Natural oscillations arise from the inherent dynamic interactions between the power system components, whereas forced oscillations are induced by sustained and periodic external inputs. For instance, this phenomenon is observed in internal combustion engines, where the torque has periodic fluctuations of various frequencies, resulting from the firing sequence of the cylinders. When the frequency of the driving torque approaches a natural mode of oscillation, there is a potential risk of resonance. In this scenario, the cyclic inputs can excite the natural oscillations and magnify their amplitudes to unacceptable levels.

The attached project includes a single-machine infinite-bus system model in which a synchronous generating unit is subject to forced oscillations introduced by periodic fluctuations in the applied mechanical torque. To simulate this behavior, a multi-sinusoidal signal is generated by a Fourier source element and injected into the turbine-governor model to account for the effect of the power stroke cycle of the engine. The Fourier source is setup to replicate the power stroke cycle of a four-stroke engine, equivalent to two shaft revolutions. Since the genset in the project is assumed to operate at 600 RPM, the lowest torque harmonic frequency is 5 Hz.

The following study cases were prepared to mirror the design phase of a real genset, in which specific parameters of the shaft and the generator are selected to minimize the potential adverse effects of forced oscillations:

• 00_Base case: This case corresponds to an initial configuration of the generating unit, in which the local mode of oscillation is located at approximately 4.9 Hz and a damping ratio of 6.8% (see “Eigenvalue Plot”). The RMS simulation results (see “RMS – Plot page”) show that the local mode of oscillation is excited by the mechanical torque's oscillation frequency, whose first component is at 5 Hz.
• 01_Shaft model stiffness: the stiffness in the shaft model is reduced from 500 to 300 kNm (parameter K_shaft). As a result, the frequency of the local mode of oscillation is reduced to approximately 4.4 Hz, and its damping ratio increases to 9.9%. The RMS simulation results show that the interaction between the engine torque and the local mode of oscillation is reduced compared to the base case.
• 02_Generator inertia: the inertia of the generator is increased from 3000 kgm2 to 6000 kgm2, thus decreasing the frequency of the local mode of oscillation down to approximately 4.1 Hz. The effect is similar to the observed in the previous case.

A similar approach can be followed to analyze the effect of injecting forced oscillations into the control loops of synchronous machines and other devices.