Transcript

Introduction

Good afternoon. My name is Landon Rhodes. I am an engineer at Power Monitors, Incorporated. I am here to talk about my white paper that I wrote about power theory in multiphase distorted systems. This is based on a research paper called Geometric Foundations of Power Theory for Multiphase AC Systems in the Frequency Domain by Francisco Montoya. My white paper explains his ideas in three steps, and walks through a calculation of those samples.

The Data File

I’m using a data file that looks like this. So you got voltage up here and current right here. The voltage has about a 4% THD, and the current has about a 16% THD. I’m using these signals to demonstrate how to use Francisco Montoya’s ideas on a distorted three-phase system.

So you see the voltage has peaks and crests, and the current has very significant dips in the center. And then this is channel four, but we’ll only be looking at channels one through three right now. This is what the harmonic content looks like. This is the PQ Canvass reading of the same data. Now this is third harmonic, fifth harmonic, seventh harmonic, ninth harmonic. I’ve suppressed the first harmonic for clarity.

Step One: Establishing Harmonic Components

So the first step in the calculation is to establish what the harmonic components are of the signal. And the way I’m doing that is I’m defining the first sequence, so sigma zero, as just a sequence of ones, of all ones. Odd sigmas, so sigma one, sigma three, sigma five, are gonna be cosines of a particular frequency. So sigma sub one is gonna be root two cosine two pi N over 2, 256. Sigma sub three is root two cosine of two pi 2N over 256, and so on. And then even sigmas are sines. This is what Francisco Montoya defined in his paper right here, in section two.

Defining the Inner Product

And then I’m defining an inner product on sequences. So the inner product is just take each element in the sequence from zero up to 255, take the element in the f sequence times the element of the g sequence, multiply them, sum them together, and then divide by 256.

The reason why I chose this basis up here is because under this inner product, sigma sub i with sigma sub j is zero if i is not equal to j, and it’s one if i equals j. So for example, if I did sigma zero with sigma one, that would be zero. But if I did sigma one with sigma one then that would equate to one. That’s the reason why sequences.

Constructing Frequency Content

So the way that I’m going to construct the frequency content is we’re going to represent our target sequence f of N as a sum of some number times sigma sub k. And we’re going to do that by calculating C sub k as the inner product of S sub N with sigma sub k of N.

And I did that here. So columns A through H in this Excel file are the sample data that I exported from PQ Canvass. So this is channel one data, this is nine cycles of channel one, this is nine cycles of channel two, nine cycles from channel three. This is channel four, which is essentially just noise, it’s all zeros or 0.1. This is current from channel one, this is current from channel two, this is current from channel three, and this is current from the neutral conductor.

And so I can change this for a number. So if I change this to zero, then this S sub k changes to ones, and then if I change this to one, then it becomes a cosine wave starting at 1.414, and then it goes down to… and then if I start at two, then it becomes a sine wave. So I calculate theta and then I calculate sine or cosine of that angle. So this is my S sub k. And then this group here, I’m just multiplying each of these sequences by this. I’m doing a point-wise multiplication of the data points times my S sub k to get this.

And then I’m calculating the inner product here by summing each set from each group of 256 samples. So I’m summing them up, and I’m dividing by 256 to get each of those numbers, and then I’m averaging to get these, and then I’m using Excel’s data table to calculate that for each of zero through 102 down here. So this is the harmonic contents of each signal.

Walkthrough of the Spreadsheet Layout

I’m going to clean that up a little bit in here. So that’s what I’m explaining here in the paper of just this table:

• Marker A is how I can change which sigma, this is a sigma index I have.
• B is the sample data.
• C is the angle that I’m computing.
• D is the sine or cosine of that angle.
• E is the point-wise multiplication of data in section B with sigma sub k in column N.
• F is doing each block of 256.
• G is them averaging them together.

And then here, I’m describing that most of the entries are not useful, and you can see that here. If I go back to sheet one, most of these entries, like this one here, contributes almost nothing to the variation. So I really only want to care about signals that contribute a lot to avoid over-complication. So what I did is I’m only considering signals where the magnitude of that frequency is more than 0.5% of the nominal.

Harmonic Analysis Results

So I did that here. So this is voltage channel one, two and three, and current channel one, two and three. So I’ve got sigma sub one, sigma sub two, sigma sub five, sigma sub six, sigma sub nine, and you can see what this looks like. So this is what channel one looks like, so voltage is here and current is here. I’ve scaled current down by a factor of 0.4 to keep it in frame nicely.

You could see that the data points that I have, the black data points, the green is just the first harmonic, just measuring just the 60 hertz part of the waveform, and then the orange part, the voltage, is measuring all of my harmonic variation that I’m calculating. And then I’m doing the same for current. So little black dots are data points. Red is my harmonic waveform, and then blue is just the 60 hertz part of that waveform. So this is just basic harmonic analysis. This is just decomposing a signal into a bunch of sums of sines and cosines. So you can see the results of that calculation here in my paper.

