Harmonic Analysis of the Line Current and Voltage of a Thyristor Reactive Power Compensator
Abstract Generated abstract
This paper analyzes harmonic distortion in the line current and bus voltage of a thyristor reactive power compensator composed of capacitor banks and line reactors switched by antiparallel thyristors. Under assumptions of sinusoidal constant supply voltage and lossless elements, it derives piecewise expressions for the regulating branch line current, obtains Fourier and rms relationships, and formulates nonlinear distortion coefficients for compensator current and network voltage as functions of control angle and first harmonic current. The results indicate that increasing the nominal control angle reduces the required reactor rating but worsens harmonic composition, while smaller nominal control angles give lower current and voltage distortion. The paper also provides practical curves for selecting the nominal control angle from the precompensation power factor so that voltage distortion remains within a 5 percent limit over the regulation range.
Full Text
Harmonic Analysis of the Line Current and Voltage of a Thyristor Reactive Power Compensator
V. N. Mishin, A. V. Savchenko
(Presented by the scientific and technical seminar of the Research Institute of AEM)
This article carries out a harmonic analysis of the line current and voltage of a thyristor reactive power compensator (Fig. 1), consisting of capacitor banks and line reactors connected through antiparallel thyristors [1]. The advantage of the compensator is the simplicity of its power circuit. The reactive power of such an installation is varied by means of the variable component of the reactive power of the inductances, which is subtracted from the constant component of the reactive power of the capacitors,
\[ Q = Q_C - Q_L . \tag{1} \]
The calculations were performed under the following assumptions: the supply voltage is sinusoidal; the amplitude of the supply voltage is constant; active losses in the circuit elements are absent.
If the origin of time is taken from the instant at which the valve in the inductance circuit is switched on, then the formula for the phase current \((i_{ab}, i_c)\) takes the form
\[ i_{\phi} = \frac{\sqrt{2}\,U_{\ell}}{x_i} [\sin(\alpha+\vartheta)-\sin\alpha] = I_m[\sin(\alpha+\vartheta)-\sin\alpha], \tag{2} \]
where \(\vartheta=\omega t\), \(\alpha\) is the control angle.
For convenience in expanding the line current \((i_\ell)\) into a Fourier series, the time origin will be taken from the beginning of the positive half-wave of the phase voltage; that is, the origin of time will pass through the center of the positive half-wave of the line current of the regulating element (Fig. 2). In ob—
Fig. 1
Fig. 2.
In this case, the expression for the instantaneous value of the line current of the regulating branch is written by three equations, valid on their respective intervals,
\[ \text{for } 0 \leq \alpha \leq \frac{\pi}{6} \]
\[ i_{\ell}=I_m\left(\sqrt{3}\cos\vartheta-2\sin\alpha\right),\quad 0\leq\vartheta\leq\frac{\pi}{3}-\alpha; \tag{3} \]
