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PEAK INRUSH CURRENTS FOR MULTIPLE-STEP CAPACITOR BANKS IN AUTOMATIC POWER FACTOR CORRECTION EMIL CAZACU*, IOSIF VASILE NEMOIANU Key words: Inrush current, Overvoltage, Switching phenomenon, Low voltage power factor correction (LV-PFC). The paper focuses on an accurate predetermination of the peak inrush current that occurs at switching the multiple step capacitor banks in automatic low voltage power factor correction systems (LV-PFC). The analytical and numerical approach rely on the exact solutions of the time second order differential equations which model the entire dynamic process. The study is also valid in low level harmonic polluted environments and for both mechanic and static commutation. Some power quality aspects are also treated along with the estimation of electric stress exerted on electrical equipment. A case study allows the comparison of the obtained results with those reported in previous works and International Electrotechnical Commission (IEC) standards. 1. INTRODUCTION The constant demand of rational use of electric energy requires its economic generation, transmission, distribution and usage with minimum losses. That implies a full control over the factors that may determine these potential losses. The lagging reactive power flow, which can occur in both industrial and public power grids, is one of the main causes of energy losses [1–4]. In order to overcome this problem, power factor correction systems (PFC) are used, destined to compensate the generated lagging reactive power by supplying leading reactive power at specific nodes in the network. The required leading reactive power is usually provided by capacitors connected in parallel and closer to the inductive load. Very often the grid conditions change due to load variation and thus the required leading reactive power should be accordingly adapted. That is commonly done by using an automatic power factor correction system (automatic PFC). This installations, thanks to a monitoring device and a power factor controller, allow the automatic switching of multiple capacitor banks and manage to follow the absorbed reactive power variations and finally to keep a constant power factor at the load [5]. The paper mainly focuses on a better evaluation of the inrush current absorbed by the * “Politehnica” University of Bucharest, 313 Splaiul Independenţei, 060042 Bucharest, Romania, [email protected] Rev. Roum. Sci. Techn. – Électrotechn. et Énerg., 57, 4, p. 341–350, Bucarest, 2012 342 Emil Cazacu, Iosif Vasile Nemoianu 2 capacitor banks during various commutation processes. An accurate value of this parameter is extremely important in designing such automatic PFC systems, because the entire selection of their main components such contactors, protective device etc., rely on the capacitors banks inrush current amplitude [6, 7]. 2. QUALITATIVE AND QUANITATIVE COMUTATION PROCESS ANALYSIS IN AUTOMATIC PFC Generally, an automatic PFC system contains, as Fig. 1 shows, the following main parts [1]: current and voltage sensors providing signals proportional to the absorbed electric energy parameters, a control and command block, an electric power board comprising switching and protection devices and the capacitor banks. The control and command block compares the measured power factor with the prescribed one and operates the connection and disconnection of the capacitor banks with the necessary reactive power (power factor regulator). Fig. 1 – Architecture of an automatic power factor correction system (PFC) with three steps. The modern PFC controllers, which are microprocessor-based, continuously analyze the signal from the current sensor and commands the commutation of contactors which add or remove capacitor bank steps in order to compensate the actual reactive power of the total load. They also ensure the economic usage of 3 Peak inrush currents for multiple-step capacitor banks 343 capacitor stages, which means a minimized number of switching operations and thus an optimized life cycle of the capacitor bank. The number of steps depends on the number of connected loads. Consequently, the smaller the inductive loads, the higher the number of steps should be foreseen. It is also useful to mention that according to IEC 60831-1 standard [8] clause 22 each capacitor bank is provided with discharge resistors that must be able to “discharge each unit in 3 min. to 75 V or less, from an initial peak voltage of 2 times rated voltage Un”. The main issues of the automatic capacitor banks switching process are the high transient overcurrent required by the banks (also called inrush current) and the transient overvoltage through the grid [5, 