Frequency stability and energy balance. A description of the interaction between frequency and grid energy flow.

In an earlier post I gave a definition of a stable power system.

A stable power system is one that continuously responds and compensates for power/ frequency disturbances, and completes the required adjustments within an acceptable timeframe to adequately compensate for the power/frequency disturbances.

This post delves into the mathematics of what happens in the first few seconds after a disturbance. So apologies – this post does involve some maths.

The first few seconds after a power balance disturbance is analysed using Newton’s laws of motion. We’re going to look at the power flow between the rotating inertia (rotational kinetic energy) of a synchronous generator and the power system. It applies for the first few seconds after the onset of a disturbance, i.e.: before the governor and prime mover have had opportunity to adjust the input power to the generator.

We begin with the rotational form of Newton’s second law of motion; Force = Mass * Acceleration.

T = J * d2A/dt2

T is the net accelerating torque applied to the rotor. T is the difference between the driving torque of the prime mover Tm, and the retarding torque from the electrical load Te. i.e.: T = Tm – Te.

J is the moment of inertia of the rotor (and the attached prime mover)

A is the angle of the rotor with respect to a fixed reference

d2A/dt2, is the second differential of angular position with respect to time, the rotational acceleration of the of the rotor

We’ll define w as the rotational speed of the rotor, i.e.: dA/dt . w is equivalent to frequency.  The shaft acceleration d2A/dt2 is equivalent to the rate of change of rotor speed dw/dt. The shaft power is the product of torque and angular speed; i.e.: P = T * w, and angular momentum M is the product of moment of inertia and angular speed; i.e.: M = J * w. Using these we can derive the following equation:

Pm – Pe = M * dw/dt

Pm is the mechanical power being applied to the rotor by the prime mover. We consider this is a constant for the few seconds that we are considering.

Pe is the electrical power being taken from the machine. This is variable.

M is the angular momentum of the rotor and the directly connected prime mover. We can also consider M a constant, although strictly speaking it isn’t constant because it depends on w.  However as w is held within a small window, M does not vary more than a percent or so.

dw/dt is the rate of change of rotor speed, which relates directly to the rate of increasing or reducing frequency.

The machine is in equilibrium when Pm = Pe. This results in dw/dt being 0, which represents the rotor spinning at a constant speed. The frequency is constant.

When electrical load has been lost Pe is less than Pm and the machine will accelerate resulting in increasing frequency. Alternatively when electrical load is added Pe is greater than Pm the machine will slow down resulting in reducing frequency. Here’s the key point, for a given level of power imbalance the rate of rise and fall of system frequency is directly dependent on synchronously connected angular momentum, M.

It should now be clear how central a role that synchronously connected angular momentum plays in power system stability. It is the factor that determines how much time generator governors and automatic load shedding systems have to respond to the power flow variation and bring correction.

In the above I have made some simplifications. I have presumed a simple two pole machine, ignored machine losses and damping, and not talked about units used. This has simplified the equations and shortened the discussion, while keeping the important concepts in place. If readers do want to understand this in more detail there are good text books on the topic; but be warned, the mathematics gets very intense.

Power System Control and Stability, PM Anderson and AA Fouad, Wiley Interscience, IEEE Press.

Power System Stability and Control, Prabha Kundur, McGraw-Hill Inc.

Electric Power Systems, Analysis and Control, Fabio Saccomanno, Wiley Interscience, IEEE Press.

Next article – governor controls.

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Generator types; synchronous versus asynchronous. What goes on inside the machines?

There are two main types of alternating current machine used for the generation of electricity; synchronous and asynchronous.  The difference between them begins with the way the magnetic field of the rotor interacts with the stator.  Both types of machine can be used as either a generator or motor.

