Voltage Biquadrate (VBR): A Fourth-Order Approach to Voltage Unbalance
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| Voltage BiquadRate |
Introduction
Voltage unbalance is an important power-quality phenomenon in three-phase electrical systems. Unequal phase or line-voltage magnitudes can produce negative-sequence voltage, increased motor losses, additional heating, torque pulsations and derating of three-phase equipment.
Traditional voltage-unbalance indicators generally use the voltage magnitudes themselves, their deviations from the average, or symmetrical components. A different mathematical perspective can be obtained by considering the square of voltage.
Since electrical power in a resistive system is proportional to \(V^2\), the variation of \(V^2\) provides a useful way of examining the distribution of voltage-related power among the three phases.
This leads naturally to a fourth-order quantity:
\[ V^4=(V^2)^2 \]
which is referred to here as the Voltage Biquadrate.
1. From Voltage to Voltage Biquadrate
For three line-voltage magnitudes
\[ V_1,\quad V_2,\quad V_3, \]
the conventional RMS-related quantity is based on the second power:
\[ V_{\mathrm{SEL}} = \sqrt{\frac{V_1^2+V_2^2+V_3^2}{3}}. \]
Here \(V_{\mathrm{SEL}}\) represents the Effective Line Voltage, obtained as the quadratic mean of the three line-voltage magnitudes.
The next mathematical order is
\[ V_i^4=(V_i^2)^2. \]
This fourth power is called the Voltage Biquadrate in this proposed formulation.
Thus:
\[ \boxed{\text{Voltage}\rightarrow\text{Voltage Square}\rightarrow\text{Voltage Biquadrate}} \]
or
\[ \boxed{V\rightarrow V^2\rightarrow V^4}. \]
2. Definition of Voltage Biquadrate Factor
The proposed Voltage Biquadrate Factor (VBR) is defined as
\[ \boxed{ VBR= \frac{\operatorname{Mean}(V^4)} {\left[\operatorname{Mean}(V^2)\right]^2} } \]
For three line voltages,
\[ \boxed{ VBR= \frac{ \dfrac{V_1^4+V_2^4+V_3^4}{3} }{ \left( \dfrac{V_1^2+V_2^2+V_3^2}{3} \right)^2 } } \]
or, equivalently,
\[ \boxed{ VBR= \frac{3(V_1^4+V_2^4+V_3^4)} {(V_1^2+V_2^2+V_3^2)^2} } \]
Since
\[ V_{\mathrm{SEL}}^2=\operatorname{Mean}(V^2), \]
the definition can also be written as
\[ \boxed{ VBR= \frac{\operatorname{Mean}(V^4)} {V_{\mathrm{SEL}}^4} } \]
which gives a particularly simple physical interpretation:
Voltage Biquadrate Factor is the ratio of the mean fourth-power voltage to the fourth power of the Effective Line Voltage.
3. Balanced Voltage
For a perfectly balanced system,
\[ V_1=V_2=V_3=V. \]
Therefore,
\[ \operatorname{Mean}(V^4)=V^4 \]
and
\[ V_{\mathrm{SEL}}^4=V^4. \]
Consequently,
\[ \boxed{VBR=1}. \]
This is an important advantage of the proposed definition.
The balanced condition is naturally represented by
\[ \boxed{VBR=1}. \]
As voltage unbalance increases, VBR increases above unity.
4. Connection with the Variance of \(V^2\)
Let
\[ X=V^2. \]
The fundamental statistical identity is
\[ E[X^2]=E[X]^2+\operatorname{Var}(X). \]
Therefore,
\[ \operatorname{Mean}(V^4) = \operatorname{Mean}(V^2)^2 + \operatorname{Var}(V^2). \]
Dividing by \(\operatorname{Mean}(V^2)^2\),
\[ \boxed{ VBR = 1+ \frac{\operatorname{Var}(V^2)} {\operatorname{Mean}(V^2)^2} } \]
or
\[ \boxed{ VBR-1= \left[ \frac{\operatorname{Std}(V^2)} {\operatorname{Mean}(V^2)} \right]^2 } \]
This is perhaps the most important mathematical interpretation of VBR.
