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Inertia in power system: We don’t actually need that much

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What does inertia have to do with controlling power system frequency?

Surprisingly, not much.

Some carefully selected analogies might make the reasons for this clearer.

Is it easier or harder to control the speed of a motor cycle or a heavily loaded truck?

Most people would correctly surmise that it is easier to control the speed of a motor cycle because it has less inertia – but we are being told by various reports (e.g. The recent Finkel review into the Australian National grid, and prior reports released by AEMO etc.) that we need to place limits on the minimum amount of inertia on power systems.

Is this emphasis on inertia misplaced? I believe it is.

Objects with large inertia want to keep moving (or rotating) at the same speed and as a result it needs large amounts of force to slow or speed things up. AC power systems are designed to operate at a constant speed, we sometimes refer to this as “frequency”, but this translates directly to the speed that rotating devices such as motors or generators operate at.

Therefore, the degree to which external forces can change the system frequency is inversely proportional to how much inertia is on the system.  So, the reasoning goes, if you want constant system frequency set up the system have as much inertia as possible, this means it takes more force to change it.

This simplistic approach seems to me to be misguided.

If implemented it could place onerous restrictions on the connection of renewable energy sources, in particular power stations of solar photovoltaic design which have no rotating parts and hence no inertia.

Perhaps we should examine this industry wide preoccupation with inertia a bit more critically.

The reason controlling the speed of a motor cycle is easier than controlling the speed of a truck is due to the responsiveness of the throttle/engine combination. Although a truck engine is more powerful than that of a motor cycle, the power to weight ratio is typically much less. This allows motor cycles to accelerate much faster than heavily laden trucks.

It is not inertia that allows constant speed, it is the effectiveness of the throttle/engine combination, the effectiveness of the control system.

In power system subjects, the terminology for “throttle” is “governor” and “engine” is usually referred to as “turbine”, although this depends on the details of the actual technology used to drive the generator. So the main thing that controls the speed, (which is the frequency) of an electrical power system are the governor controls.

Most of the points I want to make in this blog have already been made. The discussion below will be a slightly more technical and will include a few equations which will turn off some readers, but I hope you will persevere because the technicalities of an issue sometimes matter.

For experts in the subject the discussion below will be too simplistic but I ask them to remember that this is a blog, not a textbook.

Governor Control systems 101

In power systems theory, generator/turbine inertia is represented by the symbol H. The relationship between the power mismatch and the speed deviation is usually represented as 1/(2 H s) , s being a Laplace transform variable. With no governor control system if there is a power mismatch the frequency will ramp in proportion to the power mismatch, and inversely proportional to the inertia.

In calculus terms, the output (frequency deviation) signal is the integral of the input power mismatch – the inertia H being the constant which determines the slope of the ramp.

Large inertia implies lower gradient ramping relative to a small inertia.

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Note: – it is not possible to limit the frequency deviation by inertia alone, a power mismatch will cause the frequency deviation to ramp until something malfunctions and the power system collapses resulting in a blackout.

To stop the frequency from continuing to ramp, you need governor control systems.

The governor measures the speed of the turbine and feeds back a correcting power signal to the input. This is a well-known technique for control systems and is used in virtually every technology you could name, e.g. process controls, electronic amplifiers, aircraft, space craft, steam engines etc.etc.

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When you do the math represented by the inertia and the governor in the algebraic/block diagram formalism of Laplace transforms you get this:

∆f/∆P = 1/(2 H s)/(1+Gov(s) 1/(2 H s))

Translating the math to physics means the governor limits the frequency ramping – depending on the governor design parameters this can be made to return the frequency to its original value (this is known as isochronous governing), or just to limit the final variation to a finite amount.

The point of introducing the control system algebraic formalism is to show what happens if the inertia is made smaller. Below I have created three arbitrary systems and calculated the response of each. The only difference between the first and second cases of calculated simulations is that the assumed inertia is halved in the second case. For the third case I modified the governor parameters to change the response of the second case.

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As you can see, whilst the frequency initially changes more quickly in the second case (lighter inertia) the frequency excursion is slightly more, but control is established quickly and the final frequency deviation is exactly the same. This is not surprising for a control system engineer, in effect reducing the inertia has (in control system theory parlance) increased the closed loop gain. This has the effect of speeding up the response and making it more oscillatory. However this is easily fixed by making some minor changes to the governor responses.

We have ended up with a system of lower inertia which has a better response.

So in actual fact reducing system inertia and retuning the governor has actually made things a better with respect to frequency control.

The mathematics shows it is easier to control the speed of motor cycles than it is for trucks.

