Dynamic modeling, control design and stability ...

Electric Power Systems Research 160 (2018) 71–88

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Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr

Dynamic modeling, control design and stability analysis of railway active power quality conditioner Hossein Mahdinia Roudsari a , Sadegh Jamali a,b , Alireza Jalilian a,b,∗ a b

School of Electrical Engineering, Iran University of Science and Technology, Tehran 16846-13114, Iran Center of Excellence for Power System Automation and Operation, Iran University of Science and Technology, Tehran 16846-13114, Iran

a r t i c l e

i n f o

Article history: Received 11 June 2017 Received in revised form 3 January 2018 Accepted 26 January 2018 Keywords: Active railway power quality compensator (ARPQC) Electrified railway Bi-linear model Generalized averaged model Linear control system design Stability analysis

a b s t r a c t Two-phase three-wire converter has been proposed to compensate the power quality issues in AC electrified railway systems. In this paper, a step-by-step modeling procedure is presented and applied to the compensator to obtain a linear model. The model is then used in a linear control system design and stability analysis of the compensator. The first step is to obtain the bi-linear form of the exact (switched) model of the compensator representing the high and low frequency behavior of the converter. Then, the generalized state space averaging is applied to the bi-linear model to obtain the continuous time model of the compensator. The linear small signal model is derived by differentiating the averaged model around its equilibrium point. Linear control system design is established based on the linear small-signal model to improve DC-link voltage performance. The simulated model of compensator is verified by quantitatively comparing to the previously reported experimental results. The verified simulation model is used to validate the obtained models. Finally, the open loop stability analysis is performed by studying the locus of system eigenvalues with respect to the circuit parameters variations. © 2018 Elsevier B.V. All rights reserved.

1. Introduction There are a fast growing number of electrified intercity railways featuring higher energy efficiency, low air pollution, higher speed, etc. [1]. The supply voltage of electric trains can be DC (600/750/1500/3000V ) or AC (“11/15kV 16.67 Hz” or “25kV , 50/60 Hz”) [2]. Most recent electrified railways use autotransformers to provide 25-0-25 kV traction power supply for the operation of the modern 25 kV electric rolling stock. Independent from being supplied by DC or AC, electrified railway systems impose power quality (PQ) problems such as negative sequence current (NSC), harmonics [3,4], low power factor, etc. to the supply grid [5]. The PQ problems may cause maloperation of the protection relays, vibration, over-heating, etc. in the grid [6,7]. Conventional passive methods such as balanced transformers, using threephase locomotives, passive filters, phase shifting, etc. used to mitigate the PQ problems [8,9]. Furthermore, various active methods have been recently presented to mitigate the PQ problems; such as: static VAR compensators [9], static compensator (STATCOM) [10], railway power conditioner (RPC) [10], two-phase three wire compensators (TPTWC) [11], hybrid power quality conditioner (HPQC) [12], magnetic hybrid power quality compensator [1,13], etc. An extensive review of the passive and active PQ compensating approaches is also presented in [14]. The satisfactory behavior of the compensators relies on the appropriate selection of components and proper control design. Appropriate modeling is the first step in proper circuit and control design. The majority of the presented works in the literature used trial and error to design the controllers parameters. A few analytical procedures for dynamic modeling and control design of the ARPQCs are presented in the literature. In [15] a modeling procedure is presented for implementation of the nonlinear passive-based control in the current control stage of the converter. Hu et al. also proposed a harmonic model for HPQC, in order to investigate the frequency characteristics of the compensator [15]. The modeling procedure is implemented in the frequency domain and cannot be used in the control design. The majority of works have used PI controller to stabilize the DC-link voltage of the ARPQCs [16]. Regardless of the selected control approach, modeling is the first and key step in control design of the ARPQCs.

∗ Corresponding author. E-mail address: [email protected] (A. Jalilian). https://doi.org/10.1016/j.epsr.2018.01.027 0378-7796/© 2018 Elsevier B.V. All rights reserved.

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H. Mahdinia Roudsari et al. / Electric Power Systems Research 160 (2018) 71–88

Fig. 1. Traction supply system with V/v transformer and TPTWC.

The first step in the modeling of a power electronic converter is to obtain its exact (switched) model or its bi-linear model (BML); which is required for attaining other types of models. The BLM models the high-frequency behavior of the converter as well as its low-frequency characteristics [17]. The BLM is particularly suitable for designing nonlinear control laws, such as variable-structure, Lyapunov approach, matrix inequalities, etc. In general, the models used for control design and stability analysis are simpler than those used for circuit design or simulation. Generalized state space averaged model (briefly generalized averaged model (GAM)) which is also known as dynamic phasor model is widely used in the modeling of power electronic converters including HVDC system [17], doubly fed induction generators [18], quadratic boost converter [19], single-phase full-bridge inverter [20], induction generator excited by a 3-phase inverter [21], bidirectional solid-state transformer [22], matrix-reactance frequency converter [23], etc. This is because of its appropriate dynamic behavior and considerable reduction of processing requirements and simulation time [23,24]. The small signal model of the power electronic converter is obtained by linearizing the large-signal GAM around an equilibrium point. The obtained model can be used in linear control system design [17]. On the other hand, it is required to be ensured that the converter remains stable under variations in operating conditions [17]; hence, stability analysis of the power electronic converters is necessary. This paper presents a road map for dynamic modeling, control design and stability analysis of the ARPQCs used in 25 kV, 50 Hz electrified railway system. The TPTWC which is proposed by Chuanping et al. [11] is selected as a case study; but, the modeling procedure can also be applied to other compensators with linear current control system. The main contributions of this paper are summarized as: • The exact BLM of TPTWC as PQ conditioner in electrified railway systems is formulated. • In the premise of linear control system design, the GAM as the continuous time-invariant model of the TPTWC is obtained which considers only its low frequency (averaged) characteristics. • By differentiating the GAM, small-signal averaged model of the TPTWC is achieved. • Linear control system design for DC-link PI controller is established which significantly improves the performance of the TPTWC. • The stability analysis of TPTWC is performed by studying the locus of system eigenvalues.

The rest of the paper is organized as follows: The TPTWC is introduced in Section 2. Section 3 presents the modeling methodology. The large and small signal models of the TPTWC are established in Section 4. Section 5 is devoted to the linear control system design based on the obtained linear small-signal model. The models are validated in Section 6, and Section 7 is dedicated to the open loop stability analysis of the TPTWC. Finally, Section 8 gives the conclusion and the summary of the findings of this paper.

