Why is mathematics important in an electric power grid? Because alternating voltage and current are not described completely by a single magnitude. Their cycles have phase, frequency and changing relationships across many locations. Grid operators need measurements that can be compared on one time reference, even when substations are far apart.
A phasor measurement unit, or PMU, estimates voltage and current phasors and attaches precise time information. When those synchronised estimates are collected across a network, engineers can study angle differences, oscillations, frequency and rate of change. Complex numbers become a practical map of a moving electrical system.
This article is educational, not an operating, protection or control instruction. Power systems involve dangerous energy, specialised standards, cybersecurity, instrument transformers, communications and authorised procedures. Simplified calculations cannot be used to set relays, judge grid security or operate equipment.
Quick Reading Route
- Start with the rotating-vector idea to connect waves and complex numbers.
- Add a shared clock to see what makes a synchrophasor special.
- Relate angle to power flow without treating it as a complete network model.
- Follow a worked comparison with degrees, radians and time.
- Read frequency and ROCOF carefully as estimates, not magic alarms.
- Use the learning plan to build transferable signal literacy.
Why Mathematics Is Important in Synchrophasor Monitoring
Traditional supervisory measurements can report voltage magnitude, power and breaker state. A synchrophasor adds a time-referenced phase estimate, allowing measurements from different places to be aligned and compared.
The mathematics includes trigonometry, complex numbers, sampling, frequency estimation, uncertainty and network models. A PMU must infer a clean phasor from finite, noisy and sometimes distorted samples. A phasor data concentrator must align streams with delay, gaps and differing quality.
The result is useful because relationships become visible. A widening angle difference may reflect changing transfer. Oscillatory angles may reveal an electromechanical mode. Frequency measurements show system-wide imbalance, while local differences and measurement error still need interpretation.
A Sinusoid Can Be Summarised by a Phasor
A sinusoidal voltage can be written v(t) = V_m cos(ωt + φ). V_m is peak magnitude, ω is angular frequency in radians per second and φ is phase angle relative to the selected time origin.
For steady sinusoidal analysis, the common time factor can be suppressed and the signal represented by a complex phasor V = V_rms∠φ. In rectangular form, V = V_rms(cos φ + j sin φ).
RMS and peak are different
For a pure sinusoid, V_rms = V_m/√2. A peak of 325 V corresponds to about 230 V RMS. Protection, power and instrument conventions must say which magnitude is being used.
A phasor convention may use RMS magnitude, but students should verify the governing definition rather than assume. Multiplying or dividing by √2 at the wrong place creates a systematic scale error.
Polar and rectangular forms
Polar form makes magnitude and angle easy to see. Rectangular form makes addition easy. If V₁ = 100∠20° and V₂ = 60∠−10°, convert them before adding:
V₁ ≈ 93.97 + j34.20 and V₂ ≈ 59.09 − j10.42. Their sum is about 153.06 + j23.78, or 154.90∠8.83°.
Angles cannot be added by weighting them casually. Complex geometry is the correct route.
A phasor is not the waveform
A single steady-state phasor omits harmonics and detailed time-domain shape. A clipped or distorted waveform can share a fundamental phasor with another waveform while differing substantially in peaks and harmonic content.
Synchrophasor standards therefore define estimation and performance under dynamic conditions. The phasor is a designed summary, not a lossless copy of every sample.
Synchronisation Makes Distant Angles Comparable
Phase angle depends on the time origin. If two devices use different clocks, identical waveforms can appear to have different phase. Synchronisation provides a common reference so angle differences are physically meaningful within timing uncertainty.
For frequency f, a timing error Δt creates phase error Δφ = 2πfΔt radians, or 360fΔt degrees. At 50 Hz, a 1 microsecond error corresponds to 0.018°. At 60 Hz, it corresponds to 0.0216°.
Small time errors become electrical angles
If a measurement goal concerns tenths of a degree, microseconds matter. That is why precise clocks, time distribution, receiver health and time-quality flags belong to the measurement chain.
The calculation also prevents exaggeration. One nanosecond at 50 Hz is only 0.000018°. Instrument, transformer and algorithm errors may dominate long before that timing scale.
Time stamps and reporting instants
A PMU estimates values for defined reporting instants. Samples collected around an instant may be filtered over a window. The reported time therefore refers to the estimate’s convention, not necessarily the arrival time of a network packet.