There are small scale power systems in operation which have effectively no inertia, so in theory and in practice it is possible to design a large scale power system along similar lines. It won’t happen overnight, but it is possible to design and operate a national power system with no inertia. A good project for the 22nd century perhaps.

In the meantime we have a system which is changing its inertia as the mix of generation changes. Does this mean the system characteristics change? Obviously.

Is it a system security concern? Not necessarily, but it does need suitably qualified power system engineers to keep an eye on it.

What needs to be done? Well in the short term we might need to retune some governor control systems to compensate for a lighter system – that is all. An additional benefit is that this will also address many of the issues that the FCAS market has caused relating to the recent deterioration of frequency control.

In the longer term we need to ensure new renewable technologies can contribute to frequency control, but this is only important if we disconnect many of the synchronous generators which currently make up 90% of our grid. It is an upcoming issue but the time frame for this is 20 – 30 years, not one.

If we do this, we won’t need to mandate inertia limits for various regions of the system.

Bruce is an Electrical Engineer and senior advisor at Advisian. This article was first published on his LinkedIn page. Reproduced with permission of the author.  

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  • PeterT

    Thank you for this illuminating discussion of inertia. I have long thought that the comments on inertia from certain sectors (politicians) seemed misguided. We live off-grid with batteries and an inverter rated at 5 kW continuous (12 kW short-term peak). The electronics in the inverter are able to go instantly from zero to rated load and above without affecting frequency. That’s better than an engine-driven generator. I’ve often wondered why grid-scale renewable systems could not have the same performance as our little system.

    • Bruce Miller

      In theory they can. In practice this will take a long time to achieve.

  • Peter F

    There are two further issues which reinforce your point.
    1. Inertia includes all elements of the system including the transmission grid and the loads. A motor driving a big centrifugal fan has far more inertia per MW than a gas turbine generator and the capacitance of long distance transmission lines is significant so generator inertia is only a part of the system. It could be as low as 1/3rd of total system inertia
    2. A steam turbo-generator has an Inertia Constant of around 9 seconds, a gas turbine 3s and a low wind style wind turbine 4-5s. Inertia Constant of n means that the total rotary inertia is n seconds worth of rated power. Because synchronous generators can only change speed by an absolute maximum of 2% they can only supply (1-0.98^2)= 4% of their rotary energy as useful energy anyway. i.e. a 250MW gas turbine can supply the equivalent of 10MW for 3 seconds or around 30MJ.
    Variable speed wind turbines are asynchronous devices and therefore can slow down by 10% or more without affecting frequency. Thus they can supply (1-.9^2)= 19% of their rated power for 4-5 seconds. Thus 250MW of modern low wind wind turbines could supply an inertia response of around 200MJ

    • Bruce Miller

      Thanks Peter. That is correct and I understand that this is implemented on wind turbines fitted with virtual inertia. There is an energy cost that must be bourne when reaccelerating the turbine.

      • Peter F

        That is true and the more energy you give up the longer the recovery time. This also applies to any inertial response whether it is gas turbines steam turbines synchronous condensors or whatever.
        As Steve says above batteries or wind or solar plants running below peak output can supply response much faster than any thermal plant.
        One further problem with gas turbines is that as frequency falls compressor pressure drops by the square of the speed so a 2% frequency drop means a 4% output drop. So the governor response has to recover both inertia and power fall. Depending on the both the control system and thermo-mechanical time constant of the machine the energy lost to the system from the power reduction may be much greater than than energy contributed from inertia.
        If the grid is close to the limit the negative power response of gas turbines actually increases system losses and can lead to cascading failure. Such negative disturbance response does not happen with wind, solar or hydro and is unlikely with steam plant.

  • Steve Silich

    thank you Bruce for this very useful explanation of inertia in the power system.
    Inertia is not the problem. The problem is frequency response. Most solar inverters are set up to deliver maximum output of their PV into the grid and therefore have no frequency response at all. This means that when load on the grid varies, the solar inverters continue without change to output. The existing thermal and hydro (rotating) power stations must take all the load change. As the amount of renewables increases and we have less rotating capacity on the grid, the size of load changes per unit (PU) will increase for each generator. Ultimately, if there is only one rotating generator on the grid it will be taking all the load changes.
    So, we need to have the facility for solar and other renewable inverters to take a share of the load changes. This feature will change the output of each inverter with frequency and implies each inverter will be running at less than 100% during normal operation, but available to increase or decrease output in response to frequency. Adding batteries to a renewable resource will make this easier. Mathematically this is a governor response to frequency, not inertia. Traditionally inertia has been useful to bridge the gap until governors could add power to the turbines. With static inverters providing much or all of the generation, we will not need any inertia because the static inverters can respond within milliseconds. However we do need these static inverters to be capable of responding to frequency, both increasing and decreasing.