\[ i_{\ell}=I_m\left[\sin\left(\frac{\pi}{3}+\vartheta\right)-\sin\alpha\right],\quad \frac{\pi}{3}-\alpha\leq\vartheta\leq\frac{\pi}{3}+\alpha; \tag{4} \]
\[ i_{\ell}=\sqrt{3}I_m\cos\vartheta,\quad \frac{\pi}{3}+\alpha\leq\vartheta\leq\frac{\pi}{2}; \tag{5} \]
\[ \text{for } \frac{\pi}{6}\leq\alpha\leq\frac{\pi}{3} \]
\[ i_{\ell}=I_m\left(\sqrt{3}\cos\vartheta-2\sin\alpha\right),\quad 0\leq\vartheta\leq\frac{\pi}{3}-\alpha; \tag{6} \]
\[ i_{\ell}=I_m\left[\sin\left(\frac{\pi}{3}+\vartheta\right)-\sin\alpha\right],\quad \frac{\pi}{3}-\alpha\leq\vartheta\leq\frac{2\pi}{3}-\alpha; \tag{7} \]
\[ i_{\ell}=0,\quad \frac{2\pi}{3}-\alpha\leq\vartheta\leq\frac{\pi}{2}; \tag{8} \]
\[ \text{for } \frac{\pi}{3}\leq\alpha\leq\frac{\pi}{2} \]
\[ i_{\ell}=0,\quad 0\leq\vartheta\leq\alpha-\frac{\pi}{3}; \tag{9} \]
\[ i_{\ell}=I_m\left[\sin\left(\frac{\pi}{3}+\vartheta\right)-\sin\alpha\right],\quad \alpha-\frac{\pi}{3}\leq\vartheta\leq\frac{2\pi}{3}-\alpha; \tag{10} \]
\[ i_l = 0,\qquad \frac{2\pi}{3}-\alpha \leq \vartheta < \frac{\pi}{2}. \tag{11} \]
The harmonic series for the line-current curve has the form
\[ i_l=\sum_{n=1}^{\infty} a_n \cos n\vartheta, \tag{12} \]
where
\[ a_1=\frac{\sqrt{3}I_m}{\pi}(\pi-2\alpha-\sin 2\alpha),\quad 0\leq \alpha \leq \frac{\pi}{2}. \tag{13} \]
To determine the nonlinear-distortion coefficient \((K_{\text{ndi}})\) of the line current, it is necessary to know the expression for the square of the rms value of this current \((I_l^2)\).
\[ I_l^2=\frac{2}{\pi}I_m^2\left[\frac{17\pi}{12}-3\alpha-2\sin 2\alpha+\left(\alpha-\frac{2\pi}{3}\right)\cos 2\alpha\right], \quad 0\leq \alpha \leq \frac{\pi}{6}; \tag{14} \]
\[ I_l^2=\frac{2}{\pi}I_m^2\left[\frac{3\pi}{2}-\frac{7}{2}\alpha+\frac{\sqrt{3}}{2} +\left(2\alpha-\frac{\sqrt{3}}{4}-\frac{5\pi}{6}\right)\cos 2\alpha -\frac{11}{4}\sin 2\alpha\right], \quad \frac{\pi}{6}\leq \alpha \leq \frac{\pi}{3}, \tag{15} \]
\[ I_l^2=\frac{2}{\pi}I_m^2\left[\pi-2\alpha+\left(\alpha-\frac{\pi}{3}\right)\cos 2\alpha-\frac{3}{2}\sin 2\alpha\right], \quad \frac{\pi}{3}\leq \alpha \leq \frac{\pi}{2}. \tag{16} \]
The rms value of the first harmonic of the line current of the regulating branch is
\[ I_{(1)L}=\frac{a_1}{\sqrt{2}}. \tag{17} \]
The rms value of the first harmonic of the line current of the entire compensator is
\[ I_{(1)}=I_{(1)c}-I_{(1)L}, \tag{18} \]
where
\(I_{(1)c}\) is the rms value of the first harmonic of the line current of the capacitor bank.
As shown in [1], when the nominal control angle \((\alpha_n)\) is increased, the installed rating of the line reactors decreases. However, the harmonic composition of the compensator line current deteriorates.
If \(Q_c=Q_L\) at \(\alpha_n\), then
\[ I_{(1)c}=\frac{a_{1n}}{\sqrt{2}}, \tag{19} \]
where
\(a_{1n}\) is the value of \(a_1\) at \(\alpha_n\).
The nonlinear-distortion coefficient of the compensator line current is determined by the formula
\[ K_{\text{ndi}}=\frac{\sqrt{I_l^2-I_{(1)L}^2}}{I_{(1)c}-I_{(1)L}}. \tag{20} \]
All values in formula (20) are found to an accuracy of the ninth decimal place, since the values of \(I_l^2\) and \(I_{(1)L}^2\) differ very little from each other for certain values of \(\alpha\).
