Power Flow Analysis (1): The Theory
In this and the following videos, we will talk about power flow analysis. It is one of the most important and fundamental area of science when we want to examine whether a power system can be operated in a safe and stable manner.
But first we need to know what it is, why it is important, and what it usually can or cannot do. Then we will dig into the physics and math of the theory behind. In the later videos, we will then discuss methods to actually conduct a power flow analysis.
Introduction
So what is power flow analysis? Basically it is a method to quantify how much power flow or current is being transmitted on the power lines, and also what the voltage is at each node of the power system. These are obviously very important parameters we want to either monitor in real time or model for a scenario in the future, because we do not want the power lines or power electronics to break down due to overloading or under- and over-voltage.
Since these parameters, power flow, current, and voltage are perhaps the most important physical parameters in any power system, power flow analysis is the fundamental tool for any safety and stability test in both monitoring and modeling. However, you probably notice that something is missing — grid frequency was not mentioned here. This is because we usually assume the power system is at a fixed stationary point when conducting power flow analysis, so grid frequency does not change at all under this assumption and it is thus a static analysis of a time slot of the power system.
As we will learn in the later videos, we may be able to find the sensitivities of the system parameters — the voltage and the power flow — with each other, so we can get a view of how easy or difficult for the system to deviate from a desired fixed stationary point, but that is still not a dynamic analysis of the power system and we cannot model contingencies such as a sudden failure of a conventional power plant with this type of static analysis. It is of course possible to model the evolution of power flows in lines, voltage levels at nodes, and perhaps cascade failure of electronics under a contingency with a dynamic model, but that will be too much for our introduction video.
The Physics behind Power Flow Analysis
Surprisingly, the physics behind power flow analysis is quite straight forward. We all have done a simplified version of it in the middle school or high school.
Let us imagine a battery connected with a load. The voltage supply from the battery is V and the resistance of the load is R. Now as any middle school physic textbook will tell you, the current flowing around the line obeys Ohm’s law, which is I = V / R, and the power supplied from the battery to the load obeys the electric power law P = V I.
Now what happens when another load is added in the circuit, but we only know its power consumption P and not its resistance?
Well, we can still calculate the voltage difference between the second load; it is V’ = V — I R. But we also know, from the electric power law, that V’ = P / I. So we should have
So the current of the circuit I is a solution to the quadratic equation
Which has solutions
And therefore
This is actually the simplest possible case in power flow analysis which we can get, but it already shows us some important concepts for later on. The most important of all is that, there is a fundamental limit on the power consumption of the load coming out of the math; namely, if P > V²/4R，then there will be a negative value in the square root bracket, so there will be no physically meaningful solution for V’; the system breaks down and we call this situation voltage collapse.
If P happens to be exactly equal to V²/4R, we have exactly 1 solution: V’ = 2RP/V. We sometimes call this point the nose of the solution curve. Draw the the possible solutions V’ as a function of P and you can see why it is being called that way; although more formally we should call it the bifurcation point.
On the graph we can also see what happens if P is lesser than V²/4R: we will have 2 solutions. Taking the minus sign in the bracket we will get a high voltage solution, while taking the plus sign in the bracket we get a low voltage solution. The set of high voltage solutions is sometimes called the high voltage branch, and the set of low voltage solutions the low voltage branch; as seen clearly on the graph, the 2 branches meet at the bifurcation point.
Both solutions are physically possible, but they have different stability properties. On the graph we can see that the high voltage solution always has a negative slope with respect to P, while the low voltage solution always has a positive slope with respect to P. So a load controller which has a negative feedback loop for controlling P according to the deviations of V’ will only work for the high voltage solution, and the system will therefore be unstable when it is at the low voltage branch of the solution.
In reality we prefer the high voltage solution because it has less power losses on the line.
The Math to Perform Power Flow Analysis (in a Complex AC Grid)
Ok, once we know the basics physics, we are now ready to perform power flow analysis in a complex alternative current grid.
First, its AC now, so we will need to deal with complex numbers. This means that voltage, current, resistance, and power flow are all complex numbers now. In the AC world Ohm law is still the same, but we usually call the complex resistance “impedance” and write it as Z, so V = ZI; whereas we write the complex power as S and the power equation becomes S = VI*, where I* means the conjugate of the current.
We usually have three phases in an AC power system, but in our analysis we will always restrict ourselves to per phase analysis, which is an over-simplification if we want to do dynamic analysis since imbalances within the three phases might occur during a contingency, but that is not what a static analysis can cover anyway.
Secondly, we want to deal with grids that have a more complex topology than the series circuit we have just shown earlier. Obviously we need a more systematic approach to attack such a complex grid, rather than using Ohm’s law one circuit a time.
To systematically analyze a complex grid, we need another law: the Kirchhoff’s current law. What it states is that for every node on the grid, the sum of the current coming into the node is equal to the sum of the current going out.
So suppose we have a grid with m lines and n nodes. We define a m by n incidence matrix [N] as the following: the entry N_ij is 1 when the starting point of line #i is node #j, -1 when the ending point of line #i is node #j, and 0 if otherwise. The direction must be assigned for every line, but it can be arbitrarily chosen.
For example we can represent the topology of this grid as the incidence matrix shown, and we can swap the -1 and 1 for any pair of starting and ending points of the lines.
Then according to the Kirchhoff’s current law we need to have
Where {I} is the current on each line, and {I_0} is the current coming in / going out from the considered grid at each node.
Ohm’s law can be also written in the compact form, if we define a m by m admittance matrix [Y]. Admittance is basically the inverse of the impedance Z, and in most practical cases the admittance matrix will be a diagonal matrix, where the diagonal terms indicates the admittance of each line. We can write Ohm’s law as
Where {V} is the voltage at each node. Note that only on the nodes can there be voltage values assigned, so if there is a battery or any other types of voltage source on a line, we need to change it into an equivalent current source with Thevenin-Norton Law.
Combining the compact forms of Kirchhoff’s current law and Ohm’s law we get
Or
This is a linear system of equations of {V}, and we should be able to solve it if we are given sufficient information, for example if the voltage of 1 of the nodes is known. Intuitively this means that we can only calculate the relative voltage difference of each node but not the actual voltage value since the reference voltage can be arbitrarily chosen as long as the relative voltage differences among the nodes remain the same.
As in our previous single load example, it is usually more common that power consumption on each node are known and we need to determine the voltage at each node accordingly. We can therefore write the electric power law of each node in the compact form:
Where {S} is the complex power consumption / injection at each node. This is the power flow equation which we will be working with in the later videos.
Some remarks
- The Kirchhoff’s current law in 08:45 can be derived from conservation of electric charge at an infinitesimally small control volume surrounding the node considered. Since inside a conductor electric charges can move freely, the net charge must be 0 everywhere in the power line, so we can drop the time derivative term in the charge density continuity equation. Kirchhoff’s current law can then be deduced by applying divergence theorem to the divergence term of the charge density continuity equation.
- Y_eq in 10:58 is a Laplacian matrix in graph theory. In a well-connected grid where all n nodes are connected to each other by at least 1 path, the rank of the corresponding Laplacian matrix will always be n-1. We will discuss more on this in the next video.
