Winding current calculation and harmonic analysis of controllable reactor of transformer type

Transcription

1 Australian Journal of Electrical and Electronics Engineering ISSN: X (Print) X (Online) Journal homepage: Winding current calculation and harmonic analysis of controllable reactor of transformer type Yi-bin Liu, Ming-xing Tian & Jian-ning Yin To cite this article: Yi-bin Liu, Ming-xing Tian & Jian-ning Yin (216) Winding current calculation and harmonic analysis of controllable reactor of transformer type, Australian Journal of Electrical and Electronics Engineering, 13:1, 14-23, DOI: 1.18/ X To link to this article: Published online: 15 Dec 215. Submit your article to this journal Article views: 166 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at Download by: [ ] Date: 5 December 217, At: 18:34

2 Australian Journal of Electrical and Electronics Engineering, 216 VOL. 13, NO. 1, Winding current calculation and harmonic analysis of controllable reactor of transformer type Yi-bin Liu, Ming-xing Tian and Jian-ning Yin School of Automation and Electrical Engineering, Lanzhou Jiaotong University, Lanzhou, China ABSTRACT The harmonic currents in each of the windings of controllable reactor of transformer type (CRT) have a close relationship with each of its thyristors trigger angles. Based on the definition of the trigger angle vector, switching sequence and multi-winding regulation mode, the matrix expression for the windings instantaneous current in the multi-winding regulation mode is obtained by means of subsection linearization; and then a simple method that can directly generate each winding s current waveform is presented. Using Fourier series, the functional relationship in matrix form between Fourier coefficients of all the windings harmonic currents and the trigger angle vector is deduced, and then the formulae for calculating the root mean square (RMS) of each winding s harmonic current is acquired. The examples show that the proposed formulae for calculating the winding current of CRT and its harmonics in multi- winding regulation mode are correct, which can be used in more general cases, and the formula deduced in the single winding regulation mode described in other relevant literatures is merely a special case of this paper. This work expands the analysis methods of winding currents of CRT, and lays a theoretical foundation for the harmonic optimization and winding capacity configuration of CRT. 1. Introduction With the rapid development of global economy, the construction scale of extra high voltage (EHV) long transmission line continues to expand, and the reactive power balance and voltage control become more and more important (Chen, Lee, and Chen 1999; Chuang et al. 21; Froehlich et al. 1997; Rahman and Khan 27). Therefore, people are paying more attention to the manufacture of a controllable reactor that has a wide regulation range, smooth regulation process, fast response and low harmonic content (Alves, Pilotto, and Watanabe 28; Chen et al. 212; Jin, Goos, and Lopes 1994; Tian et al. 213). Controllable reactor of transformer type (CRT) is a reactive compensation equipment that can apply to EHV transmission line appropriately, which plays an important role in stabilizing the bus voltage (Aleksandrov, Al bertinskii, and Shkuropat 1995). In Tian (25, 27) and Tian and Li 24), the basic theory of CRT is studied, but it is allowed only one control winding to be regulated during a cycle. Regulating just one control winding, the control strategy is simple, however, the harmonic current injected into the power grid is uncontrollable, and which is quite obvious in some conditions. To overcome this disadvantage, a CRT is provided in Zhang et al. (29), ARTICLE HISTORY Received 27 June 214 Accepted 17 February 215 KEYWORDS Controllable reactor of transformer type; trigger angle vector; multiwinding regulation mode; harmonic optimization; winding capacity configuration the secondary windings of which are connected with pulse width modulation voltage source inverters, this reactor generates little harmonic. However, it is expensive and unpractical because of the high voltage and high power in EHV transmission system. Furthermore, the electromagnetic coupling