ResourceKraft News & Updates

News and updates from the energy sector and everything that goes with it from the latest market trends to the ramblings and insights from the brains of our energy engineers & designers.

Limerick Business Leader: Helping fellow firms keep energy bills low long-term

Limerick Business Leader: Helping fellow firms keep energy bills low long-term

Tell us a little about ResourceKraft:

ResourceKraft Shortlisted For the Limerick Chamber Business Awards

ResourceKraft Shortlisted For the Limerick Chamber Business Awards

Today it has been announced that ResourceKraft have been shortlisted for ‘Best SME Business Contribution’ in the Limerick Chamber Business awards. Also nominated in this category are Design Pro and J.J O’Toole.

ASG & ResourceKraft Announce Strategic Partnership to Deliver Energy Management Solutions to the Middle East

ASG & ResourceKraft Announce Strategic Partnership to Deliver Energy Management Solutions to the Middle East

Al Sharif Group & ResourceKraft today announced that they have formed a strategic partnership to deliver energy monitoring and management solutions to businesses across the Middle East, starting in the Kingdom of Saudi Arabia. 

ResourceKraft Awared 2018 Green Technology Award

ResourceKraft Awared 2018 Green Technology Award

ResourceKraft was named the Green Technology Award Winner at the Green Ireland Awards in Dublin on Tuesday evening last. The Green Awards, now in its tenth year honours the key innovators and leaders in best green practice in Ireland.

7 Reasons Why a BMS isn’t an Energy Management System

People often ask us whether they can use a Building Management System (BMS) to collect energy meter data and act as an Energy Management System. In theory, this sounds like a good idea – install one system to control both energy use and collect energy data. In practice, it is anything but ideal.
Whether or not you choose to use ResourceKraft Advisor Energy Management System (EnMS), the lessons of the industry are to install an EnMS that is independent of your BMS, ensures data quality and system security and is ideally Cloud based. Best practice organisations like the UK Carbon Trust and the Irish Government’s Office of Public Works recommend the same.

It’s all about data integrity, essential to implementing credible energy management systems like ISO50001, Measurement and Verification to IPMVP and best utilising Ene 02 Energy monitoring under BREEAM.

Here’s why:
1) Fit for purpose
As opposed to EnMS, BMS are not designed with continuous meter monitoring in mind, and are typically not capable of storing large amounts of real-time and historic data.
BMS software is static in design, with limited and inflexible data analysis tools, unable to produce energy monitoring reports, turn energy information into financial information or provide management and operations staff with the energy related feature set. It lacks flexibility and becoming quickly obsolete. Conversely, Cloud EnMS systems are updated regularly and can be customised if required.

2) Data loss

The primary problem with using a BMS for Energy Management is data quality.
Data loss occurs:

  • Anytime there is a power dropout.
  • Anytime the device network is reconfigured.
  • Anytime there is maintenance on the BMS.
  • Anytime there is a change in outstation connections.
  • Many BMS are powered by or backed up to one local PC or server, and often can only store energy data for a number of weeks. When the power drops or the system crashes, recording stops and often historical data is lost. Data management and backups is your problem!

EnMS data loggers are designed to time stamp meter readings immediately and store the data securely in non-volatile memory, for months if necessary, ready to send to the Cloud where it is securely stored and backed up.

3) IT Network Problems
IT networks are one of the core causes of data errors, failures and loss. Modern IP based Building Management Systems are connected and run within an organisations internal IT architecture. Where an IT error occurs, the meters cannot be read until the fault is corrected. Resulting in meter data loss.

Using Canary data logger/gateways, months of storage ensures that meter data integrity is preserved and the option of GPRS/3G connectivity avoids IT networks altogether.

4) BMS System Errors
A BMS is a complicated system with a large number of inter-dependencies between schedules, equipment and machines. Adding another layer of machine to machine communication to a BMS creates another list of possible configuration failure, points and errors. All resulting in data loss.

