Dynamic Phasor Modeling of Various Multipulse Rectifiers and a VSI Fed by 18-Pulse Asymmetrical Autotransformer Rectifier Unit for Fast Transient Analysis

To study the fast modelling of dynamic characteristics caused by power electronic switching, the dynamic phasor (DP) theory based on the time-varying Fourier decomposition and frequency shifting is firstly applied to the modelling of various multipulse rectifiers. The DP model for symmetrical 12-pulse phase-shifting reactor rectifier unit (PSR-RU) is derived with model order reduction, relations between ac and dc terminals, and Taylor series expansion. The DP model of asymmetrical 18-pulse autotransformer rectifier unit (AT-RU) is proposed based on the switching functions expressed in the DP domain. Meanwhile, the DP model of voltage source inverter (VSI) fed by asymmetrical 18-pulse AT-RU is built with the harmonic state-space (HSS) equations. Under both balanced and unbalanced conditions, the good calculation accuracy and rapid simulation speed of the developed DP models are validated by the detailed time-domain (TD) simulation.


INDEX TERMS
The modern electric power system (EPS) generally comprises many different power converters and contains magnetic devices and power electronic components. The dynamic processes of EPS are with different time constants, such as electromagnetic motion and electronic motion. The structural complexity and multi-time scale transients make the dynamic characteristic analysis of power system to be challenging [1]- [4]. To evaluate the system efficiency, power quality, and dynamic performance efficiently, a quick and precise simulation method with time-varying properties is required [5], [6]. The multipulse rectifier unit consisting of magnetic parts and power electronic components is an essential ac/dc power conversion device in the EPS [7], [8]. A large number of multipulse rectifier units have been used in the EPS because of their simple configuration, high reliability, and low electromagnetic interference [9]- [11]. Thus, this paper aims to simulate the dynamic characteristics of multipulse rectifier units with an efficient method.
The detailed time-domain (TD) modelling and the statespace averaging modelling are two general simulation methods. The TD modelling is accurate but too timeconsuming for an extensive power conversion system. The state-space averaging modelling with the quasi-steady-state (QSS) assumption is fast, but it cannot reveal the harmonic dynamics of the power conversion system. Meanwhile, the fast simulation methods of multipulse rectifier units are studied. Reference [12] proposes a functional model for the symmetrical 18-pulse autotransformer rectifier unit (AT-RU), where the 3-9-dc system is simplified to a 3-3-dc system. Reference [13] achieves the fast simulation of 12-pulse AT-RU by representing the three-phase diode bridge with a dc transformer. These two models are based on the fundamental switching function and Park's transformation, the evaluation for ac current harmonics and dc voltage ripple is unavailable, and the second harmonics dramatically decreases the simulation speed under the unbalanced condition.
The generalized averaging technique, which is also called as the dynamic phasor (DP) method, is firstly presented in [14]. This method can be applied in periodically or nearly periodically driven systems. Due to higher calculation accuracy than the QSS model and faster simulation speed than the detailed TD model, the DP method has been used in modelling of three-phase thyristor controlled reactor [15], high voltage direct current systems [16], and multi-generator power systems [17]. However, the DP simulation of multipulse rectifier units is not reported before.
Different forms to exhibit the DP model have been used in the relevant references. In [18], the DP model of a three-phase diode bridge is stated by transforming the time-domain mathematical expressions between ac and dc terminal into the DP domain. The DP models elaborated in [19]- [23] are expressed with the harmonic state-space (HSS) equations. Both approaches are adopted in this paper to make the DP modelling of multipulse rectifier units convenient.
Based on the DP theory, this paper aims to provide a solution to model the multipulse rectifiers, and extended power system with multipulse rectifiers efficiently and accurately. Here, only the effects caused by nonlinear power electronic switching are considered, the effects of nonlinear magnetic properties will be discussed in a separate work. The paper is organized as follows. Section II revises the DP theory. Section III introduces the DP model of three-phase diode bridges into the modelling of symmetrical 12-pulse phaseshifting reactor rectifier unit (PSR-RU), its applicability in the modelling of symmetrical multipulse rectifier units as a library element is also validated. Section IV proposes the DP model of asymmetrical 18-pulse AT-RU. Section V puts forward the DP model of voltage source inverter (VSI) fed by 18-pulse AT-RU, verifying that the DP model of multipulse RU is of good applicability and transplantability within the extended EPS models. Section VI shows the computation efficiency of the developed DP models, and Section VII concludes the paper.