9]. The peak inrush current, overvoltage amplitude and their frequencies are basically functions of the upstream network parameters and the characteristic of the capacitor banks. Unlike the case of a single bank (fixed correction), where the peak inrush current can reach 30 times the nominal current of the capacitor bank, the inrush current amplitude in the case of an automatic correction, depends on the power of the steps already connected, and can reach 100 times the nominal current of the step to be energized [6, 10]. Figure 2 qualitatively depicts the inrush current and voltage variation for the last switched capacitor bank (the third) in a multiple step automatic PFC system. The numerical values of the used grid parameters are detailed in Chapter 3. As we already mentioned, the inrush current (pulse) of an automatic PFC depends on numerous parameters, such as: short-circuit value of power supply transformer, remaining capacitor voltage due to fast switching banks, ohmic resistance of capacitor itself and distribution switch gears, connection cables or conductor resistance and inductance. Although is rather difficult to take into account all the network and Fig. 2 – Inrush current and voltage variation for the last switched capacitor bank in a multiple step automatic PFC whose parameters are presented in Chapter 3. 344 Emil Cazacu, Iosif Vasile Nemoianu 4 capacitor bank electric characteristics, the Standards IEC 62271-100 [11] and IEC 60831-1[8] suggest the relations for determining the inrush current amplitude in a single and multiple (different or identical) reactive power steps switching cases: ip_single_step = U n ip_multiple_steps = U n 2 S sc Q 2 C 2C ; I nc = C ; ≈ Un = I nc 3 L0 + L 3 L0 QC 3U n 2 ( C1 + C2 + " Cn −1 ) ⋅ Cn 3 ( C1 + C2 + " Cn ) L + n ip_multiple_equal_steps = U n 1 1 ; (1) 1 1 1 + +" L1 L2 Ln −1 2C n , n > 1, n +1 3 L where L0 is the network inductance. Also, for the bank step of index k, relations (1) use the following notations: Lk – the connection inductance; Qk – the reactive power; Ck – the capacity of the respective bank step; Inc – the rated bank current; Un – the network rated voltage; Ssc – the short-circuit power at the point where the capacitor are installed. Switching capacitor banks in automatic PFC is not only related to high inrush currents, but also to high voltage transients, causing degradation of power quality by generation of disturbance waves which propagates throughout the network [5]. In order to evaluate the overvoltage level, one must also consider the grid parameter alone with the reactive power QC supplied by the capacitor bank. Often, the automatic PFC system is connected in the proximity of a distribution transformer of rated apparent power Sn, percentage short-circuits voltage usc[%] and a per-phase impedance ZT. Thus, the voltage variation is given by [1, 2]: ∆U RP + XQ XQ usc [%]U n2Q usc [%]QC = ≅ 2 = = ; Q ≅ QC ; U U n2 Un S nU n2 Sn U 3Z T I n Z T S n usc [%]U n2 ZT ≅ X ; usc [%] = sc = = ⇒ Z ≅ X = . T Un Un U n2 Sn (2) Relations (2) use the fact that in a power transformer the winding resistance is negligible compared with the leakage reactance and the reactive power values ‘seen’ by the transformer, denoted by Q. In practice, the last may be considered as the reactive power of the capacitor bank QC during the switching phenomenon. These expressions are only to be used for a single step capacitor bank. Expressions 5 Peak inrush currents for multiple-step capacitor banks 345 (1) and even (2) are mostly experimentally obtained and cover almost all the practical cases, for both single and multiple steps PFC systems. In the case of automatic multiple steps capacitor banks, where the maximal values of the inrush current could reach important values, an accurate determination of transient phenomena becomes very important for the PFC systems optimal design. Beside, as we already underlined in the previous sections, the selection of all the commutation and protection devices are based on this values. 