Synchronous Machine

Let’s start by describing a synchronous generator.  The rotor is basically just a magnet on a shaft.  In practice the magnet is generally an electromagnet.  The stator consists of three coils of wire placed to intersect with the rotor’s magnetic field, equally spaced around the circumference, 120o apart.  Each coil supplies current for one phase of the grid.  As the rotor rotates past each coil the induced current in each coil rises and falls in positive and negative directions as the north and south poles of the rotor passes by.  Each revolution generates one cycle of current.  The frequency generated is directly related to the speed of rotor revolution.  For a single magnet (two pole) machine, 50 Hz is generated when rotating at 3000 RPM.  A rotor with four poles generates 50 Hz when rotating at 1500 RPM.

Therefore the operating rotational speed of a synchronous machine is essentially a constant (within a small window).  Its speed is tied to system frequency.  Synchronous machines are governor controlled.  The governor monitors the system frequency and adjusts the machine’s prime mover power to bring correction to the frequency.  This is of course subject to the power capability of the machine and whether it is operating at power setting where increases (and reductions) can readily be made.

As mechanical power is applied to the shaft the rotor advances in relation to the rotating field generated by the system voltages on the stator coils.  The machine still remains in rotational synchronism with the system voltages, but the rotor is now in advance of the system by an angle d.  The angle d varies with the power being applied and generated, where the power is proportional to Sin(d).  If d is positive the machine is in advance of the system and is acting as a generator.  If d is negative the machine is being pulled along by the system, and it is acting as a motor.  If d is zero, the machine is spinning but no energy transfer is occurring.  Observe that Sin(d) maximises at 90o.  This is the rotor advance angle limit relating to the theoretical maximum torque that the machine is capable of delivering.

Here’s a mechanical analogy of a synchronous machine that might help.  Imagine the magnetic torque between the rotor and stator as being a spring connecting two rotating wheels.  The first wheel is connected to the driving source, ie: the rotor.  The second wheel represents the power system load.  As some extra loading is applied to the second wheel the angle between the wheels begins to increase as the spring stretches.  More torque is transferred via the stretched spring and kinetic energy moves from the directly connected spinning mass of the first wheel to the second.

 

Asynchronous Machine

As would be expected by the naming, the main difference between asynchronous and synchronous machines is about rotor synchronism.  The rotor of an asynchronous generator does not run synchronism with system voltages.   An asynchronous machine operates with ‘slip’.  ‘Slip’ is a percentage measure of how much slower or faster the rotor runs compared to its synchronous speed.  When the rotor is rotating slower than synchronous speed the machine acts as a motor. When the rotor is rotated faster than synchronous speed the machine acts as a generator.

Here’s a mechanical analogy of an asynchronous machine that might help.  Imagine the magnetic torque between the rotor and stator as being a hydraulic fluid coupling between two wheels.  The first wheel is connected to driving source, ie: the rotor.  The second wheel represents the power system.  As some extra loading is applied to the second wheel the hydraulic coupling slips more, but the flow of kinetic energy from the first wheel is largely decoupled by the hydraulic coupling.

Asynchronous generators are typically used where control of the prime mover is not possible, typically wind turbines or run of river hydro.  While control systems are implemented to make best use of these resources, they cannot adjust output in response to a frequency change.

 

Summary

There are two key differences affecting their contribution to stability.

  1. The kinetic energy of the synchronous machine’s rotor is closely coupled to the power system and therefore available for immediate conversion to power. The rotor kinetic energy of the asynchronous machine is decoupled from the system by virtue of its slip and is therefore not easily available to the system.
  2. Synchronous generators are controllable by governors which monitor system frequency and adjust prime mover input to bring correction to frequency movements. Asynchronous generators are typically used in applications where the energy source is not controllable, eg: wind turbines.  These generators cannot respond to frequency movements representing a system energy imbalance.  They are instead a cause of energy imbalance.

Short -term stability

The spinning kinetic energy in the rotors of the synchronous machines is measured in megawatt seconds.  Synchronous machines provide stability under power system imbalances because the kinetic energy of their rotors (and prime movers) is locked in synchronism with the grid through the magnetic field between the rotor and the stator.  The provision of this energy is essential to short duration stability of the power system.