2. Railway electrification system and active power quality compensator The TPTWC is installed in parallel with Vv traction transformer (Fig. 1) and compensates NSC using the hybrid current control comprised of hysteresis current control (HCC) and dividing frequency control (DFC). Four-quadrant converters are used in the latest generation of electric locomotives to minimize the PQ problem. The locomotives with these converters exhibit unit power factor and minimum interference to the power supply grid. The measurements revealed that the reduction of low-order harmonics increases the emission at high frequencies (2–150 kHz), the so-called supraharmonics [25]. The supraharmonics components are not considered in this paper. Therefore, traction loads are assumed to be pure resistive and inject no harmonic into the system [11]. Loads of ˛ and ˇ feeders are:

iL˛ (t) = IL˛ · sin(ωt − 30◦ ) iLˇ (t) = ILˇ · sin(ωt − 90◦ )

(1)

To mitigate NSC in the grid side of traction transformers, secondary currents (i ˛ , i ˇ , and i c = −(i ˛ + i ˇ )) must be balanced three phases. Therefore, the injected currents by TPTWC not only must balance the consumed power of the feeders, but also shift their phases in a way that their phase difference to be 120◦ . The mean active current component is defined as Imp = 0.5(IL˛ + ILˇ ) and the reactive current


73

Fig. 2. Overall modeling procedure of the railway ARPQC.

component is defined as Imq = Imp tan(30◦ ). These components together, result in a set of three-phase currents (i ˛ , i ˇ , and i c ). Subtracting these currents from the load currents leads to the compensating reference currents as:

i C˛ (t) = iL˛ (t) − i ˛ (t)

(2)

i Cˇ (t) = iLˇ (t) − i ˇ (t) where

i ˛ (t) = Imp sin(ωt − 30) + Imq cos(ωt − 30◦ )

i ˇ (t) = Imp sin(ωt − 90◦ ) − Imq cos(ωt − 90◦ )

(3)

The above relationship is valid for Vv traction transformer which is shown in Fig. 1. If the traction transformer is impedance matching, Scott, Leblanc or Yd, then in Eq. (3) −30◦ and −90◦ should be 0◦ and −90◦ , respectively [26]. Therefore, it is very important to consider the transformer type in the reference current calculation. In order to stabilize the DC link voltage of TPTWC, a PI controller is used. The output of voltage controller (pdc ) is the required active power to stabilize the DC link voltage:

∗ (t) = i (t) + p · sin(ωt − 30) iC˛ C˛ dc ∗ (t) = i (t) + p · sin(ωt − 90) iCˇ dc Cˇ

(4)

In order to create the switching functions for the three phases bridge, the reference currents are applied to the current control unit which may be HCC, FDC, or hybrid. In this paper, the linear FDC which has a performance with zero steady-state error [11], is selected as the current control method of TPTWC. 3. Modeling methodology In order to suitably design the control system of the compensator, different models are presented which are related and exhibit different levels of accuracy. The final goal is to obtain a linear model which can be used in the linear control system design as well as the open loop stability analysis. In this section the modeling methodology is introduced and each modeling approach is explained briefly. Fig. 2 shows the overall modeling procedure. There are many circuit topologies in TPTWC, which can be modeled by the state space model (SSM). The SSM for each circuit configuration in (ti , ti+1 ) interval is defined as [17]: d x(t) = Ai x(t) + Bi e(t) dt

ti ≤ t ≤ ti+1

(5)

where Ai and Bi , x(t) and e(t) are state matrix, input matrix, state and input vectors respectively. Eq. (5) is valid for (ti , ti+1 ), and if the state of switches change, the SSM will change accordingly. Generally, if N switching states exist in a power electronic converter, system behavior can be characterized as:

d x(t) = (Ai x(t) + Bi e(t))hi dt N

(6)

i=1

Eq. (6) is the switched model and hi is a binary digit that determines the circuit topology. The model is called the “exact model”, since there is no additional approximation in it [17].

74


3.1. Bi-linear model (BLM) BLM is a compact form of (6) and uses switching function (uk ) instead of hi . With N switching states, uk is a p × 1 vector; where p is the smallest integer which fulfills 2p ≥ N. The bi-linear form of the switched model is: x˙ = Ax +

p

(Bk x + bk ) · uk + d

(7)

k=1

For each k (1 < k < p), A and Bk are n × n; and, bk and d are n × 1 matrices. uk is the control input as:

uk = u1

u2

...

up

T

(8)

The product of state variables and control inputs in (7) gives the bilinear feature. This model can be used directly in the non-linear control system design [27]. 3.2. Generalized averaged model (GAM) The GAM is used for analysis of the converters with both DC and AC stages; and it is based on complex Fourier transformation [17]. For a periodic signal x(t): +∞

x(t) =

xk (t) · ejkωt

(9)

k=−∞

where ω is the fundamental angular frequency and xk is the amplitude of kth harmonic order as: xk (t) =

1 · T

t

x() · e−jkω d

(10)

t−T

where T is the averaging period. The kth order sliding average of x(t) equals to the amplitude of kth harmonic order (x k (t) = xk (t)) [17]. The derivative of the sliding average is expressed as: d x (t) = dt k

d dt

x

k

(t) − jkω · x k (t)

(11)

Using (11) one can obtain the sliding average of the derivative of a signal. It can be used in deriving the GAM from the BLM. If the SSM of a general system is expressed as: ˙ x(t) = F(x(t), u(t))

(12)

The GAM can be obtained using Eq. (11). If the study is limited to fundamental and DC components, calculations will be simplified. In this case, it is needed to calculate xi 0 and xi ·yi 0 for DC and xi 1 and xi ·yi 1 for AC variables. Coupling factor of kth order sliding average is calculated as: x · y k =

x k−i (t) · y i (t)

(13)

i

Zero-order sliding average (ZOSA) of DC variables is calculated as [17]:

d x 0 (t) = A · x 0 + [Bk · x · uk 0 + bk · uk 0 ] + d 0 dt p

(14)

k=1

and the first order sliding average (FOSA) for AC variables is:

d x (t) = −jω · x 1 + A · x 1 + [Bk · x · uk 1 + bk · uk 1 ] + d 1 dt 1 p

(15)

k=1

Replacing (13)–(15) in the SSM of (12) results in the GAM of the converter. FOSA for AC and ZOSA for DC signals are complex and real values, respectively. Considering just fundamental and DC components, the real value of signal x(t) can be estimated as: x(t) ≈ x 0 + 2[Re{x 1 } · cos(ωt) − Im{x 1 } · sin(ωt)]

(16)

One of the main issues in modeling the converters with AC stages is their time-varying characteristic. The GAM converts the AC variables to some constant variables as presented above. 4. Large and small signal modeling of TPTWC As mentioned before, modeling is an essential step in control system design. Models used for control purposes may be most often simpler than those used for circuit design or simulation [17]. This section is devoted to obtaining a linear model, which will be used in linear control system design as well as open loop stability analysis of the TPTWC. The switched BLM is a starting point in the modeling procedure. The BLM consist of high-frequency features of the TPTWC which is parasitic and must be neglected in control designs [17]. Besides, because of the unbalance structure of TPTWC, Park transformations cannot be applied to the BLM; thus the GAM is introduced