3. MODELATING THE COMUTATION PROCESS IN AUTOMATIC PFC. NUMERICAL RESULTS For a better estimation of the peak inrush current in multiple step capacitor banks of an automatic PFC systems, we started from the schematic circuit depicted in Fig. 1 and modeled each component of the installation with a specific lumped circuit parameter [13]. The important elements that are involved in the transient process are the capacities of the capacitor banks, inductance and resistance of the connection cables between the stages and, for the first bank only, the inductance of the nearby power transformer. Figure 3 presents the per-phase equivalent circuit diagram of a three stages capacitor banks automatic PFC system. One can easily indentify the above-mentioned circuit parameters and then use them to establish the corresponding time differential transient equations. The worst case regarding the capacitor banks peak inrush current value occurs when the last capacitor bank switches on, while the others are already Fig. 3 – Main component model of the three-stage automatic PFC system. energized by lying at the maximum network voltage. That corresponds in our model to the situation when the contactors K1 and K2 are closed and K3 introduces the last bank into the circuit. The entire process could be mathematically expressed by using nodes and loops equations for the circuits. That leads to the following differential system of equations: 346 Emil Cazacu, Iosif Vasile Nemoianu d i1 (t ) 1 = [u (t ) − R1i1 (t ) − u C1 (t )] L1 dt d i 2 (t ) 1 [u C1 (t ) − u C 2 (t ) − R2 i 2 (t )] = L2 dt d i3 (t ) 1 [u C 2 (t ) − u C 3 (t ) − R3 i3 (t )] = d t L3 d u C1 (t ) 1 [i1 (t ) − i2 (t )] = C1 dt d u C 2 (t ) 1 [i2 (t ) − i3 (t )] = C2 dt d u C (t ) 1 3 i3 (t ) = d t C3 6 (3) u (t ) = U 2 sin (ωt + ϕ). Similar equations could be used for a larger number of stages, with no restriction imposed on the reactive power energy level of the capacitor bank. The main advantage of this approach is that, by solving the differential equations system, an accurate time-dependent variation for both the inrush current and all the capacitor banks overvoltages can be rapidly obtained just after the commutation process. Numerical solving of the differential equations describing the entire process can easily be assembled in a very flexible software package allowing the user to simulate a great number of parameter variations, finally leading to a minimal peak inrush current value. In order to quantitatively illustrate the accuracy of the proposed method for evaluating the inrush peak current and overvoltage variation, we consider a three stages automatic PFC system connected to the secondary windings of a 400 kVA power transformer with percentage short-circuit voltage of usc = 4% and 400 V rated voltage. Assume that the capacitor banks have different reactive power values, namely Q1 = 5 kvar, Q2 = 10 kvar, Q3 = 15 kvar, and the connection conductors are made of copper with the cross-sectional areas of 2.5, 4 and 6 mm2, but all with the same length of 1.5 m. The connection between the power transformer and the PFC system is realized by using a 5 m length copper busbar having a 10 mm2 per phase cross-sectional area. All these numerical data lead to the following value of the lumped circuit parameters: R1 = 9.4 mΩ, R2 = 7 mΩ, R3 = 9.4 mΩ, L1 = 55 µH, L2 = L3 = 5 µH, C1 = 100.28 µF, C2 = 200.57 µF and C3 = 300.86 µF. Thus, the numerical solution of the differential equations system gives the current and voltage variations over a period of only 2 ms, just after the commutation process, for all the circuit model capacitors. These variation curves are plotted in Fig. 4 and Fig. 5, respectively. The variation of the same parameters, but only for the last capacitor bank (the third), is showed in Fig. 2 during an extended time period of 20 ms. The initial values of the capacitor voltages for the both on-duty capacitor banks (already connected) at the very switching moment were considered maximal, namely the amplitude of network voltage. That forces the computation to analyze even a worse model case. Studying the inrush current variation absorbed by the last capacitor bank, one can notice that its peak values reach more than 1700 A, at a 7 Peak inrush currents for multiple-step capacitor banks 347 frequency of more than 5.2 kHz, while the bank rated current is of only 37 A at 50 Hz. The highest overvoltage exerted on the first capacitor bank terminals does not exceed more than two times its rated value. Fig. 4 – Inrush current variation through capacitors banks immediately after commutation. Fig. 5 – Overvoltage variation on the capacitors banks immediately after commutation. 348 Emil Cazacu, Iosif Vasile Nemoianu 8 The initial values of the capacitor voltages for the both on-duty capacitor banks (already connected) at the very switching moment were considered maximal, namely the amplitude of network voltage. That forces the computation to analyze even a worse model case. Studying the inrush current variation absorbed by the last capacitor bank, one can notice that its peak values reach more than 1700 A, at a frequency of more than 5.2 kHz, while the bank rated current is of only 37 A at 50 Hz. The highest overvoltage exerted on the first capacitor bank terminals does not exceed more than two times its rated value. Applying the second relation from (1) to this particular case, one will get a value