Modeling of Photovoltaic Grid Connected Inverters Based on Nonlinear System Identification for Power Quality Analysis

Photovoltaic systems are attractive renewable energy sources for Thailand because of high daily solar irradiation, about 18 MJ/m2/day. Furthermore, renewable energy is boosted by the government incentive on adders on electricity from renewable energy like solar PV, wind and biomass, introduced in the second half of 2000s. For PV systems, domestic rooftop PV units, commercial rooftop PV units and ground-based PV plants are appealing. Applications of electricity supply from PV plants that have been filed total more than 1000 MW. With the adder incentive, more households will be attracted to produce electricity with a small generating capacity of less than 10 kW, termed a very small power producer (VSPP). A possibility of expanding domestic roof-top grid-connected units draw our attention to study single phase PV-grid connected systems. Increased PV penetration can have significant [1-2] and detrimental impacts on the power quality (PQ) of the distribution networks [3-5]. Fluctuation of weather condition, variations of loads and grids, connecting PV-based inverters to the power system, requires power quality control to meet standards of electrical utilities. PV can reduce or improve power quality levels [6-9]. Different aspects should be taken into account. In particular, large current variations during PV connection or disconnection can lead to significant voltage transients [10]. Cyclic variations of PV power output can cause voltage fluctuations [11]. Changes of PV active and reactive power and the presence of large numbers of single phase domestic generators can lead to long-duration voltage variations and unbalances [12]. The increasing values of fault currents modify the voltage sag characteristics. Finally, the waveform distortion levels are influenced in different ways according to types of PV connections to the grid, i.e. direct connection or by power electronic interfaces. PV can improve power quality levels, mainly as a consequence of increase of short circuit power and of advanced controls of PWM converters and custom devices. [13] Grid-connected inverter technology is one of the key technologies for reliable and safety grid interconnection operation of PV systems [14-15]. An inverter being a power


