The Flat Earth by Jefferson Bronfeld: Source Impedance, the Other Edge
By Jefferson Bronfeld, Cadick Corporation
I have driven many mile of table top flat country; parts of Texas, Oklahoma, eastern Colorado, northern Ohio, Kansas, eastern Alberta, parts of southern Idaho... The earth's alleged roundness is easy to ignore at times like these. It seems more accurate to imagine yourself as a tiny bug, sandwiched between a huge slab of earth and an equally huge slab of sky. And assuming you are far from any of the edges, there is not much obvious evidence against a flat earth.
Perhaps that is what edges are, places where we have to abandon, or at least modify, our model for how things are and how things work. In the previous column I explored one edge of the power system, the part where it ends in the ground. In this column I will explore the other edge, the source impedance. This weird resistance like entity is used extensively in power system modeling.
Those who may not be familiar with power system modeling can imagine a circuit diagram of the power system. A circuit model simulates mathematically the behavior of an entire circuit, and allows the prediction of voltages and currents throughout the circuit, for faults and other events, which are simulated in the model. Power systems, however, are interconnected, while a circuit model can model every component in a circuit, a power system model cannot. The power system of your plant is connected to a utility power distribution system, which is connected to an interconnected transmission system, which is interconnected with most of the power facilities on the continent. It is impossible to include the entire grid in a power system model. You can only model in detail those facilities whose stability or short circuit you are studying.
It is sort of like a road map. You get a map of the area you are interested in, perhaps city roads in the vicinity of your hotel, or perhaps a map of the entire state. Some roads go right off the edge of the map, connecting to infinitely many other roads and highways all over the continent, which are not shown on your map.
The parts of the power grid outside of your immediate interest however, unlike the more distant roads and highways, have a huge impact on the behavior of those parts within your power system model. The voltages and currents experienced by components inside the model are not independent of the parts outside of the model. You can't ignore the parts outside, and yet you cannot show them. A simple arrow that says "to Boston" is not enough. This is where source impedance comes in.
Like the resistances and impedances we are comfortable with, source impedance is the impedance between two points. One point is the beginning of the power system which we are modeling, either the high side of the transformer feeding our plant, perhaps the point where "our" facilities purchase power from "their" facilities, or just any point in the power system beyond which we have no desire to model.
The source impedance connects this point, the upper edge of our power system model, with what is called "the infinite bus". And what is connected to the infinite bus? The infinite source. As cosmic a place as you could want. Didn't Xena the Princess Warrior battle somebody from the infinite source?
In radial power systems the source is usually the only part of the power system model that connects with the rest of the power grid. The source impedance combined with the infinite source, models the entire rest of the interconnected power system, everything outside the section of power system relevant to us. The behavior of everything beyond is contained in, or summarized by, that source impedance.
When I first asked about the source impedance I was told it was the Thevenin equivalent of all of the rest of the power system: the parallel/series combination of all the line impedances and all the generator impedances and all the transformer impedances "out there". This is the impedance you would see if you "looked into the power system at that point." While this is true enough, it is also a bit cosmic in scope, sounding like something from Douglas Addams's "Hitchhiker's Guide to the Universe".
We visualize an infinite bus, existing somewhere... out there... beyond the transformer high side, beyond the yard, out beyond the end of the line, just out there, somewhere. Just beyond the source impedance actually. Someday we may get to see that infinite bus, and I for one am excited about it. For that is where the infinite source is connected, and that friends, must really be something to see.
By way of a more comfortable analogy, lets look at a DC battery circuit. Picture the circuit diagram of a battery, say 12 volts, connected to a 1 ohm resistor. Draw out the circuit if you want, nobody is looking.
Your drawing should look something like Figure 1. The current flows out of the positive terminal of the battery, through the resistor and back to the negative terminal, (unless you identify positive current as the direction of electron flow, but lets not go there). The magnitude of the current is calculated by Ohms Law as 12 volts divided by 1 ohm, or 12 amps. For lower values of resistance, the current is higher, as predicted by ohms law. (I=E/R) Half an ohm would yield 24 amps, a tenth of an ohm would draw 120 amps from the battery, and so on.
In reality we know this isn't true. A real battery under load has a somewhat lower voltage across its terminals than an unloaded battery, and the voltage drops significantly for heavy load current. Our ideal battery is modeled by a "perfect" voltage source, one which has infinite energy and can provide any value of current at a rock solid 12 volts.