of only 1288 A for the peak inrush current, which does not cover the worst possible situation and may be misleading especially for the optimal design of the switching and protective devices. Moreover, the formulas presented in (1) did not consider all the network parameters and the initial voltage values on the already connected capacitor banks before commutation takes place. It indicates only an absolute value that could be reached after a half-period time of network supply while the transient phenomena occur much faster. Such extreme high values of the peak inrush currents determine important electric stresses on the capacitor banks and reduce significantly their lifetime span. It could also cause welding or fast wearing out of the main contacts of contactors and also have negative effects on power quality network by generating transit overvoltage into the grid. Consequently, the necessity of damping these currents arises. One of the simplest and costless methods of achieving this goal is to connect inductances in the circuit made by winding the cables designed to be connected to the capacitor banks, onto a cylinder shape. Only a few turns determines an additional inductance of 5–10 µH that could reduce with at least 30–50 % the inrush current amplitude. On the other hand, longer or additional cables could cause higher Joule losses within the PFC system. Another solution is offered by the usage of special design contactors equipped with pre-switching auxiliary contacts that close before the main contacts do and insert, for a very short time (2–10 ms), damping resistors into circuit [14]. Thus pre-load the capacitors and avoid the current peak values effects. This procedure decreases with more the 40 % the inrush current amplitude having also positive influence on the capacitor bank life expectancy and increases power quality by limiting transients and voltage sags to propagate through the network. When also a harmonic polluted environment is involved, and voltage and current total harmonic distortion (THD) are less than 10 % and 5 %, respectively, the common solution of damping the inrush current and also suppressing the 5th harmonic and above is to use anti-resonance inductances in the circuit. These so called detuned-reactors [15, 16] are iron base inductances of several mH connected in series with the capacitor banks and sized so that a resonance frequency is below 9 Peak inrush currents for multiple-step capacitor banks 349 the lowest frequency of the harmonic voltage in the network. Thus, a “detuned automatic PFC” manage to significantly reduce the inrush current amplitude (with more than 60 %) and also to filter the network from superior harmonics. We have to notice that other methods and procedures were also developed in order to suppress the capacitor bank switch-on transients’ consequences [17, 18]. Even with the above mention damping solution the importance of knowing an accurate value of the peak inrush current does not diminish. Only a precise value all the components of the PFC systems and damping devices could be designed in an optimal way anticipating and avoiding important electrodynamic stresses or poor power quality effects. 4. CONCLUSIONS The problem of determining an accurate evaluation for the peak inrush current and overvoltages (at switching) that occur in multiple stages LV automatic PFC systems was treated. The circuit model of the main PFC installation components allowed, by numerically solving the associated differential equations system, to extract the exact solutions for both the peak inrush currents and the overvoltages of all the involved capacitor banks during the switching phenomena. Moreover, the time-variations just after the commutation process of the capacitor banks’ electric parameters are also investigated. The proposed approach could be easily adapted to any multiple step PFC system, considering also the network parameters such as the nearby power transformers characteristics or connecting cables inductances and resistances. For a particular case, where only three steps were used, the numerical value for the peak inrush current of the last switched capacitor was compared with the one obtained according a known formula provided by IEC standards. The difference between the two values is quite considerable and appears as a consequence of different parameters considered by the analytical approach and also by the various initial value assumed by the already connected (on duty) capacitors. Inserting all the differential equations into a general purpose software environment, could be a very useful tool in designing different components of the PFC systems. 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