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Gyrator-Capacitor Modeling of a Continuously Variable Series Reactor in Different Operating Modes

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Gyrator-Capacitor Modeling of a Continuously Variable Series Reactor in Different Operating Modes Gyrator-Capacitor Modeling of A Continuously Variable Series Reactor in Different Operating Modes Mohammadali Hayerikhiyavi Aleksandar Dimitrovski Department of Electrical and Computer Engineering Department of Electrical and Computer Engineering University of Central Florida University of Central Florida Orlando, USA Orlando, USA [email protected] ADimitrovski@ ucf.edu Abstract — Continuously Variable Series Reactor (CVSR) can It can add additional impedance into the ac power circuit regulate its reactance on the ac side using the nonlinear either to damp oscillations or decrease the fault current when ferromagnetic core, shared by an ac and a dc winding for needed. In order to evaluate its functionality, it is necessary to applications like power flow control, oscillation damping, load study the CVSR under all system conditions and understand its balancing and others. In order to use a CVSR in the power grid, it impact on the power system, as well as, the ways the power is essential to know its operating characteristics. The Gyrator- system affects its operation. Capacitor approach has been applied to model the magnetic The basic concept of the CVSR is briefly reviewed in circuit coupled with the electrical circuit. In the paper, we Section II. The gyrator-capacitor (G-C) approach used for the investigate the CVSR behavior in terms of induced voltages across the dc winding, flux densities (B) through the core legs, and power modelling of the magnetic circuit is explained in Section III. exchange between the ac controlled and the dc control circuit. Section IV presents simulation results and analysis of the CVSR induced voltage, the power transferred into its dc Keywords— Gyrator-Capacitor, G-C model, Saturable reactor, windings and its equivalent inductance for different DC and AC Continuously Variable Series Reactor voltage sources. Conclusions are summarized in Section V. I. INTRODUCTION II. CVSR The existing power grids are already under considerable In the saturable-core reactor shown in Figure 1, an ac stress and strain, but as the demand for electricity continuously winding is wound on the middle leg and connected in series grows and the paradigms of operation are changing, they with the ac circuit that transfers power from the source to the become even more so. One of the most important challenges load. Typically, there is an air gap in the middle leg, necessary power system operators face is the prevention of blackouts due to achieve the desired reactance in normal conditions and not to to generation and demand volatility, transmission congestions, saturate the core with the ac flux. Two dc windings connected oscillations, etc. Most of these problems could be alleviated in series and controlled by the dc source are wound on the two with a comprehensive ac power flow control [1,2]. Typical outer legs. The ac reactance of the reactor is controlled by the devices for power flow control include phase shifting dc current [9,10]. This reactance reaches maximum value when transformers, switched shunt-capacitors/inductors, and various the core is not saturated and minimum value when it is fully types of flexible ac transmission systems (FACTS) controllers. saturated (at large enough dc current). Thus, the overall These devices either can provide only a coarse control or they reactance in the controlled ac circuit is changed by the dc. have high investment and operation costs. Saturable reactor During each half cycle of a period, the ac and dc fluxes will technology does not have these drawbacks and it provides add in one half of the core and subtract in the other half. It alternative option for power flow control with high reliability means that the reluctance of one leg will decrease and increase at low costs [3,4]. Continuously Variable Series Reactor in the second [11], when the core operates around the knee of (CVSR) is a series saturable reactor that has variable reactance Ф Ф the B-H curve. Therefore, and will be different, within the design limits. Continuous and smooth control of a large amount of power flow on the ac side can be easily and the induced voltages on the right ( ) and left dc achieved by changing the relatively small dc bias current. The winding ( ) will also be different and impose = dc source for the CVSR is low voltage/current power − in the dc circuit. It will be shown in the sequel electronics-based converter. On the other hand, FACTS that the frequency of this voltage is twice the frequency of the controllers use high voltage and current components because ac