II. DYNAMIC PHASOR THEORY
Under the QSS assumption, the average of state-variable x(t) throughout T is defined as where T is the period of the fundamental component.
To a time-domain quasi-periodic signal x(τ ) without QSS assumption, as shown in Fig. 1, it can be regarded as a periodic signal at the time moving window τ ∈ (t 1 − T , t 1 ]. Hence, x(τ ) can be represented as  where the kth Fourier coefficients X k (t) are the kth Dynamic phasors (DPs) [14] x The DPs of x(τ ) are with time-varying properties. Formula (3) equals to (1) when k is 0, that means one could retain only the 0th DP in this generalized averaging method to recover the traditional state-space averaging model.
To the signal x (t) = a(t)cos[ω s t + θ(t)] under QSS assumption, the variation of amplitude a(t) and phase angle θ (t) are minimal when compared with ω s . Here, ω s corresponds to the fundamental component of the system. As shown in Fig. 2, x (t) can be represented with its analytical signal z (t) = a(t)e jθ (t) e jωst by discarding the negative frequency component. Furtherly, z (t) is frequency shifted to the baseband signal x a (t) = a(t)e jθ (t) , and x a (t) is the fundamental phasor of x (t). The baseband representation makes it possible to use a large simulation time step, and this is the reason using phasor representation under the steadystate condition.
Unlike a(t) and θ (t) of x (t) under the QSS assumption, X k (t) in (3) are time-varying and have wider bandwidth. As shown in Fig. 3, all the frequency components of time-varying signal x(τ ) can be transformed into the baseband signals X k (t). Hence the DP representation method meets simulation requirements of the power electronics dominated power system for the high calculation accuracy and fast simulation speed.
The crucial properties of DPs are given below [16], [17]. First, the relationship between derivatives of variable x(τ ) and derivatives of kth DP X k (t) is Second, the product of two time-domain variables equals the discrete-time convolution of corresponding DPs Third, the relationship between X k (t) and its complex conjugate X * k (t) is The time-domain signal can be restored from DPs as

III. DP MODELLING FOR SYMMETRICAL 12-PULSE PSR-RU
The 12-pulse PSR-RU discussed in [11] is taken as an example to evaluate the applicability of diode bridges DP model in the modelling of multipulse rectifier unit [18]. Fig. 4 depicts the circuit topology of 12-pulse PSR-RU. PSR-RU with small volume capacity works in series. The input windings W XM , W YN , and W ZL connect with the power source, the output windings W LU2 , W LU1 , W MV2 , W MV1 , W NW2 , and W NW1 connect with diode bridges. The winding turn satisfy With this kind of connection mode, the phase of i a ' and i a '' are ahead and lag of the phase of i A with the same angle θ . Based on the principle of minimum Total Harmonic Distortion (THD), the phase-shifting angle θ is θ = π/12. The fundamental current of i A basically remain unchanged, while the nth harmonic current would be reduced effectively.
The current and voltage phasor diagram of phase A is plotted in Fig. 5, only the fundamental phasors are given here. θ = π/12 is the phase-shifting angle; ϕ is the impedance angle. The effect of 5th and 7th harmonic currents of i a and i a can be significantly reduced because of their phase differences is about π . u v1 and u v2 are with the same amplitude and π /6 phase difference. Hence, it is reasonable to transform the 12-pulse PSR-RU into two symmetrical three-phase diode bridges.
According to the voltage phasor diagram shown in Fig. 5, the relationship between voltages fed to diode bridges and VOLUME 8, 2020  voltages in the power source side can be obtained as Meanwhile, the relationship between input and output ter- The diode bridges in the 12-pulse PSR-RU can be separated into two three-phase diode bridges, as shown in Fig. 6. They are named as phase lead diode bridges and phase lag diode bridges, respectively.
Here, only DP modelling of phase lead diode bridges is analyzed due to the similarity.