5) Cost of Legacy BMS upgrade
This assumes the use of a modern IP-based BMS system. Legacy BMS often cannot acquire meter data adequately and need to be upgraded in order to do so. Upgrading a BMS is expensive, requiring significant and costly work and can take weeks if not months to complete.

The BMS does not work in isolation. Before deciding to upgrade a BMS, an organisation should monitor and analyse all of its lighting, heating, ventilation and air-conditioning systems to identify and prioritise based on Return on Investment, where the energy waste elimination opportunities are. A BMS operating well in its control function will not be one of those priorities.

6) Locked In
Most BMS companies place restrictions on the way data is collected, locking you into their proprietary systems This often means that an independent software programme cannot be used to verify the data received by the BMS. It means that organisations cannot use a 3rd party Energy Management software suite, which would provide many of energy analysis features that a BMS can’t. while denying you future access to open technologies and the advances of innovation and software.

7) Security
Upgrading to an IP-based Building Management System exposes an organisation to a number of security risks. When the environmental control systems of the building are connected to the internet they can become a target for militant, corporate or criminal hackers. Many government organisations, universities, hospitals and corporate institutions limit BMS to an analogue or silo system for this reason. The result – excluding access for operational and production management and staff to the energy management software. Thereby defeating one of the main purposes of an Energy Management System – engaging those who are actually using the energy in its management control – a requirement of ISO 50001 and the best practice approach of national energy agency management programmes – Energy Star, Carbon Trust, SEAI, etc.

Note:
It is a common misconception that BREEAM Ene 02 mandates BMS (BEMS) for metering – The core intent of issue Ene 02 is the monitoring of energy use, therefore if an alternative system can achieve this, it would be considered acceptable. Guidelines were updated recently to remove this confusion. (reference KBCN0149).
It is acceptable under BREEAM for sub-meters to connect (or have potential to connect) to automated meter readers (AMR’s) or similar systems, as an alternative to BEMS.

Introducing our NEW Partner Programme

Introducing our NEW Partner Programme

At ResourceKraft, each of our Partners have one thing in common – they all use ResourceKraft hardware and software to help their clients save energy and money.

Together, with our valued partners, we help companies drive business productivity by optimising their energy consumption and improving operational efficiency.

ResourceKraft offers a turnkey solution. You’ll be able to provide your own-branded energy monitoring solution in just a few days.

As a partner, you gain immediate access to accelerated sales team training and onboarding for your technical team. Partner Programme members are approved to sell, install and support our solutions in their region.

  • Deliver your Energy Management Solution, under your brand, to your customers.
  • Create new revenue streams with our easy to sell and deploy energy management solution.
  • Trade discount on hardware and software.
  • Increase your market reach.
  • Benefit from sales and marketing support before, during and after sale.

For more information, please click here.

Going Back to School

Going Back to School

ResourceKraft is up to the next new innovative move: teaching engineers in the making about energy. We recognize that students trying to get into the energy field need to find a balance between their academic knowledge and real on-site industrial know-how. As such, we now conduct lectures and educational programs at various Universities around the country.

Our lectures begin with an introductory session between ResourceKraft engineers and the educational faculty. Goals of the lectures are defined and an individually tailored plan of action is put into place. From here the introductory session is delivered between a ResourceKraft engineer and the students. This is typically a fun and informative way to get students engaged and excited about industrial energy use and what can be achieved through proper management. A typical course is based on an integration of software and technology tutorials delivered within the lectures. It is fast paced like a real working environment, and aims to show students the whirlwind nature of the industrial realm.

The objectives of these lectures are simple:

• Give students a taste of the industry
• Provide realistic project environments that students will face in the future, based on an array of industry issues from manufacturing, hospitals, to the typical office space
• Most importantly, allow students to understand that being an energy manager is a lot like being a detective –“it is their business to know what other people don’t know”

Perhaps you have no interest in our attempts to further the education of students, because you get a giggle out of watching the look of shock build on the faces of new recruits in your team. Or maybe you would appreciate having a fresh faced employee seemingly prepared rather than scared. Either way, we shall be assisting the educational faculties in preparing future energy experts for the real world.

