**3.1.3 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.

Fig. 7. Classification of system identification

Parametric models can be divided to two groups: linear parametric models and nonlinear parametric models. Examples of linear parametric models are Auto Regressive (AR), Auto Regressive Moving Average (ARMA), and Auto Regressive with Exogenous (ARX), Box-Jenkins, Output Error, Finite Impulse Response (FIR), Finite Step Response (FSR), Laplace Transfer Function (LTF) and Linear State Space (LSS). Examples of nonlinear parametric models are Nonlinear Finite Impulse Response (NFIR), Nonlinear Auto-Regressive with Exogenous (NARX), Nonlinear Output Error (NOE), and Nonlinear Auto-Regressive with

Modeling of Photovoltaic Grid Connected Inverters

an output static nonlinear block.

Inputnonlinear f(.)

Static

**3.2.1 Linear subsystem** 

Fig. 9. Structure of Hammerstein-Weiner Model

**3.2 Hammerstein-Wiener (HW) nonlinear model** 

Based on Nonlinear System Identification for Power Quality Analysis 61

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

B(q)/F(q)

Linear Output error model

u(t) w(t) x(t) y(t)

Dynamic

General equations describing the Hammerstein-Wiener structure are written as the Equation (1)

=

= −

=

( ) () ( ) ( )

Which u(t) and y(t) are the inputs and outputs for the system. Where w(t) and x(t) are

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, *<sup>n</sup> b* is the number of zeros. The number of coefficients in denominator polynomials F(q) is equal to the number of poles, *nf* 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). *nu* is the total number of inputs. *nk* is the delay from an input to an output in terms of the number of samples. The order of the model is the sum

> ( ) () ( ) ( ) *nu i*

1 1 ( ) ........ ( ) 1 ........

*Bq b bq bq Fq fq fq*

=+ + + =+ + +

1 2

*i i B q xt wt n*

1 1

− − + − −

*B q xt wt n F q*

( ) ( ( ))

*wt f ut*

*nu i*

( ) ( ( ))

*yt hxt*

internal variables that define the input and output of the linear block.

of bn and fn . This should be minimum for the best model.

*i i*

Outputnonlinear h(.)

Static

(1)

e(t)


*k*

*k*

*n n b n f n*

*F q* = − (2)

(3)

+

Moving Average Exogenous (NARMAX), Nonlinear Box-Jenkins (NBJ), Nonlinear State Space, Hammerstein model, Wiener Model, Hammerstein-Wiener model and Wiener-Hammerstein model [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.

### **3.1.4 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.

Fig. 8. Experimental design of a photovoltaic inverter modeling using system identification

#### **3.1.5 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.

#### **3.1.6 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.

#### **3.2 Hammerstein-Wiener (HW) nonlinear model**

60 Electrical Generation and Distribution Systems and Power Quality Disturbances

Moving Average Exogenous (NARMAX), Nonlinear Box-Jenkins (NBJ), Nonlinear State Space, Hammerstein model, Wiener Model, Hammerstein-Wiener model and Wiener-Hammerstein model [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

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

> Photovoltaic Inverter

System Identification

input output

Model

Fig. 8. Experimental design of a photovoltaic inverter modeling using system identification

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 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.

methods.

**3.1.4 Experimental design** 

**3.1.5 Model estimation** 

**3.1.6 Model validation** 

inverter is obtained, shown in Fig. 8.

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.

Fig. 9. Structure of Hammerstein-Weiner Model

General equations describing the Hammerstein-Wiener structure are written as the Equation (1)

$$\begin{aligned} w(t) &= f(u(t)) \\ \mathbf{x}(t) &= \sum\_{i}^{\frac{n\_k}{n}} \frac{B\_i(q)}{F\_i(q)} w(t - n\_k) \end{aligned} \tag{1}$$

$$\begin{aligned} y(t) &= h(\mathbf{x}(t)) \end{aligned} \tag{2}$$

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.