Communication delay affects when data are available. It should not be mistaken for the original measurement time when time stamps are valid.
Clock loss and holdover
If the external time source is lost, a local oscillator may continue in holdover. Its timing error can grow with oscillator drift and temperature.
Quality flags and clock status are essential. A smooth-looking angle stream with degraded timing may be less trustworthy than a visibly missing sample.
Sampling Turns a Continuous Wave into Data
An analogue waveform is sampled at discrete times. If the sample rate is f_s, samples are separated by T_s = 1/f_s. A 4.8 kHz sample rate gives T_s ≈ 208.3 microseconds.
At a nominal 50 Hz, that is 96 samples per cycle. At 60 Hz, it is 80 samples per cycle. Real devices may resample, track frequency or use different internal rates.
Nyquist is necessary but not sufficient
Sampling above twice the highest frequency of interest avoids ideal aliasing, but PMU performance also depends on filters, window length, leakage, noise and dynamic response.
A fundamental-only estimator must reject harmonics without becoming too slow. A fast response and strong rejection are competing design goals.
Window length and latency
A longer window averages more cycles and can improve noise rejection and frequency resolution. It also delays response and smears rapid changes.
There is no universally best window. Measurement classes and applications balance speed and accuracy differently. A protection-oriented use may value rapid response; a measurement-oriented use may prioritise accuracy under certain conditions.
Spectral leakage
If the observation window does not contain an integer number of actual cycles, a discrete Fourier estimate spreads energy across bins. A window function reduces some leakage while changing bandwidth and amplitude response.
Frequency tracking, interpolation and model-based estimation can reduce error. The correct conclusion is not that the Fourier transform “fails”, but that finite observations require a measurement model.
Phase-Angle Differences Help Describe Power Transfer
For a simplified lossless line with bus-voltage magnitudes V₁ and V₂, reactance X and angle difference δ = θ₁ − θ₂, active power can be approximated as P = (V₁V₂/X) sin δ.
For small δ in radians, sin δ ≈ δ, so P ≈ (V₁V₂/X)δ. This explains why angle difference is informative about transfer in a network dominated by reactance.
A per-unit example
Let V₁ = 1.02 per unit, V₂ = 0.99 per unit, X = 0.25 per unit and δ = 8°. Then P ≈ (1.02 × 0.99 / 0.25) sin 8° ≈ 0.562 per unit.
The small-angle approximation gives (1.02 × 0.99 / 0.25) × 0.1396 ≈ 0.564 per unit, close here. At larger angles, the approximation diverges.
The equation is deliberately simplified
Real networks have resistance, shunts, transformers, voltage-dependent loads, multiple paths and changing topology. Power does not flow through one isolated reactance unless the model boundary supports that assumption.
Angle difference is evidence, not a complete stability margin. A single large angle can mean different things on different corridors.
Direction depends on convention
Swapping bus order changes the sign of δ and P. PMU channels, current direction and phase sequence must use consistent definitions.
A negative result is not automatically an error. It may indicate reverse flow under the selected reference.
Worked Example: Two Buses and One Time Reference
Suppose PMU A reports 1.01∠12.4° per unit and PMU B reports 0.98∠5.9° at the same time stamp. The angle difference A minus B is 6.5°.
Convert to radians: δ = 6.5π/180 ≈ 0.11345 rad. If a classroom line model uses X = 0.30 per unit, estimated active transfer is P = (1.01 × 0.98 / 0.30) sin 0.11345 ≈ 0.373 per unit from A toward B under the chosen sign.
Compare the small-angle model
The linear approximation gives P ≈ (1.01 × 0.98 / 0.30) × 0.11345 ≈ 0.374 per unit. Difference is small because 6.5° is modest.
At δ = 30°, the exact sine is 0.5 while the radian value is 0.5236, a 4.7% difference before other model limitations. “Small” must be checked, not assumed.
Timing-error sensitivity
Assume relative timing uncertainty is ±2 microseconds at 50 Hz. Phase uncertainty from timing alone is ±0.036°. That is ±0.000628 rad.
Near δ = 6.5°, local sensitivity is dP/dδ = (V₁V₂/X) cos δ ≈ 3.28 per unit per radian. Timing contribution to P uncertainty is roughly 3.28 × 0.000628 ≈ 0.00206 per unit, before magnitude, transformer and estimator errors.