Introduction
Photovoltaic systems are attractive renewable energy sources for Thailand because of high daily solar irradiation, about 18 MJ/m 2 /day. Furthermore, renewable energy is boosted by the government incentive on adders on electricity from renewable energy like solar PV, wind and biomass, introduced in the second half of 2000s. For PV systems, domestic rooftop PV units, commercial rooftop PV units and ground-based PV plants are appealing. Applications of electricity supply from PV plants that have been filed total more than 1000 MW. With the adder incentive, more households will be attracted to produce electricity with a small generating capacity of less than 10 kW, termed a very small power producer (VSPP). A possibility of expanding domestic roof-top grid-connected units draw our attention to study single phase PV-grid connected systems. Increased PV penetration can have significant [1][2] and detrimental impacts on the power quality (PQ) of the distribution networks [3][4][5]. Fluctuation of weather condition, variations of loads and grids, connecting PV-based inverters to the power system, requires power quality control to meet standards of electrical utilities. PV can reduce or improve power quality levels [6][7][8][9]. Different aspects should be taken into account. In particular, large current variations during PV connection or disconnection can lead to significant voltage transients [10]. Cyclic variations of PV power output can cause voltage fluctuations [11]. Changes of PV active and reactive power and the presence of large numbers of single phase domestic generators can lead to long-duration voltage variations and unbalances [12]. The increasing values of fault currents modify the voltage sag characteristics. Finally, the waveform distortion levels are influenced in different ways according to types of PV connections to the grid, i.e. direct connection or by power electronic interfaces. PV can improve power quality levels, mainly as a consequence of increase of short circuit power and of advanced controls of PWM converters and custom devices. [13] Grid-connected inverter technology is one of the key technologies for reliable and safety grid interconnection operation of PV systems [14][15]. An inverter being a power conditioner of a PV system consists of power electronic switching components, complex control systems [16]. In addition, their operations depends on several factors such as input weather condition, switching algorithm and maximum power point tracking (MPPT) algorithm implemented in grid-connected inverters, giving rise to a variety of nonlinear behaviors and uncertainties [17]. Operating conditions of PV based inverters can be considered as steady state condition [18], transient condition [19][20], and fault condition such islanding [21][22]. In practical operations, inverters constantly change their operating conditions due to variation of irradiances, temperatures, load or grid impedance variations. In most cases, behavior of inverters is mainly considered in a steady state condition with slowly changing grid, load and weather conditions. However, in many instances conditions suddenly change, e.g. sudden changes of input weather, cloud or shading effects, loads and grid changes from faults occurring in near PV sites [23]. In these conditions, PV based inverters operate in transient conditions. Their average power increases or decreases upon the disturbances to PV systems [24]. In order to understand the behavior of PV based inverters, modeling and simulation of PV based inverter systems is the one of essential tools for analysis, operation and impacts of inverters on the power systems [25]. There are two major approaches for modeling power electronics based systems, i.e. analytical and experimental approaches. The analytical methods to study steady state, transient models and islanding conditions of PV based inverter systems, such as state space averaging method [26], graphical techniques [27][28] and computation programming [29]. In using these analytic methods, one needs to know information of system. However, PV based inverters are usually commercial products having proprietary information; system operators do not know the necessary information of products to parameterize the models. In order to build models for nonlinear devices without prior information, system identification methods are exposed [33][34]. In the method reported in this paper, specific information of inverter is not required in modeling. Instead, it uses only measured input and output waveforms. Many recent research focuses on identification modeling and control for nonlinear systems [35][36][37]. One of the effective identification methods is block oriented nonlinear system identification. In the block oriented models, a system consists of numbers of linear and nonlinear blocks. The blocks are connected in various cascading and parallel combinations representing the systems. Many identification methods of well known nonlinear block oriented models have been reported in the literature [38][39]. They are, for example, a NARX model [40], a Hammerstein model [41], a Wiener model [42], a Wiener-Hammerstein model and a Hammerstein-Wiener model [43]. Advantages of a Hammerstein model and a Wiener model enables combination of both models to represent a system, sensors and actuators in to one model. The Hammerstein-Wiener model is recognized as being the most effective for modeling complex nonlinear systems such PV based inverters [44]. In this paper, real operating conditions weather input variation, i.e. load variations and grid variations, of PV-based inverters are considered. Then two different experiments, steady state and transient condition, are designed and carried out. Input-output data such as currents and voltages on both dc and ac sides of a PV grid-connected systems are recorded. The measured data are used to determine the model parameters by a Hammerstein-Wiener nonlinear model system identification process. In the Section II, PV system characteristics are introduced. The I-V characteristic, an equivalent model, effects of radiation and temperature on voltage and current of PV are described. In the Section III, system identification methods, particularly a Hammerstein-Wiener Model is explained. In the following section, the experimental design and implementation to model the system is illustrated. After that, the obtained model from prior sections is analyzed in terms of control theories. In the last section, the power quality analysis is discussed. The output prediction is performed to obtain electrical outputs of the model and its electrical power. The power quality nature is analyzed for comparison with outputs of model. Subsequently, voltage and current outputs from model are analyzed by mathematical tools such the Fast Fourier Transform-FFT, the Wavelet method in order to investigate the power quality in any operating situations.

PV grid connected system (PVGCS) operation
In this section, PV grid connected structures and components are introduced. Structures of PBGCS consist of solar array, power conditioners, control systems, filtering, synchronization, protection units, and loads, shown in Fig. 1.

Solar array
Environmental inputs affecting solar array/cell outputs are temperature (T) and irradiance (G), fluctuating with weather conditions. When the ambient temperature increases, the array short circuit current slight increases with a significant voltage decrease. Temperature and I-V characteristics are related, characterized by array/cell temperature coefficients. Effects of irradiance, radiant solar energy flux density in W/m 2 , apart from solar radiation at sea level, are determined by incident angles and array/cell envelops. Typical characteristics of relationship between environmental inputs (irradiance and temperature) and electrical parameters (current and voltage of array/cells) are shown in Fig. 2 [45]. In our experimental designs, operating conditions of PV systems under test is designed and based on typical operating conditions.  Fig. 3. Variations of solar irradiance and temperature throughout a day conditioning PVGCS operation Firstly, under a steady state condition, input, load and utility under consideration are treated as being constant with slightly change weather condition. Installed capacities of PV systems in a steady state are low, medium and high capacity. According to the weather conditions throughout a day as shown in Fig. 3 [47][48], a low radiation about 0-400 W/m 2 is common in an early morning (6:00 AM-9:00 AM) and early evening (16:00 PM-19:00 PM), medium radiation of 400-800 W/m 2 in late morning (9:00 AM-11:00 PM) and early afternoon (14:00 PM-16:00 PM) and high radiation of 800-1000 W/m 2 around noon (11.00 www.intechopen.com AM -14:OO PM). Loads fluctuate upon activities of customer groups, for example, a peak load for industrial zones occurs in afternoon (13:00 -17:00 PM) and a peak load for residential zones occurs in evening(18:00 -21:00 PM). Variations from steady state conditions impact power quality such as overvoltage, over-current, harmonics, and so on. In case of transients, there are variations in inputs, loads and utility. Weather variations such solar irradiance and temperature exhibit significant changes. Unexpected accidents happen. Local loads may sudden change due to activities of customers in each time. A utility has some faults in nearby locations which impact utility parameters such grid impedance. These conditions lead to short duration power quality problems with such spikes, voltage sag, voltage swell. In some extreme cases, abnormal conditions, such as very low solar irradiance or abnormal conditions such islanding, the gird-connected PV systems may collapse. The PV systems are black out and cut out of the utility grid. Such can affect power quality, stability and reliability of power systems.

