Unbalanced SSSC Modelling in the Harmonic Domain - IEEE Xplore

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Unbalanced SSSC Modelling in the Harmonic Domain C. D. Collins, Non-member, N. R.Watson, Senior Member, IEEE, and A. R. Wood, Member, IEEE

Abstract— This paper presents a non-linear harmonic domain model for the Static Synchronous Series Compensator (SSSC). The model incorporates a fundamental frequency threephase power-flow and a harmonic converter representation. This combination accounts for system imbalance and the resultant non-characteristic harmonic generation. The proposed unified solution format uses a single iterative loop eliminating unnecessary assumptions regarding the operating point. The models performance is validated against time domain simulation for an unbalanced test system. This test system is then used to illustrate the impact of transmission lines on the harmonic performance of a Static Synchronous Series Compensator, particularly with respect to additional harmonic transfers. Reinforcing the importance of comprehensive three-phase modelling when considering Flexible AC Transmission System (FACTS) devices.

Via
δia

a

Rt + jX t

Vib
δib I b
θ b Vic
δic

a

-

+

Vjb
δjb

V b
δb

Rtb + jX tb I

c


c

θ V

c


Vjc
δjc

δc

Rtc + jX tc Piabc + jQabc i

Index Terms— FACTS, Harmonic Analysis, SSSC.

abc
Vconv β abc

Pjabc + jQabc j

P abc + jQabc

I. I NTRODUCTION

Rc

HE increasing prevalence of Flexible AC Transmission Systems (FACTS) devices makes accurate FACTS models an essential system planning tool [1]. FACTS devices [2], [3] are characterized by their switching nature, which in addition to providing enhanced controllability, has an impact on a systems harmonic performance. This performance is influenced by a number of factors including system imbalance and phase coupling, both of which are inherent to transmission systems. Efficient steady-state modelling and evaluation of these effects can be undertaken using the harmonic domain [4]. This technique avoids the time-variant nature inherent in any switching converter by representing the converter in terms of the harmonic transfers which occur between the ac and dc sides. As such, electrical quantities are characterized using harmonic phasors (or vectors) containing the Fourier coefficients of their spectra [5]. A range of harmonic and frequency domain models have been proposed, ranging from linear approximations [6], [7] through to comprehensive iterative solutions [8][10]. The proposed model uses the positive frequency convolution based representation proposed by Smith [9] for HVdc systems. This technique recognizes the conjugated nature of the negative frequency terms, representing each variable with a positive frequency harmonic phasor. The Static Synchronous Series Compensator (SSSC) is of particular interest since it is used in series with transmission lines, an important source of system imbalance. Detailed harmonic analysis is particularly important for transmission

C

T

The authors would like to thank the University of Canterbury for its financial support of the work discussed in this paper. C. D. Collins, N. R.Watson, and A. R. Wood are with the Department of Electrical and Computer Engineering, University of Canterbury, Christchurch, New Zealand (email: [email protected])

Vja
δja

V a
δa

I a
θa

Vdc

Fig. 1.

SSSC layout and fundamental frequency variables

lines, since transpositions do not necessarily balance the line impedances at harmonic frequencies [11]. As such this paper focuses on the derivation (Section III) and validation (Section IV) of an SSSC model, which is then used to illustrate the importance of comprehensive modelling when considering the interaction between an SSSC and transmission systems at harmonic frequencies. II. SSSC O PERATING P RINCIPLE The SSSC is in principle a synchronous voltage source, which is typically connected in series with a transmission circuit to provide line compensation. This controllability is achieved by using a controllable interface between the dc voltage source (typically a capacitor) and the ac system. The SSSC lacks the generalized series voltage source capability of the Unified Power Flow Controller (UPFC), as the steadystate nett real power transfer through the series converter must be zero (or equivalent to the switching devices losses) [12]. This limits the SSSC to either appearing inductive or capacitive, so as to increase or decrease a transmission circuit reactance, thereby providing control over the amount of power transmitted by the adjacent circuit. Fig. 1 shows the typical SSSC connection layout, where the three series transformers are connected through a converter to the dc source. This paper assumes that the converter interface is an ideal single level PWM bridge, where its losses are