If you know anyone amongst the educational community that this might help, or would like to have a potential intern of your company attend our lectures, feel free to contact us at sales@resourcekraft.com.

Things that you take for granted

Things that you take for granted

Introduction


Whilst working on modelling situations in the real world, or thinking about to how tackle a specific problem, it’s often surprising how many things I take for granted. Sometimes it’s worthwhile to consider how seemingly obvious, or ‘given’ information, is actually derived. The derivation process itself can often shed light and offer further intuition into much harder and/or complex problems.

Lately, I was working on solving a set of equations and becoming lost in a sea of inequalities and variables. Standing back for a moment and writing them out plainly on a whiteboard I realised that what I was in fact looking at was a problem that could be approached and solved quickly using fairly simple geometry rather than complex algebra. It really was a case of I couldn’t see the forest for the trees.

Geometry, in my opinion, is one of the most fascinating and often underappreciated fields of mathematics . I’d like to give you an example of something quite simple and very geometrical which has far reaching consequences in that it contributes to something that you most likely take for granted and also contributes to the fundamental understanding of one of the most important fields of mathematics.


The Area of a Circle


If you’ve made your way through basic geometry I can most likely guarantee that you know that the area of a circle can be calculated with the formula

Area = πr2

But, have you ever wondered why?

Let’s try and work it out.

How about a simple approach, enclosing the circle in a bounding square where the height and width equal the diameter of the circle.


Along with set theory. Set theory is cool.


Looks like a good start, but we have 4 sections left over – one at each corner – so the area of the circle would seem to be less than the area of the square. But we can say that,

Areacircle < diameter2

We have a square in there at least. Perhaps we could divide up the circle such that we can calculate the sum of the areas of the smaller parts?



How about if we divide up the circle into quarters, then calculate the area of one of the quarters and multiply by 4?
So, for the triangle in red we would have an.

Areatriangle = ½ * height * height
= ½ height2 * 4
= 2 * height2

Now remember that height is actually the radius of the circle so replacing height for 4 we can write the following inequality for the area of the circle as we’re missing the 4 sections of the circle from the area calculated.

Areacircle > 2 * r2

If only there were some way we could divide up the circle that allowed us to sum it.

Well, how about if we cut it into strips. Concentric strips like this,



Each of the strips divides up the circle into a series of trapezoids, which mathematically are similar to rectangles if we don’t look at them too closely. Bear with me.

Let’s consider just one of them in detail (as the principle can be applied to all of the others that make up the whole covering of the circle).



Although I’ve exaggerated it here for emphasis, the two tapering ends of the rectangle-like shape are actually very small compared with the whole shape – the section labelled width is much bigger in most cases in terms of area than the two ends that consist of triangles.

As we know the formulate for the area of a rectangle its area is very easy to calculate.

Thinking carefully, we know that the width here is approximately equal to the circumference of the circle which is given by the formula

Circumferencecircle = 2 * π * r

The height of the shape is a tricky thing though. Obviously the smaller we make this then the more accurate our measurement will be as the two triangular areas at either end of the rectangle like shape become less significant.

Let’s call this dr for the moment (those of you who’re familiar with calculus might get an inkling as to where I’m heading.)

Now the area of the shape can be said to be, approximately,

Circumferencecircle = 2 * π * dr


Step back for a second now. We’ve divided up the area of the original circle into the sum of the area of the smaller rectangle like shapes. Considering each rectangle, let’s line them up side by side, staring at the one nearest the centre of the circle.

I’ve made the ends of the rectangle like shapes quite small now so you can’t quite see them.



Each rectangle like shape has an area of
Area = 2 π dr

If we put some axis on this diagram, the x-axis is actually the radius and the y-axis, area.

Let me show this again, a little more clearly, but I’m also going to overlay the graph of

y = 2 π dr

The thing to node is that the rectangle like shapes all barely just touch the line of y = 2 πr.

And also, and more importantly, the smaller that dr becomes the smaller the gaps become between the shapes and the line y = 2 πr.

