Just thinking out loud here: From what you quoted, the imbalance appears to have been in the generator frequencies to start with? I imagine they need to stay synchronized in frequency if they are sharing loads. Otherwise you set up a beat frequency at the difference frequency between the two generating systems, and their voltages will move from being in phase to being out of phase, back to being in phase again at the rate of the beat frequency. I don't know how power grids really work, but I imagine this will cause intolerable voltage variations as they sum in phase and out of phase. Did that happen? It's a simple graph to add two sine waves of equal amplitude but with a slight difference in phase to show what happens.

But upon closer inspection of the article you mentioned it looks like the real complaint was simply that having a slightly lower frequency than the assumed 50 Hz, kitchen equipment that assumes 50 Hz and some kind of zero crossing counter as a clock reference will get a long term clock drift.

As an aside, when we talk about load impedances being a function of frequency, unless the load has a very high Q (lossless resonance), they don't change very much in the vicinity of the frequency of the stimulus. Bio-mechanical systems (a cool topic) are inherently lossy and must have low Qs. To a first order it would be surprising to see much sensitivity to the frequency of the stimulus. But I can see additions and cancellations of multiple stimulus playing a role. And our internal body clocks seem to get reset every day according to the motion of the sun.

Today in class, there was a question about AC vs. DC in homes. What did we originally use? Why did we settle on AC? Why does North America use 110V while Europe uses 220V (and why 60Hz vs. 50Hz)?

The most important question: why AC? This topic is addressed in a 4-5 page discussion in Geoffrey West's brilliant book "Scale". Edison had the first power distributions; they were small DC networks with a single generator. People loved them, and they rapidly expanded from a small number of city blocks to over a dozen. They started to have reliability problems -- problems that did not arise in those first small networks. In short, they would not scale.

Networks needed to branch to step down the voltages from the high ones produced by generators to the ones used in homes. Those branches are crucial; there must be containment/damping of changes in the demand near the endpoints of the distribution network. If there is a high-power device at a factory that is suddenly switched off, that can create a spike of the already-flowing current in the circuits. DC networks behave badly at those distribution junctures; the voltage spike can continue to travel upstream like a wave and destroy both power company equipment and customer devices.

You can see an analogue of this in the plumbing older houses when a fully-open faucet to a bathroom tub is suddenly turned off. Suddenly terminating the energy of the flowing water can create a pressure wave in the pipes: a loud booming sound. This is called water hammer; it can literally destroy the pipes in a house. In modern homes, water hammer is contained by placing water hammer arrestors in the plumbing. These devices are a vessel partially filled with water partially with pressurized air; they absorb the energy of the shockwave in the air-filled part of the chamber. The energy is dissipated as heat -- just like a resistor. The alternative means to deal with water hammer is to gently decrease the flow of water into a tub; this is probably what our grandparents did (and their children had to learn).

While Edison was developing a DC grid, Nicola Tesla was developing an AC grid -- first powered by turbines at Niagara Falls. A properly-designed AC grid does not propagate spikes upstream through the network; there is proper impedance matching at the nodes. Scaling is possible by applying Oliver Heaviside's equations for electrical impedance.

After the section on the electrical grid, author Geoffrey West describes our arterial network. Our arteries can be modeled with [mechanical] impedance; energy storage is provided by the elastic expansion and contraction of the arterial walls. [Side note: this is why "hardening of the arteries" is bad: it decreases our vital capacity for energy storage and release in our pulsing arterial network.] Back-flow must never happen, because back-flow would create eddies in the arteries and precipitate blood clots. Nature knows the precise dimensions of arteries and their branches; it maintains this precise geometric relationship as we grow from the developing embryo to adulthood.

In our circulatory network, the arteries are AC, but most of the pumping energy has petered out by the time we get to the capillaries. In other words, the capillaries of our network -- its end-nodes -- are essentially DC. This is similar to many of our electronic devices: they connect to an AC source but the electronic circuits are DC; the power is first converted by a power supply. Home design is rapidly adapting to simply provide DC to our electronic devices -- avoiding the need for a separate power supply in each device. AC outlets started adding DC USB-A sockets a few years ago. Newer outlets are now providing higher-power USB-C connectors. In the main grid, we must use AC to avoid propagating the turbulence of power transitions. In the home, we generally prefer DC for most of our electrical devices. This mirrors the design of nature.

North America standardized on 110V because we already had many incandescent electrical bulbs; those fragile filaments could not tolerate any higher voltage. Europe doubled the voltage, because it generates less heat in the grid; they figured out how to step down the voltage in the home when required. North America seems to have chosen 60Hz simply because Nicola Tesla liked that number; someone in Europe (apparently Emil Rathenau of AEG) liked 50Hz. Once a particular frequency was chosen by the "first" business, it was essential that all other installations use that same frequency. Power supplies for laptop computers and other devices -- converting AC to DC -- are fully capable of dealing with a range 110-240VAC and 50Hz-60Hz.