Power converter
There are several topologies for converting a DC to DC voltage with desired values, for example, Push-Pull, Flyback, Forward, Half Bridge and Full Bridge [49]. The choice for a specific application is often based on many considerations such as size, weight of switching converter, generation interference and economic evaluation [50][51]. Inverters can be classified into two types, i.e. voltage source inverter (VSI) if an input voltage remains constant and a current source inverter (CSI) if input current remains constant [52][53]. The CSI is mostly used in large motor applications, whereas the VSI is adopted for and alone systems. The CSI is a dual of a VSI. A control technique for voltage source inverters consists of two types, a voltage control inverter, shown in Fig. 4(a) and a current control inverter, Fig. 4(b) [54].

System Identification
System identification is the process for modeling dynamical systems by measuring the input/output from system. In this section, the principle of system identification is described. The classification is introduced and particularly a Hammerstein-Wiener model is explained. Finally, a MIMO (multi input multi output model with equation and characteristic is illustrated. www.intechopen.com

Principle of system identification
A dynamical system can be classified in terms of known structures and parameters of the system, shown in Fig.5, and classified as a "White Box" if all structures and parameters are known, a "Grey Box" , if some structures and parameters known and a "Black Box" if none are known [55]. Steps in system identification can be described as the following process, shown in Fig. 6.  Each step can be described as follows

Goal of modeling
The primary goal of modeling is to predict behaviors of inverters for PV systems or to simulate their outputs and related values. The other important goal is to acquire mathematical and physical characteristics and details of systems for the purposes of controlling, maintenance and trouble shooting of systems, and planning of managing the power system.

Physical modeling
Photovoltaic inverters, particularly commercial products, compose of two parts, i.e. a power circuit and a control circuit. Power electronic components convert, transfer and control power from input to output. The control system, switching topologies of power electronics are done by complex digital controls.

Model structure selection
Model structure selection is the stage to classify the system and choose the method of system identification. The system identification can be classified to yield a nonparametric model and a parametric model, shown in Fig 7. A nonparametric model can be obtained from various methods, e.g. Covariance function, Correlation analysis. Empirical Transfer Function Estimate and Periodogram, Impulse response, Spectral analysis, and Step response.

Nonparametric Model Parametric Model
Covariance function Correlation analysis

Empirical Transfer Function Estimate and Periodogram
Impulse response

Linear Model Nonlinear Model
Step response  [56]. In practice, all systems are nonlinear; their outputs are a nonlinear function of the input variables. A linear model is often sufficient to accurately describe the system dynamics as long as it operates in linear range Otherwise a nonlinear is more appropriate. A nonlinear model is often associated with phenomena such as chaos, bifurcation and irreversibility. A common approach to nonlinear problems solution is linearization, but this can be problematic if one is trying to study aspects such as irreversibility, which are strongly tied to nonlinearity. Inverters of PV systems can be identified based on nonlinear parametric models using various system identification methods.

Experimental design
The experimental design is an important stage in achieving goals of modeling. Number parameters such as sampling rates, types and amount of data should be concerned. Grid connected inverters have four important input/output parameters, i.e. DC voltage (Vdc), DC current voltage (Idc), AC voltage (Vac) and AC current (Iac). In experiments, these data are measured, collected and send to a system identification process. Finally, a model of a PV inverter is obtained, shown in Fig. 8.

