PMU Fundamentals

At OpenPMU, we get a lot of enquiries and encounter a lot of misconceptions about the fundamentals of phasor measurement units and synchrophasor technology. This page attempts to answer the common questions.

What is a PMU?

A Phasor Measurement Unit (PMU) is an instrument used in electrical power systems. Its job is to observe voltages and currents, over a period of time, and report a simplified description of their behaviour.

At a physical level, the PMU does not “see waveforms” or “measure phasors”. What it actually does is sample voltages and currents - real, physical quantities - at a sequence of instants in time. Each sample is a numerical value representing the instantaneous value of a voltage or current.

If samples are taken over a period of time, those samples can be used to describe how the quantity varies. This is what we often draw as a waveform. However, a waveform is not something that can be sampled directly - it is a representation, a concept, a way of visualising how a physical quantity changes over time. The PMU samples the underlying voltages and currents; the waveform is something we construct afterwards.

The purpose of the PMU is to take these samples and fit them to a measurement model.

The Measurement Model

In the IEC framework, the measurement model for the phasor is typically written as:

x(t) = Xm cos( ωt + ϕ )

Here:

x(t) is the instantaneous value of the voltage or current at time t.
Xm is the magnitude
ω is the angular frequency
ϕ is the initial phase

These quantities - Xm, ω, ϕ - are the parameters of the model. They are what the PMU calculates from the sampled data.

It is important to be clear about what a phasor is. A phasor is not a physical thing that exists in the power system, and it is not something that can be sampled. It is a concept - a compact way of describing a sinusoidal signal using magnitude, frequency and phase. Only physical phenomena, such as voltages and currents, can be sampled.

As James Clerk Maxwell expressed it, there is a distinction between the dance of molecules of the physical world, and the operations of the mind of the mathematical domain. The phasor is an operation of the mind applied to those sampled values of voltage and current.

A PMU therefore does not “measure a phasor” directly. Instead, it reports the parameters of this model, from which the phasor is defined.

Phase and Initial Phase

The phase deserves careful attention. In the expression above, the phase is the entire argument of the cosine, ωt + ϕ.

The quantity ϕ is the initial phase, meaning the value of the phase at time t = 0.

The frequency is not an independent idea - it is the rate of change of the phase with respect to time. This raises an important question: when exactly is t = 0? That question leads directly to the need for time synchronisation, which is discussed in a later section.

Phase and Frequency

You may have heard that frequency is the derivative of the phase. This section will show why this is true only if we regard the entire argument of the cosine as the phase.

We give the phase the symbol θ(t). Thus,

θ(t) = ωt + ϕ

Now, if we take the derivative of the phase with respect to time, θ(t)/dt, we are left with ω. Remember that ω = 2πf. ω is the angular frequency in radians per second (rad/s).

When t = 0, we are left with:

θ(0) = ω x 0 + ϕ

θ(0) = ϕ

This clarifies that ϕ is the phase when t = 0, i.e. it is the initial phase. It has the unit of radians (rad).

This is consistent with the IEC definitions of phase and initial phase.

From Sampled Values to Phasors

The voltage and current waveforms applied to the terminals of the PMU are sampled using an analogue-to-digital converter (ADC). An ADC takes as an input which is continuously variable in amplitude and time and transforms it into a series of numerical data which can only change at discrete instances in time, and to discrete values. This process is known as sampling, and the series of numerical output data from the ADC are known as sampled values (SV).

The difference between the numerical value of the output of the ADC and the original waveform the numerical value asserts to represent is known as quantisation error. Quantisation error may be reduced by increasing the bit-depth of the ADC.

The required sampling rate is a subject of the Nyquist-Shannon sampling theorem. In brief, one must sample at 'at least twice' the frequency of the maximum frequency one wishes to recover. In practice, it is necessary to sample at higher frequencies than the 'Nyquist rate'.

One must be conscious that the voltages and currents applied to the PMU terminals may have already been through one or more instrument transformers, the effects of which should be taken into account (i.e. frequency dependent amplitude / phase).

This section is an extremely simplistic overview of ADCs. Further reading should be considered, for example on filtering and aliasing. A good explanation of sampling is available on Youtube - D/A and A/D | Digital Show and Tell.

How are phasors estimated?

This isn't a straight forward question to answer because there are numerous methodologies in use in practice, so what follows is a superficial overview of the general theory. Essentially, the PMU uses a mathematical algorithm to calculate the values for the parameters of the measurement model which 'best' fit the applied voltage or current during the time 'window' of the measurement.

