**3.5 System indices**

148 Electrical Generation and Distribution Systems and Power Quality Disturbances

A sensitive setting will result in a large number of shallow events (with a low voltage sag

The sag event frequency index at a particular location and period is suggested as the

The System sag count index is the total number of qualified voltage sag events over the number of monitor locations. By the expression qualifying events, it implies a voltage less than 90%, with event duration limited to 15 cycles and energy greater or equal to 100.

Non-rectangular events are events in which the voltage magnitude varies significantly during the event. A method to include non-rectangular events in the voltage-sag coordination chart is also applicable according to the IEEE defined standard (IEEE Std.493, 1997). Alternatively, the function value can be defined as the number of times per year that

EPRI-Electrotek mentions that each phase of each ms variation measurement may contain multiple components (Thallam, 2000). Consequently, these phase rectangular voltage sag measurements are easily characterized with respect to magnitude and duration. Approximately 10% of the events are non-rectangular. These events are much more difficult to characterize because no single magnitude-duration pair completely represent the phase

The method suggested for calculating the indices used by EPRI-Electrotek is called the ''Specified Voltage'' method. This method designates the duration as the period of time that the rms voltage exceeds a specified threshold voltage level used to characterize the disturbance.' The consequence of this method is that an event may have a different duration

> 0.000 0.167 0.333 0.500 0.667 0.833 1.000 1.167 1.333 1.500 1.667 **Time (seconds)**

Most of the single site indices relate the magnitude and duration of the sag and the number of events. These events can be grouped in order to make their counting easier and more

**T80%**

**T50%**

**T10%**

when being assessed at different voltage thresholds as shown in Figure 5.

Measurement Event #1

Fig. 5. Illustration of "specified voltage" characterization

the RMS voltage is less than the given magnitude for longer than the given duration.

number of qualified sag events at a location and period (Thallam & Koellner, 2003).

energy) and this in a lower value for AVSEI.

**3.4.3 Non-rectangular events** 

measurement.

0

20

40

60

80

**%Volts**

100

120

140

System Indices are typically a weighted average of the single-site indices obtained for all or a number of sites within the system. The difficulty lies in the determination of the weighting factors. In order to assess any indices for the system, first monitoring of the quality of supply must take place. When the Electric Power Research Institute (EPRI)-Distribution Power Quality (DPQ) program placed monitoring equipment on one hundred feeders, these feeders needed to adequately represent the range of characteristics seen on distribution systems. This required the researchers to use a controlled selection process to ensure that both common and uncommon characteristics of the national distribution systems were well represented in the study sample. Thus a level of randomness is required. Many devices are susceptible to only the magnitude of the variation. Others are susceptible to the combination of magnitude and durationOne consideration in establishing a voltage sag index is that the less expensive a measuring device is, the more likely it will be applied at many locations, more completely representing the voltage quality electricity users are experiencing.

Power Quality and Voltage Sag Indices in Electrical Power Systems 151

Notice that SIARFIx is not defined for a threshold value of x = 10%. This is because IEEE Std.

In the same way that SIARFlx is defined for instantaneous variations, SMARFlx is defined for variations having a duration in the range of 30 cycles to 3 seconds for sags and swells,

*x*

*x* = percentage of nominal rms voltage threshold; possible values - 140, 120, 110, 90, 80, 70,

*NM =*number of customers experiencing momentary voltage deviations with magnitudes

STARFIx is defined for temporary variations, which have a duration in the range of 3 - 60

*x*

*x* = percentage of nominal rms voltage threshold; possible values - 140, 120, 110, 90, 80, 70,

*NTi* = number of customers experiencing temporary voltage deviations with magnitudes

As power networks become more interconnected and complex to analyse, the need for power quality indices to be easily assessable, and representative of the disturbance they characterise with minimum parameters, arises. This section has presented the various Voltage sag indices available in literature. Most of these indices are characterized through the sag duration and magnitude. To demonstrate the theory of equipment compatibility, with the use of the System Average RMS Variation Frequency Index, various power

Electricity distribution companies need to assess the quality of service provided to customers. Hence, a common index terminology for discussion and contracting is useful. Future voltage sag indices need to be adjustable and adaptable to incorporate future changes in technology and system parameters. This would enable implementation of indices

In this section of the chapter, the mathematical formulation of two voltage sag indices ( ξ and ζ1,2 ) is introduced as well as the results of the investigation towards their accuracy establishment. The Mathematical equations describing the development of a Combined Voltage Index (CVI) are also presented as well as the results obtained by the verification process. The index supervises the power quality of a system, through characterising voltage

*i*

*<sup>N</sup>* <sup>=</sup> (11)

*<sup>N</sup>* <sup>=</sup> (12)

*T NM*

*i*

*T NT*

1159, 1995, does not define an instantaneous duration category for interruptions. **c. System Momentary Average RMS (Variation) Frequency Indexvortage (SMARFIx)** 

*SMARFI*

**d. System Temporary Average RMS (Variation) Frequency Indexvortage ( STARFIx)** 

*STARFI*

above X% for X >100 or below X% for X <100 due to measurement event *i.* 

above x% for x >100 or below x% for x <100 due to measurement event *i.* 

into the next generation of power system planning software.