    • Bruce Miller

      Thanks Steve, I agree with your summary.

    • Peter F

      What you say is true and as the renewable share increases it may become necessary to run both solar and wind inverters at 95-97% capacity just as there are always a number of thermal/hydro plants running below capacity as spinning reserve.
      However the amount of “wasted” capacity is much smaller because the ramp rate of the inverters is much faster than gas or steam plant so less idle capacity is needed.
      Strangely enough, it is getting close to the point in the current grid where wind and solar plants should add storage anyway to improve their revenue
      a) because even a small amount of storage would allow them to participate in relatively lucrative, 6 second, one minute and 5 minute raise markets.
      b) If they added storage of about 25% of capacity for 2-3 hours, then they can shift about 20% of their sales to peak pricing. That plus FCAS revenue can increase the total annual value of their output by 40% for about 30% increase in investment and little or no increase in operating costs

    • FeFiFoFum

      But the operation of the grid is not based on all generators chasing the load and adjusting their output to maintain frequency at 50.00 Hz.
      Some units are assigned to run at 100 % ( baseload for want of a better term)
      Others are assigned to control frequency and some are assigned with spinning reserve. Usually the frequency controllers also provide the spinning reserve.

      From a market perspective, all of these generators get paid to participate in the market ( in WA at least as a capacity credit) as well as get paid for their output ( spot market price or contracted price) and those allowed to participate in FCAS get paid for that service as well.

      Its not a matter of all generators that are connected to the grid having to adjust output to maintain system frequency.

      Only exception to this is when you have a significant network event which is usually the loss of a large generator of source of generation where you end up with a shortfall of generation to meet load demand (frequency goes low), or a load rejection event due to a fault on the network, where you lose load and end up with a surplus of generation ( frequency goes high).

      In these circumstances additional generators other than those controlling frequency or providing spinning reserve may have to adjust their output for the duration of the disruption before the system is brought back into equilibrium.

  • TW

    Bruce
    You have it right as far as you have gone, but you have not gone nearly far enough.
    Just imagine a 10,000MW system with say 10 conventional machines generating. A 500MW (say) machine trips. What happens?
    First, in the network voltage will fall. This will be corrected by the voltage controllers on the generators. This will also give a power boost for a short time.
    The frequency will start to fall. The governors on each machine in the network will start to open the throttle valves. A feed forward signal will go to the boiler controls and start firing the boilers harder, but the boiler pressure will start to fall in each machine on the network.
    Over time, machines with spare capacity will load up and any quick start machines ( eg hydro) will start.
    Slowly the frequency will rise, but only after 500MW of generation has been raised to replace the 500MW that was lost.
    To all but those involved, the network carries on as usual.
    Without inertia, the frequency would fall much faster and people would notice. That is why AEMO has a price for 6 second raise, 1 minute raise, 5 minute raise etc.
    The new technology can be so fantastic – large batteries can provide load in milli second not seconds or minutes. Non synchronous wind turbines can be slowed down much more than the system frequency and therefore contribute inertia.
    System modelling which you have just put your toe into is complex. You have to model the generator transient, sub transient and static parameters, the voltage controler response, the governor response and the turbine response for each machine in the network. Add in the impedance of all the transformers and transmission lines and you have a very large model.
    There are a lot of very clever people doing this all the time. Don’t diminish their work carelessly. System modelling is a highly specialised field. If you want to dip into it, there are many good text books out there. “Power System Stability and Control” by Prabha Kundur published by EPRI is a readable text, have a look at it, but be ready for over 1000 pages of high level maths. Enjoy. It is a great subject.

    • Bruce Miller

      Thanks TW for your considered response. I am flattered you think I am clever.

      • solarguy

        Well Bruce, you are clever and so are all the guys, who are commenting on this subject here. I personally can’t add anything to this discussion, except to say, that I didn’t think a high renewable grid, especially with storage was ever going to be a major problem going forward. Clearly the answers to the problems are forth coming.

        I enjoyed reading your blog, but yes the maths, as you warned are definitely not something that I understand, as unfortunately for me that area of my brain never developed that level of capability. Perhaps a refund is in order LOL. However, as I design and install grid and off-grid SPS systems, I fully understood everything else.

        • FeFiFoFum

          Ditto. ( what he said).
          And we need practical people who construct and install these systems anyway so I think you actually can add plenty to the discussion with real feedback on coal face ( excuse pun) knowledge on how an installed system works and performs, which comes back to good design and understanding of actual conditions on the ground.

          • solarguy

            Cheers mate!