The workflow is as pictured below. The values in these diagrams are arbitraty, but time should be read in milliseconds.

The PMU begins the process by sampling the applied voltage or current, either using its own ADC or it will use sampled valve data obtained from a merging unit (IEC 61850-9-2) or a digital (numerical) instrument transformer (IEC 61869-9).

The signal, now a stream of SVs, is processed over short intervals of time (known as “windows") over which the parameters of the measurement model are to be calculated. The result is that a new set of values for the parameters is available several times every second. The exact rate of reporting results is set by a documentary standard, IEEE/IEC 60255-118-1.

A commonly employed measurement method is curve fitting, but several other methods exist. Depending on the implementation this may be described as 'least squares fitting' or 'linear regression'. At a high level, the method uses an iterative approach to adjust the estimated parameters (Xm, f, ϕ), whereby the adjustment is proportional to the error observed on the previous iteration. The SVs are compared with their corresponding points on the estimated waveform calculated using the measurement model, and the error (or residuals) is determined. When the error is below a set threshold, or a maximum number of iterations occurs, the estimator stops and returns its result.

In most PMUs, the results from several measurement windows are combined to give a single value. This process may “smooth” the stream of results, but it means that the measurements are not independent.

When the estimated parameters (Xm, f, ϕ) are accompanied with the UTC time at which the sampled values were acquired, the package of values (Xm, f, ϕ, t) is known as a synchrophasor.

Reporting - Telecommunications: Presentation & Transport

This is where we move from the domain of measurement theory, to telecommunications engineering. We must first differentiate between 'presentation' of the synchrophasor data, and 'transport' of the data.

The IEEE Std. C37.118.2 primarily describes the presentation of synchrophasor data in a bit-mapped format, and allows for transport using serial links, or using Internet Protocol (IP) using either TCP/IP or UDP/IP. A Technical Report, IEC 61850-90-5, describes a similar approach to presentation, but goes into greater depth on transport within the IEC 61850 suite.

This area is developing, but it is important to consider the cyber security aspects of synchrophasor communications systems, especially as they become used in protection and control systems. IEEE 2664 provides alternative transport mechanisms, but the cyber security aspects require caution.

Time Synchronisation - When is t = 0

In the expression

x(t) = Xm cos( ωt + ϕ )

the quantity ϕ was described as the initial phase, that is, the phase at time t = 0. This immediately raises a practical question: when is t = 0?

The answer is that t = 0 is not an arbitrary local choice. For a PMU, it is defined with respect to a global time reference. Without this, phase angles measured at different locations would have no consistent meaning, and could not be compared.

Most PMUs obtain their time reference from satellite systems such as the Global Positioning System (GPS). However, it is important to be precise in the wording. A PMU is not synchronised to GPS time. It is synchronised to Coordinated Universal Time (UTC), and uses GPS (or other systems) as a means of accessing that time reference.

Other Global Navigation Satellite Systems (GNSS), such as Galileo, as well as terrestrial time transfer methods, can also be used to achieve synchronisation to UTC. The essential requirement is not the particular technology, but that all PMUs share a common, agreed definition of time.

With this in place, t = 0 can be understood clearly. It is defined relative to UTC - typically aligned with well-defined boundaries such as the start of a second. The initial phase ϕ is therefore the phase of the signal at that specific, globally agreed instant. This is what allows phase angles measured in different substations, or even different countries, to be directly compared.

In practical PMU operation, t = 0 is tied to specific, repeating instants defined relative to UTC. According to IEEE/IEC 60255-118-1, measurements are reported at fixed rates that are related to the nominal system frequency. For a 50 Hz system, common reporting rates are 10, 25, or 50 measurement reports per second. On a 60 Hz system, the common reporting rates are 10, 30 or 60 reports per second.

Taking the 50 reports per second case for simplicity, this corresponds to one report every 20 ms. The first of these instants coincides with the UTC second boundary, and subsequent instants occur at 20 ms intervals: 20, 40, 60, 80, 100 ms, and so on, up to 980 ms, before returning to the start of the next second. Each of these instants is treated as a valid t = 0 for the purposes of the measurement model, and the reported measurements will be labelled with the full UTC timestamp including the fraction of the second. In other words, every reported phasor is referenced to its own precisely defined time origin, even though all such origins are consistently aligned to the same global time base.

There is, however, a complication. UTC is not a perfectly uniform time scale. It includes leap seconds, which are occasionally inserted to keep UTC aligned with mean solar time - so that, in a loose sense, noon remains close to when the sun is highest in the sky. While this is useful for civil timekeeping, it introduces discontinuities that can be awkward for databases and real-time systems.

