the difference between \({y}_{s}\) and \({y}_{d}\) is the model output error since \({y}_{s}\) and \({y}_{d}\) are vectors, then the error is a vector of size N X 1 which is called residuals
$$e= {y}_{s}-{y}_{s}$$
20
.
The percentage fit can be described as
$$fit=\left(1-\frac{\parallel e\parallel }{{y}_{s}-\stackrel{-}{{y}_{s}}}\right)*100\% , \left(21\right)$$
where \(\parallel .\parallel\) is the Euclidean norm, and \(\stackrel{-}{{y}_{s}}\) is the mean value of the system output data\({y}_{s}\)
4.5 stability analysis
To study the fractional system stability, we consider Matignon’s stability theorem37 :
Theorem 1
(Matignon’s stability theorem) The fractional transfer function G(s) = Z(s) = P (s) is stable if and only if the following condition is satisfied in the σ-plane:
$$\left|\text{arg}\left(\sigma \right)\right|>q\frac{\pi }{2} , \forall \sigma \in \complement , P\left(\sigma \right)=0$$
22
,
where \(0 < q < 2\)and \(\sigma ={s}^{q}\). When \(\sigma =0\) is a single root of \(P \left(s\right),\) the system cannot be stable. For q = 1, this is the classical theorem of pole location in the complex plane: no pole is in the closed right plane of the first Riemann sheet.
The algorithm for determining the system stability in (10) is shown in figure (4), according to Matignon’s stability theorem, the fractional order system stability in the complex plane is shown in figure.5.
From Fig. 1, The PV current \({I}_{PV}\left(n\right)\) and output power\({P}_{PV}\left(n\right)\)) are used data of the system identification, the measured data is collected under constant irradiation and constant dc load during a 1minute period, for the implementation of the proposed technique, an ARDUINO ATMEGA2560 has been used as a microcontroller, a circuit of the voltage divider and a hall effect current sensor ACS7-12 has been used for reading the analog date for the PV voltage and current respectively, in order to store the signals, a HANTEK 2D42 oscilloscope has been used, to measure the PV current or the output current a shunt resistor of 0.5 ohms is used, the data has been collected with a sample period of \({T}_{s}=1000\mu S\). The collected data are provided for the FOMCON toolbox in MATLAB to develop the fractional order model.
The measured data of the PV current and the output power of the boost converter collected from the previous experiment are shown in figure(6):
The PV current and the output power of the boost converter have been used as input data and output data respectively for the FOMCON toolbox in MATLAB, the Levenberg-Marquardt optimization algorithm is used for the optimization problem, and Oustaloup recursive filter approximation is used for model simulation in the time domain, the fractional order model which is estimated by the FOMCON toolbox is shown in the following equation:
Conclusion
$$G\left(S\right)=\frac{{109.57 S}^{2.5684}+{4374.9.7S}^{1.7303}+3.0654}{{8.0417 S}^{2.8057}+{226.94S}^{1.7615}+{9.347 S}^{1.2891}+1}, \left(23\right)$$
From Eq. (21) The fitting between the system output data and the output of the model (22) is 93.15%, and the error norm is 213.714, as shown in the figure.7, following theorem 1, the poles of the estimated model in (23) are satisfied with the condition (22), such that the identified model is stable for q = 0.08 as shown in Fig. 8.
Model validation
Checking the model validation by comprising the proposed model with another integer model is constructed by MATLAB System Identification TOOLBOX, with the same experimental data. the selection of the integer model was dependent on many iterative processes, the best-fitted integer model shows an accuracy of 88.74%, and the fitting between the measured output data and the output of the integer model is shown in Fig. 9. Comparing the identified fractional order model (FOM) and the integer order model (IOM), shows the accuracy of FOM is higher than the IOM accuracy. it is clear from figure.7 and figure.9 that the performance of the FOM is better than the IOM, such that the results show the validity of the proposed technique to construct the FOM of the PV system.
In this paper, a dynamical fractional order model of the PV system is driven based on the system identification technique. The measured input-output data are collected for parameters estimation based on the Levenberg-Marquardt algorithm; the identification process is based on the Least Squares technique for minimization of the errors of the model output. the fractional model stability is checked by Matignon’s stability theorem. an integer order model is constructed and its performance is estimated to demonstrate the fractional order model validation, the fractional order model shows higher accuracy (93.15%) than the other model.
Ethics approval and consent to participate: Not applicable
Competing interests: The authors declare that they have no conflicts of interest
Funding: This research received no external funding.
Consent for publication: The Authors hereby consents to the publication of the Work in the Energy, Sustainability and Society journal.
Availability of supporting data: No data were used to support this study.
Authors' contributions: R.E. and E.N. conceived and designed the experiments; R.E. and B.A. performed the experiments; R.E. and E.N. prepared figures and tables; R.E and B.A., and E.N. reviewed the manuscript; supervision B.A. E.N.; and R.E. wrote the main manuscript text.
Acknowledgments: I wish to thank Prof. Belal Abou-Zalam, Dr. Essam Nabil, and Prof. Mohmed Hamdy (faculty of electronic engineering, Minoufia University) for supporting in research.
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