Remembering that the area of the circle is just the sum of the areas of all of the shapes it’s actually very easy to calculate as the shapes considered together form a simple right angled triangle where the x-axis length is the radius and the y-axis length is 2 πr.

So, the area of the triangle is,

y = 1/2 * x * y
= 1/2 * r * 2πr
= πr2


Conclusion

Just reflect on what we’ve done for a moment. It’s pretty cool.

We divided up a circle into a series of concentric rings, laid them out side by side, summed up their areas, and derived a formula for the area of the circle.

As the width of the strips gets smaller (and closer to zero) the area of the triangles on the ends get less significant and the total area becomes a simple width x height.

The point here is that we have taken one large problem (calculating the area of a circle) and split it into a number of smaller problems – each of which in themselves are easy to solve.

Many larger problems can be broken down like this, indeed many larger problems can be reduced to simply calculating the areas under graphs. This was a simple case involving a triangle formed by a straight line equation, but this all should be sounding familiar especially when I tell you that the area under graph is known as the integral of the function that defines it. Yes, it’s in the realm of calculus.


Extra Credit

We did assume that we knew the circumference of a circle is 2 πr in order to derive the formula, but there’s an equally simple and elegant proof for that too, which I’ll leave for your investigation.

Keywords:
Problem, Limits, Sums, Calculus, Circle, Geometry, Reduction, Equation, Integral

Numbers

Numbers

Living by Numbers

The Fourier transform is one of the most used and widespread applications of mathematics in electronic engineering, decomposing a signal into its constituent sine waves. It has applications in an enormous number of areas in everything from understanding electrical oscillations in alternating current to the modulation of electromagnetic waves. Engineering just wouldn’t be the same without it.

But, don’t worry, we’re not going to discuss the Fourier transform just yet. We’re going to discuss something without which the Fourier transform would be of no use whatsoever – numbers. And, something that puzzles people about numbers is their very nature – what kinds of numbers are there? How are they derived? And what we’re leading to is the what I like to think of as the ultimate type of number – the complex number.

Although you can perform a Fourier transform without complex numbers (well, you will see why I’m talking about this later) you could only retrieve magnitude or power information. To utilise the results fully, using complex numbers, you are able to retrieve phase information as well. Phase is just as important as magnitude, especially in filtering and image processing applications.

Fourier transforms are one example of many, another one is quantum mechanics – within which complex numbers are not just a convenience, they are a necessity – in effect laying the very foundations of physics.


So, it sounds complex (no pun intended), but let’s see how complex numbers came about and how they related to the other numbers that you think you know about today.


I’m going to show you how there are different types of numbers, how they’re interrelated, and more interestingly how they’re defined in terms of each other. In order to do this I’m going to introduce a mathematical concept that you’re probably already familiar with, the set, which is described as a ‘well-defined’ collection of distinct objects.

For example, let’s take some everyday numbers 2, 3, and 4. They’re distinct objects in their own right, but when considered together like this, {2,3,4}, they form a set.
In this case the set is of size 3 and contains 3 distinct objects.

Set theory in itself is fascinating and pretty immense – in fact nearly all of mathematics can be defined in terms of set theory – but let’s leave that for another day.

Now, considering sets, I’m going to introduce another mathematical concept called closure that a set can have as a property. Formally if a set has closure under an operation (we’ll come to that in a second) then performance of that operation on the set will always produce a member of the same set.

An operation in this case is a calculation of some sort, and as we’re dealing with numbers it will be some sort of arithmetic calculation.
For example, simple arithmetic operations are addition, subtraction, multiplication, and division.


The Natural Numbers

Let’s start with the simplest numbers, the ones we use everyday for simple counting, the ones we use when we buy things, order lunch, or pay for things on the internet. Some people call them the ‘counting numbers’ but mathematically we call them the natural
numbers. Some people include zero in the natural numbers, some don’t, but for completeness I think we should too as we experience zero in our everyday counting.

If we imagine a set containing all of the natural numbers it has a special symbol, the nicely elaborate and suitably mathematical symbol ℕ.