2

Form Y Matrix

since the net real power exchange for all three phases should equate to the converter and transformer losses,  

2 0= (2) Pp − Rtp Ip2 − Vdc Rc 

3-Phase Power Flow Calculate Jacobian, [ J ]

Calculate Switching Spectrum

p=a,b,c

Calculate 3-Phase Power Flow Mismatches

Calculate DC current mismatches Midc = IdcConv - IdcCap

The final mismatches describe the control scheme, which is extremely simple in this case as a result of the current control scheme used.

pos  I − ISet 0= (3) m − mset

Calculate Updates Xi+1 = Xi - [ J-1 ] M(Xi) Combine Mismatches, M(Xi)

No

Converged Yes

Where I pos refers to the positive sequence series current, while Iset and mset are the current and modulation index set points respectively.

Calculate AC side variables

Fig. 2.

p=a,b,c

Diagrammatic overview of the iterative solution technique

B. Harmonic Frequency accounted for by a shunt dc side loss (Rc ). A basic phase angle controller has been used in this paper, it uses a variable modulation angle (α) to charge or discharge the dc side capacitor, which in turn varies the injected series voltage. This provides indirect control over the magnitude of the transmitted current. III. A U NIFIED SSSC M ODEL The proposed model maintains generality by using a nonlinear Newton type formulation, this uses a single iterative loop to solve both the fundamental and harmonic frequency variables. As a result the mismatch equations (M (Xi )) can be divided into two distinct groups, those associated with the power-flow (and controls) and those concerned with harmonic interactions. These mismatches, which are a function of the solution variables, are iteratively updated via a Jacobian to force the mismatch vector to zero (outlined in Fig. 2). A. Fundamental Frequency The proposed model uses a three-phase power-flow to represent the system at fundamental frequency, this incorporates the phase coupling and imbalance that is inherent in transmission systems. The SSSC itself is represented using a three phase adaptation of the power-flow mismatches proposed by Canizares [13]. As an example the a-phase mismatches are,         0=       

Pia − Via I a cos (δia − θa ) a a Qai − Via I a sin (δ  ia − θ a) a a a Pj + Vj I cos  δj − θ  Qaj + Vja I a sin δja − θa P a − Pia + Pja  Qa − Qai + Qaj a P a − Va2 Ga + Vconv V a Ga cos (δ a − β a ) a a a +Vconv V B sin (δ a − β a ) a a Q + Va2 B a − Vconv V a B a cos (δ a − β a ) a a a +Vconv V G sin (δ a − β a )

               

1) PWM Switching Spectra: The harmonic representation used to model the SSSC is based on the dc current mismatch principle proposed for shunt FACTS devices [14]. This convolutional representation models the converters harmonic performance by treating it as a switching modulator, with a harmonic spectra defined by Sh

Sh

Np  j  ψOF Fp − ψONp , h = 0 2π p=1 ∗ Np  jhψON p − ejhψOF Fp e , h
= 0 = hπ p=1

=

(4)

where each of the Np switching periods has an ψONp and ψOF Fp angle defined by classic bipolar PWM theory [15]. The ac ⇔ dc transfer can then be defined using the positive frequency convolution proposed by Smith [9]   nh  j  ∗  2 −2F0 S0 + Fm S m , k = 0    m=0   nh

 ∗     ∗  Fm S(m+k)     m=0  (5) (F ⊗ S)k = k     j  ∗    , k > 0 − F S m (k−m)  2      m=0    nh      ∗  + Fm S(m−k) m=k

where nh refers to the highest harmonic frequency of interest, hence the converter can be included in harmonic mismatches using the following relationships. VConvph = Vdc ⊗ Sph (1)

where G + jB refers to the connection transformer admittance, and other variables are defined in Fig. 1. In addition

Idc =

3

IConvph ⊗ Sph

(6) (7)

ph=1

2) Harmonic Current Mismatches: These transfers and basic linear circuit theory can be used to form the SSSC harmonic mismatches. Fig. 3 outlines the current injection principle used to represent the SSSC at harmonic frequencies. The magnitude of the opposing current injections (Iseh ) is

3

Numerically Derived Jacobian

Viabc

Vjabc

h

abc Ise

P and Q Mismatches

h

h = nh

Ztabc h

50Hz SSSC current (a,b,c) and mismatches

dc Current Mismatches

Harmonic lattice structure

abc Ise

h

h

Mismatches

h=2

Fig. 3.