System Identification
Model input output Fig. 8. Experimental design of a photovoltaic inverter modeling using system identification

Model estimation
Data from the system are divided into two groups, i.e., data for estimation (estimate data) and data for validation (validate data). Estimate data are used in the system identification and validate data are used to check and improve the modeling to yield higher accuracy.

Model validation
Model validation is done by comparing experimental data or validates data and modeling data. Errors can then be calculated. In this paper, parameters of system identification are optimized to yield a high accuracy modeling by programming softwares.

Hammerstein-Wiener (HW) nonlinear model
In this section, a combination of the Wiener model and the Hammerstein model called the Hammerstein-Wiener model is introduced, shown in Fig. 9. In the Wiener model, the front part being a dynamic linear block, representing the system, is cascaded with a static nonlinear block, being a sensor. In the Hammerstein model, the front block is a static nonlinear actuator, in cascading with a dynamic linear block, being the system. This model enables combination of a system, sensors and actuators in one model. The described dynamic system incorporates a static nonlinear input block, a linear output-error model and an output static nonlinear block.
Which u(t) and y(t) are the inputs and outputs for the system. Where w(t) and x(t) are internal variables that define the input and output of the linear block.

Linear subsystem
The linear block is similar to an output error polynomial model, whose structure is shown in the Equation (2). The number of coefficients in the numerator polynomials B(q) is equal to the number of zeros plus 1, n b is the number of zeros. The number of coefficients in denominator polynomials F(q) is equal to the number of poles, n f is the number of poles. The polynomials B and F contain the time-shift operator q, essentially the z-transform which can be expanded as in the Equation (3). u n is the total number of inputs. k n is the delay from an input to an output in terms of the number of samples. The order of the model is the sum of b n and f n . This should be minimum for the best model. www.intechopen.com

Nonlinear subsystem
The Hammerstein-Wiener Model composes of input and output nonlinear blocks which contain nonlinear functions f(•) and h(•) that corresponding to the input and output nonlinearities. The both nonlinear blocks are implemented using nonlinearity estimators. Inside nonlinear blocks, simple nonlinear estimators such deadzone or saturation functions are contained. i. The dead zone (DZ) function generates zero output within a specified region, called its dead zone or zero interval which shown in Fig. 10. The lower and upper limits of the dead zone are specified as the start of dead zone and the end of dead zone parameters. Deadzone can define a nonlinear function y = f(x), where f is a function of x, It composes of three intervals as following in the equation (4) () when x has a value between a and b, when an output of the function equal to () 0 Fx = , this zone is called as a "zero interval".  (5) and Fig. 11. The function is determined between a and b values. This interval is known as a "linear interval" when u is input and y is output. r is the the regressor. Q is a nonlinear subspace and P a linear subspace. L is a linear coefficient. d is an output offset. b is a dilation coefficient., c a translation coefficient and a an output coefficient. f is the sigmoid function, given by the following equation (7) 1 () 1 v. Wavelet Network (WN) activation function. The term wavenet is used to describe wavelet networks. A wavenet estimator is a nonlinear function by combination of a wavelet theory and neural networks. Wavelet networks are feed-forward neural networks using wavelet as an activation function, based on the following expansion in the equation ( In a system identification process, the wavelet coefficient (a), the dilation coefficient (b) and the translation coefficient (c) are optimized during model learning steps to obtain the best performance model.

MIMO Hammerstein-Wiener system identification
The voltage and current are two basic signals considered as input/output of PV grid connected systems. The measured electrical input and output waveforms of a system are collected and transmitted to the system identification process. In Fig. 13 show a PV based inverter system which are considered as SISO (single input-single output) or MIMO (multi input-multi output), depending on the relation of input-output under study [57]. In this paper, the MIMO nonlinear model of power inverters of PV systems is emphasized because this model gives us both voltage and current output prediction simultaneously.
Where bi n , fi n and ki n are pole, zero and delay of linear model. Where as number of subscript i are 1,2,3 and 4 which stand for relation between DC voltage-AC voltage, DC current-AC voltage, DC voltage-AC current and DC current-AC current respectively. The output voltage and output current are key components for expanding to the other electrical values of a system such power, harmonic, power factor, etc. The linear parameters, zeros, poles and delays are used to represent properties and relation between the system input and output. There are two important steps to identify a MIMO system. The first step is to obtain experimental data from the MIMO system. According to different types of experimental data, the second step is to select corresponding identification methods and mathematical models to estimate model coefficients from the experimental data. The model is validated until obtaining a suitable model to represent the system. The obtained model provides properties of systems. State-space equations, polynomial equations as well as transfer functions are used to describe linear systems. Nonlinear systems can be describes by the above linear equations, but linearization of the nonlinear systems has to be carried out. Nonlinear estimators explain nonlinear behaviors of nonlinear system. Linear and nonlinear graphical tools are used to describe behaviors of systems regarding controllability, stability and so on.