For this reason, there is an ongoing discussion about whether systems such as PMUs should instead use a continuous time scale such as International Atomic Time (TAI), which does not include leap seconds. This would simplify some aspects of system design, and indeed most of the careful handling required to relate measurements back to UTC is standard practice in database software.

In practice, regardless of the underlying time scale, the key idea remains the same: the meaning of t = 0 must be shared and unambiguous. Only then does the phase ϕ become a quantity that can be compared across a wide-area power system, and only then do synchrophasor measurements fulfil their purpose.

Stationary and Rotating Phasors

The phasor just introduced is often drawn as a fixed arrow, with a magnitude and an angle. This is useful, but it hides an important idea. In the underlying model,

x(t) = Xm cos( ωt + ϕ )

the phase is not constant. It is ωt + ϕ, which changes continuously with time. If we represent this graphically, the phasor is not stationary at all - it is rotating.

A rotating phasor is simply this idea made explicit. It is a vector of constant length Xm whose angle increases with time at a rate ω. As time progresses, the phasor turns, and the projection of this rotation onto the real axis gives the instantaneous value x(t). In this sense, the rotating phasor is another way of expressing the same model: a steady rotation in the complex plane corresponds to a sinusoidal signal in time.

From this point of view, all phasors rotate. The rotation is not an optional extra - it is the mathematical expression of frequency. If the phasor did not rotate, the voltage or current would not oscillate.

However, in power system analysis, it is often inconvenient to think about quantities that are continuously spinning. Instead, a different point of view is adopted. If all parts of the system are operating at the same frequency, then all of the phasors are rotating at the same rate. Rather than working with the mathematics of the rotating phasors, we can operate on them at a single snapshot in time.

This leads to the idea of a stationary phasor. The word “stationary” does not mean that the phasor has stopped rotating in an absolute sense. It means that it is being described in a reference frame that rotates at the nominal system frequency (or whatever the frequency happens to be in our analysis). In that rotating frame, a steady sinusoid appears as a constant vector with fixed magnitude and angle.

In this representation, the time-varying term ωt is effectively removed, and what remains is the angle ϕ, defined relative to the chosen time reference. The phasor can then be treated as a constant quantity, which greatly simplifies mathematical analysis.

This simplification relies on an assumption: that the system frequency is the same everywhere, or at least varies slowly enough that it can be treated as common over the time of interest.

When the frequency deviates from nominal, or differs between locations, the situation becomes more complicated. The phasors are no longer truly stationary in the chosen reference frame, and their angles will drift over time. In such cases, it is necessary to return to the more general idea: the phasor as a rotating quantity, whose motion reflects the underlying behaviour of the system.

The rotating phasor is the fundamental description, directly tied to the model x(t) = Xm cos( ωt + ϕ ). The stationary phasor is a convenient viewpoint, adopted by choosing a rotating frame of reference in which that motion is, for a time, made to disappear.

When studying real power systems, there have been shown to be frequency dynamics which mean that the assumption that the frequency is the same everywhere is not valid. If only the stationary phasor is reported, the frequency information is lost and can only be approximated by comparing sequential reports of the initial phase, ϕ. This can lead to a number of undesirable measurement artifacts, and should be avoided.

Final thoughts

It's really important to differentiate a PMU from a multifunction instrument, such as a disturbance recorder. Many disturbance recorders, protection relays, and other substation apparatus might include a PMU function and return synchrophasors, but the other functions of these devices (e.g. capturing waveforms during transients) are not functions of a PMU. It's an "all squares are rectangles, not all rectangles are squares" type definition.

Also, PDCs are antiquated techology in the age of high speed IP networks. PDCs do not store data (read the IEEE Standard C37.247-2019 , which only vaguely says in an appendix that they 'may also support data storage'). A microwave oven 'may have a clock'. Storage is the function of an historian, which is power systems speak for a 'database'.

PMU is the name of the instrument. Synchrophasor is the name of the measurement. These words are not interchangable.

How to cite this:

The journal publication below has a good overview of the PMU fundamentals described here. You can get paper and the IEEE formatted citation via the link below.

The OpenPMU Platform for Open-Source Phasor Measurements
IEEE Transactions on Instrumentation and Measurement
D. M. Laverty, R. J. Best, P. Brogan, I. Al Khatib, L. Vanfretti and D. J. Morrow

http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6463452

Note for students: DO NOT COPY this web page. It will be detected as plagiarism.

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