A precis of Grid Following and Grid Forming Inverter control

Introduction and motivation

Our last post link touched briefly on these topics, but given its importance to how we transition the power system, we thought it was worth following up these issues with a bit more detail.?

As we will be using some equations we risk alienating ? of our audience by daring to put mathematics into a LinkedIn article (albeit at an educational level no more advanced than taught in high school) ,? and the other ? of our audience (specifically the power system engineers) by being too simplistic.

For both groups, we ask that you bear with us.

For the first group, understanding these concepts is important if we are to make good decisions on how the grid should be transformed. Recent events in the regulatory space have not been encouraging.? The industry is struggling with new technologies and seems to make the worst decisions possible before finally (which seems yet to happen) landing on something which we can all work with.? We are reminded of similar periods in history of rapid technological change:

• people used to walk in front of cars with red flags so that the new automobile technology didn’t scare the horses,
• globally we have non-standard railway gauges,
• Different standards for electricity and communication systems. E.g. why are some of the world’s power system frequencies 60 Hz, and others 50 Hz? ?Why are some voltage levels used in one location and not in others (e.g. 220 V vs 110 V)?

Basically it was because in some jurisdictions poor design decisions were made early on, and then in some cases were stuck for ever more because it costs too much to change.

Not always, fortunately we no longer walk in front of cars with red flags.

Poor decisions are fueled by a lack of basic understanding. This seems to have been clear in the recent actions by some of our regulators.

For the second group who have strong skills in power system theory and who may find the presentation below overly simplistic, we would ask that you reconsider. It is easy to get lost in a maze of calculations and computer simulation results and lose track of a basic understanding of what is happening at a higher level. Hence the reason for this essay.?

It has been our observation that there has been far too much reliance placed on computer simulation results to inform parameter setting choices and by default not enough on establishing basic good engineering practices.

Reams of computer printouts are of little use if no one looks at them, or if those who did scan the results are too busy (or lack intuitive understanding) to pick up on key issues. Those issues might be obvious if they had a more basic understanding of the fundamentals of power system behavior, i.e. failing to see the trees because your microscope is focused on the wood.

Here end if the soap box – check out the first equation.

P = V I

Power equals voltage times current.?

The flow of Power is what we are interested in controlling in a generation or battery system. The control of voltage and current are merely the means of achieving control of power.? In simplistic terms we can change the flow of power by changing either the voltage (which is often thought of as a kind of “pressure”) or changing the current (the flow of electric charge).

This formula works for DC (direct current) systems, but it has to be modified a bit to make it work for AC (alternating current) systems.

The differences arise because DC can be adequately described with just one number, its magnitude, but AC is based on sinusoidal waveforms which need two numbers, magnitude and phase, to provide an adequate mathematical description.

Accordingly we need to update the equation to reflect these differences.

S = P + j Q = V I*

In words this is “Apparent power equals Power plus j times the reactive power which equals voltage times the complex conjugate of current”.

A bit to unpack here.

“Apparent power” is a term not often used but it is effectively just an abbreviated way of saying power and reactive power.

“Power” is what we want to control and buy or sell from our generators – the basis of the electricity market.

“j” is what electrical engineers use for the square root of minus 1 – because “i” was already used for current which is what mathematicians use. If you don’t know or have forgotten about complex numbers – don’t worry – the following descriptions won’t stray too far into this field of study.

“Reactive Power” is not what we need to ultimately operate our appliances, i.e. it is not power (which is sometimes referred to as “active power”) – it is needed so we can mathematically fully represent the “Power waveform” which needs 3 parameters – magnitude, phase and dc offset.? Some mathematics[1] allows this to be reduced to 2, P and Q, and because AC systems theory plays out in 2-dimensions, it can’t be reduced further.

“Voltage and current” in AC systems are no longer represented by single magnitudes, because they are sinusoidal waves which each need two parameters to describe (magnitude and phase, ignoring frequency which for the purposes of this article will be assumed to be constant 50 or 60 Hz).? Instead, it is a useful shorthand to represent them as arrows (the fancy terms often used are “vectors” or “phasors”).

For a given sine wave (current or voltage or apparent power), the length of the arrow is its magnitude, the direction of the arrow its phase.

