Monday, 30 October 2017

Line Loadability - Universal Line Loadability Curve

Loadability of line is defined as the extent of load which can flow through the line without exceeding the limitations. Line Loadability is expressed in percentage of Surge Impedance Loading of line. The limiting factor for line loading are: thermal limit, voltage drop limit and steady state stability. The loading of line shall not exceed these limits in any case.

Thermal limit corresponds to heating of transmission line and interposes a limit on current which can flow. Though of overhead transmission line, thermal limit is quite high but this cannot be neglected while deciding loading.

Voltage drop limit corresponds to drop in receiving end voltage. In case the loading of line is very high compared to Surge Impedance Loading (SIL), there will be a huge drop in receiving end voltage. A voltage drop of 5% is normally permissible.

Steady State limit shall be considered for long line. As the power transfer equation is inversely proportional to reactance X, increase in line length increases the reactance and hence decreases the power transfer capability.

P = VrVsSinδ /X

A steady state stability margin of 30% is usually adopted. Steady State stability margin is defined as the percentage loading of line.

Steady State stability = (Pmax – Plim) / Pmax

Pmax = Maximum Power Transfer Capability (when load angle δ is 90°)

Plim = Actual Power Transfer taking place

30% steady state stability margin will result in,

0.3 = (Pmax – Plim) / Pmax

0.7Pmax = Plim

0.7 VrVs /X = VrVsSinδ /X  (since δ = 90°)

δ = 44°

This means that for permissible steady state stability marging of 30°, load angle should be kept at 44°.

Universal Line Loadability Curve

Based on the three limitations, loadability of line is defined for different voltage and length of line. A plot of line loading with respect to length of line applicable for all voltage level was first derived by St. Clair. This plot is known as ST. Clair Curve or Universal Line Loadability Curve. In determining the line loadability curve, it is assumed that a voltage drop of 5% and steady state stability margin of 30% are permissible. Figure below shows the Universal Line Loadability Curve.

Line Loadability

As can be seen from the above loading curve, the limit to line loading are limited by the following factors:

1) Thermal limit for Short Line (Length up to 80)

2) Voltage drop limitation for medium line (length up to 320 km)

3) Steady State Stability Limitation for Long Lines (line length more than 320 km)

Once can easily suggest the maximum loading capability of transmission line for different line length using the above power transfer capability curve or Universal Line Loadability Curve. For example, if line length is 300 km, then line loadability will be around 1.5 times of Surge Impedance Laoding.

Wednesday, 18 October 2017

Surge Impedance and Surge Impedance Loading

Surge Impedance is the characteristic impedance of a lossless transmission line. It is also called Natural Impedance because this impedance has nothing to do with load impedance. Since line is assumed to be lossless, this means that series resistance and shunt conductance is negligible i.e. zero for power lines.

This means that, Series Resistance R = 0 and Shunt Conductance G  = 0

As Characteristic Impedance Zc = z/y

where z is series impedance per unit length per phase and y is shunt admittance per unit length per phase.

z = R +jwL

y = G + jwC

For lossless line, z = jwL and y = jwC

Hence according to definition,

Surge Impedance = Zs = Zc = √(jwL/jwC)

      = √(L/C)

Surge Impedance Loading SIL

A transmission line terminated with load equal to surge impedance of line is called surge impedance loading SIL. 

Surge Impedance Loading

Let us a look at the voltage profile along the line for surge impedance loading condition. We know that voltage at any point is given as

V = [(Vr +ZcIr)/2]eµx + [(Vr-ZcIr)/2]e-µx

Where Zc is characteristic impedance. Since line is assumed lossless therefore characteristic impedance and surge impedance will be equal i.e. Zc = Zs. Also, line is terminated with surge impedance therefore Vr = ZsIr

V = [(Vr +ZsIr)/2]eµx + [(Vr-ZsIr)/2]e-µx

    = ZsIr eµx
Since µ = √yz = µ = √(j2w2LC) = jw√LC

V = ZsIr ejwx√LC

Assuming wx√LC = Ɵ

V = ZsIr e = ZsIr ∠Ɵ

The above expression of voltage shows that, for surge impedance loading the voltage profile along the line is uniform or flat. This means sending end voltage and receiving end voltages are same for surge impedance loading.

Thus SIL can also be defined as,

Surge Impedance Loading is the connected load in transmission line for which reactive power generated is equal to reactive power consumed i.e. the flow of reactive power is zero. There is an exact balance between reactive power generation and consumption. Mind that reactive power is generated here by shunt capacitance and being consumed by series inductance of line.