SSSC Mismatches

SSSC representation at harmonic frequencies α

S Iac Midc

Fig. 4.

IdcConv

m=1

Yac

S Vac 1st

50Hz and Control

IdcCap

DC Voltage Harmonics

Ydc

Fig. 5.

An example Jacobian, nh = 35

VSending

DC current mismatch dependence on solution variables

VReceiving

Vinf

TL1 (100 km) 0.01 + j0.1

defined by the converter voltage (Vconvh ) and the connection transformers impedance (Zt ) (Eqn. 8). Iseh =

Vconvh Zt

35th

Variables

Vdc

TL2 (150 km) 1.05 pu

(8)

The hard switched nature of the converter leads to it being fully defined by the control variable (α), rather than by a commutation process. As such it is unnecessary to include both ac and dc quantities as solution variables, since one is a function of the other. Therefore in order to minimize the number of solution variables (and mismatch equations) the dc side voltage at each harmonic frequency (Vdch ) is used. The mismatch used to update the dc voltage is formed by comparing the dc side capacitor and converter currents, IdcCap and IdcConv respectively. This is achieved by convolving the current estimate of the dc voltage with the switching spectrum (Equation 6) to find the ac converter voltages. These voltages are substituted into the current injection representation (Fig. 3 & Eqn. 8), which in turn generates the SSSC bus voltages via the system admittance. These voltages are then used to calculate the ac series transformer currents, which are finally convolved back to the dc side (Eqn. 7), for comparison with the capacitor current. The capacitor current is calculated using the dc side voltage and admittance, this process is diagrammatically outlined in Fig. 4. C. A Numerical Jacobian The Jacobian, a matrix of partial derivatives used to update the solution variables, is from a computational perspective the most important component of the model. This is because its derivation is either complex and case specific (for an analytic formulation) or computationally expensive (for a numeric formulation). The proposed model uses a numerically derived Jacobian because of its increased flexibility, while

j0.01 pu

β

Fig. 6.

0.9 + j0.45 pu Fixed Z

Vseries fs = 750 Hz m= 1

Vdc

100µF

VBase = 110 kV SBase = 100 MVA

Single line diagram of the test system

the computational expense is minimized by not updating the Jacobian after the first full iteration. The Jacobian has three major segments accounting for the power-flow, the harmonic solution and the interaction between them. Fig. 5 highlights these segments for an SSSC Jacobian with nh = 35. Note how the real and imaginary components of each harmonic mismatch are separate variables, hence each harmonic transfer is represented by a 2 × 2 tensor (Eqn. 9).  

 (Xi ) (∆M ) J11 J12 (9) . = J21 J22 (Xi ) (∆M ) IV. M ODEL VALIDATION The model’s performance is validated through comparison with PSCAD/EMTDC, a time domain simulation technique. The test system used for comparison includes an SSSC controlled so as to regulate the current through two parallel untransposed transmission lines supplying a single fixed impedance load (Fig. 6). Both transmission circuits use identical single circuit towers, the conductor configuration and data for which is included in Fig. 7 and Table I. The electrical parameters for each transmission line are calculated by PSCAD/EMTDC’s line constant program and applied to the harmonic domain simulation as an equivalent-pi circuit. This maintains a consistent transmission

4

g1

g2 10m

−3

4

x 10

DC voltage mag, pu

PSCAD HDA

10m B

A

C

10m

3

2

1

0

0

5

10

15

20

25 30 Harmonic order

35

40

45

50

0

5

10

15

20

25 30 Harmonic order

35

40

45

50

30m

Ground Resistivity

Fig. 7.

DC voltage angle, rad

2

100 Ωm

Transmission line conductor layout for both TL1 and TL2

1 0 −1 −2 −3 −4

HDA PSCAD

Fig. 9.

