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Three-Phase Unbalance Calculation Method Based on Phase Sequence Decomposition and Measured Value

Abstract: The three-phase unbalance degree defined by the national standard needs to measure the ampli-tude and phase of the three-phase vector, and the vector operation is needed in the calculation process, which is more complicated. In practical engineering applications, the phase of the three-phase vector is not easy to be measured, which makes the calculation method based on phase sequence decomposition not applicable. In order to simplify the calculation, some international or-ganizations or groups use the measurement value of three-phase vector for calculation. Although this kind of method eliminates the complex vector operation, it also causes a large error compared with the calculation result defined by the national standard. Based on this, this paper proposes to define the three-phase unbalance degree by the square ratio of the square sum of the measured values of the three-phase zero-sequence component and the negative sequence component to the measured values of the three-phase positive sequence component. This method only needs the am-plitude of the three-phase vector to calculate, and through the calculation example analysis, this method is close to the calculation result of the national standard definition.

1. 引言

${I}_{\epsilon }=\sqrt{{I}_{0}^{2}+{I}_{1}^{2}}$ (1)

${I}_{\epsilon b}=\frac{{I}_{0}^{2}+{I}_{2}^{2}}{{I}_{1}^{2}}$ (2)

2. 文献回顾

2.1. 基于相序分解的三相不平衡度定义

IEC定义三相电压不平衡度为电压的负序分量与电压的正序分量之比

$\text{CVUF}=\frac{{U}_{2}}{{U}_{1}}×100\text{%}$ (3)

${U}_{1}=\frac{1}{3}\left({U}_{A}+\alpha {U}_{B}+{\alpha }^{2}{U}_{C}\right)$ (4)

${U}_{2}=\frac{1}{3}\left({U}_{A}+{\alpha }^{2}{U}_{B}+\alpha {U}_{C}\right)$

${U}_{0}=\frac{1}{3}\left({U}_{A}+{U}_{B}+{U}_{C}\right)$

IEC [4] 定义电压的负序分量有效值与电压的正序分量有效值之比来计算三相电压不平衡度，其表达式为：

$\text{VUF}=\frac{|{U}_{2}|}{|{U}_{1}|}×100\text{%}$ (5)

${\epsilon }_{unb}=\frac{{U}_{0}}{{U}_{1}}×100\text{%}$ (6)

2.2. 基于量测值的三相不平衡度定义

IEEEStd.936-1987定义电压不平衡度为相电压不平衡率 (PVUR)为最大均方根电压值和最小均方根电压值之差与平均相电压的比值，其表达式

${\text{PVUR}}_{936}=\frac{\mathrm{max}\left\{{U}_{A},{U}_{B},{U}_{C}\right\}-\mathrm{min}\left\{{U}_{A},{U}_{B},{U}_{C}\right\}}{{U}_{pav}}$ (7)

IEEEStd.112-1991定义电压不平衡度为相电压不平衡率 (PVUR), 即三相相电压和平均相电压值差值最大值与平均相电压的比值，其表达式

${\text{PVUR}}_{112}=\frac{\mathrm{max}\left\{{U}_{A}-{U}_{pav},{U}_{B}-{U}_{pav},{U}_{C}-{U}_{pav}\right\}}{{U}_{pav}}$ (8)

$\text{LVUR}=\frac{\mathrm{max}\left\{{U}_{AB}-{U}_{Lav},{U}_{BC}-{U}_{Lav},{U}_{CA}-{U}_{Lav}\right\}}{{U}_{Lav}}$ (9)

${\epsilon }_{G}=\sqrt{\frac{1-\sqrt{3-6\beta }}{1+\sqrt{3-6\beta }}}×100\text{%}$ (10)

$\beta =\frac{|{U}_{AB}^{4}|+|{U}_{BC}^{4}|+|{U}_{CA}^{4}|}{{\left(|{U}_{AB}^{2}|+|{U}_{BC}^{2}|+|{U}_{CA}^{2}|\right)}^{2}}$

2.3. 两大类方法的改进

3. 三相不平衡度定义的改进

Figure 1. Schematic diagram of symmetrical component method

$\left\{\begin{array}{l}{U}_{A}={U}_{1}+{U}_{2}+{U}_{0}\\ {U}_{B}={U}_{1}+\alpha {U}_{2}+{\alpha }^{2}{U}_{0}\\ {U}_{C}={U}_{1}+{\alpha }^{2}{U}_{2}+\alpha {U}_{0}\end{array}$ (11)

${U}_{A}^{2}+{U}_{B}^{2}+{U}_{C}^{2}=3\left({U}_{1}^{2}+{U}_{2}^{2}+{U}_{0}^{2}\right)$ (12)