From the above definition of SIL, we can have a second method to calculate Surge Impedance Zs.

Reactive Power Generated = Reactive Power Consumed

V2wC = I2wL

(V/I) = √(L/C)

Zs = √(L/C)

Surge Impedance Zs = √(L/C)

Hope you fully understand the concept. Still if you have any doubts please comment. Thank you!

Characteristic Impedance of Transmission Line

Characteristic Impedance of a Transmission line is defined as the square root of ratio of series impedance per unit length per phase and shunt admittance per unit length per phase. If z and y are series impedance and shunt admittance of line, the characteristic impedance Zc is given as

Zc = (z/y)

Where z = R + jωL = series impedance per unit length per phase

            y = G + jωC = shunt admittance per unit length per phase

Let us have a detailed discussion on the characteristic impedance and some of the important parameters associated with transmission line. For analysis purpose we consider a long transmission line. We know that a long transmission line have distributed Resistance (R) and Inductance (L) in series & Conductance (G) and Capacitance in shunt.

Shunt Conductance (G) of line basically accounts for loss occurring due to leakage current along insulator string and corona. In power line this effect is usually very small and can be neglected.

Therefore to model a long transmission line, associated parameters needs to be distributed over the entire length as shown in figure below. Let the length of line is ‘l’. Let us consider a small section of line dx at a distance x from the receiving end.

Characteristics Impedance
Series Impedance of section dx = (series impedance per unit length)dx

                                                    = zdx

Voltage drop in section dx = dv = (zdx)I

dV/dx = zI  ……………………………..(1)


Differential Current dI flowing in the shunt admittance can be found by multiplying Shunt admittance and Voltage V across the shunt admittance. Hence,

dI = (ydx)V

dI/dx = yV ………………………….(2)

Differentiating equation (1) and (2) w.r.t x we get,

d2V/dx2 = z(dI/dx) =yzV  …….[ from equation (2)]

and d2I/dx2 = y(dV/dx)

                  = yzI  ………[from equation (1)]

To get the profile of voltage and current along the line, we need to solve the above two differential equations of second order assuming receiving end voltage Vr and current Ir to be know. Also at receiving end x = 0 since differential section dx is considered at at distance of x from receiving end. The general solution of the differential equation is given below.

V = [(Vr +ZcIr)/2]eµx + [(Vr-ZcIr)/2]e-µx

I = [(Vr/Zc) + Ir]eµx + [(Vr /Zc) + Ir]e-µx

Where Zc = (z/y) and µ = yz

The constant Zc is called Characteristic Impedance and µ is called Propagation Constant. Characteristics Impedance and Propagation Constant are complex number.

Incident Voltage and Reflected Voltage

Carefully observe the voltage profile obtained along the transmission line. It is composed of two parts, [(Vr+ZcIr)/2]eµx and [(Vr-ZcIr)/2]e-µx. Let us draw a plot of the two components as shown in figure below.  

Incident Voltage wave

Above plot of [(Vr+ZcIr)/2]eµx clearly shows that the magnitude of voltage is increasing with increase in x. This means that voltage from receiving end to sending end is increasing. This part of voltage is known as Incident Voltage.

Reflected Voltage Wave

Above plot of [(Vr-ZcIr)/2]e-µx reveals that the magnitude of voltage is continuously reducing with increase in x. This part of voltage is known as Reflected Voltage.

Voltage at any Point = Incident Voltage + Reflected Voltage

Thus we can say that, in a transmission line voltage at any point is the sum of incident and reflected voltage. Same process of thinking applies for current and I assume that you can get the idea about current by yourself.

Flat or Infinite Transmission Line

A flat transmission line is defined as a line terminated with characteristic impedance. Such line is also called Infinite Transmission Line. Since line is terminated with characteristics impedance, hence Vr = ZcIr
Thus Incident Voltage at any point x on line = [(ZcIr +ZcIr)/2]eµx
                                                                      = ZcIreµx
Let µ = yz = a+jb where a and b are constant.

Hence, Incident Voltage at any point x on line = ZcIre(a+jb)x

                                                                       = ZcIreaxejbx

Magnitude of incident voltage = ZcIreax since magnitude of ejbx is 1.

Reflected Voltage at any point on Line = [(Vr-ZcIr)/2]e-µx

                                                              = [(ZcIr - ZcIr)/2]e-µx

                                                              = 0

Thus a flat or infinite line is having no reflected voltage but only incident voltage. This can also be though in another but ridiculous way. If line is of infinite length then how can it have reflected voltage? Where from will it get reflected? So no reflected voltage in infinite or flat line.