Longer-term stability

Longer term stability is managed by governor controls.  These devices monitor system frequency (recall that the rate of system frequency change is proportional to energy imbalance) and automatically adjust machine power output to compensate for the imbalance and restore stability.

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Energy balance, imbalance and a definition of grid stability

The previous article described the grid as having no energy storage, and indeed there is no useful storage in the grid itself.  However, there is energy storage in the form of rotating kinetic energy of certain generators, and the quantity of this stored energy is one of the key determinants of stability.  The nature of this store will be expanded upon in the next article about generators.

There are two important performance indicators for a power supply system; frequency and voltage.

Most people would presume that voltage is the key performance indicator of a stable power supply system.  But it isn’t the main one.  Voltage is an important performance indicator and it should of course be kept within acceptable tolerances.  However voltage excursions tend to be reasonably local events.  So while voltage excursions can happen from place to place and they cause damage and disruption, it turns out that voltage alone is not the main ‘system wide’ stability indicator.

The key performance indicator of an acceptably stable power system is its frequency being within a close margin from its target value, typically within 0.5 Hz from either 50 Hz or 60 Hz, and importantly, the rise and fall rate of frequency deviations need to be managed to achieve that narrow window.

So what is special about the rate of change of frequency?  It is that an increasing frequency indicates more power is entering the system than is being taken out.  Likewise, a reducing frequency indicates more is being taken out than is entering.  For a power supply system to be stable it is necessary to control the frequency.  Control systems continuously observe the frequency, and the rate of change of the frequency.  The systems control generator outputs up or down to restore the frequency to the target window.  An also as a backup, other control systems measure the frequency and rate of change of frequency and carry out staged load shedding to help restore the frequency to the target window.

Of course energy imbalances of varying size are occurring all the time.  Every moment of every day the load is continuously changing, generally following a daily load curve.  These changes tend to be gradual and lead to a small rate of change of frequency.  Now and then however faults occur.  Maybe a whole city is disconnected instantly removing say 50 MW from the grid.  Or maybe a generator trips off removing say 100 MW of generation.  A power system has to cope with these changes too.  The rate of change of frequency in these cases is far higher.  These events require a fast response (within a few seconds) if the deviation is to be corrected before system collapse.  If the system can cope with the range of disturbances thrown at it, it is described as ‘stable’.  If it cannot cope with the disturbances it is described as ‘unstable’.

A stable power system is one that continuously responds and compensates for power/ frequency disturbances, and completes the required adjustments within an acceptable timeframe to adequately compensate for the power/frequency disturbances.

In the above discussion I simply stated that changing frequency and power balance are related.  A more detailed explanation of why this relationship occurs will be covered in a later article.  For now though I’ll just state that the rate of change of frequency is directly proportional to the size of the power imbalance and inversely proportional to available ‘rotational inertia’.  Large power imbalances mean a proportionately faster frequency change occurs, and consequently the response has to be bigger and faster, typically within two or three seconds if stability is to be maintained.  If not – in a couple blinks of an eye the power is off – across the whole grid.

The next article will describe some different sorts of generators, and describe why some generators work to provide frequency stability, and some do not.

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Electric current is generated ‘on demand’. There is no stored electric current in the grid.

Here’s the second article in the series on electric power system stability:

The electric current delivered when you turn on a switch is generated from the instant you operate the switch. There is no store of electric current in the grid.  Only certain generators can provide this instant ‘service’.

The three fundamental parts of a power system are:

  • its generators, which make the power,
  • its loads, which use the power, and
  • its grid, which connects them together.

Electricity consumers turn their lights and appliances on and off whenever they want.  Factories turn on and off large industrial machinery whenever they want.   Distribution companies can even turn whole cities on or off.  This level of control at the consumer end and the almost always flawless response of the power system to deliver the required electricity, has led to a mistaken view about how the power system instantly provides this current.  People imagine the electric grid must contain a store of electric current, immediately available on the other side the switch.