#### **3.2.1 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, *<sup>n</sup> b* is the number of zeros. The number of coefficients in denominator polynomials F(q) is equal to the number of poles, *nf* 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). *nu* is the total number of inputs. *nk* 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 bn and fn . This should be minimum for the best model.

$$x(t) = \sum\_{i}^{n\_u} \frac{B\_i(q)}{F\_i(q)} w(t - n\_k) \tag{2}$$

$$\begin{aligned} B(q) &= b\_1 + b\_2 q^{-1} + \dots + b\_n q^{-b\_n + 1} \\ F(q) &= 1 + f\_1 q^{-1} + \dots + f\_n q^{-f\_n} \end{aligned} \tag{3}$$

Modeling of Photovoltaic Grid Connected Inverters

Fig. 12. Structure of nonlinear estimators

by the following equation (7)

the equation (8)

( ) ( ) (( ) )

following expansion:

scalars.

Based on Nonlinear System Identification for Power Quality Analysis 63

**iii. Piecewise linear (PW) function** is defined as a nonlinear function, y=f(x) where f is a piecewise-linear (affine) function of x and there are n breakpoints (x\_k,y\_k) which k=1,...,n. y\_k = f(x\_k). f is linearly interpolated between the breakpoints. y and x are

**vi. Sigmoid network (SN) activation function** Both sigmoid and wavelet network estimators which use the neural networks composing an input layer, an output layer and a hidden layer using wavelet and sigmoid activation functions as shown in Fig.12

Input layer Hidden layer Output layer

Input (u) Output (y)

Activation function

A sigmoid network nonlinear estimator combines the radial basis neural network function using a sigmoid as the activation function. This estimator is based on the

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

*i i i*

*<sup>y</sup> u u r PL a f u r Qb c d* =− + − −+ (6)

<sup>−</sup> <sup>=</sup> <sup>+</sup> (7)

*n*

*i*

<sup>1</sup> ( ) <sup>1</sup> *<sup>z</sup> f z e*

**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

*i ii i*

*y u r PL as f bs u r Q cs aw g bw u r Q cw d* =− + − + + −+ + (8)

Which u and y are input and output functions. Q and P are a nonlinear subspace and a linear subspace. L is a linear coefficient. d is output offset. as and aw are a scaling coefficient and a wavelet coefficient. bs and bw are a scaling dilation coefficient and a

 ( ) \*(( ) ) \*( ( ) ) *n n*

*i i*

weights weights

#### **3.2.2 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)

$$\begin{aligned} \mathbf{x} & \le a & f(\mathbf{x}) &= \mathbf{x} - a \\ a &< \mathbf{x} < b & f(\mathbf{x}) &= \mathbf{0} \\ \mathbf{x} & \ge b & f(\mathbf{x}) &= \mathbf{x} - b \end{aligned} \tag{4}$$

when x has a value between a and b, when an output of the function equal to *F x*() 0 = , this zone is called as a "zero interval".

Fig. 10. Deadzone function

**ii. Saturation (ST) function** can define a nonlinear function y = f(x), where *f* is a function of *x*. It composes of three interval as the following characteristics in the equation (5) and Fig. 11. The function is determined between a and b values. This interval is known as a "linear interval"

$$\begin{array}{cccc} \mathbf{x} > a & \quad f(\mathbf{x}) = a & \\ \mathbf{a} < \mathbf{x} < b & \quad f(\mathbf{x}) = \mathbf{x} & \\ \mathbf{x} \le b & \quad f(\mathbf{x}) = b & \\ \\ \end{array} \tag{5}$$
 
$$\begin{array}{cccc} \mathbf{a} & \quad & \mathbb{F}(\mathbf{x}) = \mathbf{b} \\\\ \mathbf{f}(\mathbf{a}) = \mathbf{a} & \\ \end{array} \qquad \begin{array}{cccc} \mathbf{F}(\mathbf{x}) = \mathbf{b} \\\\ \end{array} \qquad \begin{array}{cccc} \mathbf{b} & \quad b \le x \\\\ \end{array}$$
 
$$a \le x \le b$$
 
$$a > x & \\ \vdots & \\ \end{array}$$

Fig. 11. Saturation function

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

> ( ) () 0 ( )

≤ =− << = ≥ =−

when x has a value between a and b, when an output of the function equal to *F x*() 0 = ,

