Revised Article-Niharika
Revised Article-Niharika
Revised Article-Niharika
Niharika Agrawal
Niharika.svits@gmail.com
Synergy Institute,Bangalore.
ABSTRACT
The stability of a system refers to the ability of a system to return back to its steady state
when subjected to a disturbance. As mentioned before, power is generated by synchronous
generators that operate in synchronism with the rest of the system. A generator is
synchronized with a bus when both of them have same frequency, voltage and phase
sequence. We can thus define the power system stability as the ability of the power system to
return to steady state without losing synchronism. Usually power system stability is
categorized into steady state, transient and dynamic stability. Transient stability analysis is
the study of the system stability when subjected to server faults like three-phase to ground
short circuit or major generator outage etc.
The large-disturbance (here after will be called as transient stability) of a Single Machine
Infinite Bus (SMIB) system can be analyzed through a method called as “equal area
criterion”. Equal area criterion can be obtained from the swing equation. The equal area
criterion gives an analytic way of assessing the stability of the system. The transient stability
of the system can also be found out by numerically integrating the swing equation. Since,
swing equation is nonlinear we cannot solve for the solution of swing equation through
analytical methods. In order to solve the swing equation numerical methods have to be used.
In this article a look at a well known numerical method called as Euler’s method is given for
the numerical solution of Swing equation.
INTRODUCTION
Power system stability is understood as the ability to regain an equilibrium state after being
subjected to a physical disturbance. Synchronous stability can be divided into two main
categories depending upon the magnitude of disturbance. Steady State stability refers to the
ability of the power system to regain synchronism after small and slow disturbance, such as
gradual power changes. The transient stability is the ability of the system to regain
synchronism after a large disturbance. Transient stability studies are needed to ensure that the
system can withstand the transient conditions following a major disturbance.
Steady State stability of the power system is analyzed by the swing equation of a
synchronous machine. It represents the swings in the rotor angle𝛿during disturbances. The
change in the rotor angle 𝛿 results in change in real power, which ultimately affects the
frequency. Hence swing equation forms the basis for modelling of load frequency control
(LFC) loop of the power system. Swing equation is a rotational inertia equation describing
the effect of unbalance between the electromagnetic torque and the mechanical torque of the
individual machines.
SWING EQUATION
The behaviour of a synchronous machine during transients is described by the swing
equation. Let θ be the angular position of the rotor at any instant t. However, θ is
continuously changing with time. It is convenient to measure θ with respect to reference axis
that is rotating at synchronous speed. If 𝛿 is the angular displacement of the rotor in electrical
degrees from the synchronously rotating reference axis and 𝜔𝑠 the synchronous speed in
electrical radians ,then θ can be expressed as the sum of time varying angle 𝜔𝑠 t on the
rotating reference axis ,plus the torque angle 𝛿 of the rotor with respect to the rotating
reference axis.
In other words:
θ = 𝜔𝑠 t + 𝛿 radians. ............... 1
𝑑𝜃 𝑑𝛿
= 𝜔𝑠 + ............... 2
𝑑𝑡 𝑑𝑡
𝑑2𝜃 𝑑2𝛿
2
=
𝑑𝑡 𝑑𝑡 2
𝑑2𝜃 𝑑 2𝛿
𝛼 = = elec. rad/s2
𝑑𝑡 2 𝑑𝑡 2
Now M = J 𝜔
𝑑2𝜃
But , J = 𝑇𝑎 ;
𝑑𝑡 2
𝑑2𝛿
J = 𝑇𝑎
𝑑𝑡 2
𝑑2𝛿
𝜔J = 𝜔𝑇𝑎
𝑑𝑡 2
𝑑2𝛿
M = 𝑃𝑎 = 𝑃𝑠 - 𝑃𝑒 ............4
𝑑𝑡 2
Equation-4 gives the relation between the accelerating power and angular acceleration.
It is called the swing equation. It is a non-linear differential equation of the second order.
With this differential equation we can discuss the stability in a quantitative way,because it
describes swings in the power angle delta during transients.
The Equal Area Criterion gives an analytic way of assessing the stability of the system. The
transient stability of the system can also be found out by numerically integrating the swing
equation. Since, swing equation is nonlinear we cannot solve for the solution of swing
equation through analytical methods. In order to solve the swing equation numerical methods
have to be used.
Euler’s method
Let a non linear differential equation of the form given where f (x) is a nonlinear function of x
We can find the solution of above equation through Euler’s method. The idea is to integrate
the equation between the time instants (𝑡0 ,𝑡𝑓 ) with an initial value of 𝑥= 𝑥 0 at t= 𝑡0 with a
small time step ∆t .
Modified Euler’s method has two steps: predictor step and corrector step. These steps in
generalized form are as follows:
Where 𝑥𝑝1 is the predicted and 𝑥𝑐1 is the corrected value of the variable 𝑥 at the time instant t
= 𝑡0 + ∆𝑡. The time step ∆𝑡 has to be carefully chosen,the smaller the values the better the
accuracy. Ideally ∆𝑡 should approach zero but when the number of steps required to integrate
above equation between the limits (𝑡0 , 𝑡𝑓 ) will become infinite. Hence there is a trade off
between the accuracy and the number of steps required. Now modified Euler’s method can be
applied to integrate swing equation between the time limits (𝑡0 , 𝑡𝑓 ). The second order
differential equations of the single machine infinite bus system are given below:
Here, ∆𝜔𝑚𝑒 = 𝜔𝑚𝑒 - 𝜔𝑠 that is the change in the rotor speed from the synchronous speed.
Now the predictor step of Euler’s method when applied to eq will give:
The corrector step is given as:
By numerically integrating the swing equation for a specified period like for 5 seconds,if we
observe that the rotor angle and the rotor speed are settling to the steady state values when
subjected to a disturbace then we can say that the system is stable. But if we observe from
numerical integration of swing equation that the rotor angle and speed are either continuously
increasing or decreasing as time tends towards infinity then that means that the system has
become unstable.
To solve the swing equation by Modified Euler‟s method, it is written as two first order
differential equations:
Starting from an initial value at the beginning of any time step,and choosing a step size ,the
equations to be solved in modified Euler’s are as follows:
𝛿1 and 𝜔1 are used as initial values for the successive time step..
Euler’s method is one of the easiest methods to program for solution of differential using a
digital computer. It uses the Taylor’s series expansion, discarding all second –order and
higher order terms. Modified Euler’s algorithm uses the derivatives at the beginning of a time
step, to predict the values of the dependent variables at the end of the step.Using the predicted
values, the derivatives at the end of the interval are computed. The average of the two
derivatives is used in updating the variables.
REFERENCES:
[1] Ashfaq Husain,Electrical Power Systems,New Delhi,CBS Publishers and Distributors
Pvt . Ltd. 2007 pp 487-533.
[4] Computer Techniques in Power System Analysis Lecture notes by SJBIT Engineering
College,Bangalore pp 112-144.