Harmonic domain comparison for the dc side voltage, excluding dc

DC voltage, pu

0.07

0.065 0.8 HDA PSCAD 0.6

0.06

0.055

0.05

Fig. 8.

0

1

2

3 Angle, rad

4

5

6

a−phase current, pu

0.4

Time domain comparison for the dc side voltage, nh = 50

0.2

0

−0.2

−0.4

−0.6

line representation for both the time and harmonic domain solutions.

The proposed harmonic domain representation (HDA) converges rapidly ( 16s versus 75s for PSCAD, using a 3GHz PC) to a solution consistent with time domain simulation, the dc side voltage for instance is practically identical (Fig. 8 & Fig. 9). This similarity extends to both the series current (Fig. 10 & Fig. 11) and the fundamental frequency operating point (Table II), both of which appear consistent.

Fig. 10.

1

2

3 Angle, rad

4

5

6

Time domain comparison for the a-phase ac side current in TL2

PSCAD HDA

0.01 0.008 0.006 0.004 0.002 0

B. Test system Jacobian

0

5

10

15

20

25 30 Harmonic order

35

40

45

50

0

5

10

15

20

25 30 Harmonic order

35

40

45

50

3 AC current angle, rad

Since the harmonic transfers included in the Jacobian reflect both the ac and dc systems they can be used to illustrate the impact transmission lines have on the harmonic performance of an SSSC. Fig. 13 contrasts the numerical Jacobians for two transmission line representations, the first includes phase coupling and frequency dependence, the second ignores both using an unbalanced fixed impedance. The increase in the number of significant elements (|Jn,m | > 0.05) in the Jacobian reflects the increased magnitude/additional harmonics being generated. This simplification is mirrored in the significantly

0

0.012 AC current mag, pu

A. Simulation Results

−0.8

2 1 0 −1 −2 −3 −4

Fig. 11. Harmonic domain comparison for the a-phase ac side current in TL2, excluding fundamental

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TABLE I T RANSMISSION LINE DATA

TL1 TL2

router (mm) 20.3454 20.3454

ACSR Conductors rinner (mm) ρ(Ωm) 5.55 38.7 × 10−9 5.55 38.7 × 10−9

Ground Wires router (mm) ρ(Ωm) 5.5245 275 × 10−9 5.5245 275 × 10−9

Length (km) 100 150

TABLE II F UNDAMENTAL FREQUENCY RESULTS

PSCAD HDA

Control variables IT L2+Seq pu Vdcav pu 0.500 0.0634 0.500 0.0635

F iringAngle 24.068◦ 24.049◦

0.01

Vi pu a 0.9799 0.9799

b 0.9789 0.9789

c 0.9758 0.9758

Vj pu a 1.0129 1.0131

b 1.0122 1.0124

c 1.0100 1.0102

R EFERENCES

Realistic TL Fixed Impedance TL 0.009

0.008

a−phase ac current mag, pu

0.007

0.006

0.005

0.004

0.003

0.002

0.001

0

0

5

10

15

20

25 30 Harmonic order

35

40

45

50

Fig. 12. Harmonic domain comparison of the TL representations impact on the a-phase ac side current in TL2, excluding fundamental

less distorted series current in TL2 (Fig. 12), emphasizing the importance of including realistic transmission line representations when simulating FACTS devices.

V. C ONCLUSION This paper has described a generalized harmonic domain modelling technique for series FACTS devices, such as the SSSC. The proposed harmonic domain representation efficiently provides practically identical results to time domain simulation, for small systems including transmission circuits. By avoiding a converters piecewise nature the harmonic domain provides a valuable perspective on converter operation, permitting the development of a more in depth understanding of FACTS devices and their harmonic behaviour. Transmission circuits have been shown to have a significant impact on the harmonic performance of an SSSC, as illustrated by the Jacobian, which describes the increased number and magnitude of harmonic transfers. This mechanism which leads to an increased level of distortion in the adjacent transmission line is not necessarily obvious in the time domain; highlighting one advantage of using harmonic domain simulation. Future work will be primarily focused on extending these investigations to the interactions which occur between multiple FACTS devices across transmission line systems.

