**2. Modelling of switch mode devices**

This section studies the general specifications of switch mode devices. We simulate a CFL ballast circuit in SPICE software. We also present the device model for a personal computer. Based on these models, we develop a general circuit model to simulate the behaviour of all switch mode capacitive devices. Without circuit simulation, it is not possible to provide an accurate model representation in the power system. In contrast with the models that are based on measuring and estimating the device characteristic, this approach gives much more accurate results. The accuracy of this approach can be chosen at the desired level.

#### **2.1 Simulation of CFL ballast circuit in SPICE**

The common 220V power system voltage is not enough to start the fluorescent lamps. Therefore, CFLs include a ballast circuit for providing the starting high voltage. In traditional fluorescent lamps, inductive ballasts are widely utilized. However, electronic ballasts which are used in CFLs have much better quality (Aiello et al., 2008). Electronic ballasts are composed of a rectifier and a DC-AC converter. Fig. 1 shows the general block diagram of a ballast circuit.

Fig. 1. Block diagram of a CFL ballast circuit. Figure courtesy of (Sasaki, 1994).

Several circuits are simulated in SPICE software for this project. Fig. 2 shows one sample CFL ballast circuit model in SPICE. This circuit is similar to that of (Sasaki, 1994) with slight changes. The input full wave rectifier and the large input capacitor make the current have narrow high peaks at short intervals and almost zero value elsewhere. Fig. 3 shows the output voltage and current of the circuit in Fig. 2.

Frequency analysis shows that the CFL current is made up of odd harmonic components of the main frequency (50 or 60 Hz). The CFL is modelled by a number of current sources with

In this chapter we review our novel approach for studying the effect of switch mode devices and present a novel stochastic modelling approach for analysing the behaviour of the power system in the presence of switch mode devices. We also study the major KPI of the power system and study how these KPI will be affected by adding the current harmonics. Section 2 presents how we obtain an accurate model for CFL based on circuit simulation. This section also defines a general circuit model for the harmonic generating devices. Section 3 presents our novel approach for stochastic modelling of the power system behaviour. In section 4 we summarize the major power system KPI on both substation and consumer sides. We also discuss how the switch mode devices may affect the devices on each side. Section 5 presents our approach for simulating the power system behaviour. Conclusion and discussion are

This section studies the general specifications of switch mode devices. We simulate a CFL ballast circuit in SPICE software. We also present the device model for a personal computer. Based on these models, we develop a general circuit model to simulate the behaviour of all switch mode capacitive devices. Without circuit simulation, it is not possible to provide an accurate model representation in the power system. In contrast with the models that are based on measuring and estimating the device characteristic, this approach gives much more accurate results. The accuracy of this approach can be chosen at the desired level.

The common 220V power system voltage is not enough to start the fluorescent lamps. Therefore, CFLs include a ballast circuit for providing the starting high voltage. In traditional fluorescent lamps, inductive ballasts are widely utilized. However, electronic ballasts which are used in CFLs have much better quality (Aiello et al., 2008). Electronic ballasts are composed of a rectifier and a DC-AC converter. Fig. 1 shows the general block

Fig. 1. Block diagram of a CFL ballast circuit. Figure courtesy of (Sasaki, 1994).

Several circuits are simulated in SPICE software for this project. Fig. 2 shows one sample CFL ballast circuit model in SPICE. This circuit is similar to that of (Sasaki, 1994) with slight changes. The input full wave rectifier and the large input capacitor make the current have narrow high peaks at short intervals and almost zero value elsewhere. Fig. 3 shows the

Frequency analysis shows that the CFL current is made up of odd harmonic components of the main frequency (50 or 60 Hz). The CFL is modelled by a number of current sources with

presented in section 6.

diagram of a ballast circuit.

**2. Modelling of switch mode devices** 

**2.1 Simulation of CFL ballast circuit in SPICE** 

output voltage and current of the circuit in Fig. 2.

the proper harmonic values. Equation 1 shows the mathematical model for a CFL when the voltage is assumed to be a cosine function.

$$\begin{aligned} v\_{\text{cell}} &= V \cos(2\pi ft) \\ \dot{I}\_{\text{cell}} &= \sum\_{n=0}^{n} I\_{2n+1} \cos[2\pi (2n+1)ft + \Phi\_{2n+1}] \approx \sum\_{n=0}^{4} I\_{2n+1} \cos[2\pi (2n+1)ft + \Phi\_{2n+1}] \end{aligned} \tag{1}$$

The more the number of harmonics is, the more accurate the model will be. In this study we use the first five odd harmonics (1, 3, 5, 7, and 9). A schematic of the model is shown in Fig. 4. The power factor of this circuit is 93%. In order for having a flexible model for different market suppliers, the power factor is chosen flexible in the simulation experiments.

Fig. 2. Simulation of a sample ballast circuit in SPICE.

Fig. 3. Sinusoidal voltage and resulting current waveshape for a sample CFL ballast circuit.

Stochastic Analysis of the Effect of Using Harmonic Generators in Power Systems 199

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

In the above equation, *Φmax* is the maximum phase shift from the theoretical phase lag *Φc*. If

Where *φ* and *σ* are the mean and variance of the phase lag. The current value for a large

2 1 2 1

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

cos 2 (2 1) (2 1) *M M m n n m*

2 1 2 1 <sup>0</sup> cos 2 (2 1) (2 1) *m n <sup>n</sup> <sup>m</sup> <sup>n</sup>*

*c m c m P*

<sup>1</sup> ( ) <sup>2</sup> *P e <sup>m</sup>*

*i i I n ft n* 

*i I* 

manufacturer's datasheet. In other words for the uniform distribution:

max max

the probability distribution is Gaussian, *ΔΦm* is obtained from equation 4:

number *M* of CFLs with the above specifications is equal to *i* in equation 3:

4

obtained for different values of phase range for the electronic devices.

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

0 0 0

*m m n*

<sup>4</sup>

*n ft n*

(2)

<sup>1</sup> , ( ) <sup>2</sup>

<sup>2</sup> <sup>2</sup> /2

(5)

 

2

(4)

max

(3)

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

The values of the current and phase in equation 1 are summarized in Table 1 for the circuit in Fig. 2.


Table 1. Peak value and phase of the current harmonics for the sample CFL of Fig. 2.

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

## **2.2 Circuit model for other electronic devices**

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

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