An Impedance-Based Approach to HVDC System ... - IEEE Xplore

The 2014 International Power Electronics Conference

An Impedance-Based Approach to HVDC System Stability Analysis and Control Development Hanchao Liu,

Shahil Shah and Jian Sun

Department of Electrical, Computer and Systems Engineering RensselaerPolytechnic Institute, Troy, NY 12180-3590, USA Telephone: (518) 276-8297; Fax: (518) 276-2387; E-mail: [email protected] Abstract-

High-voltage

dc

(HYDC)

transmission

distributed power systems where multiple loads are powered from a central voltage source [7]. The method has not been used in HYDC system, partially because of the lack of proper impedance models. Impedance models of line-commutated converters have been developed by either neglecting the effects of phase control (constant firing angle assumption) [8] or with significant simplifi­ cation of control functions [9, lO]. However, such simplistic models are accurate only in a limited frequency range and are not suitable for stability analysis. Recently, the impedance-based stability criterion has been extended to current-source converters [11]. Meantime, new analytical models have been developed for VSC and LCC converters dc-terminal impedance [12] using the principle of harmonic linearization [13], making it possible now to study HYDC system stability based on impedance.

system

stability has traditionally been studied using state-space models and eigenvalue analysis. The state-space approach requires design details of each component of the HYDe grid, which is usually not available, and does not support local control development at individual terminals. This paper presents an impedance-based approach to HYDe system stability analysis that is easier to apply and supports local and adaptive control development. Represen­ tation of a HYDe system by an impedance equivalent circuit is presented first. Analytical impedance models are then developed for different HVDe converters to support system impedance analysis. The ability of impedance-based analysis to predict and mitigate different control and system instability problems are demonstrated and verified by detailed circuit simulation.

I.

INTRODUCTION

The objective of this paper is to demonstrate the ability of impedance-based approach to detect various control and system instability problems in LCC and VSC based point-to-point HYDC systems. The rest of the paper is organized as follows: Section II reviews the impedance-based approach for the stability analysis and discusses its applicability to the HYDC link stability analysis. Section III presents the small-signal impedance models for LCC based terminals and their application to evaluate the dc-link stability in LCC based systems. Section IV does the same for VSC based systems. Section V summarizes the work.

Control development for traditional point-to-point HYDC connections based on line-commutated converters (LCC) has emphasized steady-state power flow and operation under different grid conditions [1]. Stability of the dc link has not been a major concern because of the simple system topology and slow control functions. Control instability of individual terminals is a potential problem when connected to a weak grid, which has been analyzed using state-space models and eigenvalue analysis [2, 3]. This approach has influenced the more recent development in multi­ terminal HYDC systems based on voltage-source converters (VSC) [4, 5]. Although stability is considered as a much more significant problem due to the fast dynamics of VSC control, the basic approach is still eigenvalue analysis based on state-space models of the entire HVDC grid. Establishing such state-space models requires design details of each terminal but such design details are usually not available to system owners, particularly when mUltiple suppliers are involved. The complex system model also makes it difficult to associate any unstable (or marginally stable) mode with physical design parameters, which is essential for solving an instability problem. Additionally, state-space analysis can only be performed centrally and in advance. It does not support local control development at individual terminals, especially when online adaptation based on real-time grid topology and operation conditions is required.

This section uses an example to illustrates possible instability in HYDC systems and introduces the impedance-based system stability analysis method. Fig. 1 depicts a LCC-based HYDC system that will be studied. Both the rectifier and the inverter are rated for 500 kV and lOOO MW, and each uses a 12-pulse LCC converter. The ac grid at each terminal is modeled by an RL circuit having the same short circuit ratio (SCR) of 2.5. The transformer is rated for 345/211 kV and 230/211 kV, respectively. G(s) is the transfer function of the current/voltage sensing circuit. It is assumed that the inverter controls the dc link voltage, while the rectifier regulates its output current. A dq-domainPLL is used to track the ac bus voltage reference angle for phase control. The system parameters are presented in [13].

This paper presents an impedance-based approach to HYDC system stability analysis which solves the aforementioned problems. The impedance-based approach has been used in power electronics to study converter-filter interactions [6] and stability of

For the purpose of illustration, the rectifier current compensator is designed such that it operates stably when feeding current into an ideal voltage source. Similarly, the inverter voltage compen­ sator is designed such that it operates stably when fed by an ideal

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II. HVDC

967

SYSTEM INSTABILITY

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Firing Pulses Train

r--'----'--......., � w(t)

t fref

��-

Rectifier

,....----,

Cd

vdc

Zj

�

-

345 kV: 211 kV xI 0.15 p.u. =

Inverter

-

Zj

211 kV: 230 kV xI 0.15 p.u.

-

=

Fig. I. Simplified diagram of a Lee-based point-to-point HYDe system.

current source. Simulation of each converter confmned this design objective. However, it is found that when they are connected as shown in Fig. 1, the dc link voltage and current become unstable.