75

Table 1 Eight different switching states of TPTWC. ui

State 0

I

II

III

IV

V

VI

VII

u1 u2 u3

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

Table 2 The values of sub-matrices of [Ai ] and [Bi ]. Matrix

i 0 & VII

I

T

[A3i ]1×2

−1/L˛ −1/Lˇ

1/Cdc 1/Cdc

0 0

0 0

[A2i ]2×1

II

III

0 1/Lˇ 0 −1/Cdc

−1/L˛ 0

1/Cdc 0

IV

V

1/L˛ 0 −1/Cdc 0

0 −1/Lˇ

0 1/Cdc

VI

1/L˛ 1/Lˇ −1/Cdc −1/Cdc

to build a continuous time-invariant model which can be linearized around an equilibrium point to be implemented in the linear control system design. 4.1. The switching states The eight switching states of TPTWC are shown in Table 1; where, 1 (and 0) denotes that the upper (lower) switch of the leg is on. Because of eight switching states, p = 3; and switching function is u = [u1 u2 u3 ]T ; where ui determines the state of ith leg switch. Each of these switching states consists of RLC components and independent power sources. In SSM, capacitor voltages and inductor currents are selected as state variables and SSMs are formulated using these state variables. In TPTWC iC˛ + iCˇ + iCc = 0 then iCc can be rephrased in terms of iC˛ and iCˇ . iC˛ , iCˇ and DC link voltage (vdc ) is chosen as state variables; therefore, the state vector is X = [iC˛ iCˇ vdc ]T . It should be noted that the on-state resistances of the power electronic switches are considered in modeling procedure and v˛ and vˇ are the input signals. Traction load currents can be expressed based on conductance of traction loads as:

i ˛ = GL˛ v˛ − iC˛

(17)

i ˇ = GLˇ vˇ − iCˇ Independent from switching states, the above equations are always valid; hence, the output equations are:

i

˛

i

ˇ

⎡ ⎤ iC˛ GL˛ 0 v˛ −1 0 0 ⎢ ⎥ = iCˇ ⎦ + ⎣ 0 −1 0 vˇ 0 GLˇ

(18)

vdc

The output relation is unique for all eight switching states, but the SSM for each state is different from another. Using this procedure for all the eight states, eight [Ai ] and [Bi ] matrices are retrieved as follows: X˙ = [Ai ] · [X] + [Bi ] · [U]

⎡ 1 Ai

2×2

[Ai ] = ⎣ A3i

1×2

2 Ai

2×1

4 Ai

1×1

⎤ ⎦,

⎡ 1 Ai

2×2

−Rth L˛

−Ron L˛

Lˇ

Lˇ

⎢ ⎣ −Ron

=⎢

⎤

⎡

⎥ ⎢ ⎥ , A4 = 0, [Bi ]T = ⎢ i 1×1 ⎦ ⎣ −Rth

⎤

−1 L˛

0

0

0

−1 Lˇ

0

⎥ ⎥ , i = 0, I, . . ., VII. ⎦

(19)

where Ron is the on resistance, R = R˛ = Rˇ are the inductors resistance and Rth is defined as Rth = R + 2Ron ; and the elements of Ai matrix are in Table 2. Using SSMs and (6) and (7) the exact model of TPTWC is derived. 4.2. Bi-linear model of TPTWC Using the SSMs for each of the eight switching states of (19), Table 2 and the switching functions (u1 , u2 , u3 ), the model of TPTWC in the standard BLM form is expressed as: x˙ = Ax + (B1 x + b1 ) · u1 + (B2 x + b2 ) · u2 + (B2 x + b2 ) · u2 + d

(20)

76


Fig. 3. Exact equivalent circuit diagram of the TPTWC.

The matrices of the above equation are given as follows:

⎡

⎤

⎡

⎥ ⎥

B1 = ⎣

−(Rth /L˛ )

−(Ron /L˛ )

0

A = ⎢ −(R /L ) on ˇ ⎣

−(Rth /Lˇ )

0⎦,

⎢

⎡ ⎢ ⎣

B3 = ⎢

0

0

(1/L˛ )

0

0

0

0

0

⎢

0 0

0 ⎤ −(1/L˛ )

−(1/Cdc )

0

0

0

−(1/Lˇ ) ⎦ ,

b1 = b2 = b3 = 0

(1/Cdc )

(1/Cdc )

0

⎥ ⎥

0

⎡

⎤

0

0

⎢ ⎥ ⎦ , B2 = ⎢ ⎣0

0

0

⎥ ⎥

(1/Lˇ ) ⎦

0 −(1/Cdc )

0

T

,

d=

−

v˛ L˛

−

vˇ Lˇ

⎤

0

T

(21)

0

0

The above model can be used in non-linear control system design such as sliding mode, Lyapunov approach, matrix inequalities, etc. Also, the exact equivalent circuit diagram of the TPTWC is derived using above equations and shown in Fig. 3. 4.3. Large signal GAM of TPTWC In the averaging procedure, iC˛ and iCˇ are AC which are modeled by their FOSA and vdc is a DC variable and is modeled by its ZOSA. The FOSA of iC˛ and iCˇ are calculated using (15) and (20) as: a

d iC˛ 1 = −jωiC˛ 1 + (−Rth iC˛ 1 − Ron iCˇ 1 + vdc · u1 1 − vdc · u3 1 − v˛ 1 )/L˛ dt

d i = −jωiCˇ 1 + (−Ron iC˛ 1 − Rth iCˇ 1 + vdc u2 1 − vdc u3 1 − vˇ 1 )/Lˇ dt Cˇ 1

(22) (23)

The ZOSA for vdc is calculated using (14) and (20) as: d 1 v = (iC˛ u3 0 − iC˛ u1 0 + iCˇ · u3 0 − iCˇ u2 0 ) Cdc dt dc 0

(24)

Eqs. (22)–(24) are the base relations for obtaining the GAM of TPTWC. The FOSA of iC˛ , iCˇ and switching signals (u1 , u2 , u3 ) are complex numbers and ZOSA of vdc is a real number. Arbitrary notations are considered as:

iC˛ 1 = (x1 + j · x2 ), u1 1 = (a11 + j · a12 ),

iCˇ 1 = (x3 + j · x4 ),

vdc 0 = x5

u2 1 = (a21 + j · a22 ),

u3 1 = (a31 + j · a32 )

(25)

The FOSAs of v˛ and vˇ are calculated using (10) as: √ v˛ 1 = −0.25 · Vm (1 + j 3),

vˇ 1 = −0.5Vm

(26)

The ZOSA of the coupling factors of (22) and (23) are calculated by (13), using (25) notations:

iC˛ · u1 0 = 2(x1 · a11 + x2 · a12 ) iCˇ · u2 0 = 2(x3 · a21 + x4 · a22 )

iC˛ · u3 0 = 2(x1 · a31 + x2 · a32 ) iCˇ · u3 0 = 2(x3 · a31 + x4 · a32 )