  • Ian

    Very considered answers in this discussion but I have some questions. Why can’t the provision of ‘inertia’ be split into 50Hz plus correction and into 50Hz minus correction. Wind turbines are most cost effective when running at full available load most of the time. These can be off loaded and the wind energy throttled back occasionally when load drops. On the other hand, batteries and inverters can inject more energy into the system when load rises.

    The second question is this : an inverter requires a source of energy to drive synthetic inertia, usually directly from solar panels or from batteries. Is there any reason why an inverter could not derive its energy input directly from the grid it is trying to stabilise?

    Thirdly, the whole problem of grid stability can be traced to matching energy input with energy output. TW gives an example of a 10000MW grid, the typical distributed solar generator is 5KW. Each system will face a huge monolithic pool of electricity. There might be 2 million of these connected to the grid. How is the mass of these to adjust output to stabilise the grid? Has anyone modelled such a scenario?

    • Bruce Miller

      Hi Ian, I will try to answer your queries as best I can:
      Splitting inertia – in effect this is already being attempted by having different raise and lower FCAS markets, but in my view these have not been very successful.
      Investor deriving power from the grid – in conjunction with energy storage this can be and is done. Without energy storage you effectively have a “pulling your self up by the bootstraps” issue. The more power you take out of the grid in order to put it back in is like setting up two opposing forces.
      Matching energy input with output – I understand people are looking at aggregating solar PV to make it centrally controllable, there are issues with communication delays etc but in theory and in practise the technique appears to be a valid approach.

  • Ray Miller

    Thanks Bruce for the post.
    Inertia is an artifact of last century power engineering and maybe belongs in a museum.
    It is becoming very clear we need to be allot smarter at managing our energy system and drill down into the fundamentals of the whole system. Every element from all the small loads to the largest rotation machine all have various characteristics and behaviors, including the important temperature sensitivities.
    We urgently need the quality engineering to inform and guide the future. What is possible today was not possible yesterday. We are adding 100’s of MW of load and generation to the NEM each year but no one is looking at the opportunities to add extra system value at largely no cost.
    So is a better option to install electronic inverters and battery system at all substations (using the same design and mass produced to lower the cost) to provide a range of power quality, security, phase balancing and backup functions working in conjunction with the local loads and coordinated by AEMO?

    • Bruce Miller

      Thanks for your comments Ray. In future (maybe 22nd century) we may have electronic inventors or their equivalent virtually everywhere. However this would require a large economic investment which I think may happen, but only if there are sufficient economic drivers to make it happen.

      • Ray Miller

        My backup inverter and battery system, be it small 500VA provides constant 50Hz, 1500VA+ of surge with voltage regulation and low harmonic distortion.
        Anyway your idea to question the status quo is good, many of our engineers have become very lazy and with little to no imagination. They have been very complacence, seem to wear blinkers all the time while the world in changing rapidly around them. Hopefully they can see the many opportunities presented and start to solve some of the real problems instead of just complaining and finding excuses about change.

  • FeFiFoFum

    I’m not convinced you don’t need the inertia going forward. ??

    Frequency control is only on of the parameters that needs to be considered.
    Network faults cause a lot more disruption to a system than does generators tripping off.
    Depending on the electrical angle in the system and the fault location plus the location of generation relative to the fault, it is possible for the network to decouple and go out of synchronism.
    Diesel generators tend to be very tolerant of frequency deviations and fault events compared to gas turbine or steam turbine generators. Isn’t this directly related to their inbuilt inertia?

    I picture your motorcycle that can accelerate and decelerate quickly but that diesel powered prime mover will keep ploughing ahead when the going gets tough and the bike has stalled. Isn’t that torque = inertia?

    And this constant reference by ill informed pollies to ‘synchronous’ generators like they are something special.
    All generators connected to the grid need to be ‘synchronised’ or you have catastrophic results !! ( I guess the exception being wind turbine generators that are synchronised to the grid via their inverter).

    So if someone can explain how and why synthetic inertia can replace rotational or dynamic inertia I would be interested to understand this.

    • Bruce Miller

      Hi FeFiFoFum – you are correct in stating that there are other concerns than just loss of generation. We also have system faults etc to consider. With respect to your queries on synthetic inertia, that is a subject worth several articles which hopefully I will find time to write. In the meantime though I will draw your attention to the ESCRI project which is due to be commissioned early next year. This will use battery storage to provide a grid reference for residential load and hence will have only nominal inertia. This will demonstrate several principals in practice on a fairly large scale ~ 5- 10 MW which so far have only been talked about in theory.

  • Joe

    There is too much inertia in….The Liberal Government

    • Zane Alcorn

      bahahaha