**4. Voltage sag mathematical indices** 

and in the range of 0.5 cycles to 3 seconds for interruptions.

50, and 10

seconds.

50, and 10.

acceptability curves were used.

With this consideration in mind, sag monitoring devices are generally classified into l*ess expensive devices* that can monitor the gross limits of the voltage sag, and *more expensive devices* that can sample finer detail such as the voltage-time area and other features that more fully characterize the sag.

The sag limit device senses the depth, of the voltage sag. The sag area device can sample the sag in sufficient detail to plot the time profile of the sag. With this detail it could give a much more accurate picture of the total sag area, in volt-seconds, as well as the gross limits; the retained voltage, Vr, is also shown.

The developed RMS variation indices proposed by EPRI-Electrotek, are designed to aid in the assessment of service quality for a specified circuit area. The indices are defined such that they may be applied to systems of varying size (Bollen, 2001).Values can be calculated for various parts of the distribution system and compared to values calculated for the entire system.

Accordingly, the four indices presented assess RMS variation magnitude and the combination of magnitude and duration.

#### **a. System Average RMS (Variation) Frequency Indexvoltage ( SARFIx)**

 SARFIx represents the average number of specified rms variation measurement events that occurred over the assessment period per customer served, where the specified disturbances are those with a magnitude less than x for sags or a magnitude greater than x for swells. Notice that SARFI is defined with respect to the voltage threshold 'x' (Sabin, 2000).

$$SARFI\_x = \frac{\sum N\_i}{N\_T} \tag{9}$$

where

*x* = percentage of nominal rms voltage threshold; possible values - 140, 120, 110, 90, 80, 70, 50, and 10

*Ni* = number of customers experiencing short-duration voltage deviations with magnitudes above x% for x >100 or below x% for x <100 due to measurement event *i*

*NT* =number of customers served from the section of the system to be assessed

#### **b. System Instantaneous Average RMS (Variation) frequency Indexvoltage( SIARFIx)**

SIARFIx represents the average number of specified instantaneous rms variation measurement events that occurred over the assessment period per customer served. The specified disturbances are those with a magnitude less than x for sags or a magnitude greater than x for swells and duration in the range of 0.5 - 30 cycles.

$$SIARFI\_x = \frac{\sum NI\_i}{N\_T} \tag{10}$$

Where:

*x* = percentage of nominal rms voltage threshold; possible values - 140, 120, 110, 90, 80, 70, and 50

*NIi* = number of customers experiencing instantaneous voltage deviations with magnitudes above x% For x>100 or below x% for x <100 due to measurement event *i* 

With this consideration in mind, sag monitoring devices are generally classified into l*ess expensive devices* that can monitor the gross limits of the voltage sag, and *more expensive devices* that can sample finer detail such as the voltage-time area and other features that

The sag limit device senses the depth, of the voltage sag. The sag area device can sample the sag in sufficient detail to plot the time profile of the sag. With this detail it could give a much more accurate picture of the total sag area, in volt-seconds, as well as the gross limits;

The developed RMS variation indices proposed by EPRI-Electrotek, are designed to aid in the assessment of service quality for a specified circuit area. The indices are defined such that they may be applied to systems of varying size (Bollen, 2001).Values can be calculated for various parts of the distribution system and compared to values calculated for the entire

Accordingly, the four indices presented assess RMS variation magnitude and the

 SARFIx represents the average number of specified rms variation measurement events that occurred over the assessment period per customer served, where the specified disturbances are those with a magnitude less than x for sags or a magnitude greater than x for swells. Notice that SARFI is defined with respect to the voltage threshold 'x' (Sabin,

*x*

*x* = percentage of nominal rms voltage threshold; possible values - 140, 120, 110, 90, 80, 70,

*Ni* = number of customers experiencing short-duration voltage deviations with magnitudes

SIARFIx represents the average number of specified instantaneous rms variation measurement events that occurred over the assessment period per customer served. The specified disturbances are those with a magnitude less than x for sags or a magnitude

*x*

*x* = percentage of nominal rms voltage threshold; possible values - 140, 120, 110, 90, 80, 70,

*NIi* = number of customers experiencing instantaneous voltage deviations with magnitudes

*SIARFI*

**b. System Instantaneous Average RMS (Variation) frequency Indexvoltage( SIARFIx)** 

*SARFI*

*i*

*i*

*T NI*

*<sup>N</sup>* <sup>=</sup> (9)

*<sup>N</sup>* <sup>=</sup> (10)

*T N*

**a. System Average RMS (Variation) Frequency Indexvoltage ( SARFIx)** 

above x% for x >100 or below x% for x <100 due to measurement event *i NT* =number of customers served from the section of the system to be assessed

greater than x for swells and duration in the range of 0.5 - 30 cycles.

above x% For x>100 or below x% for x <100 due to measurement event *i* 

more fully characterize the sag.

system.

2000).

where

50, and 10

Where:

and 50

the retained voltage, Vr, is also shown.

combination of magnitude and duration.