<sup>a</sup> <sup>b</sup> F(x) = 0

*a* < *x* << *bxa* ≥ *bx*

**ii. Saturation (ST) function** can define a nonlinear function y = f(x), where *f* is a function of *x*. It composes of three interval as the following characteristics in the equation (5) and Fig. 11. The function is determined between a and b values. This interval is known

> ( ) ( ) ( )

> = << = ≤ =

*x a fx a a x b fx x x b fx b*

≤≤ *bxa*

F(x)=x-b

≤ *xb*

(4)

(5)

*x a fx x a*

*x b fx x b*

*a x b fx*

composes of three intervals as following in the equation (4)

F(x)=x-a

*a* > *x*

this zone is called as a "zero interval".

**3.2.2 Nonlinear subsystem** 

Fig. 10. Deadzone function

as a "linear interval"

Fig. 11. Saturation function


Fig. 12. Structure of nonlinear estimators

A sigmoid network nonlinear estimator combines the radial basis neural network function using a sigmoid as the activation function. This estimator is based on the following expansion:

$$y(u) = (u - r)PL + \sum\_{i}^{n} a\_i f((u - r)Qb\_i - c\_i) + d\tag{6}$$

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)

$$f(z) = \frac{1}{e^{-z} + 1} \tag{7}$$

**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 (8)

$$y = (\mu - r)PL + \sum\_{i}^{\mu} a s\_i \, \text{\*} \, f(bs(\mu - r)Q + c\text{s}) + \sum\_{i}^{\mu} a w\_i \, \text{\*} \, g(bw\_i(\mu - r)Q + cw\_i) + d \tag{8}$$

Which u and y are input and output functions. Q and P are a nonlinear subspace and a linear subspace. L is a linear coefficient. d is output offset. as and aw are a scaling coefficient and a wavelet coefficient. bs and bw are a scaling dilation coefficient and a

Modeling of Photovoltaic Grid Connected Inverters

Based on Nonlinear System Identification for Power Quality Analysis 65

nonlinear estimators and linear parameters. The relationships between input-output of the MIMO model have been written in the equation (10) whereas vdc is DC voltage, idc DC current, vac AC voltage, iac AC current. q is shift operator as equivalent to z transform. f(•) and h(•) are input and output nonlinear estimators. In this case a deadzone and saturation are selected into the model. In the MIMO model the relation between output and input has four relations as follows (i) DC voltage (vdc) – AC voltage (vac), (ii) DC voltage (vdc) – AC current

( ) ( ) ( ( )) ( ) ( )

= −+

= −+

 = −+⊗ −+

1 2

3 4

 

(10)

(12)

(11)

1

*bi bi*

− + − +

*n*

*n*

*fi fi*

1

*B q v t f v t n et F q*

( ) ( ) ( ( )) ( ) ( )

( ) ( ) ( ) ( ( )) ( ) ( ( )) ( ) ( ) ( )

*B q B q V t h f v t n et h f i t n et F q F q*

*B q B q I t h f v t n et h f i t n et F q F q*

1 2

*i n*

*Bq b b b q Fq f f f q*

=+++ =+++

( ) .... ( ) ....

1 2

*i n*

Where *nbi* , *nfi* and *nki* 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

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

( ) ( ) ( ) ( ( )) ( ) ( ( )) ( ) ( ) ( )

= −+⊗ −+

*B q i t f i t n et F q*

(iac), (iii) DC current (idc) – AC voltage (vac) and (iv) DC current(vdc)–AC voltage (vac).

*ac dc k*

*ac dc k*

1 2

*ac dc k dc k*

*ac dc k dc k*

1 2 3 4

3 4

describe behaviors of systems regarding controllability, stability and so on.

**4. Experimental** 

wavelet dilation coefficient. cs and cw are scaling translation and wavelet translation coefficients. The scaling function f (.) and the wavelet function g(.) are both radial functions, and can be written as the equation (9)

$$\begin{aligned} f(\boldsymbol{\mu}) &= \exp(-0.5 \, ^\ast \boldsymbol{\mu} \, ^\ast \boldsymbol{\mu}) \\ g(\boldsymbol{\mu}) &= (\dim(\boldsymbol{\mu}) - \boldsymbol{\mu} \, ^\ast \boldsymbol{\mu}) ^\ast \exp(-0.5 \, ^\ast \boldsymbol{\mu} \, ^\ast \boldsymbol{\mu}) \end{aligned} \tag{9}$$

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.