The input voltages can be written as With Euler formulate and DP definition, (12) is valid As revealed in (8), DPs of u i (t), i = V1, W1, U1, can be obtained from the power source voltages under both balanced and unbalanced conditions. Assuming u lead is the voltage vector representation of u i (t), the relationship between u lead and three input voltages in DP domain is as below Therefore, the dq components of u lead in DP domain can be derived as where, Without considering the dc voltage drop caused by commutation effect, the time-domain dc voltage u dc,lead can be represented with dq components of u lead Hence, u dc,lead can be restored with DPs by expanding (16) into Taylor series concerning u d,lead and u q,lead [18]. u dc, lead = u dc, lead 0 +2 · u dc, lead 2 e j2ωt +2 · u dc, lead 4 e j4ωt (17) where, p m (m = 0, 1, . . . , 5) are the coefficients of Taylor series at point (U d,lead,0 , U q,lead,0 ).
Formula (16) is derived without considering the affection of harmonic components. The 6th harmonic of dc voltage caused by 5th and 7th switching harmonics is given as u dc, lead 6 = r · 1/2 · u dc, lead,6 e jϕ dc, lead,6 (20) where, r = 1/2 is used to compensating the voltage ripple difference between 12-pulse rectifier and simplified 6-pulse rectifier. Finally, u dc,lead can be recovered from its corresponding DPs.
u dc, lead (t) = u dc, lead 0 +2 · Re k=2,4,6 u dc, lead k · e jkωt (22) Assuming < i dc > 0 = i dc (t), that means the 0th DP contains all information of current i dc (t) [18]. For the phase lead diode bridges, the vector of the input current is i m,lead , and it has < i m,lead > 0 = i m,lead . The relationship between i m,lead and i dc is as follows: (23) The dq components of i m,lead in DP domain are as below Besides, Similar to (16), DPs of cos ϕ lead and sin ϕ lead can be obtained by using the Taylor expansion method.
Based on (9), (23) and dq0/abc transformation, the timedomain value of i A (t), i B (t), and i C (t) can be derived.
Assuming the RMS value of the voltage source in Fig. 4 is 220V, u A becomes zero at time t = 1.0s; the frequency is 50Hz; the dc side load consists of L = 1200µH, C = 500µF, R = 25 . The parameters of core and winding for PSR are the same as [11]. The detailed TD model built with MATLAB/Simulink is used as a benchmark, where the PSR is represented with nine dimensions inductance matrix. The dc side voltage and ac side currents gained from the two models are shown in Fig. 7 and Fig. 8.
For the symmetrical 18-pulse AT-RU, three sets of voltages are produced by symmetrical AT and fed to three three-phase diode bridges. These voltages are with the same magnitude and 2π /9 phase shift. Each diode in the three-phase diode bridges has the same conducting period. The modelling of diode bridges in symmetrical 18-pulse AT-RU can be simplified to one three-phase diode bridges [12]. However, three sets of voltages produced by asymmetrical AT are with different magnitudes, and diode bridges in various three-phase diode bridges have different conducting period. Thus, the DP model of three-phase diode bridges cannot be directly used in the modelling of asymmetrical 18-pulse AT-RU.

IV. DYNAMIC MODELLING FOR THE ASYMMETRICAL 18-PULSEAT-RU
The asymmetrical 18-pulse AT-RU is shown in Fig. 9. DB-1, DB-2, and DB-3 are three three-phase diode bridges. DB-1 is directly powered by the power source (v a , v b , v c ). The primary windings of AT are triangle-connected to the power source, two taps derived from each primary winding connect the secondary windings of other two cores. Two additional threephase voltages (v a2 , v b2 , v c2 ) and (v a3 , v b3 , v c3 ) are formed by vector superposition. DB-2, and DB-3 are driven by these two auxiliary three-phase voltages. The phase-shifting angle between v a and v a2 , v a and v a3 are 37π /180, the phaseshifting angle between adjacent 18 sets of line voltage is π /9. Fig. 10 illustrates the voltages fed to diode bridges of asymmetrical 18-pulse AT-RU.
In Fig. 9, v a , v b , and v c are equal to the power source voltages v A , v B , and v C . The relationship between the voltages fed to diode bridges and the power source voltages are given as where, N p1 , N P , N p2 , and N s are the corresponding winding turn, as shown in Fig. 9.