(27)

In similar approach (assuming the FOSA of vdc is zero), the FOSA of coupling terms in (24) are:

⎧ v · u = vdc 0 · u1 1 = (x5 · a11 ) + j · (x5 · a12 ) ⎪ ⎨ dc 1 1 ⎪ ⎩

vdc · u2 1 = vdc 0 · u2 1 = (x5 · a21 ) + j · (x5 · a21 ) vdc · u3 1 = vdc 0 · u3 1 = (x5 · a31 ) + j · (x5 · a32 )

(28)


77

Substituting the above coupling terms in (22)–(24) and after mathematical simplifications a non-linear set of equations is obtained. As mentioned in Section 2, in this paper the FDC is selected as current control method. The averaged value of the PWM output is the FOSA of its input [17]. The input of the PWM is the output of a transfer function; and the input of the transfer function is the error current [11] as: ∗ iek (t) = iCk (t) − iCk (t)

k = ˛, ˇ, c

(29)

The FOSA of error currents can be calculated as: ∗ iek 1 (t) = iCk 1 (t) − iCk 1 (t)

k = ˛, ˇ, c

(30)

0 is the ZOSA of the output of the voltage The FOSA of the reference currents can be calculated using Eqs. (3), (4) and (10). Pdc 0 = Pdc controller. Substituting the FOSAs of the reference currents in (30), the FOSA of error currents can be expressed as:

⎧ √ 1 √3 3 0 1 0 ⎪ ⎪ i I P (I P = − − x + j − 2I ) − − x ⎪ e˛ L˛ L˛ 1 2 1 Lˇ ⎨ 4 4 dc 12 4 dc √ 1 ⎪ 1 0 3 ⎪ ⎪ i = − I ) − − x + j + I ) − x (I P (I ⎩ eˇ L˛ L˛ 3 4 Lˇ Lˇ dc 1

4

2

(31)

12

and the error currents of phase c can be calculated using ie˛ 1 = −ie˛ 1 − ieˇ 1 . In FDC, error currents will be applied to the transfer function of (32) and the output of transfer function gives the input of PWM. Uf (s) Ie (s)

= Kp +

2KIm s s2 + ω 2

(32)

If the Laplace transformation of iek (t) applied to the transfer function and the inverse Laplace transformation of Uf (s) is calculated; in the derived relationship the amplitude of sin(ωt) will increase with time. This is because of the big open loop gain of the transfer function [11]. If the value for KIm is assumed to be very small (KIm ≈ 0), ufk (t) can be written as: ufk (t) = KP IMk · sin(ωt − ϕMk ) = KP iek (t),

k = ˛, ˇ, c

(33)

Applying ufk (t) to the PWM controller, will lead to the switching functions as: um (t) =

1 (sgn(ufk (t) − (t)) + 1), 2

[k, m] = [˛, 1], [ˇ, 2], [c, 3]

(34)

The behavior of TPTWC is controlled using the above switching functions. The FOSA of um (t) equals to the sliding average value of the input signals (ufk 1 ) of PWM [17]:

um 1 = am1 + jam2 = 0.5Kp · (iek 1 + 1) am1 = 0.5(Kp ake + 1),

am2 = 0.5Kp bke

,

[k, m] = [˛, 1], [ˇ, 2], [c, 3]

(35)

Substituting (35) in the non-linear equations of (22)–(24), the SSM of whole TPTWC system is derived which is expressed in (36). The coefficients of the SSM are given in Appendix A.

⎧ 0 x +N x˙ 1 = A11 x1 + A12 x2 + A13 x3 + O15˛ IL˛ x5 + O15ˇ ILˇ x5 + M15 Pdc 5 15˛ x1 x5 + N35a x3 x5 + B14 ⎪ ⎪ ⎪ ⎪ 0 ⎪ x˙ 2 = A21 x1 + A22 x2 + A24 x4 + O25˛ IL˛ x5 + O25ˇ ILˇ x5 + M25 Pdc x5 + N25˛ x2 x5 + N45˛ x4 x5 + B24 ⎪ ⎪ ⎪ ⎪ ⎨ x˙ 3 = A31 x1 + A33 x3 + A34 x4 + O35˛ IL˛ x5 + O35ˇ ILˇ x5 + M35 P 0 x5 + N15ˇ x1 x5 + N35ˇ x3 x5 + B34 dc

0 x +N ⎪ x˙ 4 = A42 x2 + A43 x3 + A44 x4 + O45˛ IL˛ x5 + M45 Pdc 5 ⎪ 25ˇ x2 x5 + N45ˇ x4 x5 ⎪ ⎪ ⎪ 0 x + ··· ⎪ ⎪ x˙ 5 = O51˛ IL˛ x1 + O51ˇ ILˇ x1 + O52˛ IL˛ x2 + O52ˇ ILˇ x2 + O53˛ IL˛ x3 + O53ˇ ILˇ x3 + O54˛ IL˛ x4 + M51 Pdc 1 ⎪ ⎪ ⎩ 0 0 0 2 2 2 2

(36)

· · · + M52 Pdc x2 + M53 Pdc x3 + M54 Pdc x4 + NC x1 x3 + NC x2 x4 + NC x1 + NC x2 + NC x3 + NC x4

Consequently, the GAM is obtained which is the large signal continuous time-invariant model of the TPTWC. This model adequately represents the macroscopic behavior of the system, which can be used for control design purposes. It is worth mentioning that if the HCC is used in the compensator, because of the non-linear nature of the control scheme, the averaging procedure cannot be implemented at the current control stage. Therefore, to the authors’ knowledge, the GAM of the HCC cannot be obtained. 4.4. Small signal model of TPTWC The previously obtained BLM and GAM consist of power and current control stages of the TPTWC, which can be used in nonlinear control system designs. The models can represent the behavior of the converter, minus its voltage control system. The conventional PI controller is used to stabilize the DC-link voltage of the TPTWC [11]. In order to properly design the voltage controller of the TPTWC, it is required to obtain the linearized model of the TPTWC. In order to obtain the linearized small signal model the GAM of (36) must be differentiated around the operating point of TPTWC. The small signal model is valid just for small variation of inputs and state variables, i.e. f = f˜ + fe ; where f˜ is the small variation and fe is the equilibrium value of f. Substituting these values in (36), leads to very large system of equations. 0 are considered sufficiently small. Assuming the constant coefficients and operating points, the constant The variation of IL˛ , ILˇ , and Pdc

78


Fig. 4. Block diagram of the voltage control system.

parts of the derived system lead to a system of non-linear equations; which its solution gives the operating point of the TPWC. Using the obtained values of operating point, a system of differential equations is derived; which is the linearized small signal model of TPTWC:

⎡˙ ⎤ x˜ 1

⎡

˜ 11 A

˜ 12 A

˜ 13 A

0

⎢˙ ⎥ ⎢˜ ⎢ x˜ 2 ⎥ ⎢ A21 A˜ 22 0 ⎢ ⎥ ⎢ ⎢˙ ⎥ ⎢˜ ⎢ x˜ 3 ⎥ = ⎢ A31 0 A˜ 33 ⎢ ⎥ ⎢ ⎢ x˜˙ ⎥ ⎢ 0 A˜ ˜ 42 A43 ⎣ 4⎦ ⎣ ˜ 51 A

x˜˙ 5

˜ 52 A

˜ 24 A ˜ 34 A ˜ 44 A

˜ 53 A

˜ 54 A

⎤ ⎡ ⎤ ⎡ B˜ 11 x˜ 1 ⎥ ⎢ ⎢ ⎥ ˜ 25 ⎥ A x˜ 2 ⎥ ⎢ B˜ 21 ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ˜ 35 ⎥ · ⎢ x˜ 3 ⎥ + ⎢ B˜ 31 A ⎥ ⎢ ⎥ ⎢ ⎢ ⎢ ⎥ ⎢ ˜ 45 ⎥ A ⎦ ⎣ x˜ 4 ⎦ ⎣ B˜ 41

˜ 15 A

x˜ 5

0

B˜ 51

⎤

B˜ 12

B˜ 13

B˜ 22

B˜ 23 ⎥

B˜ 42

⎥ ⎡ ⎤ p˜ dc ⎥ ⎥ ⎢ ⎥ B˜ 33 ⎥ · ⎣ ˜iL˛ ⎦ ⎥ B˜ 43 ⎥ ⎦ ˜iLˇ

B˜ 52

B˜ 53

B˜ 32

(37)

The complete representation of small signal matrix elements is presented in Appendix B. Using this linearized SSM and the Laplace transformation, 15 input–output transfer functions are derived as: ˜ −1 · B˜ · U ˜ =H ˜ ·U ˜ X˜ = (sI − A)

⎡ x˜ ⎤ 1

⎡

˜ 11 H

˜ 12 H

˜ 13 H

(38)

⎤

⎥ ⎡ ⎤ ⎢ x˜ ⎥ ⎢ ˜ 21 H ˜ 22 H ˜ 23 ⎥ p˜ dc H ⎢ 2⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ x˜ 3 ⎥ = ⎢ H˜ 31 H˜ 32 H˜ 33 ⎥ ⎥ · ⎣ ˜iL˛ ⎦ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎣ x˜ 4 ⎦ ⎣ H˜ 41 H˜ 42 H˜ 43 ⎥ ⎦ ˜iLˇ x˜ 5

˜ 51 H

˜ 52 H

(39)

˜ 53 H

The general representation of the transfer functions is as follows: ˜ ij (s) = H

x˜ i (s) n4 s4 + n3 s3 + n2 s2 + n1 s + n0 = 5 ˜ j (s) s + d4 s4 + d3 s3 + d2 s2 + d1 s + d0 U

(40)

˜ 51 (s) which is p˜ dc (s) to V˜ dc (s) transfer function can be used The transfer functions are fifth-order and they also exhibit four zeros. The H as the plant model for the control system design of the PI voltage controller. The electric locomotives are injected to the supply system one by one; hence, the relative small variation assumption is solid. Therefore, the small signal model can be used in the PI controller design. 5. Control system design Majority of the works in the literature used PI controller to stabilize the DC-link voltage, but trial and error is the main approach in the control design. In this section, a PI controller is adapted to control the DC-link voltage of TPTWC. The p˜ dc (s) ⇒ V˜ dc (s) transfer function ˜ 51 (s)) relates small variations of the DC-link voltage (V˜ dc (s)) to the small variations of the output active power of the voltage controller. (H The voltage control block diagram is shown in Fig. 4. The kp and ki parameters of the PI controller extensively influence on the performance of the DC link voltage; therefore, the coefficients must be selected in a way to improve the time response of the DC link voltage. Some characteristics of the time and frequency responses of the closed-loop system are considered in the design procedure. Four main objectives are selected to evaluate the time domain response and closed-loop stability of the system: (1) stability index, (2) steady-state DC link voltage error, (3) the consumed energy of the controller, and (4) over/undershoot of the time response. The closed-loop transfer function of the voltage control system shown in Fig. 4 is calculated as: HCL (s) =

C(s)H51 (s) 1 + C(s)H51 (s)

(41)

The closer of HCL (s) dominant poles to the imaginary axis, the relative stability will be improved and the speed of time response will be increased; therefore, the stability index is defined as:

SI(kp , ki ) =

1 max(real(poles(HCL (s))))

(42)

The root locus of the HCL (s) for some (kp , ki ) are depicted in Fig. 5. The second objective is to minimize the steady-state error of the DC-link voltage. To do so, the RMS of edc (t) signal (Fig. 4) is calculated by:

ERMS (kp , ki ) =

1 T

T

edc 2 (t) · dt, 0

∗ −v edc (t) = Udc dc

(43)


79

Fig. 5. Root-locus of the open-loop control system.

Fig. 6. 3D surface of O.F versus kp and ki . Table 3 DC-link voltage controller parameters and corresponding characteristics. Parameter

Optimum

Case A

Case B

Case C

Proportional gain (kp ) Integrator gain (ki ) Steady state error (V) Undershoot (%) Settling time (s)

0.01 1.71 0.70 1.12 0.04

0.40 2.00 0.76 1.52 0.05

0.30 6.00 0.83 1.90 0.07

0.20 0.40 Inf 2.46 Inf

The controller must be designed in order that the energy consumed by the controller and the over/undershoot (OUSH) of the timeresponse are minimized. The consumed energy of the controller is calculated: ENC (kp , ki ) =

1 T

T

u(t) · dt, u(t) = L−1 {U(s)} = L−1 {C(s) · Edc (s)}

(44)

0

An objective function based on above-mentioned criteria is defined as follows:

O · F(kp , ki ) = W1 SI + W2 ERMS + W3 ENC + W4 OUSH

(45)

where Wi (i = 1,. . .,4) are the weighting or scaling factors which are determined based on the priority of each criterion. Selection of kp and ki leads to different values for the O.F; therefore, the values of O.F are calculated based on the pre-defined ranges for kp and ki . The 3D surface of O.F versus kp and ki for [W1 , W2 , W3 , W4 ] = [0.423, 31.92, 1968.5, 1.35] is depicted in Fig. 6. The weighting factors are selected to scale each criterion between 0 and 100. Fig. 6 shows that the minimum of O.F = 92.65 is obtained at kp = 0.01 and ki = 1.71. Unstable region presented in Fig. 6 shows the pairs of kp and ki which their selection jeopardize the stability of the voltage control. In order to compare the optimum point with other pairs of kp and ki , different values are considered for PI controller gains (Table 3). The unit step response and bode diagrams of the HCL (s) for different regulator settings are shown in Fig. 7(a,b). Fig. 7 shows that inappropriate selection of the PI controller gains will jeopardize the closed-loop system stability. As in case C, the phase margin is negative and shows the system instability. The optimum case exhibits a dominant pole near the imaginary axis of the s-plane; hence, the time response characteristics have been significantly improved compared to Cases A–C. Fig. 7(c) also shows DC-link voltage responses in the simulation model for the increment of iLˇ (t) from 10 to 150 A while iL˛ (t) is 150 A. As can be seen, similar to the unit step response, the optimum controller gains improve the DC-link performance. Undershoot and settling time are reduced compared to the cases A–C while the steady-state error remains rather unchanged. It is worth mentioning that in Case C voltage diverges from its reference value, while it is not depicted in the figure because of the time limit. It can be deduced from the results

80


Fig. 7. DC-link performance with different kp and ki of Table 3: (a) unit-step response of HCL (s), (b) bode diagram of HCL (s), (c) simulation results.