among all the windings of CRT is usually quite strong, and thus, the currents of the control windings that have been short-circuited will continuously decrease while the rest of the control windings are put into operation one by one (Tian and Zhao 25), and because the rated current of each control winding depends on its maximum current root mean square (RMS) during operation, this will result in a serious waste to the capacity of control winding. In fact, all the control windings can be regulated during a cycle, and there are many trigger angle combinations which can satisfy a given output power, but the harmonic content differs greatly among different combinations. Furthermore, if all the control windings are allowed to be regulated, the phenomenon of current decrease caused by electromagnetic coupling mentioned above can be eliminated with a reasonable control of each thyristor. Therefore, it is necessary to calculate each winding s instantaneous current of CRT and explore the functional relationship between the RMS of all the CONTACT Yi-bin Liu yanerwuming@126.com 215 Engineers Australia

3 Australian Journal of Electrical and Electronics Engineering u 1 i 1 W 1 windings harmonic currents and all the thyristors trigger angles in the case of allowing all the control windings to be regulated. For these purposes, this paper firstly defines the concepts of trigger angle vector, switching sequence and multi-winding regulation mode; then works out the piecewise matrix expression for all the windings instantaneous currents and presents a method that will directly generate each winding s current waveform; and then deduces the formulae for calculating all the windings fundamental and each harmonic current RMS. Finally, an example is designed to show how to use the formulae deduced in this paper. Comparing with the results obtained from the existing simulation model provided in the relevant literatures, it is proved that the analysis method of this paper is correct. 2. Regulation mode i 2 W 2 i 3 W 3 i n W n Figure 1. Basic circuit diagram of CRT.... The basic circuit diagram of a CRT is illustrated in Figure 1,, W 1 is the primary winding, called working winding, and W 2, W 3,, W n are secondary windings, they are often called control windings. u 1 is the instantaneous voltage across the terminals of W 1, and i 1, i 2, i 3,, i n are the instantaneous currents of W 1, W 2, W 3,, W n, respectively. X 2, X 3,, X n are current-limiting reactors connected with W 2, W 3,, W n, respectively, and T 2, T 3,, T n are thyristor switches that each of them consists of two thyristors in parallel but in opposite directions. Neglecting the resistance, saturation, and hysteresis of iron core, assume that u 1 2U 1 cos(ωt), and the starting point of trigger angle in each positive or negative half cycle is the time when the voltage of the grid reached maximum or minimum, and the trigger angle of T j (1 j n) is expressed as θ j, thus θ 1 (there is no thyristor connected with W 1, T 1 is hypothetical and in conducting state all the time) and θ 2 ~ θ n can be any value in the range [, π 2]. Arranging θ 1 ~ θ n according to the order of their subscripts, a vector θ n (θ 1, θ 2,, θ n ) is X 2 X 3 X n T 2 T 3 T n obtained, called actual trigger angle vector in this paper, θ 1 is the trigger angle of W 1, θ 2 is the trigger angle of W 2,, θ n is the trigger angle of W n. Arranging all the trigger angles in θ n from small to large, a new vector θ n (θ 1,χ1,θ 2,χ2,, θ k,χk,, θ h,χh,, θ n,χn ) is obtained, called ordered trigger angle vector in this paper, and the detailed explanation for θ n is as follows: The subscript of θ k,χk (1 k h n) is divided into two parts by,,, k is called the left subscript, and χ k is called the right subscript. The left subscript k means that the thyristor corresponding to θ k,χk is the kth one to be triggered, and the right subscript χ k is the actual subscript of the winding corresponding to the kth triggered thyristor. In other words, the thyristor connected with W χk is the kth one to be triggered, whose trigger angle equals θ k,χk. (2) When k 1, the following can be obtained: χ 1 1 and θ 1,χ1. This means that the thyristor connected with W 1 is always the first one to be