#### **3.3 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.

Fig. 13. Block diagram of nonlinear SISO and MIMO inverter model

For one SISO model, there is only one corresponding set of nonlinear estimators for input and output, and one set of linear parameters, i.e. pole bn, zero fn and delay nk , as written in the equation (9). For SIMO, MISO and MIMO models, there would be more than one set of 64 Electrical Generation and Distribution Systems and Power Quality Disturbances

( ) (dim( ) \* ) \* exp( 0.5 \* \* )

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

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

Nonlinear model

Nonlinear model

a) SISO model

Submodel Idc-Vac

Nonlinear model

Submodel Vdc-Vac

Submodel Vdc-Iac

Submodel Idc-Iac

b) MIMO model

For one SISO model, there is only one corresponding set of nonlinear estimators for input and output, and one set of linear parameters, i.e. pole bn, zero fn and delay nk , as written in the equation (9). For SIMO, MISO and MIMO models, there would be more than one set of

=− − ′ ′ (9)

Vac

Iac

Vac

Iac

*g u u uu uu*

functions, and can be written as the equation (9)

**3.3 MIMO Hammerstein-Wiener system identification** 

performance model.

( ) exp( 0.5 \* \* )

this model gives us both voltage and current output prediction simultaneously.

Vdc-Vac Vdc

Idc-Iac Idc

Fig. 13. Block diagram of nonlinear SISO and MIMO inverter model

Vdc

Idc

= − ′

*f u u u*

wavelet dilation coefficient. cs and cw are scaling translation and wavelet translation coefficients. The scaling function f (.) and the wavelet function g(.) are both radial nonlinear estimators and linear parameters. The relationships between input-output of the MIMO model have been written in the equation (10) whereas vdc is DC voltage, idc DC current, vac AC voltage, iac AC current. q is shift operator as equivalent to z transform. f(•) and h(•) are input and output nonlinear estimators. In this case a deadzone and saturation are selected into the model. In the MIMO model the relation between output and input has four relations as follows (i) DC voltage (vdc) – AC voltage (vac), (ii) DC voltage (vdc) – AC current (iac), (iii) DC current (idc) – AC voltage (vac) and (iv) DC current(vdc)–AC voltage (vac).

$$\begin{aligned} \upsilon\_{ac}(t) &= \frac{B(q)}{F(q)} f(\upsilon\_{dc}(t - n\_k)) + c(t) \\ \dot{\upsilon}\_{ac}(t) &= \frac{B(q)}{F(q)} f(\dot{\upsilon}\_{dc}(t - n\_k)) + c(t) \end{aligned} \tag{10}$$

$$\begin{aligned} V\_{\boldsymbol{\alpha}^\*} (t) &= h \left( \frac{B\_1(q)}{F\_1(q)} f(\upsilon\_{\boldsymbol{\alpha}^\*} (t - n\_{k1})) + c(t) \right) \otimes h \left( \frac{B\_2(q)}{F\_2(q)} f(\dot{\imath}\_{\boldsymbol{\alpha}^\*} (t - n\_{k2})) + c(t) \right) \bigg|\_{\boldsymbol{\alpha}^\*} \\ I\_{\boldsymbol{\alpha}^\*} (t) &= h \left( \frac{B\_3(q)}{F\_3(q)} f(\upsilon\_{\boldsymbol{\alpha}^\*} (t - n\_{k3})) + c(t) \right) \otimes h \left( \frac{B\_4(q)}{F\_4(q)} f(\dot{\imath}\_{\boldsymbol{\alpha}^\*} (t - n\_{k4})) + c(t) \right) \end{aligned} \tag{11}$$

$$\begin{aligned} B\_i(q) &= b\_1 + b\_2 + \ldots + b\_{n\_{\tilde{n}}} q^{-n\_{\tilde{n}} + 1} \\ F\_i(q) &= f\_1 + f\_2 + \ldots + f\_{n\_{\tilde{f}}} q^{-n\_{f\_i} + 1} \end{aligned} \tag{12}$$

Where *nbi* , *nfi* and *nki* 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.