Most (but not all) of the pertinent pages from "Scale" are available on the Google Books preview of the text.They start on p. 118 of the book in the section entitled "9. METABOLIC RATE AND CIRCULATORY SYSTEMS IN MAMMALS, PLANTS, AND TREES" and into the next section "10. DIGRESSION ON NIKOLA TESLA, IMPEDANCE MATCHING, AND AC/DC". It gives you a good taste of the entire text.

I had forgotten that impedance matching is the same as non-reflectivity at a boundary/interface. Did you ever see the Shive Wave Machine? This video shows John Shive himself demonstrating his machine: showing impedance matching then an impedance mismatch. Brilliant stuff -- a fantastic visualization. I once saw one of these machines tucked away in a lecture hall, but I never saw a live demonstration.

Biology uses impedance everywhere. The middle ear is an example of impedance matching: funky geometries of 3 little bones to translate vibrations in the air to a liquid medium.

It's great you asked yourself the question why the heart is a pulsatile and not a continuous pump.

Hi There,

I noticed that Modelica electrical has an additional library called Spice. This gives more advanced settings for components like capacitors etc. Is there an easy way to add this to System Modeler?

That's a very good question. I quickly created a model comparing 3 different cases: ideal (no thermal interaction), thermally isolated from the ambient and exposed to the ambient. Here is the model and results:

The model can be found here if you want to play with the parameters: https://amoeba.wolfram.com/index.php/s/TWQFSjnT9WTyxSj

The capacitor model in the Electrical and the Spice library looks the same to me. Unfortunately, we do not support it yet. For this course, you can create all the models using the Modelica.Electrical library.

class Modelica.Electrical.Analog.Basic.Capacitor
  Real v(quantity = "ElectricPotential", unit = "V", start = 0.0) "Voltage drop of the two pins (= p.v - n.v)";
  Real p.v(quantity = "ElectricPotential", unit = "V") "Potential at the pin";
  Real p.i(quantity = "ElectricCurrent", unit = "A") "Current flowing into the pin";
  Real n.v(quantity = "ElectricPotential", unit = "V") "Potential at the pin";
  Real n.i(quantity = "ElectricCurrent", unit = "A") "Current flowing into the pin";
  Real i(quantity = "ElectricCurrent", unit = "A") "Current flowing from pin p to pin n";
  parameter Real C(quantity = "Capacitance", unit = "F", min = 0.0, start = 1.0) = 1.0 "Capacitance";
equation
  n.i = 0.0;
  p.i = 0.0;
  v = p.v - n.v;
  0.0 = n.i + p.i;
  i = p.i;
  i = C * der(v);
end Modelica.Electrical.Analog.Basic.Capacitor;

class Spice3.Basic.C_Capacitor
  Real v(quantity = "ElectricPotential", unit = "V") "Voltage drop of the two pins (= p.v - n.v)";
  Real p.v(quantity = "ElectricPotential", unit = "V") "Potential at the pin";
  Real p.i(quantity = "ElectricCurrent", unit = "A") "Current flowing into the pin";
  Real n.v(quantity = "ElectricPotential", unit = "V") "Potential at the pin";
  Real n.i(quantity = "ElectricCurrent", unit = "A") "Current flowing into the pin";
  Real i(quantity = "ElectricCurrent", unit = "A") "Current flowing from pin p to pin n";
  parameter Real C(quantity = "Capacitance", unit = "F", min = -1.7976931348623157e+308, start = 0.0) = 0.0 "Capacitance";
  final constant Real WSMServices.Machine.Real_MAX = 1.7976931348623157e+308 "Maximum finite Real number.";
  parameter Real IC(quantity = "ElectricPotential", unit = "V") = 0.0 "Initial value of voltage";
  final constant Real ModelicaServices.Machine.inf = 1.7976931348623157e+308 "Biggest Real number such that inf and -inf are representable on the machine";
  evaluated parameter Boolean UIC = false "Use initial conditions: true, if initial condition is used";
  protected Real vinternal(quantity = "ElectricPotential", unit = "V") "Capacitor voltage";
  final constant Real Modelica.Constants.inf = 1.7976931348623157e+308 "Biggest Real number such that inf and -inf are representable on the machine";
equation
  n.i = 0.0;
  p.i = 0.0;
  v = p.v - n.v;
  0.0 = n.i + p.i;
  i = p.i;
  vinternal = p.v - n.v;
  i = C * der(vinternal);
end Spice3.Basic.C_Capacitor;

To understand such failures, perhaps it is better to start simple and create a model with the different components involved in the system. Here is a model that I created using a third-party free PowerSystems library with System Modeler.

The model has 2 mechanical generation units that have a PI controller to maintain the frequency at 50 Hz. Some constant loads, transmission lines and, a variable load that leads to fluctuation of frequency.

Whenever the load varies there is a change in frequency and the controller tries to adjust the rotational speed of the generation unit to get back to the desired frequency. There could be other reasons that can change the frequency as well like fault in the transmission lines or disturbances in the generation unit.