Notice that SIARFIx is not defined for a threshold value of x = 10%. This is because IEEE Std. 1159, 1995, does not define an instantaneous duration category for interruptions.

#### **c. System Momentary Average RMS (Variation) Frequency Indexvortage (SMARFIx)**

In the same way that SIARFlx is defined for instantaneous variations, SMARFlx is defined for variations having a duration in the range of 30 cycles to 3 seconds for sags and swells, and in the range of 0.5 cycles to 3 seconds for interruptions.

$$\text{SMARFI}\_{\text{x}} = \frac{\sum \text{NM}\_{i}}{N\_{r}} \tag{11}$$

*x* = percentage of nominal rms voltage threshold; possible values - 140, 120, 110, 90, 80, 70, 50, and 10

*NM =*number of customers experiencing momentary voltage deviations with magnitudes above X% for X >100 or below X% for X <100 due to measurement event *i.* 

#### **d. System Temporary Average RMS (Variation) Frequency Indexvortage ( STARFIx)**

 STARFIx is defined for temporary variations, which have a duration in the range of 3 - 60 seconds.

$$\text{STARFI}\_{\text{x}} = \frac{\sum NT\_{i}}{N\_{T}} \tag{12}$$

*x* = percentage of nominal rms voltage threshold; possible values - 140, 120, 110, 90, 80, 70, 50, and 10.

*NTi* = number of customers experiencing temporary voltage deviations with magnitudes above x% for x >100 or below x% for x <100 due to measurement event *i.* 

As power networks become more interconnected and complex to analyse, the need for power quality indices to be easily assessable, and representative of the disturbance they characterise with minimum parameters, arises. This section has presented the various Voltage sag indices available in literature. Most of these indices are characterized through the sag duration and magnitude. To demonstrate the theory of equipment compatibility, with the use of the System Average RMS Variation Frequency Index, various power acceptability curves were used.

Electricity distribution companies need to assess the quality of service provided to customers. Hence, a common index terminology for discussion and contracting is useful. Future voltage sag indices need to be adjustable and adaptable to incorporate future changes in technology and system parameters. This would enable implementation of indices into the next generation of power system planning software.

#### **4. Voltage sag mathematical indices**

In this section of the chapter, the mathematical formulation of two voltage sag indices ( ξ and ζ1,2 ) is introduced as well as the results of the investigation towards their accuracy establishment. The Mathematical equations describing the development of a Combined Voltage Index (CVI) are also presented as well as the results obtained by the verification process. The index supervises the power quality of a system, through characterising voltage sags. The voltage sags are caused by an increase in reactive demand due to induction motor starting.

A feeder can be modeled by an equivalent two-port network, as shown in Figure 6.

The sending end voltage and current of the system can be represented by equations 13 and 14.

$$\text{AL}\_s \angle \mathcal{S}\_s = \text{ALI}\_r \angle \mathcal{S}\_r + BI\_r \tag{13}$$

Power Quality and Voltage Sag Indices in Electrical Power Systems 153

Equation 18 provides a tool to calculate the voltage sag ,as a per unit value of the sending

However, since the equation is obtained through a quadratic equation, it has two solutions. ζ1 will be valid for a specific range of parameters. In the same way ζ2 will be valid for a different range of parameters. The validity of the two solutions, ζ1 and ζ2, with the use of various line X/R ratios is investigated in (Nouri et al., 2006). X/R ratio varies from Distribution to Transmission according to the cables used for the corresponding voltages. Typical values of X/R ratio are: for a 33kV overhead line -1.4, for a 132kV overhead line -2.4, for a 275kV overhead line -8.5, for a 400kV overhead line -15. A distribution line example is

According to the parameters either ζ1 or ζ2 will be the correct answer which should match

when both solutions are identical. However, in practice a gap develops when both solutions approach the Ur axis, where none of the two solutions accurately represent the receiving

Gap

The distance between the two curves at the point of the gap can be defined by equation 19.

( )

In order to fully investigate the range of accuracy of the two solutions, X/R ratio values of 1 to 15 are used for the line impedance. Since the receiving end power varies according to the load, five loading conditions are used in the investigation. Each loading consists of induction motors. The loads are switched in the system one by one to create the effect of

> θ

each X/R ratio, starting from one up to fifteen in steps of one. The results can be seen in

θδ

*r s*

*ZPCos Cos*

+− −

 δ

*Cos*

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

θ

ζ −ζ

practical

ζ1

Ur

1

cos is calculated for five different loadings and

θ

2

− = (19)

*s*

*U*

4 *<sup>r</sup>*

 θ

cos is equal to zero,

end voltage, through angles and power demand.

the IEEE34, 24.9kV overhead line with X/R ratio of 0.441.

The point of intersection of Ur with ζ1 and ζ2, occurs when ( ) 1 2

practical

1 2

supplying minimum load (one motor) and maximum load (five motors).

ζ −ζ

ζ ζ ζ2

the receiving end voltage.

voltage Ur, as seen in Figure 7.