circuit. they are also part of the primary power circuit [5-7]. Besides power flow control, there are other applications that the CVSR is suitable for, such as oscillation damping and fault currents limiting [8]. The G-C circuit model is based on the configuration of the CVSR in Figure 1. The control side is connected to an ideal dc source, while the ac circuit is connected in series with a voltage source and a load. Here, gyrators represent the windings. The fringing flux around the air gap has been taken into the account within its permeance [15, 10]. First, the CVSR is simulated under normal condition and steady state for different values of the ac voltage sources and dc bias currents. The behavior of the CVSR in transient conditions and its impact on the system have been addressed in [16]. Fig. 1. Schematic of a CVSR [1]. III. GYRATOR -CAPACITOR MODEL P, Left leg P, Middle leg P, Right leg A typical approach to modelling magnetic circuits is to use Ndc P, air gap an electric circuit analogy [12, 13]. Equivalent circuits are Nac formed using resistors to represent reluctances and voltage Load sources to represent magnetomotive forces (MMFs). However, magnetic circuits store energy and, strictly speaking, are not AC properly modelled by resistors which only dissipate energy. The G-C model shown in Figure 2 [14] keeps the power equivalence. DC Source gyrator Fig. 3. Gyrator-Capacitor model in MATLAB /SIMULINK® I Ic A. AC voltage source at 1.2 kV Vc/n V/n In this scenario, three different dc bias currents are applied V Vc C(P) in the control circuit: 0 A, 200 mA (the critical current, as explained below), and 10A. Figures 4 and 5 represent the flux densities and terminal voltages for 0 A dc bias, respectively. Since the flux densities Fig. 2. Gyrator-Capacitor model of a simple magnetic circuit. are purely sinusoidal, the terminal voltage will also be purely sinusoidal. The flux densities through the outer legs are equal In the G-C model, the analogy between MMF and voltage is at all times. The induced voltages in the two dc windings cancel kept, but electrical current is analogous to the rate-of-change of Ф each other and the equivalent voltage across both is zero. the magnetic flux , as described by (1) and (2): (1) Table I – CVSR PARAMETERS Parameter Description Value Ф (2) ) mean length of the middle leg 45.72 cm * ) mean length of the outer legs 86.36 cm These equations lead to equivalent representation of +, magnetic permeances (inverse reluctances) with capacitances. - height of the air gap 0.2014 cm 2 Hence, the nonlinear permeances of the magnetic circuit paths . cross-section area of the core 0.0103 m in Figure 1, are represented as nonlinear capacitances. The / number of turns in the dc winding 225 permeances can be approximately calculated using the / number of turns in the ac winding 150 magnetic material properties and the geometric parameters of voltage source 1.2-3.8 kV the magnetic circuit with (3): 0 load resistance 100 Ω = ! (3) 1 load inductance 130 mH " '( 23 power factor 0.9 #$= 4π×10 – magnetic permeability of free air 4 saturation point 1.34 T # – relative magnetic permeability of the core A – cross-sectional area l – mean length of the path IV. SIMULATIONS AND RESULTS The G-C model was built in MATLAB/SIMULINK ®. All the simulations were conducted using the one-line diagram shown in Figure 3, with the device and load parameters as listed in Table I. The assumed ferromagnetic core material is M36. Right Sum Left Middle 4 1 3 2 0.5 1 0 0 Current(A) -1 -0.5 -2 Flux density (Tesla) density Flux -1 -3 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Time(Second) 0 0.02 0.04 0.06 0.08 0.1 Fig. 8. Current in the ac winding ( Idc = 0 A) Time (Second) Fig. 4. Flux densities through all the legs ( Idc = 0 A) 300 With zero dc bias, the power transferred into the dc windings is 200 also zero. The imposed induced voltage in the dc circuit, Eemf , 100 is proportional to the difference between the flux rates of the 0 outer legs as in (5): Ф Ф -100 Voltage(Volt) ; = / .( − ) (5) * -200 As noted above, because the ac flux is divided equally between -300 0 0.02 0.04 0.06 0.08 0.1 the outer legs, the induced voltage across the dc winding is zero. Time(Second) Figure 9 shows the flux densities for a dc bias of 200 mA. At Fig. 5. Induced voltage across the ac winding ( Idc = 0 A) this critical current, at all times, one of the outer legs goes into Depending on the region of operation (no saturation, partial saturation and the other one is unsaturated. Hence, the induced saturation, or complete saturation), the equivalent inductance of voltage across the ac winding and ac current will change and the CVSR will change. The inductance can be expressed by (4): have some distortion as can be seen on Figures 10 and 11, 58 respectively.
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