Table 4 Test system parameters. Parameter

Value

Unit

Filter inductances (L˛ = Lˇ ) Filter resistances (R˛ = Rˇ ) DC-link capacitance (Cdc ) On-state resistance (Ron ) RMS voltage of feeders (Vm ) FDC gain (KP ) Controller proportional gain (kp ) Controller integral gain (ki ) RMS load current of ␣-feeder (IL˛ ) RMS load current of ␤-feeder (ILˇ )

1.3 20 50 1 220 1 1 0.2 259.27 129.63

mH m mF m V

A A

that the proposed control design based on the linear model obtained in the former sections, will improve the performance of the DC-link and prevent instability problems.

6. Model verification and simulation results 6.1. Validation of the reference model The suitable performance of the TPTWC is proven by simulation and experimental results by other authors in [11]. Our primary objective is to create a validated reference model, to verify the performance of the presented models. For this purpose, simulated reference model (SRM) of TPTWC (Fig. 1) is built in MATLAB/Simulink with the exact parameters presented in [11] (Table 4). In order to exactly represent the performance of the TPTWC, all the details of the power and control stages are exactly modeled. All of the simulation and experimental results of [11] are implemented in the SRM; but for brevity, two of the most important experimental scenarios are presented here; which are FDC and HCC. In order to validate the SRM, the results of the model are compared to the experimental results (ER). The ˛ and ˇ feeders loads are 40 and 20 kW respectively. For t < 0.1 s, the ˛ feeder load, and for t > 0.1 s, the ˇ feeder load is also connected. The SRM and experimental results of TPTWC are compared in two ways: (1) visual comparison, (2) quantitative comparison. In the former, Fig. 8(a,c) shows the experimental results of iA , iB , iC and also vDC presented in [11] for HCC and FDC, respectively. Fig. 8(b,d) shows the same results obtained with SRM. Scales of Fig. 8(b,d) are selected in accordance with Fig. 8(a,c); and visual comparison of the results shows the close resemblance in transient behavior of the compensator. For the sake of quantitative comparison, two approaches are considered here: (1) absolute steady-state error, (2) feature selective validation (FSV). The former is related to the RMS values of the signals in steady state, which does not give any information about dynamics of the response; but, the latter gives the information about not only the amplitude of the signals but also, the other curve features. This method is widespread and is currently being developed as a standard of validation for computational electromagnetics [28]; but, FSV method is the means to consistently compare the model being validated to the reference data [29]. Therefore, FSV is used here as a quantitative comparison tool to validate the SRM. The experimental results that expressed graphically were converted to numerical values using graph digitizer. The SRM and digitized experimental data are sampled at 10−4 s. The FSV comprised of three phases [30]: (1) the datasets are decomposed to DC, low frequency and high frequency components; (2) amplitude difference measure (ADM) and feature difference measure (FDM) calculation; (3) obtaining global difference measure (GDM) based on ADM and FDM. The DC, low frequency and high frequency components are obtained using


81

Fig. 8. Comparison between SRM results and ER of [11] for iA , iB , iC and vDC (scale of ER: 20 ms/div for time, 100 A/div for iA , iB , iC , 300 V/div for vdc ). (a) ER with HCC, (b) SRM results with HCC, (c) ER with FDC, (d) SRM results with FDC. Table 5 Comparison between SRM results and ER of [29]. Results

Control mode

iA (A)

iB (A)

iC (A)

NSC

vdc (V)

Experimental [11] Reference model Absolute error ADM FDM GDM

HCC

88.6 89.7 1.1 0.11 0.21 0.25

91.3 91.5 0.2 0.11 0.2 0.24

94.5 93.8 0.7 0.12 0.2 0.25

3.32 2.33 0.99 – – –

500 500 0 0.18 0.63 0.73

Experimental [11] Reference model Absolute error ADM FDM GDM

FDC

92.3 92.1 0.2 0.14 0.22 0.35

91.7 91.4 0.3 0.14 0.23 0.36

91.8 92.2 0.4 0.16 0.25 0.38

0.39 0.32 0.07 – – –

500 500 0 0.25 0.59 0.69

discrete Fourier transformation. The procedure of obtaining the FSV coefficients and criteria are based on IEEE 1597.2. If N is the total number of data samples, the ADM for the i th sample (ADMi ) is calculated as:

SRMDC,i − ERDC,i SRMLO,i − ERLO,i + ODM e|ODMi | ODM = ADMi = N−1 , i i 1 N−1 1 SRM LO,j + ER LO,j SRMDC,j + ERDC,j N N j=0 j=0

(46)

The FDM for i th sample (FDMi ) is calculated as:

⎧ ⎪ ⎪ SRM LO,i − ER LO,i SRM HI,i − ER HI,i ⎪ ⎪ ⎪ FDM1i = N−1 , FDM = ⎪ 2i ⎪ 2 6 N−1 ⎪ ⎨ SRM LO,j + ER LO,j SRM HI,j + ER HI,j N j=0 j=0 N ⎪ ⎪ SRM HI,i − ER HI,i ⎪ ⎪ , FDM = 2 FDM + FDM + FDM ⎪ DM = ⎪ 3i 7.2 N−1 i 1i 2i 3i ⎪ ⎪ SRM HI,j + ER HI,j ⎩ N

(47)

j=0

where prime and double prime superscripts denote to the first and second derivative of samples. The total ADM and FDM equals to the average of ADMi s and FDMi s, respectively. The GDMi is calculated as: GDMi =

ADMi2 + FDMi2 ,

DM =

N DMi | i=1

: (A, F, G)

(48)

All of the SRM results are compared to the ER using FSV indices. The results of these quantitative comparisons are given in Table 5. The plots of FSV time indices (ADMi , FDMi , and GDMi ) and their corresponding confidence results are depicted in Fig. 9. The confidence results are categorized into six levels: excellent (EX), very good (VG), good (G), fair (F), poor (P) and very poor (VP). The smaller values for the FSV indices represent the higher similarity. Fig. 9 shows that the ADM is better than FDM; because the information corresponding to the low-frequency oscillations of vdc had not been available. Therefore the feature difference which is related to the shape of the signals is relatively high. As can be seen in Table 5, the FDM for current waveforms are lower than the FDM of vdc ; because more information is available in the ER currents. Comparing the results