The mistaken view is that the instantly available on demand electric energy has already been produced and ready and waiting on the other side of their switch.  There is voltage on the other side of the switch, but the additional current required for your load is generated only from the instant the switch is closed.  There is no store of current in the grid.  It’s the generators that instantly provide that current, and only some generators at that.

The current carrying components of an electric grid comprise only three types of things:

  • conductors, typically either overhead wires and underground cables,
  • transformers and several other sorts and voltage conversion devices, and
  • switches, circuit breakers and fuses.

None of these store electric current.  They transmit electricity over a distance, convert voltages up and down, and control the flow of current, but they don’t store it.

So if there is no storage in the grid the amount of electric power being put into the grid has to very closely match that taken out.  If not, voltage and frequency will move outside of safe margins, and if the imbalance is not corrected very quickly it will lead to voltage and frequency excursions resulting in damage or outages, or both.

The next essay will discuss grid energy balance, imbalance and a definition of grid stability.

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Electric Power System Stability

I intend to do a few essays about electric power system stability.  I aim to share some concepts about power generation and grids, and in particular how electrical stability is achieved.

Some types of generation provide grid stability, other types undermine it.  Grid stability is an essential requirement for a power supply reliability and security.  However there is insufficient understanding of what grid stability is and the risk that exists if stability is undermined to the point of collapse.  Increasing grid instability will lead to power outages.  The stakes are very high.

I will discuss various engineering concepts with readers, and I’ll try to do it in a way that is accessible.   I’ll do my best to keep the discussion reasonably simple, sticking mainly to mechanical analogies rather than maths.

Here are some of the topics that I’ll be covering:

  • Control of the prime mover, governors, how it is done and why it matters:

I aim to do one of these a week.

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The Decadal Global Climate Bet – Now at the half way mark

Updated to December 2015 (month 60/120):

We’re now at the half way mark; and the present decade is still running cooler than the preceding decade. The average temperature anomaly for the previous decade is 0.202 C. The average temperature anomaly for the past 5 years is 0.184 C.

The recent uptick at months 58 through 60 is the beginning of the atmospheric warming caused by the current El Nino. It’s barely visible but definitely there. But I wonder how long it will last? If it continues along the lines of the 2010 El Nino, (which lasted about 9 months with an average anomaly of about 0.47 C) the lines will be very close in July 2016, and the lead could pass to the warmists. However, a La Nina typically follows an El Nino. La Nina brings a cooling, which would likely pass the lead back to the coolists.

Climate Bet, Dec 2015More detail here:  Climate Bet Page

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The Decadal Global Climate Bet – October 2015 Update

Updated to October 2015 (month 58/120): The present decade is still running cooler than the preceding decade. I wonder when we’ll begin to see the heating effect of the big El Nino? There’s not much sign of it yet.

Climate Bet, Oct 2015More detail here: Climate Bet Page

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Global Energy Emissions (during the climate bet)

Having just updated the climate bet, which confirms the lack of recent warming (ie: the ‘pause’ or ‘hiatus’ as some call it), I thought next we should take a look at global energy emissions on the same time comparative basis.

For those who are not sure of the background, take a look at a few of my previous posts on ‘The Bet’ and you’ll soon catch on. We are comparing two decades; a past decade Jan 2001 to Dec 2010, and a current decade Jan 2011 to Dec 2020. Of course we are only part way through the current decade, so we’re not making a full comparison.

Here’s the (very unexciting … even boring) graph.

OLWIR, accumulating comparison, Jan01 to Dec14

The data is monthly outgoing long wave infrared radiation (OLWIR) provided by the National Oceanographic and Atmospheric Administration (NOAA). I have totalled the energy emitted (OLWIR) in W/m2 units from each 2.5 by 2.5 degree latitude and longitude area to arrive at a global monthly average for each month. Then, in order to show emitting energy over the decade I have divided each monthly figure by 120 (the number of months in a decade) and produced an accumulating total.