Fig. 7. Developed gap of inaccuracy

Using MathCad, the value of ( ) 1 2

Figure 8 (Nouri et al., 2006).

$$I\_s = \mathbb{C}\mathbb{U}I\_r\mathbb{Z}\delta\_r + \text{DI}\_r\tag{14}$$

Where Us is the sending end voltage, Is the sending end current, Ur the receiving end voltage, Ir the receiving end current, δs the sending end voltage angle, δr the receiving end voltage angle, and A,B,C, D are the two port network constants. For a short length line, corresponding to distribution network, the two port network parameters can be approximated as: A=D=1, B= *Z*∠θ , C=0.Where Z is the transmission line impedance vector magnitude, and θ the transmission line impedance vector angle.

Fig. 6. The equivalent two port network model.

The line power flow, for the active power at the sending and receiving end of the line, can be described by (15) and (16).

$$P\_s = \frac{\text{U}\_s}{Z} \text{Cov}(\theta) - \frac{\text{U}\_s \text{U}\_r}{Z} \text{Cov}(\theta + \delta\_s - \delta\_r) \tag{15}$$

$$P\_r = \frac{\text{Lls} \, l \, l \, r}{Z} \text{Cost}(\theta + \delta r - \delta s) - \frac{\text{L} \, l \, r^2}{Z} \text{Cost}(\theta) \tag{16}$$

#### **4.1 'ζ' index**

If the index ζ signifies the voltage magnitude during the sag as a per unit function of the sending voltage (Ur=ζUs), and is substituted in equation 16, equation 17 yields (Polycarpou & Nouri, 2005).

$$\frac{\left(\mathcal{L}\boldsymbol{U}\_{s}\right)^{2}}{Z}\text{Cov}(\boldsymbol{\theta} + \boldsymbol{\delta}\_{r} - \boldsymbol{\delta}\_{s}) - \frac{\left(\mathcal{U}\_{s}\boldsymbol{\zeta}\right)^{2}}{Z}\text{Cov}(\boldsymbol{\theta}) - P\_{r} = 0\tag{17}$$

Thus the solution of the second order equation, resulting from (17), can be calculated using equation (18).

$$\mathcal{L}\_{1,2} = \frac{\text{Cov}\left(\theta + \delta\_r - \delta\_s\right) \pm \left(\text{Cov}^2\left(\theta + \delta\_r - \delta\_s\right) - \frac{4Z P\_r \text{Cov}\theta}{\text{U}\_s}\right)^{\frac{1}{2}}}{2 \text{Cov}\theta} \tag{18}$$

sags. The voltage sags are caused by an increase in reactive demand due to induction motor

The sending end voltage and current of the system can be represented by equations 13 and 14.

*U AU BI ss rr r* ∠= ∠+

*s rr r I CU DI* = ∠+ δ

Where Us is the sending end voltage, Is the sending end current, Ur the receiving end voltage, Ir the receiving end current, δs the sending end voltage angle, δr the receiving end voltage angle, and A,B,C, D are the two port network constants. For a short length line, corresponding to distribution network, the two port network parameters can be

θ

( ) <sup>2</sup>

*U UU P Cos Cos Z Z*

( ) *<sup>r</sup> UsUr Ur P Cos r s Cos Z Z* = +− − θδ

2 2

 δ+− −

*Z Z*

θδ

 δ

θδ

Is Ir Load

The line power flow, for the active power at the sending and receiving end of the line, can be

= − +− θ

If the index ζ signifies the voltage magnitude during the sag as a per unit function of the sending voltage (Ur=ζUs), and is substituted in equation 16, equation 17 yields (Polycarpou

> ( ) 0 *<sup>s</sup> <sup>s</sup> r s r*

Thus the solution of the second order equation, resulting from (17), can be calculated using

2

*Cos*

θ

+− ± +− −

 θδ

*r s r s*

*ZPCos Cos Cos*

ζ

*<sup>U</sup> <sup>U</sup> Cos Cos P*

( )( )

( ) *<sup>s</sup> s r s s r*

> δ

 θδ

( ) ( )

θ

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

<sup>=</sup> (18)

 δ 4

 δ

( ) <sup>2</sup>

 θ

(15)

(16)

− = (17)

1

θ

2

*s*

*r*

*U*

*s Ur*∠

 δ

> δ*r*

(13)

(14)

, C=0.Where Z is the transmission line impedance vector

A feeder can be modeled by an equivalent two-port network, as shown in Figure 6.

δ

θ

magnitude, and θ the transmission line impedance vector angle.

*Us*∠δ

Fig. 6. The equivalent two port network model.

ζ

1,2

ζ <sup>E</sup> *Z*∠

starting.

approximated as: A=D=1, B= *Z*∠

described by (15) and (16).

**4.1 'ζ' index** 

& Nouri, 2005).

equation (18).

Equation 18 provides a tool to calculate the voltage sag ,as a per unit value of the sending end voltage, through angles and power demand.

However, since the equation is obtained through a quadratic equation, it has two solutions.