Step Two: Multi-Vector Power Quantity

So now the interesting part is here. We have a vector for voltage as a sum of sigmas, a sum of different sigmas. So sigma one, sigma two, up to sigma 38 for voltage, from voltage channel one. And current is sigma one up to sigma 46.

And you can see that here where I’m just doing minus 273.86 for sigma one, so that’s sigma one here, and then plus 6857, sigma two, minus 0.99, sigma five, plus 1.72, sigma six, and then all the way down to here, which is 1.48 times sigma 38 for voltage. And then this is the current, and that goes down to sigma 46.

Clifford Product and Geometric Algebra

Kritschke-Matoya defines a multi-vector quantity, which is taking these two vectors and I’m performing a multiplication on the vector. These are both vectors in 52-dimensional space, and we are multiplying them together as vectors, and then we’re computing a multi-vector power quantity. So normal power in ordinary DC is just voltage times current. And so we’re gonna take inspiration from that, and we’re going to define a multi-vector quantity as just the product of these two.

And these just distribute like normal multiplication, but multiplication is not commutative. You can’t flip powers, but you can distribute over multiplication. And so there’s 16 terms in voltage channel one and 18 terms in current channel one. And so this product is gonna have 288 terms total. So you’re just gonna multiply each. So this is 27386, or 0.86 minus 437.25 times sigma one, sigma one for getting this element times this element. And then you do this element times this element, and then this element times this, and then you just distribute over the whole product.

What’s special about a Clifford product or a geometric algebra product, as opposed to just this, what I’ve defined up here, this is just a tensor product. This is just a free tensor product. In order to apply some geometric meaning to it, we need to apply a restriction. And that is that the product of two vectors that are the same is the inner product between two vectors. And this is for a general inner product. The star, this is just a symbol that doesn’t have any outside meaning. It’s just there is a vector space V with an associated inner product, and I’m gonna multiply these two vectors.

And then when you take a vector multiplied by itself, that’s the same as taking the inner product of a vector with itself. And then this means that whenever we see a sigma i, sigma i, so like up here, this term right here, sigma one, sigma one, that’s gonna be just one because inner product of sigma one with sigma one is just one. That was defined up here. And then this other calculation that we walked through is that when two vectors are perpendicular to each other, which means their inner product is zero, we can flip the order and multiply by negative one. The product is anti-symmetric when they’re perpendicular to each other. So when you have sigma i, sigma j, you can flip it and multiply by negative one.

Computing Real and Reactive Power

And then that is this calculation here on sheet four. And so I’ve got, let me make this channel one. So I have voltage down here, and this is current across. So this is just the initial part, and then the real power for each harmonic. The way I’m calculating that is I’m just doing this part plus this part for harmonic number one, this part plus this part for harmonic number three, this part plus this part for harmonic number five. And then for reactive power, what I’m doing is this component minus this component for harmonic one, that goes right here, this component minus this component for harmonic number two.

And so what this does is, the standard power triangle, you have a complex plane where you have on the real axis is real power, and then on the imaginary axis is reactive power. And that’s symbolized by how much energy is being sent back and forth and not being absorbed by the user. Instead of having just one dimension of reactive power, we have on the order of 150 different directions that we can send energy back and forth. That energy can ring with itself and it’s not absorbed by the user. So real power describes how much energy actually gets to the user, and reactive power describes how much energy stays in the system and it rings back and forth.

So this is just pure first harmonic part of the reactive power. This is pure third harmonic part. This is pure fifth harmonic part. This is pure seventh harmonic part. And then what I’m doing here is I’m computing each, I’m applying this rule of being antisymmetric when they’re perpendicular, and I’m adding each of these added together with its transpose on the other side. So this minus that, for example, and then this minus this term here. So this term minus this term and this term minus this term. So it’s just like each element subtracted from its elements on the other side of the matrix to get this table here.

So this table describes each of the particular elements in each of the particular reactive powers, like the direction that it’s rotating and how much power is being reflected at a particular direction. And these describe interactions between different harmonics. For example, this element here describes the interaction of the cosine terms, of the cosine of the first harmonic terms with the cosine of the fifth harmonic terms. And so each particular harmonic contributes some reflection to the other harmonics. They interact non-linearly with each other.

Summarizing the Results

And then what I’m doing here is I’m just summarizing all of this information. So fundamental real power is just this element here. Total real power is the sum of all of these columns of first harmonic, third harmonic, fifth harmonic, seventh harmonic. Fundamental reactive power is this element. Other single frequency reactive power is summing, undoing a square, square sum. And then these are just other summarizations of this table down here. And then I can do the same for channel two, and then I can do the same thing for channel three.

So I’ve described all of this in my paper of how I computed all this, and this is the channel metrics. So this is the real power and reactive power for each of the three channels, and then this is the totals that I have for each of the three channels. So we have about 144 bi-vector terms that we’re kind of averaging together. So this contains a lot more information than what the normal power triangle has, because a normal power triangle only has just two pieces of information. You have the real part and you have the reactive part.

So this is a way to analyze and describe distorted signals in a lot more coherent way. It allows you to build devices and filters and stuff to target particular problems. Like, if you’re getting a really, really bad resonance between particular harmonics, you can use this method and target that rather than just relying on power factor correction like with the normal power triangle analysis.