In summary – to adapt the fundamental DC equation P = V I into its AC equivalent, we have to represent V and I in two dimensions (magnitude and angle), and using a special product rule[2] ,this replaces the one dimensional P into its two dimensional equivalent S, which can then be represented by components P and Q – each of which is still one dimensional.?

Reactive power “Q” naturally appears because we need another quantity to represent a second dimension. ?This is the S = P + j Q identity. The j operator being used algebraically rotate Q into the second dimension.?

All calculations in AC power systems operate in two dimensions for which complex numbers are ideally suited, and because of this they are widely used in electrical engineering.

The following sketch summarizes the relationships described above.

Control of Power – Grid Following

If the grid that the inverter is connected to is “strong” (i.e. low impedance), the voltage won’t change.? Accordingly, to change the power – we just need to change the current. The angle between the voltage and the current determines the power and reactive power components. Because the voltage is fixed, apparent power is directly proportional to the magnitude of the current - no worries.

Change the angle of the current relative to the voltage angle, and the angle of the apparent power also changes – this gives us the components of power and reactive power.

Effect of a weak grid on Grid Following

Problems can arise when the grid is “weak” i.e. high impedance. In this case as current is pushed into the grid, the voltage changes (because of ohms law) which causes the apparent power (i.e. power and reactive power) to also change.

The following chart shows the trajectory of the apparent power for an infinitely strong grid compared to “weak” grid with a short circuit ratio of 1.2, and an X/R ratio of 3.

As you can see, there is quite a large gap between the apparent power delivered to the strong system compared to the apparent power delivered to the weak system if the current is not adjusted to make up the difference.

To make up the difference, both the angle and the magnitude of the current has to be changed, the following chart shows by how much.

This gets worse the weaker (i.e. the lower the fault level at the point of connection) the system is.

In the extreme case of an islanded grid (i.e. infinite impedance, zero fault level) – it can’t work. But even before this point is reached – larger and larger currents have to be controlled in order to control the output power to its set point. This can lead to control system instability as we discussed in our previous article.

The incremental change in current for a change in Power is plotted below, (gain = dP/dI):

For a “strong” zero impedance system, the effective gain does not change, but for the “weak” system the effective gain increases with increased power output in a non-linear way.

Control theory tells us that the risk of instability increases as the control loop gain is also increased – this was discussed in our previous post.

Effect of a strong grid on Grid Forming

There are a lot of different “types” of grid forming inverters, which on closer inspection seem to be more influenced by marketing than by engineering.

In this over simplified description, we are defining grid forming inverters to be the dual to grid forming described above. Specifically, to control the power output, the voltage is controlled and the current is left uncontrolled, i.e. it will flow as required to satisfy the equation S = P + j Q = V I*.

For an infinitely strong, i.e. zero impedance system, this won’t work, any change in voltage will result in an infinite change in current which is not physically possible.

Therefore, grid forming inverters need some system impedance (which limits how much the current will change by) to work.? In the extreme case, they can even operate when the grid is not present to supply local loads, i.e. when the impedance is infinite.

Accordingly there are two modes of operation, when it operates as an island, and when it operates as part of a grid.

Grid forming on an islanded system (three cases)

The simplest mode of operation is the islanded case.? Basically the load determines the behavior of the system and the only way that power can be controlled is by varying the voltage magnitude output of the inverter. ?

How the power is affected depends on the load, of which there are three mathematically distinct cases which are easy to analyze. ?Actual loads are a complex combination of all three types with often some dynamic characteristics thrown in, but this is best left to computer simulation.

The three types of load are:

·???????? Constant impedance – e.g. incandescent lights, resistor based heaters

·???????? Constant current – which fluorescent lighting approximates

·???????? Constant power – which motors driving pumps or fans approximate

?

Changing the voltage magnitude has some effect on the power delivered to constant impedance or current loads but no impact on constant power loads.

Power systems are also designed to be operated at constant voltage close to 1 per unit, the scope to effect changes is thus severely limited.

Therefore, in islanded systems it is the load which necessarily determines the inverter power output – not the control systems of the inverter. ?

Grid forming control is only needed to provide a stable voltage source – the load then determines what the current is going to be.