triggered, whose trigger angle always equals. Although no thyristors are connected with W 1 in reality, we suppose that W 1 is connected with two hypothetical thyristors like control windings, the trigger angle of which always equals. This method doesn t change the basic working principle of CRT, but it is quite convenient for the unified description of the following formulae. (3) The thyristors corresponding to the trigger angles behind θ h,χh in θ n are turned off during a cycle, namely, θ h+1,χh+1 θ h+2,χh+2 θ n,χn π 2. That is to say, h of all the n windings are put into operation, and the other n h are always open-circuited during a cycle. Listing all the right subscripts of θ n one by one, a sequence is obtained, written as X n (χ 1 χ 2 χ n ). Considering all the switching sequences of the thyristors, there will be (n 1)! possible results for X n (χ 1 1), and among them, (1 2 3 n) is the standard sequence, written as X n. If X n X n, we say that θ n conforms to the standard switching sequence, or the current switching sequence is the standard switching sequence, while if X n X n, we will say that θ n doesn t conform to the standard switching sequence, or the current switching sequence is the non-standard switching sequence. Since CRT can automatically regulate the output power in case the loads change, the line voltage can remain constant with the changes of transmitted power. Therefore, the output power of CRT can be directly measured by the RMS of the working winding s fundamental current. When all the trigger angles equal π 2, CRT operates in the condition of zero- load, and when

4 16 Y-B. Liu et al. all the trigger angles equal, CRT operates in the condition of full-load. Setting appropriate values for all the elements of θ n, CRT can be controlled to output any power between no-load and full-load. Furthermore, the number of vectors that can satisfy a given output power is much more than one. For convenience, the variation trends of all the trigger angles while the output power of CRT changes continuously from no-load to full-load can be called regulation mode. Thus, it can be easily seen that there are a lot of regulation modes for CRT, but the operation performance differs greatly among different modes. For any given output power, if only one of all the trigger angles of θ n can be set to a value between and π 2, while others are just allowed to be or π 2, it will be called single winding regulation mode, and if two or more trigger angles are between and π 2 during operation, it will be called multi-winding regulation mode. 3. Instantaneous current The variables shown below are explained as follows, and if there is no special explanation, the meaning of each variable appearing in the following discussion of this paper stays the same as that explained here. L k (1 k h n) is the self-inductance of W χk (W χk is the winding corresponding to the element θk,χk in θ n, may not be W k ), M kq (1 q h n) is the mutual inductance between W χk and W χq, L xk is the inductance of current-limiting reactor connected with W χk (no current-limiting reactor is connected with W 1 ; hence, L x1 ), i k (1 k h n) is the instantaneous current of W χk. Since all the windings current waveforms of CRT are symmetric about ωt π 2 in [, π], and if the trigger pulses in [, π] and [π,2π] are symmetrical, there will be an easily provable fact that the waveform of the instantaneous current of each winding is half-wave symmetric. Thus, all the windings current waveforms of CRT are symmetric among four quarter periods. Therefore, we just have to work out the expression for the instantaneous current in [, π 2]. According to the explanations of θ n in Section 1, h of n windings (including W 1 ) are put into operation during a cycle, the thyristors of which will conduct according to the corresponding trigger angles sequentially. Therefore, the circuit topology of CRT will change when its thyristors conduct one by one, but the circuit topology of CRT is a certain linear circuit during the period formed by any two adjacent trigger times. According to the above analysis, piecewise linearization should be adopted, we can divide [, π 2] into h subintervals according to the switching sequence of θ n, and