ζ1 will be valid for a specific range of parameters. In the same way ζ2 will be valid for a different range of parameters. The validity of the two solutions, ζ1 and ζ2, with the use of various line X/R ratios is investigated in (Nouri et al., 2006). X/R ratio varies from Distribution to Transmission according to the cables used for the corresponding voltages. Typical values of X/R ratio are: for a 33kV overhead line -1.4, for a 132kV overhead line -2.4, for a 275kV overhead line -8.5, for a 400kV overhead line -15. A distribution line example is the IEEE34, 24.9kV overhead line with X/R ratio of 0.441.

According to the parameters either ζ1 or ζ2 will be the correct answer which should match the receiving end voltage.

The point of intersection of Ur with ζ1 and ζ2, occurs when ( ) 1 2 ζ −ζ θ cos is equal to zero, when both solutions are identical. However, in practice a gap develops when both solutions approach the Ur axis, where none of the two solutions accurately represent the receiving voltage Ur, as seen in Figure 7.

The distance between the two curves at the point of the gap can be defined by equation 19.

$$\mathcal{L}\_1 - \mathcal{L}\_2 = \frac{\left(\text{Cov}^2\left(\theta + \delta\_r - \delta\_s\right) - \frac{4Z P\_r \text{Cov}\theta}{\text{U}\_s^{-2}}\right)^{\frac{1}{2}}}{\text{Cov}\theta} \tag{19}$$

In order to fully investigate the range of accuracy of the two solutions, X/R ratio values of 1 to 15 are used for the line impedance. Since the receiving end power varies according to the load, five loading conditions are used in the investigation. Each loading consists of induction motors. The loads are switched in the system one by one to create the effect of supplying minimum load (one motor) and maximum load (five motors).

Using MathCad, the value of ( ) 1 2 ζ −ζ θ cos is calculated for five different loadings and each X/R ratio, starting from one up to fifteen in steps of one. The results can be seen in Figure 8 (Nouri et al., 2006).

Power Quality and Voltage Sag Indices in Electrical Power Systems 155

and to the accuracy of each solution. The relationship between the slope of ζ1 and ζ2 with the index accuracy and choice of solution is described by equation 20. The value of 'i' is 1for

( ) <sup>0</sup> *<sup>i</sup> i valid*

If ξ signifies the voltage magnitude during the sag as a per unit function of the sending

*ZP Cos U*

−

θδ

ξ and ζ1,2 signify the voltage magnitude during the sag as a per unit function of the sending voltage. When the two equations are combined, the resulting Combined Voltage Index (CVI), described by equation 22 features improved accuracy (Polycarpou & Nouri,2005). The

> 1 *<sup>a</sup> CVI*

Where 'a' is the value of the scaling factor (Polycarpou & Nouri, 2009) and is defined as

2

+ −− + − <sup>=</sup>

For simplicity, the value of scaling factor setting is 1.6 for the entire range of line X/R ratios investigated in the next sections of the chapter. Equations 18, 21 and 22 provide a tool to calculate the load voltage, as a per unit value of the sending end voltage. The equations are functions of receiving end variables such as the the receiving end voltage angle, δr, and the receiving end power Pr. The receiving end power can be described by <sup>2</sup> cos *P P IZ r s* = −

The angle δr of the receiving end voltage can be represented by sending end quantities

*a* ξ ζ

*n kl* <sup>=</sup>

θ

*Cos*

ζ

2

 δ

*s s s r*

( )

<sup>∂</sup> ∀ < <sup>=</sup> (20)

<sup>=</sup> + − (21)

<sup>+</sup> <sup>=</sup> <sup>+</sup> (22)

2

(23)

*hl wl wl hl*

ζ ζ− cos

θ

θ.

negative slope. Thus their slope is directly related to the minimum value of ( ) 1 2

*dLoading* ζ

voltage (Ur= ξ Us), and is substituted in equation 15, equation 21 yields.

ξ

value of CVI is the value of the receiving end voltage of the system power line.

1

*n l*

*a*

1 1 <sup>1</sup>

ζ1 or 2 for ζ2.

**4.2 Combined voltage index** 

shown in equation 23.

*w* = +− cos( ) θδ

cos( ) *s r j* = +− θδ

2 4 *<sup>r</sup> s ZP K <sup>h</sup> <sup>U</sup>* <sup>=</sup>

 δ*r s*

and n= Number of loads supplied.

through equation 24 (Nouri & Polycarpou, 2005).

 δ

Where:

*k* = cosθ

Fig. 8. Mathematical results obtained for ( ) 1 2 ζ ζ − cosθ

It can be observed from Figure 8, that the minimum values of ( ) 1 2 ζ ζ − cosθ occur within X/R ratio values of 3 to 8, for all test cases. Therefore during those points, the gap of inaccuracy for the index can be expected for the two solutions. Taking under consideration Figure 7, solution ζ1 should cover the ranges less than three and solution ζ2 should cover X/R greater than eight. Between those X/R values the gap position varies according to the loading and the X/R ratio of the line, thus it cannot be generalized. The accuracy of the defined location of the gap is and verified through application on a two-bus system within Power system Computer Aided Design software. The resulting data for a test system of X/R ratio equal to five, shown in Figure 9, verifies the mathematical theory concerning the gap.