According to Kirchhoff's current law and principle of magnetic potential balance, the relationship among the power side currents and the three sets of currents fed to diode bridges can be obtained as Based on Fig. 10, the switching functions of diode bridges are plotted in Fig. 11. For each diode in DB-1, the conducting period is 4π /9. For each diode in DB-2 and DB-3, the conducting period is π /9.
The corresponding switching functions of DB-1, DB-2, and DB-3 can be derived in (30), as shown at the bottom of the next page. Thus, the relationship between the dc-link voltage and the three sets of input voltages can be written as shown in the following (31). 3 (31) 43150 VOLUME 8, 2020 Then, u dc can be transformed into the DP domain as below Based on (27) and Euler formulate, the voltages fed to diode bridges in the DP domain can be derived. Similarly, the DPs of switching functions of three three-phase diode bridges can be obtained from (30).
For the asymmetrical 18-pulse AT-RU, u dc mainly contains the dc component and 18th component under the balanced condition. The disturbance caused by unbalanced fault will appear in the form of harmonics, which are integer multiple of the 2nd harmonic component. The dominant kth(k = 0, ±2, ±4, ±18) harmonics are retained to achieve fast and accurate simulation, while the other insignificant terms are neglected. The relationships among dc current and input currents of three three-phase diode bridges in DP domain are gained as The DPs of i dc is available after getting < u dc > k , the DPs of currents can be derived from (33), then the input currents in the power source side can be achieved from (29).
Assuming the RMS value of voltage source is 105V; the frequency is 400Hz; the dc side resistor load is R = 6 ; the core type of AT is SR20kW; N p1 = 67; N p2 = 125; N s = 35; v A becomes zero at time t = 1.0s. The AT in the detailed TD model is represented with fifteen dimensions mutual inductance matrix.
The dc side voltage u dc (t) and ac side current i A (t) obtained from the detailed TD model and the DP model are compared in Fig. 12 and Fig. 13, respectively.    It can be known that the DP model can efficiently reflect the dynamic characteristic of asymmetrical 18-pulse AT-RU, under both balanced and unbalanced conditions.

V. DP MODELLING FOR VSI FED BY 18-PULSE AT-RU
To verify the interfacing of the DP model of multipulse RU within extended EPS models, a voltage source inverter (VSI) fed by the asymmetrical 18-pulse AT-RU is studied. The circuit topology is shown in Fig. 14. Fig. 15 illustrates the principle of the two-level Sinusoidal Pulse Width Modulation (SPWM) method.
By comparing the reference signal U r sin(ω r t + ϕ) to the triangular carrier signal u c (t) with much higher frequency, the switching functions of inverter bridges in a period of the triangular carrier signal can be derived. Only s U (t) is given as where θ 1 can be derived by properties of similar triangular ABC and ADE, θ 2 can be gained with the same way.
where m is the modulation index. The Fourier series expansion of (34) is shown in (36), as shown at the bottom of this page.
Where the Bessel Function can be represented by The DP modelling of the rectifier part is detailed in Section IV. Thus, this section is mainly illustrating the DP modelling of the inverter part.
According to Fig. 14, the state-space equation at the inverter side can be written as (38).
where u dc is the input of this state-space equation, and it is directly derived from (31). In Fig. 14, supposing the frequency of power source u A (u B , u C ) is 400Hz, the frequency of output terminal current i U (i V , i W ) is 50Hz. It is knows that u dc (t) mainly contains dc component and 18th (7200Hz) harmonic component under balanced condition, and the 2nd harmonic (800Hz) will appear under unbalanced condition. To the inverter, the impact of high frequency 18th harmonic voltage in u dc (t) can be neglected. Here, the 2nd harmonic (800Hz) of u dc is  renamed as 16th harmonic since the frequency of reference signal U r sin(ω r t + ) in Fig. 15 is 50Hz. Therefore, the DPs of u dc (t) are < u dc > 0 , < u dc > 16 and < u dc > −16 ; the DPs of u dc (t) are <u dc > 0 , <u dc > 16 , and <u dc > −16 ; the High order harmonics will be eliminated by inductive loads, thus, the considered DPs of s U (t), s V (t), and s W (t) are < s U > 1 , < s U > −1 , < s V > 1 , < s V > −1 , < s W > 1 , and < s W > −1 . The 16th component of u dc (t) and the fundamental component of switching function are retained, since i U (t) is based on u dc (t) and s U (t), the DPs of i U (t) are Based on (4), (5), and (6), the state-space equation in the DP domain can be gained by replacing the time-domain variables in (38) with their DPs [19], [20]. The real and imaginary parts of DPs are furtherly separated to improve the calculation speed. Since the whole state-space equation in the DP domain is too large to show here, only the DP domain equations of di U /dt are presented in (39).