Mathematically, ℕ = {0,1,2, …}

*Sometimes ℕ is written as ℕ0 when zero is included and ℕ* when zero is not

Now, here’s the thing, let’s explore whether this set is closed for some simple operations.


Addition

If we add any two natural numbers together, do we always get another natural number?
Well, how about 2 + 3, that’s 5 which is also a natural number. If we consider things carefully we see that we always get a bigger (or the same if we’re using zero) number which is always positive and a whole number so, yes, we always get a natural number – therefore we say that ℕ is closed under (the operation of) addition.


Multiplication

Let’s do multiplication next as it’s really just a case of repeated addition and so is the natural next step from addition. It’s also my favourite, but you know.
So, back to the script, if we multiply any two natural numbers together then what we’re really doing is lots of repeated additions and since we know that addition is closed for ℕ then we can also conclude that so is multiplication.


Subtraction

Let’s do subtraction next as it’s pretty simple and really is kind of the opposite of addition whilst not being as complicated as division (which is the opposite of multiplication, right?)
So, taking any two natural numbers, say 5 and 3, 5 – 3 = 2. Great, a natural number. Let’s try another, how about 10 and 20? Well, 10 – 20 is, well -10. Oops.

When we subtract a natural number from a smaller natural number we get a negative number. That’s strange (I was going to say that’s odd, but that would confused things…). We can’t have -2 cupcakes or have -5c in change at the local shop.

We’ve produced a number that isn’t natural!


The Integers

Okay, so we’ve produced a negative number – something that not everyone in history actually acknowledged existed! In order to cater for negative numbers we need to create a new set of numbers that includes both ℕ and what is essentially a negative reflection of ℕ.

We call this set of numbers the Integers and they also have a special symbol, ℤ. (The Z actually comes from the German word Zahlen meaning, suitably, “numbers”).

Mathematically, ℤ = {…, -3, -2, -1,0,1,2,3,…}

(With zero in this case being defined as neither negative nor positive.)

We can also say that ℤ is what’s termed a ‘superset’ of ℕ meaning that ℤ contains all of ℕ. Essentially, everything that is in ℕ is also in ℤ, and we can represent the relation like this:

This kind of representation is known as a Venn diagram and is used to represent relations between items in sets. (I love the history of things, so it’s worth mentioning that Venn diagrams are named for John Venn who came up with the idea around 1880. They usually use circles but sometimes you’ll see generally curved lines.)

Let’s do some closure tests on ℤ.

Addition: we know that ℕ is closed so let’s trying adding some negative numbers – they can add up to either a negative or positive number which is inside of ℤ, so addition is closed.

Multiplication: we know that ℕ is closed so let’s trying multiplying some negative numbers or combinations of negative and positive numbers (integers) – there’s a rule for this kind of multiplication, remember your school days?

Positive * Positive = Positive
Positive * Negative = Negative
Negative * Positive = Negative
Negative * Negative = Positive

Nothing unusual there, we always get a whole number and it’s always positive or negative (or zero). So we can conclude that ℤ is closed under multiplication.

Subtraction: Well, this is how we got here in the first place and we’ve now covered the issue that subtracting two numbers from ℕ can become negative by extending our set to include the negative numbers. We can say that ℤ is closed under subtraction.

Operation + *
Closed Yes Yes Yes

Division: Now, we come for the first time to division. Take any two numbers from ℤ and divide them by each other and what do you get? Well, how about 10 and 5? 10/5 is 2 so that’s ok. How about -2 and -2, well -2/-2 = 1 so we’re good there. But what happens with 5 and 10, 5/10 = ½ which isn’t in ℤ.Looks like we need to extend our set again.


The Rational Numbers

Okay, so we have a puzzle – we took an integer and divided it by another integer and got something that (sometimes) wasn’t an integer so we had to extend our set of numbers.

We have to extend our set to include numbers that are the result of two integers from ℤ being divided by each other.