then solve the corresponding circuit equations of the h subintervals one by one, thus, the expressions of all the windings instantaneous currents in [, π 2] can be deduced. According to the definition of θ n, in the rth (1 r h n) subinterval (θ r,χr ], r of h windings (W χ1 ~ W χr ) are conductive, while the other h r haven t conducted yet, thus, the circuit equations for all the conductive windings can be rewritten as L r (pi r )u r p d dt; u r [ u 1 ] T ] Ț ; ir [i 1r i 2r i kr i rr i kr (1 k r)represents the value of i k in the interval (θ r,χr ]; L r is the submatrix of, which can be obtained by extracting the former r lines and r columns from,, is a nth-order square matrix, called inductance matrix in this paper, which can be derived as follows: M L 1 M 12 M 1k M 1r M 1h M 1n M 21 L 2 M 2k M 2r M 2h M 2n M k1 M k2 L k M kr M kh M kn M r1 M r2 M rk L r M rh M rn M h1 M h2 M hk M hr L h M hn M n1 M n2 M nk M nr M nh, L x diag(l x1, L x2,, L xk,, L xr,, L xh,, L xn ), and the meaning of the subscript for each element in both M and L x stays the same with what was mentioned at the beginning of this section. Although all the windings self and mutual inductances and current-limiting reactors inductances are determined by the structure of CRT, their actual position in entirely depends on the switching sequence of the current cycle according to (2). Hence, it is the structure of CRT and the switching sequence in the current cycle that determine the final result of. from and (2), (3) can be obtained L 1 r represents the first column of L 1 r ), assume that (L 1 r is the inverse matrix of L r L 1 r [ M + L x pi r L 1 r u 1 1 l 1,r 1 l 2,r 1 l k,r 1 l r,r ] T. (2) (3) To solve each of the instantaneous currents in (3), the initial value of the corresponding current in the current period should be given. Since the initial value in the current subinterval is the final value in the previous subinterval, and the initial value of i k in its first conductive subinterval (θ k,χk, θ k+1,χk+1 ] is i kk ωtθ k,χk,

5 Australian Journal of Electrical and Electronics Engineering 17 any winding s instantaneous current can be worked out by means of recursive algorithm, as long as its thyristor is conductive in (θ r,χr )(1 r h n). Through proper simplification, the corresponding formulae can be written in the following matrix form: i r (4) ω C r H r S r H r (, E) (E, ), in which, E is an rth-order identity matrix, is a r 1 zero vector; [ ] Ț S r sin(θ 1,χ1 ) sin(θ 2,χ2 ) sin(θ k,χk ) sin(θ r,χr ) sin(ωt) C r is the submatrix of C n, which can be obtained by extracting the former r lines and r columns from C n, and C n is a nth-order square matrix as shown below: C n 1 l 1,1 1 l 1,2 1 l 1,k 1 l 1,r 1 l 1,h 1 l 1,n 1 l 2,2 1 l 2,k 1 l 2,r 1 l 2,h 1 l 2,n 1 l k,k 1 l k,r 1 l k,h 1 l k,n 1 l r,r 1 l r,h 1 l r,n 1 l h,h 1 l h,n 1 l n,n C n is an upper triangular matrix, and the nonzero part in each of its columns from the first column to the last column is L 1 1, L 1 2 k r, h, L 1 n, respectively, and L 1 1, L 1 2 k r,, L 1 h n are the first columns of L 1 1, L 1 2, k, L 1 r h n, respectively, and L 1 1, L 1 2, k, L 1 r h n are the inverse matrices of L 1, L 2,, L k,, L r,, L h,,, respectively, and L 1, L 2,, L k,, L r,, L h,, are the sub-matrices of, which can be obtained by extracting the former 1, 2,, k,, r,, h,, n lines and columns from respectively. In the above analyses, the M and L x corresponding to the standard switching sequence can be obtained easily, and according to (2), the corresponding is easy to derive, called standard inductance matrix and written as L n in this paper. Based on L n, the corresponding to the non-standard switching sequence (called non-standard inductance matrix) can be derived as follows: TL n T T (6) T is an nth-order square matrix, whose element in row i and column j (written as t ij ) is shown as follows: { 1, j X t ij n (i) (7), j X n (i) X n (i) is the ith element of X n. According to (6) and (7), is derived from L n by means of elementary transformation, and T is the corresponding transformation matrix, whose final result is determined by X n. For example, when X n X n, namely, the current switching sequence is the standard sequence, the following can be obtained: T E and L n. The expression for each winding s instantaneous current of CRT in (θ r,χr ](1 r h n) expressed in matrix form is shown as (4), whose simple matrix form is quite convenient for the subsequent deduction in this paper, however, which is not intuitive. In fact, after a further derivation a**bout (4), the expression for the instantaneous current of W χk (1 k r) in (θ r,χr ] can be written as follows: r i kr i k(1 r) δ k,j (8) δ k,j When θ 1,χ1 θ 2,χ2 θ k,χk θ j,χj (1 k j r h n) and θ j+1,χj+1 θ j+2,χj+2 θ n,χn π 2, can be obtained from (4). Obviously, the corresponding current waveform of W χk is not distorted in the above conditions, called the (j k + 1)th boundary waveform of W χk in this paper (when j k, the first boundary waveform of W χk is obtained). The δ k,j in (9) represents the difference between the (j k + 1)th and (j k)th boundary waveforms of W χk at θ j,χj, called as the (j k + 1)th difference of W χk in this paper (when j k, δ k,j equals the value of the first boundary waveform of W χk at θ k,χk ). According to (8), the waveform of W χk in (θ r,χr ] can be obtained by translating its (r k + 1)th boundary waveform, and the translational distance equals the sum from the first to the (r k + 1)th differences of jk i k(1 j) ωtθj,χ, j k j (9) i k(1 j) ωtθj,χ i k(1 j 1), k + 1 j r j ωtθj,χj i k(1 j) sin ωt ωl k,j (5)

6 18 Y-B. Liu et al. W χk. Hence, when θ n is given, we can firstly deduce all the boundary waveforms and differences of each winding from (9) and, and then the waveform of each winding in any subinterval of [, π 2] can be obtained by translating the corresponding boundary waveform directly. Tian (25) proposes three different kinds of regulation modes: step-single-branch mode, fixed-single-branchmode, and transfer-single-branch-mode. According to the concept of the regulation mode analysed above, these three regulation modes belong to the single winding regulation mode, namely, θ 2,χ2 θ 3,χ3 θ h 1,χ h 1, θ h,χ h π 2, θ h+1,χh+1 θ h+2,χh+2 θ n,χn π 2 in θ n. In this condition, [, π 2] is divided into only two subintervals ([, θ h,χh ] and, π 2]), thus, the expression for each winding s instantaneous current can be derived from(4). i h 1 i h ω ω Input Start θ n Arrange all the trigger angles in from small to large, gain θ n and end L 1 h 1 sin ωt, L 1 h sin ωt ωt θ h,χ h (11) ω (L 1 h L 1 ) sin θ h 1[1] h,χ h, θ h,χh <ωt π 2 [ ] Ț L 1 h 1 1 l 1,h 1 1 l 2,h 1 1 l k,h 1 1 l h 1,h 1 L 1 [ ] T h 1[1] 1 l 1,h 1 1 l 2,h 1 1 l k,h 1 1 l h 1,h 1, and the only thing to note here is that L 1 h 1[1] has one more element,, than L 1 h 1, especially, the expression for the working winding is as follows: and * According to (6) and (7), * Derive from L and X n X n get the sub-matrixes L1, L2,..., L, h from L, n and then deduce the inverse matrixes of them Generate C n Calculate each harmonic current RMS from (15)~(17) n θ n Figure 2. Process for calculating each winding s fundamental and harmonic current RMS. ( ωl 1,h i 1h 1 sin ωt, ωt θ ωl h,χh 1,h 1 i 1h sin ωt ωl 1,h sin θ h,χh sin θ ωl h,χh ), θ h,χh <ωt π 2 1,h 1 (12) 4. Harmonic analysis According to the above analyses, the waveform of i k (1 k h n) is symmetric among four quarter periods, which contain only fundamental and odd harmonics. Hence, i k can be expressed as a Fourier series like this: i k {b k2m 1 sin[(2m 1)ωt]} m1 b k2m 1 4 π 2 i π k (ωt) sin[(2m 1)ωt]d(ωt) (m 1, 2, 3, ) (13) (14) The expression for i k in any subinterval (θ r,χr ] on [, π 2] can been derived from (4), substitute i k into (14) and simplify, then b k2m 1 (m 1, 2, 3, ) is obtained, the final result can be written in the following matrix form: b n2m 1 (15) πω C n H n G n m [ ] T; b n2m 1 b 12m 1 b 22m 1 b k2m 1 b h2m 1 b n2m 1 C n is the same as (5); H n (, E) (E, ), in which E is an nth-order identity matrix, and is an n 1 zero vector; [ ] Ț G nm g m (θ 1,χ1 ) g m (θ 2,χ2 ) g m ) g m (θ n,χn ) g m (π 2) in which, g 1 (θ) 2θ + sin 2θ g m (θ) 1 sin 2(m 1)θ [ + 2m 1 m 1 (m 2, 3, 4, ) sin 2mθ m ] According to the explanation in Section 1, θ h+1,χh+1 θ h+2,χh+2 θ n,χn, thus, b h+12m 1 b h+22m 1 b n2m 1. This is because the control windings corresponding to θ h+1,χh+1, θ h+2,χh+2,, θ n,χn are always open-circuited in one cycle, the fundamental and each harmonic current of which equal in whole cycle. [ ] T, Assume that I n2m 1 I 12m 1 I 22m 1 I h2m 1 I n2m 1, I k2m 1 (1 k h n) represents the RMS of the (2m 1)th harmonic current of W pk, then (17) can be derived as follows: [ I n2m 1 b 12m 1 b 2 2m 1 b h2m 1 ] T b n2m 1 2 (17) Equations (15) (17) are deduced in accordance with the multi-winding regulation mode; therefore, they can reflect the quantitative relationship between each winding s harmonic current RMS and all the trigger angles in any switching sequence. The results of (15) depend } (16)