### **4. 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

Modeling of Photovoltaic Grid Connected Inverters

study power quality as required.

**4.1 Steady state conditions** 

Current) as shown in Fig.17.

conditions

conditions

VacFV(V)

VacFA(V)

<sup>0</sup> <sup>1000</sup> <sup>2000</sup> <sup>3000</sup> <sup>4000</sup> <sup>5000</sup> <sup>6000</sup> <sup>7000</sup> <sup>8000</sup> <sup>9000</sup> <sup>10000</sup> -400

Time(msec)

<sup>0</sup> <sup>1000</sup> <sup>2000</sup> <sup>3000</sup> <sup>4000</sup> <sup>5000</sup> <sup>6000</sup> <sup>7000</sup> <sup>8000</sup> <sup>9000</sup> <sup>10000</sup> -400

Time(msec)

Based on Nonlinear System Identification for Power Quality Analysis 67

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

To emulate working conditions of PVGCS systems under environment changes (irradiance and temperature) affecting voltage and current inputs of inverters, six conditions of DC voltage variations and DC current variations. The six conditions are listed as Table 1. They are 3 conditions of a fixed DC current with DC low, medium and high voltage, i.e. , FCLV (Fixed Current Low Voltage), FCMV (Fixed Current Medium Voltage) and FCHV (Fixed Current High Voltage) which shown in Fig. 16. The other three corresponding conditions are a DC fixed voltage with DC low, medium and high current, i.e., FVLC (Fixed Voltage Low Current), FVMC (Fixed Voltage Medium Current), and FVHC (Fixed Voltage High

IacFV(A)

Fig. 16. AC voltage and current waveforms corresponding to FCLV, FCMV and FCHV

Fig. 17. AC voltage and current waveforms corresponding to FVLC, FVMC and FVHC

IacFA(A)

<sup>0</sup> <sup>1000</sup> <sup>2000</sup> <sup>3000</sup> <sup>4000</sup> <sup>5000</sup> <sup>6000</sup> <sup>7000</sup> <sup>8000</sup> <sup>9000</sup> <sup>10000</sup> -20

Time(msec)

<sup>0</sup> <sup>1000</sup> <sup>2000</sup> <sup>3000</sup> <sup>4000</sup> <sup>5000</sup> <sup>6000</sup> <sup>7000</sup> <sup>8000</sup> <sup>9000</sup> <sup>10000</sup> -40

Time(msec)

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.

Fig. 14. Experimental setup

Fig. 15. An inverter modeling using system identification process

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.

#### **4.1 Steady state conditions**

66 Electrical Generation and Distribution Systems and Power Quality Disturbances

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

Idc(t) Iac(t) Vac(t)

Pdc(t) <sup>X</sup> <sup>X</sup> Pac(t)

Load

Parameter Adjust Model Evaluation

MIMO Modeling

Model Estimation & Validation

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.

Vac\*(t)

Iac\*(t)

Vac(t) e(t) <sup>+</sup> -

Iac(t) <sup>+</sup> -

e(t)

voltage and current waveforms.

Fig. 14. Experimental setup

DC Supply Inverter

Vdc(t)

Fig. 15. An inverter modeling using system identification process

To emulate working conditions of PVGCS systems under environment changes (irradiance and temperature) affecting voltage and current inputs of inverters, six conditions of DC voltage variations and DC current variations. The six conditions are listed as Table 1. They are 3 conditions of a fixed DC current with DC low, medium and high voltage, i.e. , FCLV (Fixed Current Low Voltage), FCMV (Fixed Current Medium Voltage) and FCHV (Fixed Current High Voltage) which shown in Fig. 16. The other three corresponding conditions are a DC fixed voltage with DC low, medium and high current, i.e., FVLC (Fixed Voltage Low Current), FVMC (Fixed Voltage Medium Current), and FVHC (Fixed Voltage High Current) as shown in Fig.17.