To model a real battery requires the addition of another resistor. This resistor is connected to the positive side of the battery, in series with the load, and is considered to be "inside" the battery, i.e. "behind" the battery terminals, as shown in Figure 2. This resistor corresponds to the internal resistance of the battery.
Assume for this example, an internal resistance of a tenth of an ohm. For a 12 volt battery, and a 1 ohm load, we would have a total current of 12 volts divided by 1.1 ohms, or 10.9 amps. The voltage across the battery terminals, which is the voltage across the load resistor, is also calculated by Ohms law, (E=IR) as 10.9 amps times 1 ohm, or 10.9 volts. As predicted, the voltage is lower than the nominal 12 volts when the battery is under load.
If the load resistance is one half ohm, the total current flow would be 12 volts divided by 0.6 ohms (0.5+0.1), which equals 20 amps. The voltage across the battery terminals for this current flow is 20 amps times 0.5 ohms equals 10 volts, lower than last time.
For a very low load resistance, (modeling a very high load on the battery), let's use say 0.2 ohms. The total current drawn would be 12 volts divided by 0.3 ohms which equals 40 amps. The battery terminal voltage would drop to 8 volts. (40 amps times 0.2 ohms.)
Note the "perfect" voltage source, behind the internal resistance, never changed in value. It stayed rock solid at 12 volts for all current values.
By the use of the internal resistance we have modeled behavior more reminiscent of actual batteries, where the terminal voltage drops for heavy load currents. The combination of the "perfect" voltage source in series with an internal resistance behaves more like a real battery. The internal resistance is not an actual resistor inside the battery of course, it is a circuit model element added to "capture" the voltage dropping behavior of real batteries. And the "perfect" voltage source is a fiction too, it is the voltage "behind" the internal resistance of the battery, but does not correspond to any particular voltage source in the battery. You cannot separate out the perfect voltage source or connect anything directly to it, or even measure it. The chemistry of the battery requires two discrete components to model its behavior, the internal resistance and the voltage source "behind" it, and neither actually exists separately inside the battery.
The internal resistance and its perfect voltage source, I think, is a better way to understand the power system's source impedance. They behave in similar ways. (The power system is AC, and so both reactance and resistance are important, hence we use the term impedance. Don't let this detail get you off track.) Like the battery's perfect voltage source behind the internal resistance, we have an "infinite bus" behind the source impedance. This "infinite bus" has an infinite source, with enough energy to supply any amount of fault current while maintaining full power system voltage.
The infinite source, infinite bus, and the source impedance are as much a fiction as the battery internal resistance. They do not depict any particular power system element; instead they capture the behavior of the entire external power system at the point where it connects the power system model. Applying a fault directly to the infinite bus does not correspond to a fault anywhere in the power system. It is meaningless. You can, however, apply a fault at the other end of the source impedance, the near end, where the model of our power system begins, and the resultant current flow predicts what would happen in the real power system if you applied a fault there. And when we apply loads or faults deep within our power system model, the voltage at the source of our system will drop somewhat, as we know it would in reality.
The source impedance depicts the relative "strength" or "stiffness" of the entire power system at its point of interconnection with our modeled system. A strong or stiff system is one whose voltage does not drop much for heavy current flow. A weak system's voltage, on the other hand, drops dramatically for large current flow. The ratio of voltage drop for a given current flow, (be it load or fault current), like any other voltage to current ratio, is, per Ohm's law, accurately modeled by a resistance.
In a future column I will talk about some other resistance like entities, equivalent impedances, transfer impedances, and the like. Until then, don't go near the edge.
Questions? Comments? Send them to editor@electricnet.com.
Jefferson Bronfeld is the Chief Engineer for Cadick Corporation, an electric power engineering consulting and training company. A licensed professional engineer with two patents, Jeff has over 19 years experience in the power industry. Jeff makes his home in Binghamton, NY, where, when not working, Jeff is active in the IEEE, and tutors mathematics to local junior high and high school students. When not being a nerd Jeff can be found playing fiddle tunes on the mandolin, fly fishing, or asking someone to take him sailing on their boat. Jeff's column appears every Monday on ElectricNet. (Back to top)