**6.2.1 Root mean square**

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

$$\begin{aligned} V\_{rms} &= \sqrt{\frac{1}{T} \int\_0^T v^2(t)dt} \\ I\_{rms} &= \sqrt{\frac{1}{T} \int\_0^T v^2(t)dt} \end{aligned} \tag{17}$$

Modeling of Photovoltaic Grid Connected Inverters

power Q (Var) are related through the equations

**6.2.4 Harmonic calculation** 

fundamental current

and the transient step down condition

Equations 25 and 26, respectively.

Based on Nonlinear System Identification for Power Quality Analysis 75

The power factor, the apparent power S (VA), the active power P (W), and the reactive

*P W PF S VA* = = φ

*P S* = cos

*Q S* = sin

Total harmonic distortion of voltage (THDv) and current (THDi) can be calculated by the

2 ( ) 2 1( ) % 100% *h rms*

*I THD x I*

*rms*

2 ( ) 2 1( ) % 100% *h rms*

*V THD x V*

*rms*

Where Vh (rms) is RMS value of h th voltage harmonic , Ih (rms) RMS value of h th current harmonic, V1 (rms) RMS value of fundamental voltage and I1 (rms) RMS value of

**Parameter Steady state FVHC condition Transient step down condition** 

Vrms (V) 218.31 218.04 0.12 217.64 218.20 -0.26 Irms (A) 23.10 23.21 -0.48 4.47 4.45 0.45 Frequency (Hz) 50 50 0.00 50.00 50.00 0.00 Power Factor 0.99 0.99 0.00 0.99 0.99 0.00 THDv (%) 1.15 1.2 -4.35 1.18 1.24 -5.08 THDi (%) 3.25 3.12 4.00 3.53 3.68 -4.25 S (VA) 5044.38 5060.7 -0.32 972.85 970.99 0.19 P (W) 4993.94 5010.1 -0.32 963.12 961.28 0.19 Q (Var) 711.59 713.85 -0.32 137.24 136.97 0.19 V p.u. 0.99 0.99 0.00 0.98 0.99 -1.02 Table 4. Comparison of measured and modeled electrical parameters of the FVHC condition

 **Experimental Modeling % Error Experimental Modeling % Error** 

*h*

*h*

<sup>=</sup> = 

∞

<sup>=</sup> = 

∞

*i*

*v*

( ) cos

( )

φ

φ

(21)

(23)

(24)

(25)

(26)

*S VI* = ′ (22)

**6.2.3 Power factor, apparent power, active power and reactive power** 

$$\begin{aligned} V\_m &= \sqrt{2}V\_{rms} \\ I\_m &= \sqrt{2}I\_{rms} \end{aligned} \tag{18}$$

Fig. 23. Prediction and experiment results of AC output current under a transient step down condition

#### **6.2.2 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).

$$
\phi = \frac{\Delta t (ms) \cdot 360^{\circ}}{T \text{ ms}} \tag{19}
$$

$$f = \frac{1}{T} \tag{20}$$

Δ*t* is time lagging or leading between voltage and current (ms), T is the waveform period.

#### **6.2.3 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

$$PF = \cos\phi = \frac{P\text{(W)}}{S\text{(VA)}}\tag{21}$$

$$S = VI'\tag{22}$$

$$P = S \cos \phi \tag{23}$$

$$Q = S \sin \phi \tag{24}$$

#### **6.2.4 Harmonic calculation**

74 Electrical Generation and Distribution Systems and Power Quality Disturbances

2 0

(17)

Prediction Experimental

= (18)

<sup>1</sup> ( )

*T*

*V v t dt T*

*rms*

=

=

*rms*

2 0

<sup>1</sup> ( )

*T*

2 2 *m rms m rms*

AC current

<sup>0</sup> 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 -15

We calculate a phase shift between voltage and current from the equation (19), and the

Δ ⋅ <sup>=</sup>

*T ms*

Δ*t* is time lagging or leading between voltage and current (ms), T is the waveform period.

(19)

<sup>1</sup> *<sup>f</sup> <sup>T</sup>* <sup>=</sup> (20)

Fig. 23. Prediction and experiment results of AC output current under a transient step

φ

time(msec)


down condition

**6.2.2 Period, frequency and phase angle** 

*t ms* ( ) 360

frequency (f) from equation (20).