82


Fig. 9. FSV indexes calculated in comparing vdc SRM and experimental results. Right plots are ADMi , FDMi , and GDMi . Left plots are the confidence results for ADM, FDM and GDM.

shows that the SRM properly represent the behavior of the compensator; therefore, the model can be used for verification of the proposed models. 6.2. Verification of BLM In order to verify the BLM of (20), a script corresponds to control system of TPTWC based on (3) and (4), and HCC with HB = 1 A is written in MATLAB. The flowchart of the mentioned script is shown in Fig. 10. The LPF in [11] is substituted with a “mean” function and the PI controller is simply modeled with a gain and integral of edc (t). Based on the eight switching states, one of the eight switching values of Table 2 must be determined based on HCC. For HCC implementation, the current slopes of each phase must be calculated based on: source voltages (v˛ (t) and vˇ (t)), present switching states, and load currents). The current slope (mj ) of each phase is calculated using (20). Considering a small constant sample time (T = t(k) − t(k − 1)) and constant slope between each two samples, the linear current path can be approximated using: ij (t = k) = mj (k) · T + ij (t = (k − 1)) j = ˛, ˇ, c

(49)

Eq. (49) uses the value of previous sample (t(k − 1)) and the calculated slopes of currents (m˛ , mˇ , and mc ). As shown in Fig. 11, if the current in the present sample t(k) is between UHB and LHB, the switching state of the leg does not change, otherwise, the switch state changes. After approximation of the three phase currents in kth sample, switching function is obtained and the corresponding SSM is selected. The obtained system of linear equations of (20) is solved and the state variables are obtained. In order to compare the results of the BLM with the previously mentioned model, the same scenario is applied to the BLM and SRM. The load switching occurs at t = 0.095 s instead of t = 0.1 s and the circuit elements values are as given in Table 4. It can be seen in Fig. 12 that the BLM compensating currents, iC˛ and iCˇ , overlay the SRM results. The maximum error between these two models is 5.07% (maximum error is 7.6 A). The high frequency characteristic of the TPTWC is also reflected into the BLM as well as SRM results. 6.3. Verification of GAM In order to validate the obtained GAM, the non-linear set of (36) is solved using the Tustin integration. The PI controller gains are selected based on the optimum values obtained in Section 5. A number of scenarios are studied and the results seem to be promising for the GAM; but the same scenario of the previous sections is given here. Study time is 0.2 s and after t = 0.1 s, ˇ phase load is connected to the circuit. The outputs of the GAM are x1 to x5 which are converted to time domain signals of iC˛ , iCˇ and vdc , using (16). The results of the GAM compared to reference currents and SRM results are depicted in Fig. 13. Fig. 14 compares the ZOSA of DC link voltage (solid black) to its reference (dashed black) and SRM results (gray). As can be seen in Fig. 14, the DC link voltage fluctuations as a result of reactive power exchange between phases of TPTWC are not reflected in ZOSA. In the average modeling procedure, state variables are assimilated to their fundamental or DC values; therefore, the GAM has less accuracy in the switching moment. Truncation errors are 7.8%, 54.7% and 5.4% for iC˛ , iCˇ and vdc , respectively. These errors are specific to switching moments; while these errors are reduced to 0.12%, 0.33% and 0.21% in steady state. Therefore, the GAM has an acceptable accuracy in the steady state. On the other hand, the GAM cancels the high frequency characteristics of the TPTWC. 6.4. Verification of small-signal model In order to validate the small signal model of TPTWC, a number of scenarios are studied; but for brevity, only two cases are presented. In these cases: (1) relatively small step changes are considered in load, (2) both increase and decrease are considered for the load changes, (3) different changes are selected to evaluate the effect of variations in the operating point. The dynamic responses of the step changes are calculated and expressed as follows.


Fig. 10. Flowchart of the BLM implementation.

Fig. 11. HCC linear current path approximation for BLM implementation.

83

84


Fig. 12. Compensating currents derived from the BLM, Simulink and errors. (a) ␣ phase (b) ␤ phase.

Fig. 13. Compensating currents of ˛ and ˇ; upper figures: comparison between GAM and reference, lower figures: comparison between GAM and Simulink.

Fig. 14. DC link voltage: comparison between reference, Simulink, GAM.

Case 1: iL˛ (t) steps down from 259 to 239 A and iLˇ (t) remains constant on 130 A. Before and after the step change, the operating points are calculated as xe,(t0.1) = [59.83 −2.88 −27.42 53.25 500], respectively. These are the equilibrium points of the TPTWC which are shown with gray lines in Fig. 15(a). ˜ 12 to H ˜ 42 are also illustrated in Fig. 15(a). iC˛ (t) and iCˇ (t) are derived using (16) and The dynamic responses of the transfer functions H compared to the SRM results in Fig. 15(b). The figure shows that the small signal derived signals approximately overlay with the SRM ˜ 52 , reference voltage and SRM result of vdc are demonstrated. In this case, iL˛ steps down and iC˛ , results. In Fig. 15(c) the step response of H iCˇ are reduced accordingly; therefore, the DC link voltage increases temporarily but the PI controller stabilizes the voltage. Case 2:


85

Fig. 15. Small signal model, Case 1: time response to step down in ˛-phase load. (a) Reference vs. small-signal of xi s. (b) Small-signal derived vs. Simulink of iC˛ and iCˇ . (c) vdc small signal vs. Simulink vs. reference.

Fig. 16. Small signal model, Case 2: time response to step up in ˇ phase load. (a) Reference vs. small signal of xi s. (b) Small signal derived vs. Simulink of iC˛ and iCˇ . (c) vdc small signal vs. Simulink vs. reference.