Experimental
In this work, we model one type of a commercial grid connected single phase inverters, rating at 5,000 W. The experimental system setup composes of the inverter, a variable DC power supply (representing DC output from a PV array), real and complex loads, a digital power meter, a digital oscilloscope, , a AC power system and a computer, shown www.intechopen.com schematically in Fig 14. The system is connected directly to the domestic electrical system (low voltage). As we consider only domestic loads, we need not isolate our test system from the utility (high voltage) by any transformer. For system identification processes, waveforms are collected by an oscilloscope and transmitted to a computer for batch processing of voltage and current waveforms. Major steps in experimentation, analysis and system identifications are composed of Testing scenarios of six steady state conditions and two transient conditions are carried out on the inverter, from collected data from experiments, voltage and current waveform data are divided in two groups to estimate models and to validate models previously mentioned.
The system identification scheme is shown in Fig.15. Good accuracy of models are achieved by selecting model structures and adjusting the model order of linear terms and nonlinear estimators of nonlinear systems. Finally, output voltage and current waveforms for any type of loads and operating conditions are then constructed from the models. This allows us to study power quality as required.

Results and discussion
In the next step, data waveforms are divided into the "estimate data set" and the "validate data set". Examples are shown in Fig. 20, whereby the first part of the AC and DC voltage waveforms are used as the estimate data set and the second part the validate data set. The system identification process is executed according to mentioned descriptions on the Hammerstein-Wiener modeling. The validation of models is taken by considering (i) model order by adjusting the number of poles plus zeros. The system must have the lowest-order model that adequately captures the system dynamics.(ii) the best fit, comparing between modeling and experimental outputs, (iii) FPE and AIC, both of these values need be lowest for high accuracy of modeling (iv)

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Nonlinear behavior characteristics. For example, linear interval of saturation, zero interval of dead-zone, wavenet, sigmoid network requiring the simplest and less complex function to explain the system. Model properties, estimators, percentage of accuracy, final Prediction Error-FPE and Akaikae Information Criterion-AIC are as follows [58]: where y* is the simulated output, y is the measured output and y is the mean of output. FPE is the Akaike Final Prediction Error for the estimated model, of which the error calculation is defined as equation (14) 1 where V is the loss function, d is the number of estimated parameters, N is the number of estimation data. The loss function V is defined in Equation (15) where N θ represents the estimated parameters. Criterion (AIC), as shown in equation (16), is used to calculate a comparison of models with different structures.
Waveforms of input and output from the experimental setup consist of DC voltage, DC output current, AC voltage and AC output current. Model properties, estimators, percentage of accuracy, Final Prediction Error -FPE and Akaikae Information Criterion -AIC of the model are shown in Table 3 In Table 3 Table 3, is the accuracy of the voltage output, the second the current output from the model. From the results, nonlinear estimators can describe the photovoltaic grid connected system. The estimators are good in terms of accuracy, with a low order model or a low FPE and AIC. Under most of testing conditions, high accuracy of more than 85% is achieved, except the case of FVLC. This is because of under such an operating condition, the inverter has very small current, and it is operating under highly nonlinear behavior. Then complex of nonlinear function and parameter adjusted is need for achieve the high accuracy and low order of model. After obtaining the appropriate model, the PVGCS system can be analyzed by nonlinear and linear analyses. Nonlinear parts are analyzed from the properties of nonlinear function such as dead-zone interval, saturation interval, piecewise range, Sigmoid and Wavelet properties. Nonlinear properties are also considered, e.g. stability and irreversibility In order to use linear analysis, Linearization of a nonlinear model is required for linear control design and analysis, with acceptable representation of the input/output behaviors. After linearizing the model, we can use control system theory to design a controller and perform linear analysis. The linearized command for computing a first-order Taylor series approximation for a system requires specification of an operating point. Subsequently, mathematical representation can be obtained, for example, a discrete time invariant state space model, a transfer function and graphical tools. www.intechopen.com