7 Australian Journal of Electrical and Electronics Engineering 19 on the C n and G nm, in which, each of the elements of G nm is a function of the corresponding trigger angle, and C n is deduced from that is determined by the structure of CRT and the current switching sequence mentioned above. Therefore, (15) gives the quantitative relationship not only between all the trigger angles but also each inductance (especially the inductance of each current-limiting reactor) of CRT and each winding s harmonic current RMS, which is significant for the optimization of the structure parameters of CRT and its regulation mode. According to the above analyses, the proposed method for calculating the fundamental and each harmonic current of CRT can be expressed as in Figure 2. Letting θ 2,χ2 θ 3,χ3 θ h 1,χ h 1, θ h,χh π 2, θ h+1,χh+1 θ h+2,χh+2 θ n,χn π 2 in θ n, the (2m 1) th harmonic current RMS of W χk in the single winding regulation mode can be derived from (15) to (17) I k2m 1 U g (18) πω l m (π 2)+( 1 )g k,h l k,h 1 l m ) k,h From (18), the (2m 1)th current harmonic ratio of working winding can be derived κ 2m 1 I 1 2m 1 g m ) (19) I 11 g 1 )+πl 1,h 1 (l 1,h l 1,h 1 ) In particular, for the step-single-branch mode (Tian 25; Tian and Li 24) ξ h is the so-called stepwise power increment coefficient mentioned in Tian and Li (24). Simplifying (2) with (16), we can find that (2) is the same as (13) derived in Tian and Li (24). Formula (21) clarifies the relationship between the stepwise power increment coefficient and the inductance of CRT, which is very important for the design of CRT. According to the above explanation, the formulae for calculating the instantaneous current of CRT and its harmonics in multi-winding regulation mode can fully apply to the analyses of single winding regulation mode, and the single winding regulation mode described in relevant literatures is merely a special case of multi-winding regulation mode. 5. Example g m ) κ 2m 1 g 1 )+πξ h (1 ξ h ) ξ h l 1,h 1 l 1,h (2) (21) A CRT with six windings is given in Tian (25, 27) and Tian and Li (24), whose rated voltage is 5 3 kv, the frequency is 5 Hz, and all the self and mutual reactances of which have been calculated in Tian (27) (all the parameters of all windings have been referred to the working winding in Tian 27, units: Ω): Z j thus, M can be obtained(units: H), M Z jω The inductances of current-limiting reactors are as follows (units: H). L x diag(, , , , , ) According to (2), L n is obtained from M and L x (in this example, n 6). L Assume that θ 6 (, π 6, π 4, π 12, π 3, π 2) (units: rad), according to the explanations of the multi-winding regulation mode analysed above, the following results are obtained: h 5, θ 6 (, π 12, π 6, π 4, π 3, π 2), X X 6. Obviously, this switching sequence is non-standard, from (7), T is derived as follows T According to (6), L 6 can be deduced from L 6, L 6 is the inductance matrix that corresponds to the current switching sequence just mentioned in this example

8 2 Y-B. Liu et al. - ACu 1 + i 1 l 12 l x 2 i 2 T 2 l 26 l16 l 13 l23 T 6 i 6 l 15 l 14 l 36 l x3 l x6 l 56 l 46 l 25 l 35 l 24 l 34 i 3 T 3 l 45 i 5 i 4 l x4 T 5 l x5 T 4 Figure 3. Equivalent circuit of CRT. L 6 According to the analysis in Section 2, C 6 is obtained from L 6 C Based on C 6, all of the boundary waveforms and differences of each winding can be derived from and (9) respectively, and then each winding s current waveform in [, π 2] can be obtained by translating each of its boundary waveforms one by one. Tian (27) presents the equivalent circuit for CRT, shown in Figure 3, and the core part of which is the polygon-type equivalent circuit of