Magnitude sensitivity
If V₁ and V₂ each have ±0.1% uncertainty and are treated as independent for illustration, relative product uncertainty is approximately √(0.1² + 0.1²)% ≈ 0.141%.
That contributes roughly 0.00053 per unit to the 0.373 estimate. Correlation and systematic calibration can change the combination, so the calculation is illustrative rather than a device specification.
What the result does not say
It does not prove the line is secure, predict a stability limit or account for alternate network paths. It shows how synchronised magnitude and phase can feed a transparent network calculation.
Frequency and ROCOF Describe Different Motion
Frequency describes how rapidly electrical phase advances. If phase angle θ(t) is unwrapped and expressed in radians relative to a nominal reference, frequency deviation is related to its derivative:
Δf = (1/2π)dθ/dt.
Rate of change of frequency, or ROCOF, is df/dt and has units hertz per second. It is a second derivative of phase in that sense.
Why differentiation amplifies noise
Small sample-to-sample angle noise can become large when divided by a small time interval. Differentiating again for ROCOF amplifies it further.
Filters and fitting windows reduce noise but introduce delay and shape dynamic response. A ROCOF number is meaningful only with its estimator and quality conditions.
A phase-ramp example
If unwrapped phase relative to nominal increases by 18° over 0.20 s, that is 0.3142 rad / 0.20 s = 1.571 rad/s. Frequency deviation is 1.571/(2π) ≈ 0.25 Hz.
Phase wrapping must be handled. A change from 179° to −179° represents a +2° step under one continuous path, not −358°.
Frequency is broadly shared but not perfectly uniform
In an interconnected AC system, electromechanical coupling keeps frequency close across the network, but local estimates can differ during transients and because of measurement windows, noise and angle dynamics.
Treating one PMU as universal truth can hide local or device-specific effects. Wide-area comparison is one of the strengths of synchrophasors.
Symmetrical Components Organise Unbalance
Three-phase voltages can be decomposed into positive-, negative- and zero-sequence components. The transformation uses complex rotation operator a = e^(j120°).
Positive sequence has the normal phase order, negative sequence reverses it and zero sequence has three phasors aligned. This turns an unbalanced set into structured components that relate to different physical conditions.
A linear transformation
The conversion is matrix multiplication. Because it is linear, components can be transformed back and summed to reconstruct the original phases.
Students should not treat sequences as additional physical wires. They are a mathematical basis for representing the same measurements.
Phase-sequence mistakes
If channels are labelled in the wrong order or polarity, negative sequence can appear artificially large and angles can flip.
Commissioning tests and channel maps are therefore part of measurement mathematics. Metadata errors can mimic electrical events.
Oscillations Become Patterns Across Space
Generators and controls can participate in electromechanical oscillations. A local mode may involve one plant; an inter-area mode may have groups of machines swinging against each other.
PMU angle and frequency streams across many locations can reveal coherent groups, amplitude, frequency and damping. The pattern in space helps distinguish a wide-area mode from local noise.
Damped sinusoid model
A simple ring-down can be represented as x(t) = Ae^(−ζω_nt) cos(ω_dt + φ) under a second-order model. The envelope decays exponentially when damping is positive.
Estimating damping from a short, noisy record is uncertain. Ambient-data methods and event-based methods have different assumptions.
Mode shape
Complex mode shape records relative amplitude and phase at measurement locations. Two regions nearly 180° apart may swing in opposite directions for that mode.
Missing PMUs or poorly placed sensors can obscure the pattern. Observability is a geometry problem on the network as well as a signal-processing problem.
Correlation is not causation
Two angle traces can move together because both respond to a third event. Correlation can identify coherence but not necessarily the source.
Topology, disturbance records, models and controlled studies are needed to make causal claims.
Data Quality Is Part of the Measurement
A wide-area stream may contain missing frames, late packets, duplicates, bad time, saturation, clipping or channel changes. Silently filling every gap can fabricate smooth behaviour.
Good practice retains quality flags and distinguishes measured, interpolated and estimated values. Analysis windows should state how gaps were handled.
Latency and completeness
Real-time applications value low latency; offline analysis may wait for late data to improve completeness. These objectives conflict.
A data concentrator can align frames by time stamp and release them after a wait. A longer wait captures more late packets but delays the output.