[1] IEEE task force,“Characteristics and modeling of harmonic sourcespower electronic devices”, IEEE Trans. on Power Delivery, vol. 16, pp. 791-800, Oct. 2001. [2] N.G. Hingorani,“High Power Electronics and flexible AC Transmission System”, IEEE Power Engineering Review, pp. 3-4, Jul. 1988. [3] N.G. Hingorani,“Flexible AC transmission”, IEEE Spectrum, vol. 30, pp. 40-45, 1993. [4] J.G. Mayordomo, L.F. Beites, R. Asensi, F. Orzaez, M. Izzeddine, and L. Zabala,“A contribution for modeling controlled and uncontrolled AC/DC converters in harmonic power flows”, IEEE Trans. on Power Delivery, vol. 13, pp. 1501-1508, Oct. 1998. [5] E. Acha, and M. Madrigal,Power Systems Harmonics:Computer Modelling and Analysis. New York:Wiley, 2001. [6] M. Madrigal, and E. Acha,“Modelling of custom power equipment using harmonic domain techniques”, Proceedings of the 9th ICHQP, vol. 1, pp. 264-269, 2000. [7] A. Wood, and C. Osauskas, “A linear frequency-domain model of a STATCOM”, IEEE Trans. on Power Delivery, vol. 19, pp. 1410-1418, Jul. 2004. [8] A. Semlyen, E. Acha, and J. Arrillaga,“Newton-type algorithms for the harmonic phasor analysis of nonlinear power circuits in periodical steady state with special reference to magnetic nonlinearities”,IEEE Trans. on Power Delivery, vol. 3, pp. 1090-1098, Jul. 1988. [9] B.C. Smith, N.R. Watson, A.R. Wood, and J. Arrillaga,“Steady state model of the AC/DC convertor in the harmonic domain”, IEE Proc., Gener. Transm. Distrib., vol. 142, pp. 109-118, Mar. 1995. [10] G.N. Bathurst, B.C. Smith, N.R. Watson, and J. Arrillaga,“A modular approach to the solution of the three-phase harmonic power-flow”, Proceedings of 8th ICHQP, vol. 2, pp. 653-659, 1998. [11] J. Arrillaga, E. Acha, T.J. Densem, and P.S. Bodger,“Ineffectiveness of Transmission Line Transpositions at Harmonic Frequencies”, IEE Proc., Part C, vol. 133, pp. 99-104, Mar. 1986. [12] N.G. Hingorani, and L. Gyugyi, Understanding FACTS Concepts and Technology of Flexible AC Transmission Systems. New York:IEEE Press, 1999. [13] C.A. Canizares,“Power flow and transient stability models of FACTS controllers for voltage and angle stability studies”, IEEE Power Engineering Society Winter Meeting, Vol. 2, pp. 1447-1454, 2000. [14] C.D. Collins, A.R. Wood, and N.R. Watson,“Unbalanced STATCOM Analysis in the Harmonic Domain”, Proceedings of the 11th ICHQP, Lake Placid, 2004. [15] N. Mohan, T.M. Undeland, and W.P. Robbins,Power Electronics:Converters, Applications, and Design. New York:Wiley, 1989.

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Jacobian : Fixed Impedance TL

Mismatches

Mismatches

Jacobian : Transmission Lines Modeled

Variables

Fig. 13.

Variables

Comparison of two transmission line models impact on the SSSC Jacobian

Christopher Collins completed his B.E. in electrical and electronic engineering at the University of Canterbury, Christchurch, New Zealand in 2003. He is presently a Ph.D. student studying non-linear harmonic domain analysis of FACTS devices.

Neville Watson received his B.E. and Ph.D. degrees in electrical and electronic engineering from the University of Canterbury, Christchurch, New Zealand. His research interests include steady-state and dynamic analysis of ac/dc power systems.

Alan Wood completed his B.E. and Ph.D. at the University of Canterbury, Christchurch, New Zealand, in 1981 and 1993. From 1982 to 1989 he worked for the Electricity corporation of New Zealand, and then for Mitsubishi Electric in the UK and the Middle East. He is presently a senior lecturer at the University of Canterbury.