VBR is not simply an arbitrary fourth-power quantity. It is a normalized measure of the dispersion of the squared voltages.
5. Relationship with \(\Sigma\)
Define the proposed squared-voltage variation parameter
\[ \boxed{ \Sigma= \frac{1}{\sqrt2} \frac{\operatorname{Std}(V^2)} {\operatorname{Mean}(V^2)} } \]
Then
\[ \frac{\operatorname{Std}(V^2)} {\operatorname{Mean}(V^2)} =\sqrt2\,\Sigma. \]
Consequently,
\[ \boxed{ VBR=1+2\Sigma^2 } \]
and
\[ \boxed{ \Sigma=\sqrt{\frac{VBR-1}{2}} } \]
Thus VBR provides a direct fourth-order representation of the squared-voltage dispersion.
6. Connection with Voltage Unbalance
The significance of this formulation becomes clearer when considering a three-phase system with unequal line voltages.
Suppose
\[ V_1,\;V_2,\;V_3 \]
are unequal.
Then
\[ V_1^2,\;V_2^2,\;V_3^2 \]
are also unequal.
For an approximately identical resistive load on each phase,
\[ P_i\propto V_i^2. \]
Thus, the dispersion of \(V_i^2\) corresponds to the dispersion of phase power.
The fourth-order quantity
\[ V_i^4=(V_i^2)^2 \]
therefore emphasizes the unequal distribution of the squared-voltage, or power-related, quantities.
This provides a useful interpretation:
\[ \boxed{ VBR-1 \;\sim\; \text{normalized squared dispersion of phase-power-related voltage} } \]
It should, however, not be interpreted as a direct measurement of electrical power, because \(V^4\) has units of \(V^4\). The normalized VBR itself is dimensionless.
7. Relation to Conventional Voltage-Unbalance Factors
VBR is not intended to replace the established sequence-component definition of VUF.
For example, the true voltage-unbalance factor can be expressed through negative- and positive-sequence voltages:
\[ VUF=\frac{|V_2|}{|V_1|}. \]
Magnitude-based approximations such as line-voltage deviation methods are also commonly used.
VBR approaches the same physical phenomenon from a different mathematical direction:
\[ \boxed{ \text{Conventional approach} \rightarrow \text{sequence/deviation relationships} } \]
whereas
\[ \boxed{ \text{VBR approach} \rightarrow V^2 \rightarrow V^4 \rightarrow \text{dispersion of }V^2 } \]
Therefore, VBR can be investigated as a fourth-order voltage-unbalance indicator and compared against conventional VUF methods.
8. Why Fourth Order?
The progression has a useful physical interpretation:
First order
\[ V \]
represents voltage magnitude.
Second order
\[ V^2 \]
is directly associated with electrical power in a resistive relationship:
\[ P=\frac{V^2}{R}. \]
Fourth order
\[ V^4=(V^2)^2 \]
represents the square of the power-related voltage quantity.
The normalized fourth-order ratio then measures how unevenly \(V^2\) is distributed:
\[ \boxed{ VBR= \frac{\operatorname{Mean}[(V^2)^2]} {\operatorname{Mean}(V^2)^2} } \]
This makes VBR conceptually different from simply calculating the fourth power.
9. A Useful Determinant Representation
The fourth-order formulation can also be represented using the second-order voltage matrix
\[ \mathbf M_V^{(2)} = \begin{bmatrix} 1&1&1\\ V_1^2&V_2^2&V_3^2\\ V_3^2&V_1^2&V_2^2 \end{bmatrix}. \]
Its determinant is
\[ \det(\mathbf M_V^{(2)}) = V_1^4+V_2^4+V_3^4 -V_1^2V_2^2 -V_2^2V_3^2 -V_3^2V_1^2. \]
It can also be written as
\[ \boxed{ \det(\mathbf M_V^{(2)}) = \frac12 \left[ (V_1^2-V_2^2)^2+ (V_2^2-V_3^2)^2+ (V_3^2-V_1^2)^2 \right]. } \]
Thus the determinant is directly related to the variance of \(V^2\).