Step Three: Symmetrical Components for Three-Phase Systems

After we’ve defined this basis, step two was multiplying these two vectors together and getting a multi-vector. Step three is combining this with the ideas of symmetrical components to look at three-phase systems instead of just a single phase, because before, in step two, we were just looking at just channel one individually and then just channel two individually and then just channel three individually.

For step three, we’re going to look at symmetrical components. So the way we’re going to define this is that we have a vector for channel one, vector for channel two, vector for channel three, and then we have a positive sequence vector, a negative sequence vector, and a zero sequence vector. Symmetrical components is a way to describe three-phase unbalanced systems in a sum of balanced systems because balanced three-phase systems are much easier to deal with.

This is the transformation that we use to represent the positive sequence, negative sequence, and zero sequence. So we have A, which is a rotation by 120 degrees, or a phase shift by 120 degrees, and then A star is a phase shift in the other direction by 120 degrees. So this just means the positive sequence is one-third times one times vector for channel one plus positive 120-degree rotation times U2 plus negative 120-degree rotation times U3. That’s positive sequence, and then we multiply that by one over three. And then negative sequence is one times U1 plus A star times U2 plus A times U3, and then the zero sequence is just the sum. So U1 plus U2 plus U3 times one over three. So this is the forward transformation. This is the backward transformation.

Applying Symmetrical Components to Harmonics

So Montoya, in his paper, proposes to perform this symmetrical component computation on each harmonic in the voltage and current signals. So what we’re going to do is come over here. This is the harmonic components we computed before. I’m going to multiply E-H of channel two and three by a rotation in one direction and a rotation in the other direction for A and A star. So this is A V2, A star V2, A V3, A star V3, A I2, A star I2, and so on.

And then I’m going to sum them in the particular way that I need. And I’m choosing A to be the direction such that the positive sequence dominates. So you see how positive sequence is minus 474 and 120, whereas negative sequence is almost nothing. So I chose A in the particular way that made positive sequence dominate and negative sequence almost nothing. If you flipped the direction of A and A star, then you would just flip positive and negative sequence, and you would just get positive sequence would be almost nothing and negative sequence would dominate.

So this is the positive sequence for voltage, negative sequence for voltage, zero sequence for voltage, and this is positive sequence for current, negative sequence for current, and zero sequence for current. And this describes the cosine terms in the first harmonic, sine terms in the first harmonic. This is cosine terms in third harmonic and sine terms in third harmonic. This is cosine terms in fifth harmonic and sine terms in fifth harmonic, and then so on down to here, which is cosine terms and sine terms in the 25th harmonic for current.

Computing the Multi-Vector for Three-Phase

Once we have that defined, this screenshot is exactly what I showed before. Positive sequence, negative sequence, zero sequence, and positive, negative, and zero sequence for current. Now we’re computing multi-vector M as voltage. So now we’re just gonna create a vector of summing all of these together. So this is interaction between each sequence with each other.

So I’m defining a matrix of this. Up to here is positive sequence current. So this is positive sequence in this red block. This is negative sequence, and this is zero sequence. So we get a nine-by-nine matrix of smaller matrices. And we’re just doing the same calculation that we did on sheet four, but we’re doing it kind of all together in this big calculation.

So if I zoom out, you see how we’ve got this group here is positive sequence voltage times positive sequence current. This group here is negative sequence voltage times positive sequence current. This group here is zero sequence voltage times positive sequence current. And then so on down to this is zero sequence voltage and zero sequence current. And then we’re computing everything in almost exactly the same way that we did in step two. We just group all of the terms, all the real power terms, and group all the reactive power terms.

So we get a positive sequence real power of 355 kilowatts in the first harmonic. Third harmonic is negative 0.71 kilowatts. Positive sequence in the fifth harmonic is 0.97 kilowatts. So this is positive sequence real, positive sequence reactive, negative sequence real, negative sequence reactive, zero sequence real, and zero sequence reactive. And then these are some summary terms that I’ve collected together.

Summary and Validation

So that’s my paper, and that describes the calculation that was proposed by Francisco Montoya. This is a very interesting way to analyze three-phase distorted signals and give a fairly accurate picture. And also, if you notice, you get 355.66 kilowatts for the real, for just channel one, and then 355.96 kilowatts for total, for the individual channel summary. And then for the other way to calculate with symmetrical components, we get basically the same. So 355.69 for fundamental and 355.85. So this is calculating just a little bit lower in the tens of watts range than what the individual sequence was.

So this is a helpful way to describe power theory in multiphase AC distorted systems. You just collect everything together as sums of harmonics and multiply them together as vectors and then use the ideas of geometric algebra to simplify that product together and interpret each. When the same vector multiplies together, then that gives a real number. When different vectors multiply together, that gives a reactive power.

Closing

Well, thank you for attending today’s webinar. If you have any questions, you can reach us anytime by calling 800-296-4120 or emailing support@powermonitors.com. Thank you and have a good afternoon.