Outliers
An isolated angle jump may be a real switching event, a timing step or a bad sample. Detection based only on magnitude cannot decide which.
Cross-check neighbouring channels, breaker status, frequency and time quality. Context turns anomaly detection into diagnosis.
Calibration chain
The PMU receives secondary signals from instrument transformers. Ratio error, phase error, wiring and burden contribute to the full chain.
Device accuracy alone does not equal system measurement accuracy. Traceability should follow the signal from primary quantity through transformer, cabling, acquisition, time and algorithm.
Total Vector Error and Other Metrics
Total vector error, or TVE, compares an estimated complex phasor with a reference and normalises the vector difference by reference magnitude.
If the estimate has small magnitude error ε and small phase error δ radians, TVE is approximately √(ε² + δ²). A 0.2% magnitude error and 0.1° phase error, or 0.001745 rad, give about √(0.002² + 0.001745²) ≈ 0.00265, or 0.265%.
One metric combines two errors
TVE is convenient, but the same value can arise from different magnitude and phase combinations. Applications sensitive to angle may need the components separately.
Frequency error and ROCOF error require separate metrics because a phasor can be close at one instant while its derivative estimate is poor.
Dynamic tests
Steady-state accuracy does not fully describe response during modulation, ramps or steps. A windowed estimator has transient delay and overshoot.
Testing should match the application’s relevant dynamics. A device excellent for steady metering may not be ideal for a rapid control signal, and vice versa.
State Estimation and Model Checking
Traditional state estimation combines measurements with a network model to estimate bus voltage magnitudes and angles. PMUs add direct angle-related information and can improve observability and bad-data detection.
The estimation problem often minimises a weighted residual. Measurements with smaller justified variance receive larger weight. Giving a PMU infinite weight is not appropriate because every measurement chain has uncertainty.
Residuals are evidence
A large residual can indicate bad data, wrong topology, parameter error or model mismatch. Automatically deleting the largest residual may conceal a topology problem.
Inspect patterns. Several coherent residuals around one substation may point to a common configuration issue.
Observability
A system is observable if the available measurements and model allow the state to be determined under the chosen formulation. PMU placement is therefore a combinatorial optimisation problem.
The minimum number of devices is not the only objective. Redundancy, outage conditions, communication paths, critical buses and cost also matter.
Complex Power Connects Voltage and Current Phasors
For an RMS single-phase convention, complex power is S = VI*, where I* is the complex conjugate of current. Write S = P + jQ: P is active power in watts and Q is reactive power in vars under the selected sign convention.
If V = 230∠0° V and I = 10∠−30° A, then I* = 10∠+30°. Therefore S = 2300∠30° VA = 1992 + j1150 VA. Power factor magnitude is cos 30° ≈ 0.866.
Why the conjugate appears
Without the conjugate, voltage and current angles would add rather than subtract. Average active power for sinusoids depends on their phase difference. The conjugate makes the complex product encode that difference.
Sign conventions for reactive power and current direction must be declared. A current channel reversed at the transformer can shift its phasor by about 180° and reverse calculated power.
Balanced three-phase power
For a balanced three-phase system using line-to-line RMS voltage V_LL and line current I_L, apparent-power magnitude is |S| = √3 V_LL I_L. Active power is P = √3 V_LL I_L cos φ.
Suppose V_LL = 400 kV, I_L = 800 A and power factor is 0.95. Active power is about √3 × 400,000 × 800 × 0.95 ≈ 526.5 MW. The result assumes balance and consistent RMS quantities.
A PMU can supply phase or positive-sequence phasors, but the calculation must match the channel type. Applying the √3 formula to phase voltage while also multiplying by three incorrectly can introduce a √3 error.
Power as a cross-check
If separate metering reports P and Q, phasor-based power can be compared after aligning transformer ratios, direction and time. A steady mismatch may indicate calibration or mapping; a transient mismatch may reflect different filtering windows.
Agreement does not prove every channel is correct because common errors can cancel. It is one consistency check among several.
An Event-Analysis Scenario
Imagine a disturbance record from four PMUs. At 14:05:12.000, one transmission-line breaker opens. PMU A and B angle difference rises from 4° to 11° over one second, system frequency falls by 0.12 Hz, and one data stream contains three missing frames.
