# How to use the rate of change of frequency equation to calculate frequency drop in a grid based on lost generation

My question involves calculating the frequency drop in a power grid when a load is lost. The closest question I have found to mine is, Change in frequency in a grid that has lost generation?

Based on my research I have found the following equations.

I am interested in how I use the second equation, which would be the rate of change of the frequency based on how much load is lost. If the equation works the way I believe it does, then the units on the left side are Hz/s and the units on the right side would be MW/s. I am not sure how I can calculate the frequency drop using this equation, especially because each side seems to have different units. Maybe I am looking at the equation wrong, and the right side is supposed to be a sort of "scalar"? I think I am looking at this too much from a number standpoint and not enough from the stand point of what is "physically" happening when a load is lost. Here is the link for the National Grid power point presentation that presents the equation on page 6, the slide is titled, "The maths behind inertia", (https://www.nationalgrid.com/sites/default/files/documents/16890-Meeting%208%20-%20Inertia%20presentation.pdf).

I am trying to apply the equation to the figure below:

I believe in the thesis, the author used a value of 9 seconds for H, as they state that, "It was found that an inertia constant of 9 s gives a good fit and that value was used in subsequent simulations." I obtained the figure from an online article titled "Measuring grid inertia accurately will enable more efficient frequency management" (http://watt-logic.com/2017/10/12/inertia/). The original source for the figure is a thesis titled, "Use of Smart Meters for Frequency and Voltage Control" by Kamalanath Bandara Samarakoon. The figure itself is on page 73 of the thesis, Figure 4.11.

If I am reading the figure correctly, for the first generator loss (345MW), it looks like the time interval is about 25 seconds, 11:33:50 to 11:33:75 (HH:MM:SS), and the frequency drops from what looks like 49.95Hz to 49.8Hz.

Another source I have been using to understand the equation is a NREL paper titled "Grid Frequency Extreme Event Analysis and Modeling" that has similar equations (Equations 2 and 3) to the ones I presented earlier, however, I am still having trouble understanding how the equations were used to produce the figures for frequency drop as a result of lost load, https://www.nrel.gov/docs/fy18osti/70029.pdf

Further Questions

• If the equations can be used to calculate frequency drop from a lost load, can they also be used to calculate how the frequency would increase if a load was added? Is it a matter of a positive, or negative sign in front of the delta P?
• How does MW*s/MVA (inertia constant) on the right side of the equation "cancel" to just seconds? Why do you have MW over the rating of the machine MVA? I think this is meant to be a "ratio"?

I am hoping someone can give me an example calculation on how much the frequency would drop based on how much generation was lost using the aforementioned equation. And explain how a change of rate in frequency (HZ/s) on the left side can equal a change in power per second (MW/s) on the right side.

If clarification is needed on my question, please let me know. And let me know if I am using the wrong equation for what I am trying to solve.

• I haven't seen many question or answers of this type here, so you may have to wait for an answer, if any. It also seems a pretty particular question. Unfortunately, I also don't know where to refer you for better chances. I guess what I'm saying is you shouldn't expect too much. But, looking at your formulas, what I'd do is try to translate the graph to a set of discrete points, then perform a derivative on it (x[k+1]-x[k]), which will whow the variation of frequency in terms of delivered power (itself a function of time). See if that gets you somewhere. Feb 7, 2021 at 9:05

If the equations can be used to calculate frequency drop from a lost load, can they also be used to calculate how the frequency would increase if a load was added? Is it a matter of a positive, or negative sign in front of the delta P?

First of all, when load (demand) is added, frequency goes down and vice versa. Having that said, yes, you can use the equation for both cases. The inertia itself will change slightly because added/dropped load has inertia itself, but you can neglect it.

How does MW*s/MVA (inertia constant) on the right side of the equation "cancel" to just seconds? Why do you have MW over the rating of the machine MVA? I think this is meant to be a "ratio"?

Ratio is "unitless". MW and MVA are essentially the same units. You use one or another to make it clear what physical parameter you are talking about.

And explain how a change of rate in frequency (HZ/s) on the left side can equal a change in power per second (MW/s) on the right side.

Well, in fact they do not say that $$\\frac{df}{dt}\$$ is in Hz/s. I suppose they just left out some constants in the left part of the equation.

