Let's review what we have learned in the lesson thus far:

- Power, voltage and resistance are all related, through Ohm's Law: $\text{V=I\xd7R}$
- Series resistances are additive.
- Parallel resistances are inversely additive.

While the complex physics of AC power flow are well beyond the scope of this course (see Alexandria von Meier's excellent book *Electric Power Systems: A Conceptual Introduction* if you must know more), we now know enough to be able to talk about very simplified analysis of the flow of electricity on electric transmission lines. The model we'll develop in the rest of this lesson will take us almost all the way through the rest of the course.

In this section we'll introduce some concepts to use in our simplified power grid models. You may be surprised to know that the little "toy grids" that we are developing here are used in basically the same form to train people working for the biggest power companies in the world. So our little model does go a long way!

An electric transmission grid is basically a network that connects power plants ("generators") with customers ("loads"). A simple example, which is typical of one that we'll use in this class is shown in the figure below.

In our simple transmission models, nodes represent substations where generators, customers (aka "loads") or transmission lines interconnect. Nodes with a "G" are generator nodes and nodes with an "L" are load (customer) nodes. Sometimes we'll have both generation and load at the same node. A node with neither a G nor an L would represent a substation or switchyard, where multiple transmission lines intersect.

Transmission lines are represented by "branches" connecting two nodes. We will use the term "line" and "branch" synonymously.

A "path" in our model is a series of branches that connect two nodes. A path may be a single branch or a path may consist of multiple branches in series. For example, in the three-node network above there are two "paths" that connect "nodes" G and L. Path A consists solely of one branch. Path B consists of two branches in series (one branch from G to the substation at the top of the triangle, then another branch from the substation to L).

Many students confuse "path" and "branch," especially when calculating resistances.

In real electric power networks, current can flow in any direction (although in our models we will generally have power flowing from generators towards loads). So we use a numerical convention to assign direction to flow.

The way that we do this is to assign one direction a positive number and one direction a negative number. The reason for this will become clear very soon, but it's very important to remember that a negative number on a power flow magnitude does not mean "negative flow." It means "flow in the direction that we have defined as negative."

Which direction we define as positive and which we define as negative does not actually matter, as long as we are consistent.

Here's an example. In the two-node network below let's say that we have 100 MW of power transferred between A and B. So we know the flow magnitude (100 MW) but we haven't said anything about direction.

Let's say that we define A to B as the positive direction and B to A as the negative direction. So we would write our 100 MW of flow as:

- A to B: + 100 MW
- B to A: -100 MW

Our numerical convention for direction turns out to be convenient because electricity obeys the *law of superposition*. This says that flows in opposite directions (known as "counterflows") cancel each other out. So, if we have flow in one direction (which we call positive) and flow in another direction (which we call negative) then we can add up the flows using positive and negative numbers, and get the right answer.

Two examples will help to make this clear. They'll both use the two node network above, and for both we'll define A to B as the positive direction, and B to A as the negative direction.

Example 1: There is 100 MW of flow from A to B, and 100 MW of flow from B to A. The total flow on the branch is:

- A to B: + 100 MW
- B to A: -100 MW
- Total flow: 100 MW - 100 MW = 0 MW.

Example 2: There is 100 MW of flow from A to B, and 50 MW of flow from B to A. The total flow on the branch is:

- A to B: + 100 MW
- B to A: -50 MW
- Total flow: 100 MW - 50 MW = 50 MW.

We are now ready to state a couple of additional things that we know about electric power flow:

- Power is
*injected*at generation nodes and*withdrawn*and load nodes. Power tends to flow from "source" to "sink" (i.e., supply to demand). - Power injected into an AC transmission network flows along all parallel paths, apportioned by the relative resistance of each path.
- Power cannot be stored inside electric transmission lines.

This second statement is extremely important, and is one reason that electricity is a harder commodity to manage than, say, oil or natural gas. This statement says that when we put electricity into a power grid, we can't control where it goes. It eventually winds up being used by a home, business or factory, but we can't control the path that the electricity takes to get there.

So while we use networks and paths to visualize electric power flow, this can be a misleading way of understanding how power grids actually work. In fact, the power grid is a lot more like a lake than like a network of pipelines or railways. A generator providing electricity is like someone pouring water in the lake. A consumer using electricity is like someone taking water out of a lake. When water is put into the lake, the droplets disperse and are mixed with the droplets from all of the other people putting water into the lake. When water is taken out of the lake, it's impossible to figure out which "source" that water actually came from.

And the fact that electricity can't be stored in transmission lines is equivalent to saying that the lake has to be kept at the same depth constantly. Any demand to take water out of the lake must be simultaneously met with the addition of water to the lake.

These facts have implications for the purchase and sale of electricity that can be hard to understand. All contracts in electricity are essentially financial. Buying electricity is not like buying pizza. When electricity is being bought and sold, the commodity being traded is the right to withdraw some electricity from the grid and the obligation to inject the same amount of electricity (plus losses) into the grid. When I withdraw electrons from the grid, I can't say for sure whose electrons I'm getting - the generator that I paid, or the generator that someone else paid.

You may live in a place where your electric utility offers you to purchase "green power" from renewable power generation facilities. Or maybe you've seen businesses advertise that they run on 100% renewable energy. What do these claims really mean? No one attached to a big power grid is getting 100% renewable energy. So when you or a business pay more for "green power," you are getting the same mix of electrons as everyone else. What you're paying for is some change in that mix of electrons, which are then consumed by everyone on the power grid. This can give rise to a type of "free rider" problem in making power grids more environmentally friendly. We'll come back to this problem later in the course.