In case 2 iLˇ (t) increases from 130 to 173 A, while iL˛ (t) is 259 A. The operating points for t < 0.1 s, is xe,(t 0.1 s, is xe,(t>0.1) = [64.81 −12.47 −21.61 62.37 500]. Dynamic responses of H up in iLˇ are illustrated in Fig. 16(a). The time domain waveforms of iC˛ and iCˇ derived from small signal model are presented in Fig. 16(b). In the figure small signal derived waveforms (black) are compared to their corresponding Simulink results (gray). Fig. 16(c) shows the DC

86


Fig. 17. Locus of system poles vs. variations of the system parameters: (a) L variations (b) C variations (c) ω variations (d) IL˛ variation with different ILˇ s.

link voltage response to the step versus reference and Simulink model output. Because of the step up in iLˇ and consequently increment in the compensating currents, the DC link voltage drops temporarily but the PI controller stabilizes the voltage to its reference value. The load step of the ˇ phase in Case 2 is considered to be higher in comparison with ␣ phase load change in Case 1. Consequently, the small signal derived signals in Case 2 are poorly overlaid with the Simulink results compared to Case 1. This is in accordance with the inherent characteristic of small signal model, which is its validity for small variations in operating point of the converter. Since the operating point variation in second case in relatively bigger than the one in the first case, the small signal model has lost its accuracy in Case 2 compared to Case 1. 7. Eigenvalues and stability analysis The base parameters of the whole system are as Table 4. Using these parameters, the operating point of the converter is calculated as xe = [64.81 0 −32.41 56.13 500]. The eigenvalues of TPTWC are: 1 = −286.92, 2,3 = −318.03 ± j404.27, and 4,5 = −1361.76 ± j330.64. These five poles are located in the left half s-plane which means that the system is stable; but, the locations of these poles change respect to the system and load parameters variations. This may worsen the relative stability of the system; therefore, the open loop stability analysis of the TPTWC in respect of different variations is studied and presented here. 7.1. Stability to L variation L is the value of output inductances of the single-phase bridges. In order to analyze the stability of TPTWC with respect to L = L˛ = Lˇ variations, the locus of system eigenvalues is obtained for L of 1 to 1.5 mH with 10 ␮H steps (Fig. 17(a)). System eigenvalues movement by increasing the inductance is shown by arrows in the figure. The eigenvalues shifts toward the imaginary axis and the system time constant increases whilst its response speed decreases. Although the margin of system stability decreases, the whole system remains stable. 7.2. Stability to Cdc variation The compensator works properly for DC link capacitors between 10 and 70 ␮F. Fig. 17(b) shows the locus of the system eigenvalues when Cdc varies in the above range in steps of 2 ␮F. The arrows show the movement of system eigenvalues while Cdc is increasing. When 1 moves to the right, system damping factor decreases and while the 2,3 move to the left the amplitude of the system fluctuations mitigates. In the whole range of Cdc variations, the whole system remains stable. 7.3. Stability to ω variation Adjustable speed drives of induction machines (which are used in electric trains) inject the harmonic components from 10 to 400 Hz [11]. This range with step of 20 Hz is considered in the investigation of locus of system eigenvalues (Fig. 17(c)). Increasing ω leads to 1 approaches to the origin and 2,3 move away from the imaginary axis whilst increasing their imaginary parts. This leads to increase in system fluctuations and decrease in damping ratio for higher order harmonics. In can be concluded from the locus that 4,5 have no significant impact on the system dynamics; therefore, the system order can be reduced to the 3rd order. 7.4. Stability to load variation In order to investigate the impact of loads variation on the stability of TPTWC, loads are changed simultaneously. The locus of the system poles is shown with different values of ILˇ . As can be seen in Fig. 17(d), the conductance of each ILˇ for each locus is presented. The results show the simultaneous increase of ILˇ and IL˛ leads to a prominent reduction in the system stability margin, whilst the overall system remains stable. The system damping ratio increases and the system response becomes faster.


87

Table 6 Parameters of the small-signal model. m

ı

1 2 3 4

3 4 1 2

1 2 0 0

0 0 1 2

3 4 0 0

0 0 3 4

8. Conclusion In this paper, a modeling procedure is presented to obtain a linearized model which can be used in linear control design and stability analysis of the railway active power quality compensators (TPTWC is selected as a case). In the first step, the BLM of the TPTWC which is an exact model is derived. Then by applying generalized state-space averaging to the BLM, the GAM is obtained. The GAM properly models the low frequency characteristics of the TPTWC. This makes it an appropriate tool for stability and control design purposes. It is worth mentioning that averaging procedure can be implemented on the linear current control systems. The small signal model of TPTWC is derived by linearization of the GAM around an equilibrium point. A simulation reference model (SRM) is validated by extensively comparing its results to the existing experimental data. The SRM is selected as base model to verify the obtained models. The outputs of the BLM, GAM and small signal model are compared to the SRM; and the results show that the BLM and GAM have an acceptable level of precision. The performance of the small signal linear model is also evaluated for two different step changes in loads. The test results show that the model has an acceptable accuracy in small variations but it loses its accuracy when the input changes are relatively large. The obtained linear model is used in linear control design of the DC-link PI voltage controller of the TPTWC. The results of the optimal control system are compared to the non-optimal settings of the voltage controller and the comparison reveals that the considerable improvement in the stability of the DC-link voltage. Finally, the stability of TPTWC with respect to variation of circuit parameters and inputs is investigated. In the whole range of variations, the TPTWC remains stable, but the stability margins change extensively. It can also be concluded that, for the sake of simplicity and without scarifying any significant accuracy, the system can be reduced to the 3rd order. Appendix A. A11 = A22 = A33 = A44 = −

Rth , L

A12 = A34 = −A21 = −A43 = ω,

A13 = A24 = A31 = A42 = −

Ron L

O25ˇ O35ˇ Kp Kp 4M35 2M25 O25˛ 4M45 O45˛ = = − , O15ˇ = O15˛ = √ = − √ = −O35˛ = = √ = √ = M15 = √ 5 2L 2 8L 3 3 3 3 3 O52ˇ O53ˇ Kp O52˛ O54˛ O51˛ = O51ˇ = √ = − √ = −O53˛ = , O53ˇ = 0 = √ =− 2 4Cdc 3 3 3 Kp N15˛ = N25˛ = N35ˇ = N45ˇ = 2N35a = 2N45˛ = 2N15ˇ = 2N25ˇ = − L 1 1 Vm , B14 = √ B24 = B34 = 2 4L 3

(A.1)

Kp M51 M52 M54 M53 NC = √ = = √ = = 2 5 4 2Cdc 3 3

Appendix B. ˜ 11 = A ˜ 22 = A ˜ 33 = A ˜ 44 = (A11 + N15˛ x5e ), A ˜ 13 = (A13 + N35a x5e ), A

˜ 55 = 0, A

˜ 31 = (A31 + N15ˇ x5e ), A

˜ 12 = A12 , A

˜ 21 = A21 , A

˜ 24 = (A24 + N45˛ x5e ), A

˜ m5 = Om5˛ IL˛e + Om5ˇ ILˇe + Mm5 Pdce + Nı5˛ xıe + N 5ˇ x e + N5a xe + N A ˜ 5m = O5m˛ IL˛e + O5mˇ ILˇe + M5m Pdce + 2NC xme + NC x e A B˜ m1 = Om5ˇ x5e ,

B˜ 51 =

4 p=1

M5p xpe ,

B˜ 52 =

4 p=1

O5p˛ xpe ,

4

˜ 43 = A43 A

˜ 42 = (A42 + N25ˇ x5e ) A 5ˇ x e

, B˜ m1 = Mm5 x5e , B˜ 53 =

˜ 34 = A34 , A

B˜ m2 = Om5˛ x5e

(B.1)

O5pˇ xpe

p=1

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