For three voltages,
\[ \boxed{ \det(\mathbf M_V^{(2)}) = \frac92\operatorname{Var}(V^2) } \]
using population variance.
This provides an interesting algebraic connection between:
\[ \boxed{ \text{Matrix determinant} \leftrightarrow \text{variance of }V^2 \leftrightarrow VBR \leftrightarrow \Sigma. } \]
10. Proposed VBR–TAU Relationship
If the derived relationship in the proposed ASC formulation is
\[ \boxed{ \Sigma=\frac12\sin(2\tau) } \]
then
\[ VBR=1+2\Sigma^2 \]
gives
\[ VBR = 1+ 2\left[\frac12\sin(2\tau)\right]^2. \]
Therefore,
\[ \boxed{ VBR=1+\frac12\sin^2(2\tau) } \]
and using
\[ \sin^2(2\tau)=\frac{1-\cos(4\tau)}{2}, \]
we obtain
\[ \boxed{ VBR=\frac54-\frac14\cos(4\tau) } \]
or
\[ \boxed{ \cos(4\tau)=5-4VBR. } \]
Hence,
\[ \boxed{ \tau= \frac14\cos^{-1}(5-4VBR) } \]
within the applicable range and assumptions of the proposed model.
This is an interesting result because a fourth-order voltage statistic produces a fourth-angle relationship.
11. Example
Consider
\[ V_1=99,\qquad V_2=105,\qquad V_3=110. \]
Then
\[ VBR= \frac{3(99^4+105^4+110^4)} {(99^2+105^2+110^2)^2}. \]
Numerically,
\[ \boxed{VBR\approx1.0021} \]
and therefore
\[ \Sigma = \sqrt{\frac{VBR-1}{2}} \]
giving approximately
\[ \boxed{\Sigma\approx0.0324}. \]
Thus the departure of VBR from unity provides a compact measure of the unequal distribution of \(V^2\).
12. VBR Interpretation
The proposed interpretation can therefore be summarized as:
\[ \boxed{ \begin{array}{ccl} VBR=1 &\rightarrow& \text{Perfect voltage balance}\\ VBR>1 &\rightarrow& \text{Voltage unbalance}\\ VBR-1 &\rightarrow& \text{Squared relative dispersion of }V^2 \end{array}} \]
The factor is dimensionless and independent of the absolute voltage scale. Multiplying all three voltages by the same constant does not change VBR.
Conclusion
Voltage Biquadrate (VBR) provides a fourth-order statistical approach to analysing three-phase voltage unbalance.
Its recommended definition is
\[ \boxed{ \displaystyle VBR= \frac{\operatorname{Mean}(V^4)} {\left[\operatorname{Mean}(V^2)\right]^2} = \frac{\operatorname{Mean}(V^4)} {V_{\mathrm{SEL}}^4} } \]
with the important properties
\[ \boxed{VBR\ge1} \]
and
\[ \boxed{VBR=1\quad\text{for balanced voltage}.} \]
Furthermore,
\[ \boxed{ VBR-1= \left[ \frac{\operatorname{Std}(V^2)} {\operatorname{Mean}(V^2)} \right]^2 } \]
and, under your proposed normalization,
\[ \boxed{ VBR=1+2\Sigma^2. } \]
If the proposed ASC relationship
\[ \Sigma=\frac12\sin(2\tau) \]
is adopted, then
\[ \boxed{ VBR=1+\frac12\sin^2(2\tau) } \]
and
\[ \boxed{ \tau=\frac14\cos^{-1}(5-4VBR). } \]
Note: VBR should presently be described as a proposed analytical indicator, not as an established IEC/IEEE voltage-unbalance standard. Its equivalence, accuracy, sensitivity, and practical usefulness relative to sequence-component VUF should be validated across different unbalance patterns before presenting it as an exact VUF replacement.
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