In an earlier post I gave a definition of a stable power system.
A stable power system is one that continuously responds and compensates for power/ frequency disturbances, and completes the required adjustments within an acceptable timeframe to adequately compensate for the power/frequency disturbances.
This post delves into the mathematics of what happens in the first few seconds after a disturbance. So apologies – this post does involve some maths.
The first few seconds after a power balance disturbance is analysed using Newton’s laws of motion. We’re going to look at the power flow between the rotating inertia (rotational kinetic energy) of a synchronous generator and the power system. It applies for the first few seconds after the onset of a disturbance, i.e.: before the governor and prime mover have had opportunity to adjust the input power to the generator.
We begin with the rotational form of Newton’s second law of motion; Force = Mass * Acceleration.
T = J * d2A/dt2
T is the net accelerating torque applied to the rotor. T is the difference between the driving torque of the prime mover Tm, and the retarding torque from the electrical load Te. i.e.: T = Tm – Te.
J is the moment of inertia of the rotor (and the attached prime mover)
A is the angle of the rotor with respect to a fixed reference
d2A/dt2, is the second differential of angular position with respect to time, the rotational acceleration of the of the rotor
We’ll define w as the rotational speed of the rotor, i.e.: dA/dt . w is equivalent to frequency. The shaft acceleration d2A/dt2 is equivalent to the rate of change of rotor speed dw/dt. The shaft power is the product of torque and angular speed; i.e.: P = T * w, and angular momentum M is the product of moment of inertia and angular speed; i.e.: M = J * w. Using these we can derive the following equation:
Pm – Pe = M * dw/dt
Pm is the mechanical power being applied to the rotor by the prime mover. We consider this is a constant for the few seconds that we are considering.
Pe is the electrical power being taken from the machine. This is variable.
M is the angular momentum of the rotor and the directly connected prime mover. We can also consider M a constant, although strictly speaking it isn’t constant because it depends on w. However as w is held within a small window, M does not vary more than a percent or so.
dw/dt is the rate of change of rotor speed, which relates directly to the rate of increasing or reducing frequency.
The machine is in equilibrium when Pm = Pe. This results in dw/dt being 0, which represents the rotor spinning at a constant speed. The frequency is constant.
When electrical load has been lost Pe is less than Pm and the machine will accelerate resulting in increasing frequency. Alternatively when electrical load is added Pe is greater than Pm the machine will slow down resulting in reducing frequency. Here’s the key point, for a given level of power imbalance the rate of rise and fall of system frequency is directly dependent on synchronously connected angular momentum, M.
It should now be clear how central a role that synchronously connected angular momentum plays in power system stability. It is the factor that determines how much time generator governors and automatic load shedding systems have to respond to the power flow variation and bring correction.
In the above I have made some simplifications. I have presumed a simple two pole machine, ignored machine losses and damping, and not talked about units used. This has simplified the equations and shortened the discussion, while keeping the important concepts in place. If readers do want to understand this in more detail there are good text books on the topic; but be warned, the mathematics gets very intense.
Power System Control and Stability, PM Anderson and AA Fouad, Wiley Interscience, IEEE Press.
Power System Stability and Control, Prabha Kundur, McGraw-Hill Inc.
Electric Power Systems, Analysis and Control, Fabio Saccomanno, Wiley Interscience, IEEE Press.
Next article – governor controls.