As illustrated in Fig. 3, the negative DPs do not contain any additional information of the time-domain signal. Here, only positive orders of DPs are applied.
Assuming the parameters of asymmetrical 18-pulse AT-RU are the same as that given in section IV, u A becomes zero at time t = 0.3s. The smoothing reactor Z is with R 0 = 0.02 and L 0 = 20mH; C 0 = 300µF; the RL load consists of R = 3 and L = 1mH. To the carrier-based two-level SPWM method, the frequency of carrier signal is f c = 10kHz; the frequency of reference signal is f r = 50Hz; the minimum and maximum values of the carrier signal are −1 and 1; the modulation index is 0.8.
By using the detailed TD simulation model and the DP simulation model, the waveforms of state-space variables i U , i V , i W , u dc , and i dc1 are plotted in Fig. 16 and Fig. 17, respectively. Meanwhile, the power source side current i A and the dc-link voltage u dc in the rectifier side are drawn in Fig. 18.
All these results reveal that the DP model can accurately reflect the dynamic characteristics of VSI fed by the 18-pulse asymmetrical AT-RU, under both balanced and unbalanced conditions. VOLUME 8, 2020

VI. COMPUTATION EFFICIENCY OF DP MODELLING
The DP models are coded with m-file in MATLAB, and the detailed TD models are built with MATLAB/Simulink. The computation time is recorded by the software automatically.
In TABLE 1, t TD and t DP are the computation time of the two models. The DP model for symmetrical 12-pulse PSR-RU achieves 16.9 times faster computation time than the detailed TD model, and this acceleration factor becomes 11.3 times and 25.1 times for the asymmetrical 18-pulse AT-RU and the VSI fed by asymmetrical 18-pulse AT-RU. The total harmonic distortion (THD) of i A under the balanced condition is also analyzed, and the maximum error between the DP simulation and the detailed TD simulation is about 0.33%. ρ(i A ) and ρ(u dc ) are Pearson's correlation coefficient of waveforms i A and u dc obtained by DP and TD models. All of them are close to 1, showing that the DP models achieve a good calculation accuracy.

VII. CONCLUSION
The accurate and fast simulation models of the various multipulse rectifier units and the VSI fed by asymmetrical 18-pulse AT-RU are proposed in this paper. The diode bridges of the 12-pulse PSR-RU can be simplified to two three-phase diode bridges due to the symmetrical properties, and derivation of the DP model is based on the relations between ac and dc terminals. This approach is useful for the DP modelling of symmetrical multipulse rectifiers. The DP modelling of asymmetrical 18-pulse AT-RU is mainly based on the switching functions expressed in the DP domain. This way applies to all kinds of multipulse rectifiers. To the VSI fed by asymmetrical 18-pulse AT-RU, the dynamic performance is revealed by solving the state-space equations in the DP domain. This method may result in huge size equations for an extensive power conversion system.
The efficiency of proposed three DP models are verified by the detailed TD models, respectively. The proposed DP models achieve at least 11.3 times faster computation time than the detailed TD model. The maximum error of ac side current harmonic distortion between the DP model and the detailed TD model is 0.33%.
The proposed DP models cover the various multipulse rectifier units, and can be used as library elements interfacing within extended EPS models such as the VSI given in this paper. Predictably, the efficiency of the DP model will become increasingly apparent for a much more complicated power electronics dominated power system. She is also the Deputy Director of the DOE/NSF co-funded Engineering Research Center (CURENT). Prior to joining UTK/ORNL, she was a Professor at Virginia Tech. She led the effort to create the North American power grid frequency monitoring network at Virginia Tech, which is currently operated at UTK and ORNL as GridEye. Her current research interests include power system wide-area monitoring and control, large interconnection-level dynamic simulations, electromagnetic transient analysis, and power transformer modeling and diagnosis. She was elected as a member of the National Academy of Engineering, in 2016. VOLUME 8, 2020