The first four years of the comparison shows hardly any difference. The current decade (green line) sits right on top of the past decade (red line).   The blue line zooms in on the difference, which is just the difference multiplied by 100. The heat is going into space in very nearly the same quantity as usual.

It is also worth noting (some would think worth highlighting) that the current decade shows slightly higher emissions, while temperature shows slight cooling. Higher outgoing energy, at the same time as a lower temperature, is exactly opposite to what the global warming models tell us will happen.

Also – all that talk of the missing heat hiding in the oceans. It’s just scary story. The missing energy is speeding away at 300,000,000 m/s, past Alpha-Centauri and beyond … and will never be seen again.

The spreadsheet workings are available for viewing here:

https://drive.google.com/folderview?id=0BwCJWmtRR6xeOFZSNzVrVW5rcVE&usp=sharing

The link to the source data from NOAA is here:

http://www.esrl.noaa.gov/psd/cgi-bin/db_search/DBSearch.pl?Variable=Outgoing+Longwave+Radiation&group=0&submit=Search

 

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The Decadal Global Climate Bet – June 2015 Update

The Decadal Global Climate Bet is now 4.5 years into the race.  Here is a quick update of progress.

An interesting change has taken place with the data.  UAH has extensively revised their dataset to version 6.  I have updated my spreadsheet using UAH’s updated v6 data.

[UAH v6 is still a Beta version as at June 2015]

(Note: The above link is to Beta2 data, and it is no longer valid.  Here is a link to UAH’s  Beta3 data, UAH Beta3 data)

The difference between Beta2 and Beta 3 is very small.  The graph below is based on Beta2 data.

Notice the gap between the decades.  This decade (starting Jan 2011) is tracking noticeably cooler than the previous decade (starting Jan 2001).

Climate Bet, June 2015

The spreadsheet is available here.

Refer to this earlier post for further details.

Oh.  And by the way – there’s still no sign of that El Niño as far as I can see.

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The Decadal Global Climate Bet – Dec 2014 update, 4 years into the race

Four years ago, early 2011, a decadal global temperature bet was made. And with all the hype about 2014 being the hottest year, now is a good time to check out progress with ‘the bet’. But first a little background for any newcomers.

At the time the bet was made (early 2011) December 2010 marked the end of the previous ten year period. January 2011 marked the beginning of the next. The transition stimulated the formation of a decadal global climate bet.

And so the coolists (led by Pierre Gosselin – NoTricksZone) and the warmists (led by Rob Honeycutt – Skeptical Science) are having a bet.  They agreed to use a composite of Remote Sensing Systems (RSS) and University of Alabama Huntsville (UAH) lower troposphere temperature – close to the earth’s surface.  They agreed that the result would be accepted without quibbling, as it was agreed between them that those series are the best that we’ve got.  The average of the two series will decide the bet!

I have shown ‘the race’ as it progresses by graphing an accumulating total, ie: adding 1/120th of the average of the UAH and RSS monthly global anomalies month by month.  The sum of these numbers after 120 months (10 years) is of course the global decadal average.

Climate bet at 4 years

We are now 40% through the race … and clearly it is still close with the coolists in the lead by half a nose.

Now back to the hype we’re hearing about the ‘hot’ 2014. Notice that during 2014 (months 37 to 48) the green line just keeps trucking along at the same basic slope as it has for the past few years and much of the previous decade too. There was even a small El Nino in 2014, whose effect is conspicuous by its absence. Maybe the heat that is missing in the observed atmospheric temperature trend is hiding in the ocean?  😉

A spreadsheet with all the data and graph is available for download from this folder: https://drive.google.com/folderview?id=0BwCJWmtRR6xeMnVaOHBrTWRVRmc&usp=sharing

Earlier posts:

http://www.kiwithinker.com/2013/05/the_decadal_global_climate_bet/

http://www.kiwithinker.com/2013/11/a-decadal-global-climate-bet-a-second-view-of-the-race/

http://www.kiwithinker.com/2014/01/a-decadal-global-climate-bet-three-years-down-the-track/

 

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