On the Reduction of the Compensating Effect of Power Factor Correction Capacitor Banks in the Presence of Higher Harmonics
Abstract Generated abstract
The paper examines how higher harmonics in industrial power supply networks reduce the reactive power compensating effect of power factor correction capacitor banks, especially in systems with substantial rectifier converter loads. Using harmonic power factor relations and the method of equivalent sinusoids, it derives modified expressions for the required capacitor bank capacity when voltage and current waveforms are nonsinusoidal and when bank connection further increases distortion or creates resonant conditions. The analysis shows that conventional sizing formulas can underestimate required capacity, and field measurements at an industrial plant indicated cases where compensating capability fell by more than 50 percent, with some banks disconnected because of resonance related current overloads.
Full Text
ON THE REDUCTION OF THE COMPENSATING EFFECT OF COSINE CAPACITOR BANKS IN THE PRESENCE OF HIGHER HARMONICS
G. Z. MARKMAN
(Presented by the scientific seminar of the Department of Electrical Systems and Networks)
The operating conditions of capacitor installations in power-supply systems with a large relative share of rectifier-converter load are complicated by the danger of phenomena close to resonance. The selection of bank capacity for such networks is carried out according to the method proposed in [1]. However, it does not take into account the substantial reduction in the compensating effect of the banks at a high level of higher harmonics.
Figure 1 shows a section of the power-supply scheme of an enterprise, where a capacitor bank of capacity \(Q_k\) is connected to the buses of one of the substations supplying a rectifier converter (RC) and a certain equivalent active-inductive load \((Z_n)\).
Fig. 1.
The voltage \(U^*\) on the substation buses, in addition to the fundamental harmonic of the voltage, contains a whole series of higher harmonics caused by the consumption by the RC of nonsinusoidal current \(i_{\mathrm{vp}}\). It is natural to assume that the waveform of the load current must also be substantially distorted, i.e., contain higher harmonics.
The active power of the first harmonic (fundamental wave), consumed by the load, is equal to
\[ P_{1\mathrm{n}}=U_1\cdot I_{1\mathrm{n}}\cdot \cos\varphi_{1\mathrm{n}} =U^*\cdot I_{\mathrm{n}}^*\cdot \cos\varphi_{1\mathrm{n}}\cdot \frac{U_1\cdot I_{1\mathrm{n}}}{U^*\cdot I_{\mathrm{n}}^*}, \]
where \(U_1, I_{1\mathrm{n}}\) are the voltage and current of the fundamental frequency;
\(U^*, I_{\mathrm{n}}^*\) are the rms values of the load voltage and current with higher harmonics taken into account,
\[ U^*=\sqrt{U_1^2+U_2^2+U_3^2+\cdots+U_\nu^2}, \]
\[ I_{\mathrm{n}}^*=\sqrt{I_{1\mathrm{n}}^2+I_{2\mathrm{n}}^2+I_{3\mathrm{n}}^2+\cdots+I_{\nu\mathrm{n}}^2}. \]
Let us denote
\[ \gamma_1=\frac{U_1\cdot I_{1\mathrm{n}}}{U^*\cdot I_{\mathrm{n}}^*}, \]
then
\[ P_{1\mathrm{n}}=U^*\cdot I_{\mathrm{n}}^*\cdot \gamma_1\cdot \cos\varphi_{1\mathrm{n}} . \tag{1} \]
The value \(\gamma_1\) proves to be less than unity by an amount that increases with the harmonic level.
Analogously to (1), for all the remaining harmonics,
\[ P_2=U^*\cdot I_{\mathrm{n}}^*\cdot \gamma_2\cdot \cos\varphi_{2\mathrm{n}}; \]
\[ \cdots \]
\[ P_\nu=U^*\cdot I_{\mathrm{n}}^*\cdot \gamma_\nu\cdot \cos\varphi_{\nu\mathrm{n}}. \]
The load power factor is then equal to
\[ \alpha_{\mathrm{n}}= \frac{ U^*\cdot I_{\mathrm{n}}^*\cdot \gamma_1\cdot \cos\varphi_{1\mathrm{n}} + U^*\cdot I_{\mathrm{n}}^*\cdot \gamma_2\cdot \cos\varphi_{2\mathrm{n}} +\cdots+ U^*\cdot I_{\mathrm{n}}^*\cdot \gamma_\nu\cdot \cos\varphi_{\nu\mathrm{n}} }{ U^*\cdot I_{\mathrm{n}}^* } \]
or
\[ \alpha_{\mathrm{n}}=\gamma_1\cdot \cos\varphi_{1\mathrm{n}} +\gamma_2\cdot \cos\varphi_{2\mathrm{n}} +\cdots+ \gamma_\nu\cdot \cos\varphi_{\nu\mathrm{n}}. \tag{2} \]
It follows from (2) that the value of the power factor in the presence of higher harmonics proves to be smaller than if only voltages and currents of frequency 50 Hz were acting, and is determined by the harmonic level.