the multi-winding transformer. In accordance with the formulae presented in Tian (27), the branch impedances in the polygon-type equivalent circuit can be calculated from M, the detailed calculation process is not repeated here, which has been fully introduced in Tian (27). Figure 4 shows the process that generates each winding s current waveform in [, π 2] using the direct generating method presented in this paper. Furthermore, based on the equivalent circuit of CRT illustrated in Figure 3, we construct the corresponding simulation model for CRT in MATLAB/ SIMULINK, and then all the windings current waveforms are obtained by simulation. To confirm the correction of the calculating method presented in this paper, the waveforms obtained by simulation and the direct generating method are compared, as shown in Figure 4. It isimportant to notice that the subscript of each current labelled on each vertical axis in Figure 4 represents the actual mark number of each winding, which is different from what is stated at the beginning of Section 2. Furthermore, the sub-pictures in Figure 4 are arranged in the switching sequence (Figure 4 always represents the waveforms of W 1 and its generating process), and the numbers in the rightmost () of each sub-picture are adopted to identify the boundary waveform of the corresponding winding, for example, the (3) in Figure

9 Australian Journal of Electrical and Electronics Engineering 21 i 1 /A i 4 /A i 2 /A i 3 /A i 5 /A ω t/rad working winding (3) -8 (2) ω t/rad (2) the first conducted control winding ω t/rad (3) the second conducted control winding (5) (4) (3) (2) (4) (3) (2) ω t/rad (4) the third conducted control winding ω t/rad (5) the fourth conducted control winding Boundary waveform Moved subsection Actual waveform by calculating Actual waveform by simulation Figure 4. Current waveform of each winding. (2) 4(2) corresponds to the third boundary waveform of W χ2 (χ 2 4). The actual calculating waveforms in Figure 4 are obtained by the direct generating method mentioned above, specifically, the moved subsection (shown by the red dashed line in Figure 4) of each period should be firstly selected from the corresponding boundary waveform, and then the actual waveform by calculating (shown by the black dashed line in Figure 4) can be obtained by translating each of the moved subsections one by one. Figure 4 shows that the waveforms obtained by simulation and the direct generating method coincide approximately, which indicates that the proposed formulae for calculating all the windings current and the direct generating method proposed in this paper are correct. Based on the trigger angles given in this example, each winding s fundamental and harmonic current RMS can be worked out from (15) to (17), comparing them with what was obtained by simulation, the results are shown in Table 1. Here C means the values in the corresponding line are obtained by calculation, and S means the values in this line are obtained by simulation. Table 1 shows that the results obtained by calculation is approximately consistent with those obtained by simulation, which indicates that the formulae for calculating the RMS of each winding s fundamental and each harmonic current are correct. As previously mentioned, the number of trigger angle combinations that can satisfy a given output power is much more than one. To compare the differences of the harmonic contents in different combinations, two sets of actual trigger angle vectors that have the same fundamental current are given. θ 6 (,,,,45,9 ) (2) θ 6 (,28,8.7,54,,9 ) According to the process shown in Figure 2, the RMS of working winding s fundamental and each harmonic current in the above two different vectors can be calculated, shown as Table 2. Both of the working winding s fundamental current RMS corresponding to the two sets of trigger angles are A,, the second one belongs to the multi-winding regulation mode, and the first one belongs to the single winding regulation mode. However, Table 2 shows that the harmonic currents in the second combination are less than those in the first one, which indicates that it is possible to decrease the harmonic current when CRT operates in the multi-winding regulation mode.