${U}_{1}^{2}+{U}_{2}^{2}+{U}_{0}^{2}=\frac{1}{3}\left({U}_{A}^{2}+{U}_{B}^{2}+{U}_{C}^{2}\right)$ (13)

${\epsilon }_{unb}=\frac{{U}_{0}^{2}+{U}_{2}^{2}}{{U}_{1}^{2}}$ (14)

$\left\{\begin{array}{l}{U}_{AB}=\sqrt{3}\left({U}_{B}-{U}_{A}\right)=\sqrt{3}\left({U}_{1}\angle {30}^{\circ }+{U}_{2}\angle {\theta }_{1}-{30}^{\circ }\right)\\ {U}_{BC}=\sqrt{3}\left({U}_{C}-{U}_{B}\right)=\sqrt{3}\left({U}_{1}\angle -{90}^{\circ }+{U}_{2}\angle {\theta }_{1}+{90}^{\circ }\right)\\ {U}_{CA}=\sqrt{3}\left({U}_{A}-{U}_{C}\right)=\sqrt{3}\left({U}_{1}\angle {150}^{\circ }+{U}_{2}\angle {\theta }_{1}-{150}^{\circ }\right)\end{array}$ (15)

$\left\{\begin{array}{l}{U}_{AB}^{2}=3\left({U}_{1}^{2}+2{U}_{1}{U}_{2}\mathrm{cos}\left({\theta }_{1}-\frac{\pi }{3}\right)+{U}_{2}^{2}\right)\\ {U}_{BC}^{2}=3\left({U}_{1}^{2}+2{U}_{1}{U}_{2}\mathrm{cos}\left({\theta }_{1}+\pi \right)+{U}_{2}^{2}\right)\\ {U}_{CA}^{2}=3\left({U}_{1}^{2}+2{U}_{1}{U}_{2}\mathrm{cos}\left({\theta }_{1}+\frac{\pi }{3}\right)+{U}_{2}^{2}\right)\end{array}$ (16)

Figure 2. The relationship between three phase voltage and three phase line voltage

${U}_{AB}^{2}=3\left({U}_{1}^{2}+{U}_{2}^{2}+2{U}_{1}{U}_{2}\left(\frac{1}{2}\mathrm{cos}{\theta }_{1}+\frac{\sqrt{3}}{2}\mathrm{sin}{\theta }_{1}\right)\right)$ (17)

${U}_{BC}^{2}=3\left({U}_{1}^{2}+{U}_{2}^{2}-2{U}_{1}{U}_{2}\mathrm{cos}{\theta }_{1}\right)$ (18)

${U}_{BC}^{2}=3\left({U}_{1}^{2}+{U}_{2}^{2}-2{U}_{1}{U}_{2}\mathrm{cos}{\theta }_{1}\right)$ (19)

${U}_{AB}^{2}+{U}_{BC}^{2}+{U}_{CA}^{2}=9\left({U}_{1}^{2}+{U}_{2}^{2}\right)$ (20)

${U}_{AB}^{2}+{U}_{CA}^{2}-2{U}_{BC}^{2}=18{U}_{1}{U}_{2}\mathrm{cos}{\theta }_{1}$ (21)

${U}_{AB}^{2}-{U}_{CA}^{2}=6\sqrt{3}{U}_{1}{U}_{2}\mathrm{sin}{\theta }_{1}$ (22)

$\frac{{U}_{AB}^{2}+{U}_{CA}^{2}-2{U}_{BC}^{2}}{9}=2{U}_{1}{U}_{2}\mathrm{cos}{\theta }_{1}$ (23)

$\frac{{U}_{AB}^{2}-{U}_{CA}^{2}}{3\sqrt{3}}=2{U}_{1}{U}_{2}\mathrm{sin}{\theta }_{1}$ (24)

$4{U}_{1}^{2}{U}_{2}^{2}=\frac{4{U}_{AB}^{4}+4{U}_{BC}^{4}+4{U}_{CA}^{4}-4{U}_{AB}^{2}{U}_{BC}^{2}-4{U}_{AB}^{2}{U}_{CA}^{2}-4{U}_{BC}^{2}{U}_{CA}^{2}}{81}$ (25)

$2{U}_{1}{U}_{2}=\frac{2}{9}\sqrt{{U}_{AB}^{4}+{U}_{BC}^{4}+{U}_{CA}^{4}-{U}_{AB}^{2}{U}_{BC}^{2}-{U}_{AB}^{2}{U}_{CA}^{2}-{U}_{BC}^{2}{U}_{CA}^{2}}$ (26)