Applications: Power quality problem analysis
A power quality analysis from the model follows the Standard IEEE 1159 Recommend Practice for Monitoring Electric Power Quality [59]. In this Standard, the definition of power quality problem is defined. In summary, a procedure of this Standard when applied to operating systems can be divided into 3 stages (i) Measurement Transducer, (ii) Measurement Unit and (iii) Evaluation Unit. In comparing operating systems and modeling, modeling is more advantageous because of its predictive power, requiring no actual monitoring. Based on proposed modeling, the measurement part is replaced by model prediction outputs, electrical values such as RMS and peak values, frequency and power are calculated, rather than measured. The actual evaluation is replaced by power quality analysis. The concept representation is shown in Fig.22 Fig.23 shows an actual and predictive output current waveforms of the transient step down condition. We see good agreement between experimental results and modeling results.

Electrical parameter calculation
In this stage, output waveforms are used to calculate RMS, peak and per unit (p.u.) values, period, frequency, phase angle, power factor, complex power (real, reactive and apparent power) Total Harmonic Distortion -THD.

Root mean square
RMS values of voltage and current can be calculated from the following equations: www.intechopen.com

Period, frequency and phase angle
We calculate a phase shift between voltage and current from the equation (19), and the frequency (f) from equation (20).
t Δ is time lagging or leading between voltage and current (ms), T is the waveform period. www.intechopen.com

Power factor, apparent power, active power and reactive power
The power factor, the apparent power S (VA), the active power P (W), and the reactive power Q (Var) are related through the equations We next demonstrate accuracy and precision of power quality prediction from modeling.

Conclusions
In this paper, a PVGCS system is modeled by the Hammerstein-Wiener nonlinear system identification method. Two main steps to obtain models from a system identification process are implemented. The first step is to set up experiments to obtain waveforms of DC inverter voltage/current, AC inverter voltage/current, point of common coupling (PCC) voltage, and grid and load current. Experiments are conducted under steady state and transient conditions for commercial rooftop inverters with rating of few kW, covering resistive and complex loads. In the steady state experiment, six conditions are carried out. In the transient case, two conditions of operating conditions are conducted. The second stage is to derive system models from system identification software. Collected waveforms are transmitted into a computer for data processing. Waveforms data are divided in two groups. One group is used to estimate models whereas the other group to validate models. The developed programming determines various model waveforms and search for model waveforms of maximum accuracy compared with actual waveforms. This is achieved through selecting model structures and adjusting the model order of the linear terms and nonlinear estimators of nonlinear terms. The criteria for selection of a suitable model are the "Best Fits" as defined by the software, and a model order which should be minimum. After obtaining appropriate models, analysis and prediction of power quality are carried out. Modeled output waveforms relating to power quality analysis are determined from different scenarios. For example, irradiances and ambient temperature affecting DC PV outputs and nature of complex local load can be varied. From the model output waveforms, determination is made on power quality aspects such as voltage level, total harmonic distortion, complex power, power factor, power penetration and frequency deviation.
Finally, power quality problems are classified. Such modelling techniques can be used for system planning, prevention of system failures and improvement of power quality of roof-top grid connected systems. Furthermore, they are not limited to PVGCS but also applicable to other distributed energy generators connected to grids. The utilization of renewable energy sources such as wind energy, or solar energy, among others, is currently of greater interest. Nevertheless, since their availability is arbitrary and unstable this can lead to frequency variation, to grid instability and to a total or partial loss of load power supply, being not appropriate sources to be directly connected to the main utility grid. Additionally, the presence of a static converter as output interface of the generating plants introduces voltage and current harmonics into the electrical system that negatively affect system power quality. By integrating distributed power generation systems closed to the loads in the electric grid, we can eliminate the need to transfer energy over long distances through the electric grid. In this book the reader will be introduced to different power generation and distribution systems with an analysis of some types of existing disturbances and a study of different industrial applications such as battery charges.

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