Fig. 16. AC voltage and current waveforms corresponding to FCLV, FCMV and FCHV conditions

Fig. 17. AC voltage and current waveforms corresponding to FVLC, FVMC and FVHC conditions

Modeling of Photovoltaic Grid Connected Inverters


**Electrical parameters Voltage, current and power for**

Table 2. Inverter operations under step up/down conditions

**5. Results and discussion** 

Hammerstein-Wiener modeling.


Iacdown(A)

0

5

10

Vacdown(V)

Based on Nonlinear System Identification for Power Quality Analysis 69

<sup>0</sup> <sup>1000</sup> <sup>2000</sup> <sup>3000</sup> <sup>4000</sup> <sup>5000</sup> <sup>6000</sup> <sup>7000</sup> <sup>8000</sup> <sup>9000</sup> <sup>10000</sup> -400

Time(msec)

<sup>0</sup> <sup>1000</sup> <sup>2000</sup> <sup>3000</sup> <sup>4000</sup> <sup>5000</sup> <sup>6000</sup> <sup>7000</sup> <sup>8000</sup> <sup>9000</sup> <sup>10000</sup> -15

Time(msec)

**Voltage, current and power for transient step up conditions** 

Fig. 19. AC voltage and current waveforms under the step down transient condition

**transient step down conditions** 

AC output voltage (V) 220 220 220 220 AC output current ( A) 7 2 2 7 AC output power (W) 1540 440 440 1540

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

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)


Table 1. DC and AC parameters of an inverter under changing operating conditions

#### **4.2 Transient conditions**

Transient conditions are studied under two cases which composed of step up power transient and step down power transient. The step up condition is done by increasing power output from 440 to1,540 W, and the step down condition from 1,540 to 440 W, shown in Table 2. Power waveform data of the two conditions are divided in two groups, the first group is used to estimate model, the second group to validate model. Examples of captured voltage and current waveforms under the step-up power transient condition (440 W or 2 A) to 1540 W or 7A) and the step-down power transient condition (1540 W or 7A) to 440 W (2A) are shown in Fig. 18 and 19, respectively.

Fig. 18. AC voltage and current waveforms under the step up transient condition

1 FCLV 12 210 2520 11 220 2420 2 FCMV 12 240 2880 13 220 2860 3 FCHV 12 280 3360 15 220 3300 4 FVLC 2 235 470 2 220 440 5 FVMC 10 240 2,400 10 220 2,200 6 FVHC 21 245 5,145 23 220 5,060

Transient conditions are studied under two cases which composed of step up power transient and step down power transient. The step up condition is done by increasing power output from 440 to1,540 W, and the step down condition from 1,540 to 440 W, shown in Table 2. Power waveform data of the two conditions are divided in two groups, the first group is used to estimate model, the second group to validate model. Examples of captured voltage and current waveforms under the step-up power transient condition (440 W or 2 A) to 1540 W or 7A) and the step-down power transient condition (1540 W or 7A) to 440 W

<sup>0</sup> <sup>1000</sup> <sup>2000</sup> <sup>3000</sup> <sup>4000</sup> <sup>5000</sup> <sup>6000</sup> <sup>7000</sup> <sup>8000</sup> <sup>9000</sup> <sup>10000</sup> -400

Time(msec)

<sup>0</sup> <sup>1000</sup> <sup>2000</sup> <sup>3000</sup> <sup>4000</sup> <sup>5000</sup> <sup>6000</sup> <sup>7000</sup> <sup>8000</sup> <sup>9000</sup> <sup>10000</sup> -15

Time(msec)

Fig. 18. AC voltage and current waveforms under the step up transient condition

**Pdc (W)**

**Iac (A)**

**Vac (A)**

**Pac (VA)**

**Vdc (V)**

Table 1. DC and AC parameters of an inverter under changing operating conditions

**No. Case Idc** 

(2A) are shown in Fig. 18 and 19, respectively.



0

Iacup(A)

5

10

15

Vacup(V)

**4.2 Transient conditions** 

**(A)**

**Electrical parameters Voltage, current and power for transient step down conditions Voltage, current and power for transient step up conditions**  AC output voltage (V) 220 220 220 220 AC output current ( A) 7 2 2 7

AC output power (W) 1540 440 440 1540


Table 2. Inverter operations under step up/down conditions