Considering the division operation carefully the result could be another integer or something that we now call a rational number (a number that has something other than zero after its decimal place) that belongs to the set ℚ.

For example, 10/5 = 2 which is an integer but as it’s a subset of the rational numbers is also a rational.
And with something like 5/2 = 2.5 we have something that isn’t an integer but is a rational. See what I mean?

So now we have a set of number, the rational numbers or ℚ that is closed under additional, multiplication, subtraction, and division. Great, so we’re done, right? Well, no. There are in fact more operations we can do to numbers as so far we’ve only considered the simple arithmetic ones. How about powers and roots?

Let’s try squaring some rational numbers, how about (4/9)2 for instance?

(4/9)2 = 4/9 * 4/9 = 16/81

Looks good, it’s still a rational number – i.e. one integer divided by another. Let’s try something a bit more esoteric, how about 2½ (or √2 as it can also be written).

It so happens that √2 cannot be represented as a rational number and written as one integer divided by another integer. There are many proofs for this (my favourite is the proof by infinite decent which uses a proof by contradiction). Check it out on Wikipedia.

Now, oops, √2 isn’t rational.
In fact, it’s a decimal that goes on forever, starting 1.41421356…

So now we have to make yet another set, as the numbers in the new set are not rational they will be called… wait for it… irrational numbers!


The Irrational Numbers (or ℝ \ ℚ)

Now we have another set of numbers, but there’s something very special about this set as they can’t be written as a complete ratio of two integers, i.e. a/b where a and b are both integers, something which makes them unique.

Due to this property they don’t include the rational numbers as a subset. In our diagram they have to go to the side and don’t overlap with the rational numbers.

Irrational numbers themselves cannot be written down fully and actually go on forever after their decimal place and their digits do not repeat. (This too can be proven, but we don’t have space here today.)

Inside the irrational numbers are also another set of special numbers that are called transcendental numbers.

Inside the irrational numbers are also another set of special numbers that are called transcendental numbers. Examples are numbers such as π and ε and various other irrational powers of reals. Proving that a number is transcendental is actually quite hard, but all transcendental numbers are themselves irrational therefore we must mention them here.

We don’t have the space to go into the complete definition, but suffice it to say that transcendental numbers are not algebraic numbers (the set of algebraic numbers are denoted by the letter A).

What this means is that they are not roots of a (non-zero) polynomial equation with rational coefficients.

For example, an algebraic number is defined to exist as the solution to a non-zero polynomial equation such as,

0 = a0 + a1x + a2x2 + … + anxn

whereas a transcendental number cannot be expressed in this way. A little like the way that an irrational number cannot be expressed as a/b where a and b are integers.

You may think that this makes transcendental numbers a minority but in fact almost all complex (see later) and real numbers are transcendental numbers! Mind blown.

The irrationals are sometimes denoted by P or I but we’ll use the common nomenclature using the set complement operator “\” where A \ B is defined as the set of all elements that are members of A but not members of B. So they are in fact defined as A \ ℝ.

You’re probably wondering where the ℝ came from, right? Well, apologies, we had to leap ahead just one section to define what the irrationals and transcendental numbers were in terms of sets. You have probably guessed, but will find out in the next section.

Wow, this section took a little longer than I thought, I think we’ll have to go into transcendental numbers again in a future post. So, as we now have two distinct sets of numbers again, let’s group them all together…


The Real Numbers

Now we’re getting somewhere. We are able to define a set of numbers that includes all other sets of numbers – each having a distinct definition in terms of closures (or in the terms of the transcendental numbers algebraic operations).

So, we’re there. Well, not quite. We still have further operations to consider – we did powers but what about roots? Are the reals closed when it comes to performing roots?

Let’s try a few examples. Firstly, √4, well that’s ok it’s just 2 or -2 which are both real. How about √2? Well, that’s irrational but also a real. Ok, how about √-1? Oops. We broke the real numbers – what exactly is the square root of -1? We can’t express it as a real number.

We need another set and we’re finally approaching the last pieces of the puzzle.