0

Iac A)

5

10

15

*V V I I* =

*I v t dt T*

> Total harmonic distortion of voltage (THDv) and current (THDi) can be calculated by the Equations 25 and 26, respectively.

$$\sqrt[n]{THD\_i} = \frac{\sqrt{\sum\_{h=2}^{2} I\_{h(rms)}^2}}{I\_{1(rms)}} \ge 100\% \tag{25}$$

$$\%THD\_v = \frac{\sqrt{\sum\_{h=2}^{n} V\_{h(rms)}^2}}{V\_{1(rms)}} \ge 100\,\% \tag{26}$$

Where Vh (rms) is RMS value of h th voltage harmonic , Ih (rms) RMS value of h th current harmonic, V1 (rms) RMS value of fundamental voltage and I1 (rms) RMS value of fundamental current


Table 4. Comparison of measured and modeled electrical parameters of the FVHC condition and the transient step down condition

Modeling of Photovoltaic Grid Connected Inverters

**Type Typical** 

**1.Transient**  - Impulsive - Oscillation - low frequency - medium frequency - high frequency

**2.Short Duration**  - voltage sag - voltage swell

**3. Long Duration**  - overvoltage (OV) - undervoltage (UV) - voltage Interruption

**5.Waveform distortion** 


**6.Voltage fluctuation** 

**7.Frequency variation** - Overfrequency - Underfrequency

**7. Conclusions** 

system electromagnetic phenomena

**Duration** 

5 ns – 0.1ms

0.3-50 ms 5-20 ms 0-5 ms

10 ms-1 min 10 ms-1 min

Steady state Steady state Steady state Steady state Steady state Steady state

< 10 s -

> 1 min > 1 min > 1 min

Based on Nonlinear System Identification for Power Quality Analysis 77

**Typical Spectral Content** 

< 5 kHz 5-500 kHz 0.5–5 MHz



0-100th 0-100th 0-6 kHz < 200 kHz

Broad band

± 3 Hz > 53 Hz < 47 Hz **Steady State FVHC**

**Transient Step down**

0.99 pu. 0.99 pu. Pass

0.99 pu. 0.99 pu. Pass

1.24 % 3.68 % - - - -

50 50 Pass

Pass Pass - - - -


1.20 % 3.25 % - - - -

**Result** 

**Typical Voltage Magnitude**

0.4 pu. 0-8 pu. 0.4 pu.

0.1-0.9 pu. 1.1-1.8 pu.

> 1.1 pu. < 0.9 pu. 0 pu.

**4. Voltage Unbalance** Steady state 0.5-2% - - - -

< 5% THD < 20% THD 0-2% 0-0.1% - 0-1%


Table 5. Comparison modeling output with Categories and Typical Characteristics of power

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

We next demonstrate accuracy and precision of power quality prediction from modeling. Table 4 shows the comparisons. Two representative cases mentioned above are given, i.e. the steady state Fix Voltage High Current (FVHC) condition, and the transient step down condition. Comparison of THDs is shown in Fig. 23. Agreements between experiments and modeling results are good.

Fig. 24. Comparison of measured and modeled THD of AC current of the transient step down condition

#### **6.3 Power quality problem analysis**

The power quality phenomena are classified in terms of typical duration, typical voltage magnitude and typical spectral content. They can be broken down into 7 groups on transient, short duration voltage, long duration voltage, voltage unbalance, waveform distortion, voltage fluctuation or flicker, frequency variation. Comparisons of the Standard values and modeled outputs of the FVHC and the transient step down conditions are shown in Table 5. The results show that under both the steady state and the transient cases, good power quality is achieved from the PVGCS.

We next demonstrate accuracy and precision of power quality prediction from modeling. Table 4 shows the comparisons. Two representative cases mentioned above are given, i.e. the steady state Fix Voltage High Current (FVHC) condition, and the transient step down condition. Comparison of THDs is shown in Fig. 23. Agreements between experiments and

> Prediction Model Experiment

0 50 100 150 200 250 300

Frequency (Hertz)

The power quality phenomena are classified in terms of typical duration, typical voltage magnitude and typical spectral content. They can be broken down into 7 groups on transient, short duration voltage, long duration voltage, voltage unbalance, waveform distortion, voltage fluctuation or flicker, frequency variation. Comparisons of the Standard values and modeled outputs of the FVHC and the transient step down conditions are shown in Table 5. The results show that under both the steady state and the transient cases, good

Fig. 24. Comparison of measured and modeled THD of AC current of the transient step

modeling results are good.

THDexp is 3.53 %

THDmodel is 3.68 %

0

down condition

**6.3 Power quality problem analysis** 

power quality is achieved from the PVGCS.

1

2

M

agnitude current (A)

3

4

5

6


Table 5. Comparison modeling output with Categories and Typical Characteristics of power system electromagnetic phenomena
