No, it' doesn't. It depends on the characteristics of the load. If resitive, for example, then it presents the same impedance to all harmonics. Since the impedance of the electric transmission system generally goes up with frequency, a resistive load will actually reduce the harmonics more than the fundamental.
When the new load comes onto the grid, a large load current will flow through it.
This load current has to be sourced by the generators. The increased current in the generators throws an increased torque requirement onto the engine or turbine that's driving the generators.
As the engine control at that instant has no way of knowing that more torque is needed, then the turbine+generator combo will start to slow down. The increased output power is delivered from the fall in kinetic energy of the large rotating mass. It's this energy stored in the rotating machinery that is the buffer that allows the grid to handle unplanned load variations.
Spread out over the grid, there is a lot of rotating machinery, so not much change in speed is required to source the power. However, the grid is not infinitely stiff, so the machines local to the load will slow down more quickly. This creates a phase shift between the local machines and the remote grid, which creates a power flow into the local grid area. This distributes the load across the grid, and maintains phase synchronization between the local and remote areas.
The interconnects between parts of the grid have to be big enough to handle these transient power flows without overloading. For a really big power requirement, say a generating station suddenly goes offline, then the interconnect can drop out, leading to cascading blackouts.
Eventually the engine or turbine controllers will detect the fall in speed, and increase fuel or water flow to match the output power required. The grid controllers will use the instantaneous grid frequency as a proxy for its relative loading, and bring other sources of power online if the frequency is too low.
Over the course of 24 hours, most grids will control the frequency so that there will have been 24 * 3600 * grid_frequency cycles, so that motor-driven electric clocks keep good time from day to day. During the course of a day however, their hour to hour timekeeping could be out due to the load-related frequency variations.
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For one thing, turbine plants use much higher voltages than 120. 4140V or 12kV are more typical for a large turbine, with that voltage stepped up further for long lines distribution.
For another, a turbine like you describe would be only for a backup generator. Large turbines for a municipal-size grid use steam heated by the fuel, be it natural gas, coal or nuclear fission.
Anyway, what happens with any generator linked to the grid is that they vary the torque to match the load, while keeping RPM constant. This is because the generator's waveforms not only need to be the same frequency, but the same phase.
For example, windmills in groups run at the same RPM together, varying their power output by feathering the props and thus varying their torque, but all in lockstep with the grid frequency and phase.
Think of it like what happens when your car climbs a hill at a constant speed: the engine RPM doesn't change, but to maintain speed you have to give it more accelerator to increase the torque, and thus, the power (power being basically torque x RPM.)