Saturday, 23 September 2017

Laplace Transform – How Time Domain Converted to Frequency Domain

In Control System, we often encounter Laplace Transform. Transfer Function is expressed as the ratio of Laplace Transform of output and input to a system. Laplace Transform is used to ease the calculation as solving differential equation is quite tiresome. Suppose there is a second order differential equation, then solving of the second order differential equation will be very tiresome but if we convert this differential equation into Laplace Transform then it will reduce to a quadratic equation which is easy to solve. Laplace Transform converts the time domain function f(t) is into frequency domain.




Origin of Laplace Transform


Let us consider a time domain function e-jwt .

e-jwt = Coswt + jSinwt

The magnitude of this function is obviously UNITY. But the important feature of this function is that it is a rotating vector in anticlockwise direction with angular speed w as shown in figure below.



The vector OM can be divided into two component, one along the X axis and another along the Y axis. Thus we can write

OM = OMCosƟ + j(OM)SinƟ where Ɵ = wt

But the magnitude of OM = 1, hence

OM = CosƟ + jSinƟ = Coswt + jSinwt
      = e-jwt

Therefore if we multiply any time domain function f(t) by e-jwt, the magnitude of f(t) is not going to change. But the function is changed from time domain to frequency domain. Well, now how to find the aggregate value of this function?




To get the answer, we consider a function g(t) = e-jwt Sinwt. The maximum and minimum value of g(t) are e-jwt and - e-jwt respectively. Therefore the graph of g(t) will be as shown below.



From the above graph we can see that the value of function when time tends to infinite is zero. This means that the function gets converges. Therefore whenever function is multiplied by e-jw it becomes converging in nature. Therefore to find the aggregate value of g(t) over the whole time domain i.e. zero to infinity, we can use integration as below.



Isn’t the above integration Laplace Transform of function f(t)??? Hehehe….Yes it is. Thus Laplace Transform of function f(t) for t >=0 is denoted as L[f(t)] and defined as




Hope you enjoyed. Thank you!

Thursday, 21 September 2017

Steady State & Transient State Stability


Power System instability is basically of two forms: a) Instability of Load which means stalling of Synchronous Load connected to the system. b) Pole slipping or out of step of Generators connected to the Synchronous Grid known as Synchronous Instability. Synchronous Stability is categorized into two:

1) Steady State Stability

2) Transient State Stability

Steady State Stability


Steady State Stability is defined as the stability of system under slow or gradual change in load. It is assumed that the rate of change of load is comparatively low when compared with the rate of change of field flux of machine/generator in response to change in load.





Note: There are two types of Grid, Synchronous Grid and Asynchronous Grid. In Synchronous Grid all the Generators connected operate at same speed viz. 3000 rpm. This is necessary to maintain a constant frequency of 50 Hz of Grid. On the other hand, in asynchronous grid, different generators operate at different speed. This means that frequency of two Asynchronous Grids will be different. Two asynchronous grid are connected to each other via HVDC link.


Basically any change in system condition leads to swing. Swing is nothing but the relative motion between the connected machines. Whenever there is any change, all the connected machines try to cope up the change. This give rise to relative motion i.e. Swing. Swing can be either in power or other electrical quantity. Now for system to be stable, these swings should be damped out as quickly as possible. The damping in Generator is mainly provided by Inertia of Rotor H, Damper winding, Field Winding etc. Figure below shows a stable swing. As can be seen from the figure, the swing/disturbance dies out.


stable-power-swing


But there are some conditions / operating points of Generator in which such swings/disturbance do not die out. Such kind of swing/disturbance is known as unstable swing. If no proper measure is taken then the Generator will definitely loose synchronism and become unstable.

unstable-power-swing


Transient State Stability


Transient State Stability is capability of system to be stable when there is a sudden or large change in system condition like load, fault at a point, change is mechanical input to the rotor shaft of Generator etc.

Dynamic Stability


Apart from the steady state and transient state stability, there is one more kind of stability called Dynamic Stability. Modern Generators are equipped with fast acting Voltage Regulators. These regulators are capable of changing the field flux very quickly. Therefore the rate of change of flux is quite more when compared to rate of change of load. This is why the stability limits are increased by the fast acting voltage regulators.

Dynamic Stability is same as steady state stability but the main difference lies in enhanced rate of change of flux capability due to regulators. This is the origin of Power System Stabilizer (PSS) which forces the voltage regulator to respond to transients quicky to change field flux. I will be elaborating System Stabilizer (PSS) in next post.

Principle of Automatic Voltage Regulator



Automatic Voltage Regulator (AVR): 

Automatic Voltage Regulator is a device which maintains the Generator output terminal voltage. To be more accurate, AVR is a controller which always compares the Generator output terminal voltage Vt with the set reference voltage Vref and as per the error signal i.e. (Vref - Vt) it changes the filed excitation of Generator to maintain constant terminal voltage Vt





Principle of Automatic Voltage Regulator:


For better understanding of principle of Automatic Voltage Regulator i.e. AVR, we will first have a brief look on Generator Excitation System. I am here taking static excitation system for example. As we know that in static excitation system Generator output is fed to a thyristor bridge rectifier. This thyristor bridge rectifier converts the AC current to DC current. Note that the DC current output by the Thyristor Bridge can be controlled by controlling the firing angle of the Thyristor. The DC current output of Thyristor Bridge is then fed to the filed winding of Generator as shown in figure below.

Static-Excitation-System


Let us assume that filed current at any time is If. Then the air gap flux of Generator may be written as Ø = KIf where K is some constant.

But we are interested in maintaining the Generator Output Terminal Voltage Vt which is given as

Vt = 1.414πfNØ where symbols have their usual meaning.

From the above, it is quite obvious that changing If will change the terminal voltage Vt

Thus voltage regulation can be achieved by controlling field current. Automatic Voltage Regulator AVR performs this action by changing its firing angle. Figure below shows a simplified diagram of AVR.

Automatic-Voltage-Regulator-AVR


AVR takes three inputs namely reference voltage Vref, terminal voltage Vt and limiting signals. For simplicity we will assume only two inputs Vref and Vt. Reference voltage Vref is manually set in AVR. This reference voltage is also dynamically changed around the manually set Vref by Power System Stabilizer (PSS). But for this discussion we will eliminate the effect of PSS and assume that Vref is constant. The Error Signal (Vref-Vt) is fed to Controller. The controller in the diagram is denoted by its Transfer Function. The output of Transfer Function is the fed to Thyristor Bridge Rectifier to change firing angle and hence field excitation.

Suppose Vref = 21 kV and due to some reason the terminal voltage Vt = 25 kV. Thus AVR will reduce the field current If to reduce the value of air gap flux. This in turn will reduce the terminal voltage and will try to make it stable at 21 kV.