Fig. 9. ζ1 , ζ2 and Ur for line X/R ratio of five

As shown in Figure 9, the plot of ζ1 has a negative slope until loading two, and then it becomes positive. Whereas the plot of ζ2 has a positive slope for the initial loadings and becomes negative when the third load is switched in.

Throughout the investigation of various X/R ratios a pattern was established regarding the slope of ζ1 and ζ2. When the slope of ζ1 is negative it is the accurate solution. When ( ) 1 2 ζ ζ − cosθreaches minimum, ζ1 deviates and ζ2 becomes the correct answer with negative slope. Thus their slope is directly related to the minimum value of ( ) 1 2 ζ ζ − cosθ and to the accuracy of each solution. The relationship between the slope of ζ1 and ζ2 with the index accuracy and choice of solution is described by equation 20. The value of 'i' is 1for ζ1 or 2 for ζ2.

$$\forall \frac{\partial(\mathcal{L}\_i)}{\text{dLoading}} < 0 \Rightarrow \mathcal{L} \text{'i} = \text{valid} \tag{20}$$

#### **4.2 Combined voltage index**

154 Electrical Generation and Distribution Systems and Power Quality Disturbances

maximum loading

minimum loading

0 2 4 6 8 10 12 14 16 XR ratio

θ

ζ ζ− cos

*Ur*

θ

occur within

ζ ζ− cos

X/R ratio values of 3 to 8, for all test cases. Therefore during those points, the gap of inaccuracy for the index can be expected for the two solutions. Taking under consideration Figure 7, solution ζ1 should cover the ranges less than three and solution ζ2 should cover X/R greater than eight. Between those X/R values the gap position varies according to the loading and the X/R ratio of the line, thus it cannot be generalized. The accuracy of the defined location of the gap is and verified through application on a two-bus system within Power system Computer Aided Design software. The resulting data for a test system of X/R ratio equal to five, shown in Figure 9, verifies the mathematical theory

> 0123456 Loading

As shown in Figure 9, the plot of ζ1 has a negative slope until loading two, and then it becomes positive. Whereas the plot of ζ2 has a positive slope for the initial loadings and

Throughout the investigation of various X/R ratios a pattern was established regarding the slope of ζ1 and ζ2. When the slope of ζ1 is negative it is the accurate solution. When

reaches minimum, ζ1 deviates and ζ2 becomes the correct answer with

ζ1

It can be observed from Figure 8, that the minimum values of ( ) 1 2

ζ2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Fig. 9. ζ1 , ζ2 and Ur for line X/R ratio of five

becomes negative when the third load is switched in.

Voltage pu

Fig. 8. Mathematical results obtained for ( ) 1 2

(

concerning the gap.

( ) 1 2 ζ ζ− cos

θ

ζ

ζ


 )

θ

cos 2 1

> If ξ signifies the voltage magnitude during the sag as a per unit function of the sending voltage (Ur= ξ Us), and is substituted in equation 15, equation 21 yields.

$$\xi = \frac{\cos \theta - \frac{\mathcal{Z} \mathcal{P}\_s}{\mathcal{U}\_s}}{\cos(\theta + \delta\_s - \delta\_r)} \tag{21}$$

ξ and ζ1,2 signify the voltage magnitude during the sag as a per unit function of the sending voltage. When the two equations are combined, the resulting Combined Voltage Index (CVI), described by equation 22 features improved accuracy (Polycarpou & Nouri,2005). The value of CVI is the value of the receiving end voltage of the system power line.

$$\frac{a\xi + \xi}{a+1} = CVI\tag{22}$$

Where 'a' is the value of the scaling factor (Polycarpou & Nouri, 2009) and is defined as shown in equation 23.

$$a = \frac{1}{n} \sum\_{l=1}^{n} \left[ \frac{1 + \sqrt{1 - hl} - \left[ vol + \sqrt{vol^2 - hl} \right]}{2kl} \right] \tag{23}$$

Where:

$$\begin{aligned} w &= \cos(\theta + \delta\_r - \delta\_s) \\ j &= \cos\left(\theta + \delta\_s - \delta\_r\right) \\ k &= \cos\theta \\ h &= \frac{4ZP\_rK}{\left(L\right)^2} \end{aligned}$$

and n= Number of loads supplied.

For simplicity, the value of scaling factor setting is 1.6 for the entire range of line X/R ratios investigated in the next sections of the chapter. Equations 18, 21 and 22 provide a tool to calculate the load voltage, as a per unit value of the sending end voltage. The equations are functions of receiving end variables such as the the receiving end voltage angle, δr, and the receiving end power Pr. The receiving end power can be described by <sup>2</sup> cos *P P IZ r s* = − θ . The angle δr of the receiving end voltage can be represented by sending end quantities through equation 24 (Nouri & Polycarpou, 2005).

$$\mathcal{S}\_r = a \tan \left[ \frac{\mathcal{U}\_s \sin(\mathcal{S}\_s) - ZI \sin(\theta + i)}{\mathcal{U}\_s \cos(\mathcal{S}\_s) - ZI \cos(\theta + i)} \right] \tag{24}$$

Power Quality and Voltage Sag Indices in Electrical Power Systems 157

switched in while the previous four are in steady state operation. The Combined Voltage

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Load X/R

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Load X/R

Observing the figures above it can be concluded that as loading levels (M1 to M5) and X/R ratio increases, the receiving end voltage naturally decreases. This is due to the voltage drop occurring on the impedance of the transmission line. The accuracy of the proposed index is well within acceptable limits(0.7% in the worst case). Thus for line X/R ratio of 0.120817 the index is capable of calculating the receiving end voltage for variation of load X/R ratio.