Grid forming when connected to a power system

When grid forming inverters are connected into a power system shared by other generation, the control is a bit more complex.? As soon as you have two or more generators on a power system, you need a way of controlling the power output of both so duty can be shared.

Changing the voltage magnitude is not an effective way to control the power ouput, and besides power systems are designed to operate close to a nominal voltage level, so we would like this to be fixed close to 1 per unit.

So what to do?

Fortunately we are working with AC power systems, if we can’t change the voltage magnitude, we can always change the voltage angle.

In the following we are assuming the inverter is connected into a power system which is much larger than the inverter rating. Specifically changes in the inverter P, Q output have no significant impact on the rest of the generation in the power system. This is a variant of the single machine infinite bus (SMIB) assumption which is commonly used in power system analysis.

Changing the angle over the full 360 deg range (which is never done in practice for reasons discussed below) results in circle loci in the PQ apparent power plane, as shown below.

The amount of power that can be transferred depends on the impedance of the power system. Weak, high impedance systems are only able to accept limited amounts of power – whereas strong, low impedance power systems can accept larger amounts of power.

This is not a limitation of grid forming inverters, rather it is a limitation of the system itself. All types of generation, grid following, grid forming, and traditional synchronous machines are subject to the same limitations.

If you want to be able to transfer large amounts of power from one point on the system to another – you still need low impedance transmission lines to do it.

In order to keep the voltage magnitude at the point of connection constant, the reactive power has to change with the power, which is the other dimension required to make the circular loci.?

How do grid forming inverters perform with respect to control system stability? Well, it’s a bit more complicated than the case for grid following inverters.

To change power – grid forming mainly changes the voltage angle, whereas recall grid following changes mainly the current magnitude.

Depending on the X/R ratio and the impedance of the power system, the relationship between the change in power to the change in voltage angle is very non-linear as shown below.? This makes the speed of response of a grid forming inverter highly dependent on its operating point.

The following are some key points to note:

• The highest control loop gain is at low voltage angles
• “Strong” systems have higher gain than “weak” systems
• Beyond a certain angle – the gain can switch to negative

Recall that higher gains tend to give faster responses, but they are also more prone to control system instability.? Grid forming inverters are thus more likely to be unstable at low power levels, and when connected to strong power systems – this is the exact opposite to the situation for grid following inverters.

Another feature is if you push the angle too far, the control loop gain reduces to zero (which means no feedback control) or even negative. Negative feedback control loop systems with negative gain become positive feedback control loop systems – which don’t work.? Their outputs diverge from the control set point in an exponential fashion.? This is another point of potential failure that needs to be guarded against for grid forming control. ?We need to keep the angle range within reasonable bounds.

This is in fact nothing new to power systems, existing synchronous generation can fail in an analogous way, where it sometimes referred to as “pole slipping instability”. If the angle between the generator and the system is too high, the direction of the synchronizing torque changes and the generator can pull out of synchronism.

Grid following combined with grid forming

The power equation S = P + j Q = V I* has two terms we can change, V and I. With grid following inverters we change I (the current) and let V (voltage) be whatever it needs to be, whereas for grid forming we change V (voltage) and let I (current) be whatever it needs to be. ?In both cases the uncontrolled voltage or current signal is fed back to provide a reference for the control system.

However could we change both the voltage and the current at the same time? Can we walk and chew gum at the same time?

The authors know of inverters that can be both grid following and grid forming but so far they have tended to be either one or the other but may be able to switch from one to the other. ?So far we don’t know of an inverter on the market which is both grid forming and grid following at the same time. In theory it is possible.?

Whether it would be useful or not is an open question. Simplicity is often a virtue.

In another article, link , we stated about inverters that :

• their strength is they can do virtually anything
• their weakness is they can do virtually anything

Conclusion

Thank you very much if you have managed to read this far. For a LinkedIn post we realize this has been very long but we hope you found the journey enlightening.

Feel free to add your comments below.

There is a lot more technical detail we could have discussed, but time and the format imposes its constraints.

We hope that these summaries will be enablers for the energy transition to be as smooth as possible.

[1] via complex conjugation of the current – this has the end product of subtracting the angle of the current from the angle of the voltage, i.e. two angle parameters are combined into one which is referenced to the current angle

[2] The Hilbert Inner product