10 22 Y-B. Liu et al. Table 1. Fundamental and harmonic current RMS (units: A). I 1 I 3 I 5 I 7 I 9 W 1 C S W 2 C S W 3 C S W 4 C S W 5 C S Table 2. Working winding s fundamental and harmonic current RMS (units: A). I 1 I 3 I 5 I 7 I (2) Conclusions In this paper, the multi-winding regulation mode of CRT is proposed, and the expression for the windings instantaneous current in the multi-winding regulation mode is obtained by means of subsection linearization, and then a simple method that can directly generate each winding s current waveform is presented. Using Fourier series, the functional relationship between all the winding s harmonic currents RMS and the trigger angle vector is deduced, and then the detailed calculation process is established. The example shows that the proposed calculation method in multi-winding regulation mode are correct. This work shows that each winding s current waveform can be obtained by moving the corresponding boundary waveforms directly, and the multivariate function between each winding s harmonic current RMS and the trigger angles is a linear combination of several univariate functions that have the same form in multi-winding regulation mode. The multi-winding regulation mode contains all the regulation modes of CRT, the single winding regulation mode analysed in Tian (25) is merely a special case of this paper. Disclosure statement No potential conflict of interest was reported by the authors. Funding This work was supported by the National Natural Science Foundation of China [grant number ]; the National Natural Science Foundation of China [grant number ]; Science and Technology Program of Gansu Province, China [grant number 134WCGA181]; Science and Technology Program of Lanzhou, China [grant number ]; Basic Scientific Research Foundation of Gansu Province Department of Finance [grant number 21352]. Notes on contributors Yi-bin Liu was born in Gansu Province, China, in Currently, he is pursuing an MS degree in the Power Electronics and Power Drives in the School of Automation and Electrical Engineering, Lanzhou Jiaotong University, Lanzhou Gansu, China. His main research interests concern electrical machines, drives and transformers; yanerwuming@126.com. Ming-xing Tian was born in Gansu Province, China, in He received a DE degree from Xi an Jiaotong University (Xi an, China) in 25. Currently, he is a DE supervisor in Lanzhou Jiaotong University. His main research interests concern electrical machines, drives and reactors; tianmingxing@mail.lzjtu.cn. Jian-ning Yin was born in Gansu Province, China, in Currently, he pursuing an MS degree in the Power Electronics and Power Drives in the School of Automation and Electrical Engineering, Lanzhou Jiaotong University, Lanzhou Gansu, China, code: 737. His main research interests concern magnetic integrated technique, design of controllable reactor; yinjianning@126.com. References Aleksandrov, G. N., B. I. Al bertinskii, and I. A. Shkuropat Operating Principles of a Controllable Shunting Reactor of Transformer Type. Russian Electrical Engineering 66 (11): Alves, J. E. R., L. A. S. Pilotto, and E. H. Watanabe. 28. Thyristor-controlled Reactors Nonlinear and Linear Dynamic Analytical Models. IEEE Transactions on Power Delivery 23 : Chen, X. X., B. C. Chen, C. H. Tian, J. X. Yuan, and Y. Z. Liu Modeling and Harmonic Optimization of a Two-stage Saturable Magnetically Controlled Reactor for an Arc Suppression Coil. IEEE Transactions on Industrial Electronics 59 (7): Chen, J. H., W. J. Lee, and M. S. Chen Using a Static VAR Compensator to Balance a Distribution System. IEEE Transactions on Industry Applications 35 (2): Chuang, C. L., Y. C. Wang, C. H. Lee, M. Y. Liu, Y. T. Hsiao, and J. A. Jiang. 21. An Adaptive Routing Algorithm over Packet Switching Networks for Operation Monitoring of

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