Since the values of the coefficients \(\gamma_2, \gamma_3, \ldots, \gamma_\nu\) are an order of magnitude smaller than the coefficient \(\gamma_1\), with an accuracy sufficient for practical calculations one may write
\[ \alpha_{\mathrm{n}}=\gamma_1\cdot \cos\varphi_{1\mathrm{n}}. \tag{3} \]
Using the method of equivalent sinusoids (MES), we replace the nonsinusoidal voltage and load-current curves by equivalent sinusoidal ones in such a way that the rms values of the curves being replaced and of the sinusoids replacing them are equal. In making the replacement it is necessary to satisfy the condition that \(\cos\varphi_\mathrm{e}\) of the equivalent sinusoids be equal to the power factor \(\alpha_{\mathrm{n}}\) [2], that is,
\[ \cos \varphi_3=\alpha_{\mathrm{n}} . \tag{4} \]
The connection of capacitor banks leads to a change in the frequency characteristics of the network, which, as shown by the calculations performed in [3], causes the nonsinusoidality of the network voltage to increase by a factor of two or more.
In addition, the capacitance of the banks and the inductance of the network form parallel resonant circuits at the frequency of one of the higher harmonics; this leads not only to overloading of the capacitor banks (and sometimes to their failure), but also, owing to resonance, to an increase in the nonsinusoidality of the load current and, as a consequence, to an additional increase in the nonsinusoidality of the voltage on the supply buses.
All of the above ultimately reduces the value of the coefficient \(\gamma_1\) still further, which leads to additional loading of the consumer with reactive power.
As is known, the capacity of a capacitor bank for raising the power factor from the value \(\cos\varphi_1\) to the value \(\cos\varphi_2\) is determined from the expression
\[ Q_k=P_{\mathrm{cp}}(\operatorname{tg}\varphi_1-\operatorname{tg}\varphi_2). \tag{5} \]
But since the connection of the bank causes an additional increase in the nonsinusoidality of the voltage and of the load current, it will now be necessary to raise the value of the power factor not from \(\cos\varphi_1\), but from a power-factor value equal to
\[ \cos \varphi'_3=\gamma'_1\cdot \cos \varphi_1, \]
where \(\gamma'_1<\gamma_1\) owing to the additional distortion of the waveform of the voltage and of the current loading the capacitor banks.
Consequently, \(\cos\varphi'_3<\cos\varphi_3<\cos\varphi_1\), and
\[ \varphi'_3>\varphi_3>\varphi_1 . \]
Thus, in order to obtain a power-factor value equal to \(\cos\varphi_2\), a greater capacity of cosine-capacitor banks will be required than the capacity calculated by formula (5). This new value of the reactive power of the banks will be determined as
\[ Q'_k=P_{\mathrm{cp}}(\operatorname{tg}\varphi'_3-\operatorname{tg}\varphi_2). \tag{6} \]
Thus, owing to the nonsinusoidality of the network voltage and the additional influence of the bank capacitance on the waveform of the load current, in order to obtain the specified value \(\cos\varphi_2\), an additional bank capacity will be required, equal to
\[ \Delta Q_k=Q'_k-Q_k=P_{\mathrm{cp}}(\operatorname{tg}\varphi'_3-\operatorname{tg}\varphi_1). \tag{7} \]
The ratio
\[ \frac{\Delta Q_k}{Q_k} = \frac{\operatorname{tg}\varphi'_3-\operatorname{tg}\varphi_1}{\operatorname{tg}\varphi_1-\operatorname{tg}\varphi_2} = \frac{\operatorname{tg}[\arccos(\gamma'_1\cdot\cos\varphi_1)]-\operatorname{tg}\varphi_1}{\operatorname{tg}\varphi_1-\operatorname{tg}\varphi_2} \tag{8} \]
characterizes the reduction in the compensating capability of cosine-capacitor banks and makes it possible to construct the dependence of the change
the compensating effect of banks under the influence of higher harmonics on the coefficient \(\gamma'\), which, in turn, characterizes the level of nonsinusoidality of voltage and current. Measurements of the spectral composition of currents and voltages in the power-supply system of the West Siberian Metallurgical Plant, carried out by the author of this article, as well as the calculations performed, show that in a number of cases the compensating capability of cosine capacitor banks decreased by more than 50%. Cases were also observed in which capacitor banks were disconnected because of current overloads due to resonance phenomena.
Conclusions
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Switching cosine capacitor banks into a nonsinusoidal voltage leads to an additional increase in the nonsinusoidality of the voltage and current curves of the network. This adversely affects the operating conditions not only of the banks, but also of other electrical receivers.
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When capacitors are installed in networks with higher harmonics, the compensating effect of the banks proves to be lower than when operating in networks with sinusoidal currents and voltages. In solving questions of reactive-power compensation, one should either increase the installed capacity of the banks or take measures to reduce the harmonic level. This question is best resolved on the basis of a techno-economic calculation.
References
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I. V. Zhezhelenko, K. K. Shevtsov. Design of installations of cosine capacitor banks at substations with harmonic sources. Instructional Guidelines for the Design of Industrial Electrotechnical Installations. Tyazhpromelektroproekt. Moscow, 1970, No. 9.
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L. R. Neiman, P. L. Kalantarov. Theoretical Foundations of Electrical Engineering. Moscow, GEI, 1959.
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S. I. Gamazin, V. V. Cherepanov. Influence of static capacitor banks on voltage nonsinusoidality in the power-supply systems of industrial enterprises with harmonic sources. Proceedings of the conference “Compensation of Reactive Loads in the Electrical Networks of Industrial Enterprises.” Moscow, MDNTP named after Dzerzhinsky, 1972.