$\begin{array}{l}{U}_{AB}^{4}+{U}_{BC}^{4}+{U}_{CA}^{4}-{U}_{AB}^{2}{U}_{BC}^{2}-{U}_{AB}^{2}{U}_{CA}^{2}-{U}_{BC}^{2}{U}_{CA}^{2}\\ ={U}_{AB}^{4}+{U}_{BC}^{4}+{U}_{CA}^{4}+2{U}_{AB}^{2}{U}_{BC}^{2}+2{U}_{AB}^{2}{U}_{CA}^{2}+2{U}_{BC}^{2}{U}_{CA}^{2}-3{U}_{AB}^{2}{U}_{BC}^{2}-3{U}_{AB}^{2}{U}_{CA}^{2}-3{U}_{BC}^{2}{U}_{CA}^{2}\\ ={\left({U}_{AB}^{2}+{U}_{BC}^{2}+{U}_{CA}^{2}\right)}^{2}-3\left({U}_{AB}^{2}{U}_{BC}^{2}+{U}_{AB}^{2}{U}_{CA}^{2}+{U}_{BC}^{2}{U}_{CA}^{2}\right)\\ ={A}^{2}-3B\end{array}$ (27)

${U}_{1}^{2}+{U}_{2}^{2}=\frac{{U}_{AB}^{2}+{U}_{BC}^{2}+{U}_{CA}^{2}}{9}=\frac{A}{9}$ (28)

$\begin{array}{c}2{U}_{1}{U}_{2}=\frac{2}{9}\sqrt{{U}_{AB}^{4}+{U}_{BC}^{4}+{U}_{CA}^{4}-{U}_{AB}^{2}{U}_{BC}^{2}-{U}_{AB}^{2}{U}_{CA}^{2}-{U}_{BC}^{2}{U}_{CA}^{2}}\\ =\frac{2\sqrt{{A}^{2}-3B}}{9}\end{array}$ (29)

${\left({U}_{1}+{U}_{2}\right)}^{2}={U}_{1}^{2}+{U}_{2}^{2}+2{U}_{1}{U}_{2}=\frac{A}{9}+\frac{2\sqrt{{A}^{2}-3B}}{9}$ (30)

${\left({U}_{1}-{U}_{2}\right)}^{2}={U}_{1}^{2}+{U}_{2}^{2}-2{U}_{1}{U}_{2}=\frac{A}{9}-\frac{2\sqrt{{A}^{2}-3B}}{9}$ (31)

${U}_{1}^{2}=\frac{A}{18}+\frac{\sqrt{3}\sqrt{-{A}^{2}+4B}}{18}$ (32)

${U}_{2}^{2}=\frac{A}{18}-\frac{\sqrt{3}\sqrt{-{A}^{2}+4B}}{18}$ (33)

${U}_{1}^{2}+{U}_{2}^{2}+{U}_{0}^{2}=\frac{{U}_{A}^{2}+{U}_{B}^{2}+{U}_{C}^{2}}{3}$

${U}_{0}^{2}=\frac{{U}_{A}^{2}+{U}_{B}^{2}+{U}_{C}^{2}}{3}-\left({U}_{1}^{2}+{U}_{2}^{2}\right)=\frac{{U}_{A}^{2}+{U}_{B}^{2}+{U}_{C}^{2}}{3}-\frac{1}{9}A$ (34)

$\begin{array}{c}{\epsilon }_{unb}=\frac{{U}_{0}^{2}+{U}_{2}^{2}}{{U}_{1}^{2}}\\ =\frac{\frac{{U}_{A}^{2}+{U}_{B}^{2}+{U}_{C}^{2}}{3}-\frac{A}{18}-\frac{\sqrt{3}\sqrt{-{A}^{2}+4B}}{18}}{\frac{A}{18}+\frac{\sqrt{3}\sqrt{-{A}^{2}+4B}}{18}}\\ =-1+\frac{\frac{{U}_{A}^{2}+{U}_{B}^{2}+{U}_{C}^{2}}{3}}{\frac{A}{18}+\frac{\sqrt{3}\sqrt{-{A}^{2}+4B}}{18}}\\ =-1+\frac{3\left({U}_{A}^{2}+{U}_{B}^{2}+{U}_{C}^{2}\right)\left(A-\sqrt{3}\sqrt{-{A}^{2}+4B}\right)}{2\left({A}^{2}-3B\right)}\end{array}$ (35)

4. 算例分析

Table 1. Three groups of voltage data used in simulation

Table 2. Three groups of line voltage data used in simulation

Table 3. Three phase unbalance degree calculated by different methods

5. 总结

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