The Imaginary Numbers

A whole branch of mathematics is devoted to the study of imaginary numbers, stemming from the original solution to the question of just what is the square root of -1, or √-1.

Just to say that √-1 is defined as i where i 2 = -1.
So, if you wanted to calculate √-25 your answer would be 5i .

Imaginary numbers crop up in all kinds of places, especially in science and engineering, and are used both to simplify complex problems and of themselves have opened up many branches of mathematics in their own right.

Their very existence makes equations that were previously unsolvable now solvable in such a way that the non-existence of negative numbers made equations such as x + 5 = 0 previously impossible to solve.

They’re distinct in their own right and don’t have any subsets of our existing numbers so they will exist separate and distinct from the reals.

Once again we need to group things together, just like we did with the real numbers, and come to the last piece of the puzzle, the all defining class of numbers.

(For completeness, the number zero is consider both real and imaginary. Think on that.)


The Complex Numbers

Complex numbers contain the sets of all other sets of numbers. Our final diagram, though remember that the sizes of the sets are another matter altogether – we’re just showing how they relate to each other in terms of (set) membership.


Consider some interesting facts that come from our diagram.

  • Every number in mathematics is a complex number
  • An integer cannot be transcendental
  • An imaginary number isn’t a real number (kind of makes sense…)

And so on…

All complex numbers are written as,
a + bi
where a and b are real numbers and i is an imaginary number.

In your everyday life when you pay 1 euro for something, although you’re using an integer (or a rational if it’s say 1.05 euro, or 1 euro and 5 cent) you’re really using a complex number – with b equal to zero – thus removing the imaginary component of the complex number by making it zero!

Now, that’s cool and although it took a thousand years or so to get to this stage with the help of some very clever mathematicians I guess it was always meant to be.

Living by numbers, now…


Extra Credit

Oh, wait, maybe you’re thinking there’s something even more encompassing, perhaps you’re thinking maybe we need something to explain something like the square root of an imaginary number like √-i ?

Well, that’s actually √2/2 – (√2/2) i.
I’ll leave the working out to you.

Complex numbers are closed for simple arithmetic operations as well as powers and roots and for algebraic expressions. The proofs are fun. Probably a definition of fun that you’ve not come across before, but fun nonetheless.

You can also think of complex numbers as two dimensional numbers that extend a traditional one dimensional number line out into two dimensional space. This is a fascinating way to think about complex numbers and will most likely the the subject of a future post too.

Keywords:

Fourier, transform, complex, imaginary, numbers, rational, irrational, engineering, mathematics, transcendental

Benfords Law and Energy Data: A Mathematical Aside

Benfords Law and Energy Data: A Mathematical Aside

Here is what Wikipedia has to say about Benfords Law:

Benford’s law, also called the First-Digit Law, refers to the frequency distribution of digits in many (but not all) real-life sources of data. In this distribution, 1 occurs as the leading digit about 30% of the time, while larger digits occur in that position less frequently: 9 as the first digit less than 5% of the time. Benford’s law also concerns the expected distribution for digits beyond the first, which approach a uniform distribution.

It has been shown that this result applies to a wide variety of data sets, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, physical and mathematical constants,[1] and processes described by power laws (which are very common in nature). It tends to be most accurate when values are distributed across multiple orders of magnitude.

So, does it apply to energy data? Well I ran a little SQL query on 150 million records of 15-minute energy data I had lying around. Here is the distribution of the digits:

“1” 49647887
“2” 27419091
“3” 18099660
“4” 14904116
“5” 13496329
“6” 10929017
“7” 9671925
“8” 7708455
“9” 6564295

and here is the distribution chart:

The vast majority of the data analysed are electricity usage data measured in watt hours. However, there are also gas and water usage data in there also. I left out negative numbers for simplicity (negative numbers can legitimately occur when energy is being produced, i.e. exported).

So yes! Benfords law is alive and well in 15 minute energy data. But why does that matter? In the case of our energy data, it demonstrates its veracity and the effectiveness of our data acquisition and data management on a large scale….and its just kind of interesting.

Tweets