You may start with a swing equation of a synchronous machine $$\frac{2H}{\omega_s}\frac{d^2\delta}{dt^2}=P_m-P_e$$ where Pm, Pe are mechanical and electrical power in p.u.; $$\\delta\$$ is load angle in radians and $$\\omega_s\$$ is angular velocity in radian/s.

Mechanical power is provided by turbines and electrical power is drawn by loads.

Then substitute $$\ f=\frac{1}{2\pi}\frac{d\delta}{dt}; f_s=\frac{\omega_s}{2\pi}\$$ and switch from p.u. powers to MW and you can get the equation

$$\frac{S_{rated}}{f_s}\frac{df}{dt}=\frac{\Delta P}{2H}$$

where $$\\Delta P\$$ is power imbalance caused either by load change or by generation tripping; $$\S_{rated}\$$ total rated power of all synchronous machines under consideration.

I am hoping someone can give me an example calculation on how much the frequency would drop based on how much generation was lost using the aforementioned equation.

Things are a bit more complex. Let's suppose you have load drop in your system. Synchronous machines start to accelerate and frequency starts rising according to the swing equation above. At the same time load starts to increase because of rising frequency (this is called load response; in the National Grid presentation they state load response 2% per Hz). After some time frequency goes out of turbine governor's deadband and they start to reduce power generation; governors may have different droop - the rate at which they change power with change of frequency; typical droop is 5%. This stops frequency rising. After some more time power station target power output is adjusted and frequency returns to the normal value.

In general, you would need more or less precise power system model and power system simulation software to calculate frequency change. If you try to use one single equation for the whole system you in effect represent the whole system as one equivalent machine; this is not very precise.

What you may try to estimate is the frequency at which rising/falling is stopped at first stage, that is when new balance is reached. The balance is reached due to load response and governor action. Thus

$$\Delta P = \frac{k_{LR}}{100}P_L\Delta f + \frac{100}{d}P_{rated}\frac{\Delta f}{f_s}$$

where $$\k_{LR}\$$ is load response in percents per Hz (e.g. 2%); d - droop in percents (e.g. 5%); $$\P_L\$$ total load power; $$\P_{rated}\$$ total rated power of generators in work; $$\f_s\$$ - average voltage frequency (Hz) before the accident; positive $$\\Delta P\$$ corresponds to load loss, negative - generation loss.

You can solve this equation for $$\\Delta f\$$ frequency deviation in Hz. It would be good to test the equation on real data and adjust equivalent droop and load response accordingly.

• Is fs the frequency that the generators are trying to maintain? For example, in the US the standard would be 60Hz. And in an “ideal condition”, would load match generation, so PL would be equal to Prated? Is Prated the total combined power rating of all the generators in the system put together, so if I had a 20MW generator, a 50MW generator and a 35MW generator in my system, my Prated would be 105MW? And PL would be all the loads combined that are using the generators in the system (Prated)? Feb 24, 2021 at 23:00
• @W.Churchill Yes, fs is synchronous frequency: 50 or 60 Hz depending on the country. Prated is an arithmetic sum of rated, or nominal, powers of generators that have frequency control (have droop); some generators produce power less than rated, so it can be that PL < Prated. PL is total power consumption in the system at the moment just after disturbance. Feb 25, 2021 at 7:33
• Can this equation be used in a feedback loop where delta P is adjusted to maintain grid frequency? The change in power can either be from a loss of generation, or load loss, so if that happens at t=0, would either the PL, or Prated, have to take this into account at t=1? For example, if we assume a negative change in delta P (generation loss) at t=0, would that mean Prated at t=1 would need to decrease by the same amount that delta P changed at t=0? From a physical standpoint, if Prated decreased, does this mean grid inertia decreases as well? Mar 27, 2021 at 22:29
• I didn't get the first question. Also keep in mind that this is a rough estimation of frequency deviation. It doesn't account for secondary frequency control, for example. As for your example, if a generator is triped at t=0, then you shall adjust Prated accordingly at this moment. But usually a single generator have small rated power compared to the whole system. Mar 28, 2021 at 5:17
• @W.Churchill The last one? It is derived from definition of load response and droop (turned inside out). Nov 4, 2021 at 22:09