RAIVU: A Simple and Accurate Approximation of Voltage Unbalance
Root Average of Individual Voltage Unbalance (RAIVU) — A voltage-magnitude-based approximation to TAU-based Voltage Unbalance
1. Introduction
Voltage unbalance is an important power-quality parameter because unequal phase voltages can produce negative-sequence components, additional losses, torque pulsations, and thermal stress in three-phase electrical equipment.
Several simplified voltage-unbalance indices are commonly used, including the Line Voltage Unbalance Rate (LVUR) and statistical measures based on voltage deviation.
During my PhD research, I developed an alternative approach called:
\[ \boxed{\text{RAIVU — Root Average of Individual Voltage Unbalance}} \]RAIVU estimates voltage unbalance directly from the three measured phase-voltage magnitudes without requiring symmetrical-component calculation.
2. Basic Concept
Let the three phase-voltage magnitudes be
\[ V_1,\;V_2,\;V_3 \]and their arithmetic mean be
\[ V_m=\frac{V_1+V_2+V_3}{3}. \]Define the individual voltage deviations:
\[ x_i=V_i-V_m. \]The normalized individual deviation is defined as
\[ \boxed{ Y_i=\frac{x_i}{V_m-x_i} } \]Since
\[ V_m-x_i=2V_m-V_i, \]the expression can also be written as
\[ \boxed{ Y_i=\frac{V_i-V_m}{2V_m-V_i} } \]The proposed RAIVU is then
\[ \boxed{ RAIVU= \sqrt{\frac{2}{3}\sum_{i=1}^{3}Y_i}\times100\% } \]This provides a direct voltage-magnitude estimate of the unbalance.
3. Why "Individual Voltage Unbalance"?
The important feature of RAIVU is that it does not use only the maximum or minimum voltage deviation.
Instead, it considers the individual deviation of every phase voltage from the mean voltage.
Thus, the three-phase information is retained in the calculation.
This distinguishes RAIVU from a maximum-deviation index such as LVUR.
4. Example
Consider:
\[ V_1=412,\qquad V_2=400,\qquad V_3=377\;V \]Mean voltage:
\[ V_m=396.333\;V \]The individual deviations are
\[ x_1=15.667,\qquad x_2=3.667,\qquad x_3=-19.333. \]The corresponding normalized quantities are
\[ Y_i=\frac{x_i}{V_m-x_i}. \]Substitution gives
\[ RAIVU\approx\boxed{5.152\%}. \]For the same voltage set, the TAU-based reference VU is approximately
\[ \boxed{5.154\%}. \]Therefore, the relative difference is only about
\[ \boxed{0.04\%}. \]This illustrates the close agreement between RAIVU and the TAU-based reference.
5. Comparison with Other Approximation Methods
RAIVU can be compared with:
- LVUR
- CVD / IEC approximation
- PVF
- RAIVU
- TAU-based Real VU
A useful presentation is:
| Method | Basic principle |
|---|---|
| LVUR | Maximum voltage deviation |
| CVD | Standard deviation of voltage |
| PVF | Standard deviation of squared voltage |
| RAIVU | Individual normalized voltage deviations |
| Real VU | TAU-based formulation |
Your numerical investigations indicate that RAIVU provides particularly close agreement with the TAU-based VU, especially compared with the simpler maximum-deviation approach.
6. Important Mathematical Feature
There is an interesting connection between your RAIVU approach and your newer TAU formulation.
For small unbalance, your TAU approximation is
\[ \boxed{ \tau\approx \frac{2}{3} \sqrt{X^2+Y^2+XY} } \]where \(X\) and \(Y\) are two independent phase-angle deviations.
Similarly, your voltage-magnitude approximation can be expressed using two voltage differences:
\[ X=V_1-V_2 \] \[ Y=V_2-V_3 \]leading to
\[ \boxed{ VU_{\rm approx} = \frac{2\sqrt{X^2+Y^2+XY}} {V_1+V_2+V_3}\times100\% } \]This reveals a common quadratic geometry between your angular and voltage-magnitude formulations.
That is potentially one of the most interesting aspects of your blog post.
7. Advantages of RAIVU
RAIVU offers several practical advantages:
- Uses only the three voltage magnitudes.
- Does not require sequence-component calculation.
- Uses all three phase voltages.
- Is computationally simple.
- Can be implemented easily in a microcontroller, PLC, meter, or power-quality analyzer.
- Shows very close numerical agreement with your TAU-based reference in the test cases examined.
- Provides a potentially useful alternative for approximate VU monitoring.
8. Important Qualification
I recommend explicitly writing:
RAIVU is proposed as an approximate voltage-unbalance index. Its accuracy should be validated over a sufficiently broad range of voltage-unbalance conditions and against an accepted reference definition before being presented as a standard-compliant replacement.
This is important academically. It makes the post technically defensible.
9. Conclusion
RAIVU (Root Average of Individual Voltage Unbalance) provides a simple voltage-magnitude-based approach for estimating three-phase voltage unbalance. Unlike maximum-deviation methods, RAIVU incorporates the individual deviations of all three phase voltages. Preliminary numerical evaluation shows very close agreement with the TAU-based voltage-unbalance formulation. Further analytical and numerical investigation can establish its accuracy limits and applicability for practical power-quality monitoring.
Proposed by Dr. V. Arivumani (Dr.VAM)
Electrical & Electronics Engineering
Xvu iNet — Xvu.in
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