**3. Stochastic modelling of switch mode devices in power system**

Phase of a harmonic generating device is not a constant value. But it is a random variable that varies in a specific range that can be provided by the manufacturer. Therefore, the model in equation 1 will be modified to that of equation 2.

The values of the current and phase in equation 1 are summarized in Table 1 for the circuit

Personal computers and other electronic equipment such as printers, etc. generate current harmonics in the power system too, because they all include a rectifier. The harmonic components of personal computers are calculated and provided in the literature (Key et al., 1996). Fig. 5 shows the relative value of these components. Therefore, we can use a similar

**Current Harmonic First Third Fifth Seventh Ninth Peak I2n+1 (A)** 0.2 0.182 0.162 0.138 0.112 **Phase Φ2n+1 (Rad)** 0.260 3.499 0.609 4.000 0.799 Table 1. Peak value and phase of the current harmonics for the sample CFL of Fig. 2.

Fig. 4. Circuit model of a switch mode device.

We name the overall current phase lag as central phase lag *Φc*.

model to that of Fig. 4 for modelling such electronic devices.

Fig. 5. Relative values of the current harmonics for a personal computer.

model in equation 1 will be modified to that of equation 2.

**3. Stochastic modelling of switch mode devices in power system** 

Phase of a harmonic generating device is not a constant value. But it is a random variable that varies in a specific range that can be provided by the manufacturer. Therefore, the

**2.2 Circuit model for other electronic devices** 

in Fig. 2.

$$\dot{q}\_{\pi} = \sum\_{n=0}^{4} I\_{2n+1} \cos[2\pi(2n+1)\theta + \Phi\_{2n+1} + (2n+1)\times\Delta\Phi\_{\pi}] \tag{2}$$

In this model, *m* is the device number in the network. In practice, the total number of devices *M* is a large number. For each device we have a phase shift *ΔΦm* from the central phase. Distribution of *ΔΦm* can be assumed to be uniform or Gaussian depending on the manufacturer's datasheet. In other words for the uniform distribution:

$$
\left(\Phi\_c - \Phi\_{\text{max}} < \Delta\Phi\_m < \Phi\_c + \Phi\_{\text{max}}\right) \qquad P(\Delta\Phi\_m = \phi) = \frac{1}{2\Phi\_{\text{max}}}\tag{3}
$$

In the above equation, *Φmax* is the maximum phase shift from the theoretical phase lag *Φc*. If the probability distribution is Gaussian, *ΔΦm* is obtained from equation 4:

$$P(\Delta\Phi\_w = \phi) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{\left(\phi - \overline{\phi}\right)^2 / 2\sigma^2} \tag{4}$$

Where *φ* and *σ* are the mean and variance of the phase lag. The current value for a large number *M* of CFLs with the above specifications is equal to *i* in equation 3:

$$\mathbf{i} = \sum\_{m=0}^{M} \mathbf{i}\_m \cdot \sum\_{n=0}^{M} \sum\_{n=0}^{4} I\_{2n+1} \cos \left[ 2\pi (2n+1) \, ft + \Phi\_{2n+1} + (2n+1) \times \Delta \Phi\_n \right] \tag{5}$$

Finding the probability density function, expectation and variance of current in the above equation is complicated (if possible!). Instead, we rely on numerical simulation to find the *pdf* of power system current. In a sample experiment, our power system is composed of a thousand CFLs. The average phase lag of these CFLs is fifteen degrees and has a uniform distribution of range *15±10* degrees. We use a Parzen window (Duda et al., 2007) approach to find the *pdf* of the peak output current. Fig. 6 shows the result. The mean value and standard deviation of the current in this experiment are 611.5 and 2.45. We may notice that if the variance in the phase shift is not considered, meaning that we do not assume the CFL phase to be a random variable, the peak current will be equal to 701 Amperes. When we dimension the network, this means at least 13% more capacity. Different results can be obtained for different values of phase range for the electronic devices.

Fig. 6. Relative values of the current harmonics for a personal computer.

Stochastic Analysis of the Effect of Using Harmonic Generators in Power Systems 201

This factor may be calculated for voltage, current or power. Current THD is normally much

Crest Factor is the indicator that shows the ratio of the peak value of the wave shape to its

*x*

This factor is especially important for the analysis of the effect on protective and control

Power Factor is caused by the input rectifying capacitor of the ballast circuit. The capacitive

This section studies the effect electronic devices on the substation equipments, which provide the low voltage power to the end users and can be classified to the following

Transformers are used in the distribution system in order to change the levels of voltage and current in the low voltage scales. These may also include the power and instrumentation transformers and Auto-Boosters. In transformers, both the core and the wires are sensitive to the change of the power KPI. The harmonics have the following different effects on

In equation 8, *PN* is the "no-load" loss and *PLL* is the "full-load" loss. The no-load loss depends on the voltage and core material. The full load loss is defined as in equation 9:

In equation 9, *PDC* is the DC resistance loss, *PEC* is the eddy current loss and *POSL* is the stray loss. Eddy current, which is proportional to the square of frequency, is caused by skin effect and proximity effect. Therefore, the current harmonics increase the eddy current loss dramatically. This increase results in the increase in temperature and hence reducing the

*Lifetime* of a transformer depends on the functioning situations such as loading percentage and functioning temperature. Current harmonic components can increase the RMS value of

*C*

*peak rms*

*<sup>x</sup>* (7)

*PPP T N LL* (8)

*PPPP LL DC EC OSL* (9)

higher than the voltage THD for electronic devices and is more than 100%.

characteristic of CFL is an advantage in systems with a high inductive load.

average value in a time interval (rms)1, as shown in equation 7.

**4.1.2 Crest Factor** 

equipment such as relays.

**4.2 Power substation equipment** 

*Transformer loss*, which is obtained as in equation 8:

**4.1.3 Power Factor** 

**4.2.1 Transformers** 

transformer lifetime (Ashok).

1 Root Mean Square

categories:

transformers.