The resulting receiving end voltage, obtained through variation of the load X/R ratio for the

M5 M4

M1

M2

M3

M4

M5

M3

M2

M1

Index (CVI) deviation corresponding to this scenario can be seen in Figure 12.

Fig. 11. Ur for various Load X/R ratios and loadings whilst Line X/R is 0.120817

Fig. 12. CVI deviation for load X/R variation whilst line X/R=0.120817

0.86 0.88 0.9 0.92 0.94 0.96 0.98

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008

**b. Distribution Line X/R ratio is 1** 

specific line X/R ratio can be seen in Figure 13.

Ur(pu)

CVI Dev

1

Assuming the presence of an infinite bus at the sending end, equation 24 can be reduced to equation 25.

$$\delta\_r = a \tan\left[\frac{Zl\sin(\theta + i)}{Zl\cos(\theta + i) - 1}\right] \tag{25}$$

#### **4.2.1 Combined voltage index accuracy investigation**

Most distribution power system loads have a power factor of 0.9 to 1. Industrial companies have to keep their power factor within limits defined by the regulatory authorities, or apply power factor correction techniques, or suffer financial penalties. In order to cover a wider area of investigation it is decided to simulate loads of power factor 0.8 to 0.99.

The relationship between the power factor and the X/R ratio of a load is: X/R ratio = tanθ , where <sup>1</sup> θcos *pf* <sup>−</sup> = .

In order to achieve the load X/R ratio variation the circuit model of the double cage induction motor, used within the PSCAD environment, is considered. The Sqc100 Motor circuit diagram is shown in Figure 10 (Polycarpou &Nouri, 2002).

Fig. 10. The Double cage Induction motor model circuit diagram

The motor circuit parameters are:

Slip: 0.02, Stator resistance(Rs) 2.079 pu, First cage resistance(Rr1) 0.009 pu, Second cage resistance(Rr2)0.012 pu, Stator reactance (Xs)0.009 pu, Magnetizing reactance (Xm) 3.86 pu Rotor mutual reactance (Xmr) 0.19 pu, First cage reactance (Xr1)0.09 pu

The resistance of the stator winding is varied in order to achieve the required power factor and X/R ratio. Load X/R ratios of 0.1 to 0.75 are investigated. Two distribution line X/R ratios are used in the investigation in order to observe the accuracy of the index while varying both load as well as line X/R ratio for distribution system lines. The line X/R ratios are 0.12087, and 1. The amount of loading is varied through introducing five identical motors for each investigated case. The results of this investigation are presented in the following subsections.

#### **a. Distribution Line X/R ratio is 0.120817**

The per unit receiving voltage, obtained with variation of the load X/R ratio while line X/R ratio is 0.120817, can be seen in Figure 11. M1 Signifies the minimum loading with the first motor being switched in. As any switched in motor reaches rated speed, the next load is switched in the system. M5 corresponds to the Maximum loading with the fifth motor being

δ

δ

Assuming the presence of an infinite bus at the sending end, equation 24 can be reduced to

sin( ) tan cos( ) 1 *<sup>r</sup>*

<sup>+</sup> <sup>=</sup> + −

Most distribution power system loads have a power factor of 0.9 to 1. Industrial companies have to keep their power factor within limits defined by the regulatory authorities, or apply power factor correction techniques, or suffer financial penalties. In order to cover a wider

The relationship between the power factor and the X/R ratio of a load is: X/R ratio = tan

In order to achieve the load X/R ratio variation the circuit model of the double cage induction motor, used within the PSCAD environment, is considered. The Sqc100 Motor

Slip: 0.02, Stator resistance(Rs) 2.079 pu, First cage resistance(Rr1) 0.009 pu, Second cage resistance(Rr2)0.012 pu, Stator reactance (Xs)0.009 pu, Magnetizing reactance (Xm) 3.86 pu

The resistance of the stator winding is varied in order to achieve the required power factor and X/R ratio. Load X/R ratios of 0.1 to 0.75 are investigated. Two distribution line X/R ratios are used in the investigation in order to observe the accuracy of the index while varying both load as well as line X/R ratio for distribution system lines. The line X/R ratios are 0.12087, and 1. The amount of loading is varied through introducing five identical motors for each investigated case. The results of this investigation are presented in the

The per unit receiving voltage, obtained with variation of the load X/R ratio while line X/R ratio is 0.120817, can be seen in Figure 11. M1 Signifies the minimum loading with the first motor being switched in. As any switched in motor reaches rated speed, the next load is switched in the system. M5 corresponds to the Maximum loading with the fifth motor being

Xmr Rr1

Xr1

Rr2

*ZI i*

− + <sup>=</sup> − +

sin( ) sin( ) tan

*U ZI i*

*U ZI i*

*ZI i*

θ

θ

*s s*

*s s*

*r*

δ

**4.2.1 Combined voltage index accuracy investigation** 

equation 25.

where <sup>1</sup> θ

cos *pf* <sup>−</sup> = .

