Real power/Frequency Balance, Reactive Power /Voltage Balance, Node Equations ,Bus Impedance & Admittance Matrices with it's Applications

Real power/frequency balance:

fig:Closed loop control system of real power/frequency balance

Above figure shows the block diagram of closed loop control system of real power/frequency balance.In this system real power demand P_D and P_G is the real power generated or power output of generator.When the real power demand P_D changes the governor and turbine have to be respond quickly to change input to the generator.Otherwise,speed of the generator changes with the change in frequency of the system.The frequency must be maintained for the better operation of the electrical appliances.

Figure shows the interconnected power system with generators G1 & G2 supplying 3 loads PD1, PD2 PD3. For the balanced operation,these generators must be meet the total demand of load.

where G1 is generator at bus 1 & G2 is generator at bus 2.

PG1+PG2 =PD1+PD2+PD3+P_loss

We have, Regulation=(Change in frequency/change in output power)

=(ðf/ðP_G)

For the two generators,

R1=(ðf/ ðP_G1) & R2=(ðf/ ðP_G2)

So, ðP_G1= ðf/R1 & ðP_G2= ðf/R2

For the steady state condition,

ðP_G1+ ðP_G2 = ðP_D­

or, (ðf/R1) + ( ðf /R2) = ðP_D­

Hence, ðf= ðP_D­/((1/R1)+(1/R2))

This is the required expression for change in common frequency with the change in power demand.

Reactive Power/Voltage balance:

Consider a transmission line supplying a load from synchronous generator as shown in above figure.

This means that the voltage drop depends on reactive power.So for maintaining voltage flow,reactive power should be maintain.Flow of the reactive power can be maintained by using reactive compensators or by controlling the excitation at the generator but the excitation has some limit.

There are several methods to control voltage improving stability of the interconnected system are:planning,excitation system control & protection,maintenance,reactive power compensation,HVDC transmission,use of capacitor banks,Tap changing, voltage reduction and so on.

Nodal equation and bus admittance matrix:

Nodal admittance is an N x N matrix describing a power system with N buses . It represents the nodal admittance of the buses in a power system.It contains thousands of buses, the Y matrix is quite sparse. Each bus in a real power system is usually connected to only a few other buses through the transmission lines. The Y Matrix is also one of the data requirements needed to formulate a power flow study.

The nodal admittance matrix form:

and the nodal equation in matrix form is given by,

Where,

Y₁₁=y₁₀+y₁₂+y₁₃= Sum of admittances connected at bus-1.

Y₁₂= -y₁₂ =mutual admittances between node 1 & node 2.

Y₁₃= -y₁₃ =mutual admittances between node 1 & node 3. So admittance matrix for 3 bus system is;

This is always a square matrix of order equals to number of buses.

It can be concluded that,

The diagonal element of each node is sum of admittance connected to it and the off diagonal element is the negated admittance between nodes.

The inverse of bus admittance matrix is known as bus impedance matrix.

Applications of bus admittance matrices:

The main application is to analyse the given system and to calculate the voltages & currents at each of the buses. It is also useful in calculating total active and reactive power looses of the system.