The analysis should begin with a timeline. Use the breaker event time, PMU time stamps and data-concentrator receipt times as different fields. Confirm whether the topology model changes at the breaker’s effective time and whether all devices report valid clock status.
Separate step, ramp and oscillation
The breaker opening may create an immediate angle step from topology change, followed by a slower ramp as power redistributes, then a damped oscillation. Fitting one straight line across the entire second would mix three mechanisms.
Segment the record using physical events, not just statistical convenience. For the oscillatory part, subtract a slowly varying trend and examine several locations. A true inter-area mode should show coherent frequency and relative phase patterns, not a single noisy trace.
Handle missing frames honestly
Three missing frames at 60 reports per second span about 50 ms. Linear interpolation may be acceptable for a visual guide in a slow trend, but it can suppress a rapid step. Mark the gap and run key estimates both with and without interpolation.
If frequency is estimated by differentiating angle, interpolation can create an artificial constant slope. A device-reported frequency channel, raw waveform record or neighbouring PMU may provide independent evidence.
Compare predicted and observed power
Use the updated topology and pre-event state to predict the qualitative direction of flow change. Compare PMU voltage–current complex power and supervisory power measurements after accounting for time alignment and filtering.
If the angle difference rises but measured transfer falls, the simple two-bus interpretation may be wrong because alternate paths, line status or reference angle changed. The disagreement should improve the model, not be edited away.
Quantify damping with a range
Fit a damped sinusoid over several plausible windows and report the spread of estimated frequency and damping. Short windows can confuse a changing trend with weak damping; long windows can mix later controls.
The final event note should distinguish direct facts—breaker status, time quality, samples—from calculated quantities and interpretations. This is mathematics as evidence management: align clocks, preserve gaps, test a network hypothesis and state uncertainty before drawing an operational conclusion.
Before publishing a chart, add a small data-quality panel showing valid-frame percentage, maximum gap, clock status and topology version. This compact summary prevents a polished graphic from outrunning its evidence. It also gives a later reviewer enough context to reproduce the same window and understand why particular samples were included.
Common Misconceptions Worth Correcting
“A phasor is the complete waveform”
It summarises a component under a defined model. Harmonics and transients require additional information.
“All angles use an absolute universal zero”
Angles depend on a time and convention reference. Synchronisation makes comparisons possible within uncertainty.
“A bigger phase angle always means instability”
Angle must be interpreted with network topology, reactance, transfer, operating limits and dynamics.
“Frequency and ROCOF are directly measured without a model”
They are estimated from samples. Windowing, noise and algorithms affect the result.
“A time stamp removes communication delay”
It preserves measurement time. Data can still arrive late, out of order or not at all.
“More PMUs automatically solve observability”
Placement, channel configuration, redundancy, data quality and network model all matter.
Did You Know?
At 50 Hz, one full electrical cycle lasts 20 ms. One degree lasts about 55.6 microseconds. The phase scale turns tiny time differences into visible angles.
An angle that crosses from +179° to −179° may have moved only two degrees. Phase unwrapping is a small algorithm with a large effect on frequency estimates.
The power-flow sine relation is nonlinear, but its small-angle linearisation helps explain why grid models often use matrices. Local simplicity can emerge from a nonlinear physical system.
What Students Should Practise
- Convert sinusoidal equations to RMS phasors and back.
- Add phasors in rectangular form, then check magnitude and angle.
- Convert timing error into phase error at different frequencies.
- Use radians, not degrees, in derivatives and small-angle formulas.
- Unwrap phase before estimating a slope.
- Compare exact sin δ with δ over several angles.
- Build an uncertainty budget that separates magnitude, phase, time and model terms.
- Mark missing, late and interpolated data rather than silently smoothing them.
A spreadsheet investigation
Generate two 50 Hz sinusoids with a known phase difference, sample them and estimate phase with sine and cosine correlations. Add noise and alter the window length.
Plot the bias and spread. The aim is not to build a compliant PMU; it is to see how finite samples, frequency mismatch and noise influence an estimate.
A Four-Week Learning Plan
Week 1: Complex numbers and sinusoids
Practise Euler’s formula, polar–rectangular conversion, RMS, phase and vector addition.
Week 2: Sampling and time
Create sampled waves, calculate samples per cycle, examine leakage and convert clock error into electrical angle.