The motor circuit parameters are:

following subsections.

**a. Distribution Line X/R ratio is 0.120817** 

*a*

δ

circuit diagram is shown in Figure 10 (Polycarpou &Nouri, 2002).

Zs

Fig. 10. The Double cage Induction motor model circuit diagram

Rotor mutual reactance (Xmr) 0.19 pu, First cage reactance (Xr1)0.09 pu

Xm

*a*

area of investigation it is decided to simulate loads of power factor 0.8 to 0.99.

cos( ) cos( )

 θ

> θ

(24)

(25)

θ, switched in while the previous four are in steady state operation. The Combined Voltage Index (CVI) deviation corresponding to this scenario can be seen in Figure 12.

Fig. 11. Ur for various Load X/R ratios and loadings whilst Line X/R is 0.120817

Fig. 12. CVI deviation for load X/R variation whilst line X/R=0.120817

Observing the figures above it can be concluded that as loading levels (M1 to M5) and X/R ratio increases, the receiving end voltage naturally decreases. This is due to the voltage drop occurring on the impedance of the transmission line. The accuracy of the proposed index is well within acceptable limits(0.7% in the worst case). Thus for line X/R ratio of 0.120817 the index is capable of calculating the receiving end voltage for variation of load X/R ratio.

#### **b. Distribution Line X/R ratio is 1**

The resulting receiving end voltage, obtained through variation of the load X/R ratio for the specific line X/R ratio can be seen in Figure 13.

Power Quality and Voltage Sag Indices in Electrical Power Systems 159

The first part of this chapter presents various statistical voltage sag indices proposed by electrical association organisations and indices suggested by recent researchers. These indices are used to characterise any voltage sag, according to the individual index point of view. The procedure to evaluate the quality of supply, reference to non-rectangular events and equipment compatibility issues are also presented. To demonstrate the theory of equipment compatibility, with the use of System Average RMS Variation Frequency Index,

Furthermore the formulation defining a set of Mathematical voltage sag indices, leading to the Combined Voltage Index, is presented. Various motor power factors and loading levels are used in order to establish the behavior of the index for a wide range of loads. Mathematical description of voltage angle characteristics, relating to line X/R ratio variation is also illustrated. A relationship is established between the slope of ζ and the range of accuracy for each solution of the quadratic index. The CVI has proven to be easy to assess, accurate and representative of the disturbance it characterizes at distribution level. If better accuracy is required for distribution system applications, the scaling factor can be varied to achieve it. The described CVI index can be used in conjunction with optimization techniques for power quality improvement as well as power system operation optimization. The index is adaptable to incorporate future changes in technology and system parameters. This enables its implementation into the next generation of power

The author would like to express his appreciation to the University of the West of England, UK, and to Prof. Hassan Nouri, for the opportunity to carry out significant portion of the research work presented in this chapter at their establishments, as a member of the Power

Bergeron,R. (1998). Canadian electrical association approved quality indices. *IEEE Power* 

Bollen, M. (2000). Voltage sag indices-Draft 2. *Working document for IEEE P1564and CIGRE* 

IEEE Std. 1250 (1995) IEEE Guide for Service to Equipment Sensitive to Momentary Voltage

IEEE Std. 493 (1997). Gold book, *IEEE recommended practice for the design of reliable industrial* 

IEEE Std. 1100, (1999). IEEE Recommended Practice for Powering and Grounding Electronic

Bollen, M. (2001).Voltage Sags in Three-Phase Systems*. IEEE Power Eng. Review*, pp. 8-15. Bollen, M.&Styvaktakis,S. ( 2000). Characterization of three phase unbalanced sags, as easy

IEEE Std. 1159 (1995). Recommended practice for monitoring electric power quality.

Disturbances –Description, Corrected Edition Second Printing

as one, two, three. *IEEE Power summer meeting*.

**5. Conclusion** 

system planning software.

Systems and Electronics Research Group.

*and commercial power systems*

*summer meeting*

*WG* 36-07

Equipment

**6. Acknowledgment** 

**7. References** 

various power acceptability curves were used.

Fig. 13. Ur for load X/R variation whilst line X/R=1

Comparing Figure 11 to Figure 13, it is concluded that as the X/R ratio of the line increases, any load increment has more severe impact on the receiving end voltage due to the impedance magnitude of the line. The CVI deviation corresponding to this investigation case can be seen in Figure 14.

Fig. 14. CVI deviation for load X/R variation whilst line X/R=1

As the X/R ratio of the line increases the index accuracy did not decrease homogeneously. Between the values of 0.3 and 0.5 for load X/R an area of decreased inaccuracy can be observed. This is due to the scaling factor setting being 1.6 for the entire range of line X/R ratios investigated. The accuracy of the index is within acceptable margins as the largest deviation is within 2.5% (Polycarpou & Nouri, 2009). The index is within acceptable accuracy limits.
