# Power systems and inertia

The Power system is the largest man made structure in the world. It supplies the world with electrical power, which is used for many appliances, in industry and mobility. It is a very complex system, but one simple rule in the system is: At all time, the generated power has to be identical to the consumed power! The power in the system has to come from somewhere and be used somewhere. Now, what happens if we turn on a light switch and increase the consumed power? How is the power system able to still provide power? After all, the system is not able to predict the time you turn on a light.

In fact, the system uses inertia to provide power. Inertia is the property of masses to, once moving, stay moving. For example, stopping a large rolling rock is not easy. But what is the inertia in a power system? The electrical power is generated in large turbines, which rotate at a certain speed. The rotation in caused by the mechanical input, for example wind in wind turbines. If we increase the load, the frequencies of this rotations will be reduced and the loss of rotational energy is the power used for the device.

The power can only decrease to a certain amount. If the frequency in the system is too low, not enough energy is transmitted and a blackout occurs. If the frequency is too large, too much power is transmitted and the lines can be damaged.

Now, the following applet allows to simulate what happens when a generator suddenly decreases in power output. This behavior, called a perturbation, can be triggered by clicking the button at the top. Play around with the options of the simulations and investigate the effects!

This virtual lab decreases the power output of a selected generator in a simple, small power system simulation. The simulation options are explained here:

• The system is perturbed by the button on top

• The inertia of the system can be change by the second button

• The generator which is perturbed can be chosen

• The relative amount of power generated during the perturbation can be changed

• The length of the perturbation can be changed changed

• A security margin can be selected

The influence of the inertia is not just purely academical, but a real world problem. The power system is undergoing many changes to transition to renewable energy sources. Some of this sources, like PV panels or off-shore wind farms, have a lower intrinsic inertia. This loss of inertia can destabilize the system and has to be taken into account for future planning and running of the power system.

It should be obvious that in general, the low inertia results in a larger deviation from the default system frequency.

The simulation is based on the IEEE 39 bus system. All the loads and buses are removed, so that only the ten generators remain. A diagram showing the full system is shown here [taken from here]: We use a model called the synchronous machines model. The details are given in:

1.
Nishikawa, T. & Motter, A. E. Comparative analysis of existing models for power-grid synchronization. New Journal of Physics 17, 015012 (2015).

The ordinary differential equations to calculate the generator phases $$\theta$$ are given as:

\begin{aligned} M_i \ddot \theta_i + D_i \dot \theta_i = \omega_i - \sum_j A_{ij}\sin \left( \theta_i - \theta_j\right)\end{aligned}

With the inertia $$M$$, the damping $$D$$, the generated power $$\omega$$ and the coupling matrix $$A$$   Inertia and power systems - A virtual Lab by Felix Koeth