Week 3: Network relationships
Use the two-bus power equation, compare exact and linear models, and explore how reactance and voltage change transfer.
Week 4: Estimation and evidence
Estimate frequency from unwrapped phase, study noise sensitivity, flag bad data and write a short note separating measurement, model and operational interpretation.
Guidance for Parents and Teachers
Begin with two rotating clock hands. If they turn at the same speed but one leads, their angular separation represents phase. Then ask what happens when the clocks themselves disagree. That creates a natural reason for synchronisation.
Let students switch between a graph and a complex number. The waveform shows time; the phasor shows magnitude and angle. Neither view replaces the other.
Reward careful language. “The angle difference increased under this model” is better than “the grid became unstable”. Students learn scientific responsibility when they know the boundary between observation and inference.
Frequently Asked Questions
What is a phasor?
It is a complex-number representation of the magnitude and phase of a sinusoidal component under a defined convention.
What makes a synchrophasor different?
Its phase estimate is referenced to a common time standard, so measurements from different locations can be compared.
What does a PMU measure?
It estimates voltage and current phasors and commonly reports frequency, ROCOF and quality information from sampled signals.
Why are phase angles useful?
Differences relate to power transfer in network models and reveal coherent spatial and dynamic patterns.
Is GPS the only possible time source?
No. Systems can use other satellite or network timing architectures, but the required accuracy, resilience and status monitoring must be engineered.
What is ROCOF?
It is the rate of change of frequency, measured in hertz per second. Because it involves differentiation, it is sensitive to noise and estimator design.
What is TVE?
Total vector error is the normalised complex difference between estimated and reference phasors. It combines magnitude and phase error.
Can PMUs predict a blackout?
They provide high-value measurements. Prediction and prevention require models, thresholds, operator processes, controls and other evidence; no single sensor guarantees the outcome.
Useful Next Reading
- Read the US Department of Energy’s overview of grid modernisation and the smart grid, including the role of PMUs.
- Explore the DOE’s Big Data Synchrophasor Analysis page for wide-area research context.
- Use the IEEE Technology Navigator on phasor measurement units to identify current governing standards and applications.
- Connect hardware timing to Why Mathematics? | PCB Traces, Controlled Impedance and Crosstalk.
- Compare distributed timing and agreement with Why Mathematics? | Distributed Consensus, Quorums and Fault Tolerance.
Twelve Checks Before Trusting a Synchrophasor Trend
1. Confirm the channel
Verify phase, polarity, ratio, sequence and engineering units. A label error can imitate an event.
2. Inspect time quality
Check clock lock, holdover, leap handling and time-quality flags. Do not trust angle comparisons with unknown timing.
3. Identify the phasor convention
State RMS or peak, phase reference, reporting rate and positive-sequence or phase quantity.
4. Review the measurement chain
Include instrument transformers, burden, wiring, acquisition and calibration, not only the PMU device.
5. Check data completeness
Count missing, late and duplicate frames. Preserve the difference between missing and interpolated data.
6. Match the estimator to the event
Window length and filtering affect dynamic response. Do not compare devices without their measurement class and settings.
7. Unwrap angles carefully
Handle ±180° boundaries and real switching jumps. Plot raw and unwrapped versions.
8. Keep topology current
An open breaker changes the network model. An angle pattern interpreted with old topology can mislead.
9. Separate measurement and model uncertainty
Quantify phase, magnitude and time error separately from line-parameter and network-model error.
10. Cross-check neighbouring evidence
Compare voltage, current, power, frequency, breaker state and nearby PMUs before diagnosing an anomaly.
11. Protect the data path
Authentication, access control, monitoring and resilient communications are part of trustworthy operation.
12. Preserve provenance
Archive device configuration, firmware, calibration, time source, data-quality flags, topology version and analysis code.
Closing Perspective
Synchrophasors show how abstract mathematics becomes shared situational awareness. A complex number summarises a wave; a precise time reference makes distant summaries comparable; derivatives reveal changing frequency; network equations connect angle to transfer.
The subject also teaches restraint. A clean trace can contain clock error, transformer bias, missing data or a model mismatch. More samples do not remove the need for provenance and interpretation.
That is why mathematics matters in grid monitoring: it allows many local electrical cycles to be read as one evolving system, while keeping uncertainty and assumptions visible.