Figure 3 shows the dependences \(K_{\mathrm{ni}} = f(I_{(1)}^*)\) for \(\alpha_{\mathrm{n}} = 0, \dfrac{\pi}{6}, \dfrac{\pi}{4}\),
where
\[ I_{(1)}^* = 1 - \frac{I_{(1)L}}{I_{(1)c}} . \]
This dependence is more convenient for consideration. It is easily related to the control range. From the graph it is seen that a compensator having \(\alpha_{\mathrm{n}}=0\) has the best harmonic composition of the line current.
Fig. 3.
The higher harmonics of the voltage on the AC buses of the compensator are caused by the voltage drop from the higher harmonics of the alternating current across the inductive reactance of the network. If the assumption is made that after compensation \(\cos\varphi=1\) and the voltage drop from the first current harmonic across the equivalent reduced inductive reactance of the network \((x_s)\) is equal to zero, then the coefficient of nonlinear distortions of the network voltage \((K_{\mathrm{niu}})\) at the point where the compensator is installed can be represented in the form
\[ K_{\mathrm{niu}} = \sqrt{6}\,\frac{x_s}{x_L}\sqrt{f_1 - f_2}, \tag{21} \]
where
\(x_s\) and \(x_L\) are the inductive reactances for the first current harmonic,
\[ f_{(1)} = \frac{2}{\pi I_m^2}\int_0^{\pi/2}\left(\frac{d i_{\mathrm{l}}}{d\vartheta}\right)^2 d\vartheta , \tag{22} \]
\[ f_2 = \frac{I_{(1)L}^2}{I_m^2}. \tag{23} \]
\[ f_1 = \frac{2}{\pi}\left(\frac{3\pi}{4} - 2\alpha - \sin 2\alpha\right), \quad 0 \le \alpha \le \frac{\pi}{6}; \tag{24} \]
\[ f_1 = \frac{2}{\pi}\left(\frac{2\pi}{3} - \frac{3\alpha}{2} - \frac{\sqrt{3}}{4}\cos 2\alpha - \frac{3}{4}\sin 2\alpha\right), \quad \frac{\pi}{6} \le \alpha \le \frac{\pi}{3}; \tag{25} \]
\[ f_1 = \frac{2}{\pi}\left(\frac{\pi}{2} - \alpha - \frac{1}{2}\sin 2\alpha\right), \quad \frac{\pi}{3} \le \alpha \le \frac{\pi}{2}. \tag{26} \]
Figure 4 presents the dependences of the relative coefficient of nonlinear voltage distortion \((K^*_{\text{nli}u})\) on the relative current of the compensator first harmonic \((I^*_{(1)})\) for \(\alpha_{\mathrm{n}} = 0;\ \frac{\pi}{6},\ \frac{\pi}{4}\).
\[ K^*=\frac{K_{\text{nli}u}}{\dfrac{x_s}{x_L}} =\frac{\sqrt{6}}{N}\cdot\sqrt{f_1-f_2}, \tag{27} \]
where
\(N\) is a coefficient taking into account the dependence \(x_L=f(\alpha_{\mathrm{n}})\).
\[ N=0 \text{ for } \alpha_{\mathrm{n}}=0;\quad N=0.391 \text{ for } \alpha_{\mathrm{n}}=\frac{\pi}{6};\quad N=0.181 \text{ for } \alpha_{\mathrm{n}}=\frac{\pi}{4}. \]
It is evident from the graph that the harmonic composition of the network voltage is also better for a compensator having a smaller nominal control angle.
Fig. 4.
For practical use, the graph of the dependence \(\alpha_{\mathrm{n}}=f(\cos\varphi)\) is convenient, where \(\cos\varphi\) is the power factor before compensation (Fig. 5). In the shaded region, \(K_{\text{nli}u}\le 0.05\) over the entire regulation range \((D=\infty)\).
Fig. 5.
Knowing $\cos \varphi$ before compensation, from this graph one can determine the nominal control angle of the compensator at which the coefficient of nonlinear voltage distortion does not exceed 5%, which satisfies the GOST standard for voltage quality.
References
- V. V. Khudyakov, V. A. Chvanov. A controlled static source of reactive power. Elektrichestvo, 1969, No. 1.