**5. 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)

Modeling of Photovoltaic Grid Connected Inverters

97.03% and 91.7 % are shown in Fig 21.

**Type I/P O/P Linear model** 

with different structures.

**Steady state condition**s

FCLV DZ DZ

FCMV PW PW

FCHV ST ST

FVLC SN SN

FVMC WN WN

FVHC WN WN

Step Up DZ DZ

Step Down PW PW

**Transient conditions**

Based on Nonlinear System Identification for Power Quality Analysis 71

Criterion (AIC), as shown in equation (16), is used to calculate a comparison of models

<sup>2</sup> log *<sup>d</sup> AIC V*

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. Examples of voltage and current output waveforms of multi input-multi output (MIMO) model in steady state condition (FVMC) having accuracy

> **parameters [nb1 nb2 nb3 nb4] poles [nf1 nf2 nf3 nf4] zeros [nk1 nk2 nk3 nk4] delays**

> > [4 4 3 5]; [5 5 3 6]; [3 4 4 2]

> > [5 2 4 4]; [4 2 3 4]; [2 2 4 3];

> > [2 2 3 4]; [1 2 1 2]; [2 1 3 2];

> > [3 6 3 2]; [8 5 4 3]; [2 4 3 5];

> > [3 4 2 5]; [4 2 3 4]; [2 3 2 4];

> > [1 4 3 5]; [5 2 3 5]; [1 3 2 4];

[3 4 2 4]; [4 5 4 3]; [2 3 5 5]; [ 4 5 2 2];

[3 5 5 3]; [3 5 4 3]; [3 5 5 4]; [4 4 4 1];

Table 3. Results of a PV inverter modeling using a Hammerstein-Wiener model

*N*

= + (16)

**% fit Voltage Current**

87.3

84.5

89.5

56.8

97.03

88

91.75

85.99

**FPE AIC** 

85.7 3,080.90 10.9

86.4 729.03 6.59

88.7 26.27 3.26

60.5 0.07 2.57

91.7 254.45 7.89

<sup>94</sup>3,079.8 10.33

87.20 3,230 7.40

85.12 3,233 10.0

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]:

Fig. 20. Data divided into Estimated and validated data

#### **Criteria for Model selection**

The percentage of the best fit accuracy in equation (13) is obtained from comparison between experimental waveform and simulation modeling waveform.

$$Best\,\,\text{fit} = 100\,\,\text{\*}\,\left(1 - norm(y\,\,\, \text{'} - y) \,\,/\,\, norm(y - y)\right) \tag{13}$$

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)

$$FPE = V \left(\frac{1 + \bigvee\_{N}}{1 - \bigvee\_{N}}\right) \tag{14}$$

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 θ *<sup>N</sup>* represents the estimated parameters.

$$V = \det\left(\frac{1}{N} \sum\_{1}^{N} \varepsilon\left(t, \theta\_N\right) \left(\varepsilon\left(t, \theta\_N\right)\right)^{\top}\right) \tag{15}$$

The Final Prediction Error (FPE) provides a measure of a model quality by simulating situations where the model is tested on a different data set. The Akaike Information

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

Input and output signals

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

Time

The percentage of the best fit accuracy in equation (13) is obtained from comparison

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

> 1 1

 <sup>+</sup> <sup>=</sup> <sup>−</sup>

*<sup>N</sup> FPE V <sup>d</sup>*

where V is the loss function, d is the number of estimated parameters, N is the number of

<sup>=</sup>

The Final Prediction Error (FPE) provides a measure of a model quality by simulating situations where the model is tested on a different data set. The Akaike Information

*d*

*N*

( )( ) ( )

 εθ

*<sup>N</sup> <sup>T</sup>*

\* *Best fit* =− − − 100 \* (1 ( ) / ( )) *norm y y norm y y* (13)

(14)

θ

(15)

*<sup>N</sup>* represents the

Error-FPE and Akaikae Information Criterion-AIC are as follows [58]:


200

calculation is defined as equation (14)

estimated parameters.

Fig. 20. Data divided into Estimated and validated data

between experimental waveform and simulation modeling waveform.

estimation data. The loss function V is defined in Equation (15) where

1 <sup>1</sup> det , ,

*V tt N N <sup>N</sup>* εθ

220

Vdc

**Criteria for Model selection** 

240

260

0

Vac

500

Criterion (AIC), as shown in equation (16), is used to calculate a comparison of models with different structures.

$$AIC = \log V + \frac{2d}{N} \tag{16}$$

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. Examples of voltage and current output waveforms of multi input-multi output (MIMO) model in steady state condition (FVMC) having accuracy 97.03% and 91.7 % are shown in Fig 21.


Table 3. Results of a PV inverter modeling using a Hammerstein-Wiener model

Modeling of Photovoltaic Grid Connected Inverters

**6. Applications: Power quality problem analysis** 

analysis. The concept representation is shown in Fig.22.

measured

Voltage/current waveform

Measuremnent Transducer

> Model Prediction

experimental results and modeling results.

power) Total Harmonic Distortion - THD.

**6.2 Electrical parameter calculation** 

**6.2.1 Root mean square** 

Electrical input signal

Electrical input signal

**6.1 Model output prediction** 

Based on Nonlinear System Identification for Power Quality Analysis 73

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

> Measuremnent unit

Electrical value calculation

IEEE 1159 Recommend Practice Monitoring Electric Power Quality

Proposed Modeling

In this stage, the model output prediction is demonstrated. From the 8 operation conditions selected in experimental, we choose two representative case. One is the steady state Fix Voltage High Current (FVHC) condition, the other the transient step down condition. To illustrate model predictive power, Fig.23 shows an actual and predictive output current waveforms of the transient step down condition. We see good agreement between

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

RMS values of voltage and current can be calculated from the following equations:

Fig. 22. Diagram of power quality analysis from IEEE 1159 and application to modeling

unit Input signal to

Measurement result

RMS, Peak, frequency, Power Evaluation

Power Quality analysis

Measurement Evaluation

> Power quality Problem

Fig. 21. Comparison of AC voltage and current output waveforms of a steady state FVMC MIMO model

In Table 3 *nbi* , *nfi* and *nki* are poles, zeros and delays of a linear model. The subscript (1, 2, 3 and 4) stands for relations between DC voltage-AC voltage, DC current-AC voltage, DC voltage-AC current and DC current-AC current respectively. Therefore, the linear parameters of the model are 1234 [,,,] *nnnn bbbb* , 1234 [,,,] *nnnn ffff* , 1234 [,,,] *nnnn kkkk* .

The first value of percentages of fit in each type, shown in the 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.