Fig. 2 shows the simulated unstable behavior when the dc current ramps up to its rated value during start-up. To identifY the cause of system instability, Fig. 3 shows an equivalent circuit model of the system where the rectifier is modeled as current source in parallel with an output impedance Zrec' while the inverter is modeled as voltage source in series with an input impedance Zinv' Note that this equivalent circuit resembles that of an inverter connected to a non-ideal grid [11]. Therefore, the impedance-based stability criterion developed in [11] for grid-connected inverters can also be applied to detennine point-to-point HYDC system stability. Specifically, an HYDC system consisting of a current-controlled rectifier and voltage­ controlled inverter is stable if

2.2 = 2.0

�

u

1.8

1) The rectifier control is stable when feeding current into an ideal voltage source; 2) The inverter control is stable when being fed by an ideal current source; and 3) The ratio of the inverter input impedance to the rectifier output impedance satisfies the Nyquist stability criterion.

7 >

r--

..:.:: N 0 ...... '-'
In the following two sections, we will use this impedance­ based criterion to study LCC and VSC HYDC system stability. Since the method requires an impedance model for each converter

6 5

�

�

4 3

Fig. 2. De link current and voltage responses of the example Lee point-to­ point HYDe system depicted in Fig. I.

Fig.3. Small-signal impedance model of a rectifier-inverter system.

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The 2014 International Power Electronics Conference

tenninal, each section will start from the development of the rectifier output impedance and the inverter input impedance, followed by numerical simulation of possible instability and its characterization using the impedance models. III. LCC HYDC

Fig. 5 shows the validation of the impedance model of the rectifier. The impedance model of the inverter and its validation are presented in (5) and Fig. 6 in the following page. B.

SYSTEMS

Stability Analysis

As explained in Section II, the dc link stability of the HYDC system depends on the ratio between the inverter impedance Zinv and the equivalent impedance Ze v which is the dc network q impedance seen by the inverter including the dc link impedance and and the rectifier output impedance. In the unstable case (Case I) presented in Section II, the dc current compensator transfer function is designed to be

A. Impedance Modeling Fig. 4 depicts detailed ac side equivalent impedance of the LCC HYDC rectifier tenninal. The ac grid is modeled by an RL circuit with short circuit ratio (SCR) set to 2.5 and 75 degrees at the fundamental frequency (60 Hz). The phase control design is presented in Fig.l. The ac filters are designed to absorb 11th and 13th characteristics hannonics current and provide low impedance path at 3rd harmonics to avoid low-oder harmonics resonance problems.

(Ld

Cd)

H(s)

=

(3)

l.6+ 19.3/s,

and the dc voltage compensator transfer function is H(s)

The modeling process and circuit parameters have been presented in [13] and shown in (1), where L(s) H(s)G(s), kT is the voltage ratio of the converter transfonner; Zac represents the ac side impedance seen by the converter; and a ao + 5 7t / 6 , ao is steady-stage firing angle. The dynamic of thePLL is charac­ terized by the loop gain

=

(4)

2+ 0.2/S.

=

=

is' 60 � � 50 "0

(2)

.a .i:J � :;8

where VI is the amplitude of the grid voltage and HplJ (s) is the PLL compensator transfer function.

Lgrl

I Crl �

I Cr2 Lrl

b)

Lr2

Rrl Cr4

30 10Hz

Rgr2

a)

40

Rr2

oed

"""'

0 �

200

II

8150 (]) '" oj

Rr3

..c: p..

-

100

�

�

-r

f\ \

8

�-rI ��I

V II

1 kHz

100Hz

II

10 Hz

100Hz

1 kHz

Frequency Fig. S. Lee rectifier output impedance responses. Solid lines: model predic­ tion; Dots: point by point simulation results.

Fig. 4. Lee rectifier ac side impedance: a) ac grid equivalent circuit; b) recti­ fier filters circuit.

(1)

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L(s)/ J3

_

1 +Jla Tpll(s) [Zac(S +)2rrjj)e-ju -Zac(s - )2rrjj)ju]

- 3 J3 cosa VI + {18)Ia[Zac(s +)2rr/l) -Zac(S -)2rr/I)] }/(kTrr)

(5)

10 Hz II 300

.......

§' 200 I-B (])

� 100

..c: �

=Ftr--- ��

�

o I--- ----+

-

10Hz

100Hz

1 kHz

10Hz

100Hz

1 kHz

I

,...... -

I I

1 k Hz

100Hz

r

10 Hz

-f-

I

I I

V' 1

r-\J

-

.

r� r-�

100 Hz

H I

/0' 1 k Hz

Frequency

Frequency

Fig. 6. LCC inverter input impedance response. Solid lines: model prediction; Dots: point by point simulation results.

Fig. 7. Impedance analysis of the LCC system. Dashed line: Case I. Solid lines: Case II.

margin of 25 degrees which is corresponding to the resonance frequency of 30 Hz in Fig.2. In Case II, the two impedances Zinv and Zeqv intersect at 28 Hz, with a phase margin of 45 degrees which shows a stable system design. In order to show the effect of the rectifier control, the passive dc link impedance Zdc(s) is also shown in Fig. 7, where

In addition to the unstable case, a stable system design case is also presented as Case II in which the dc voltage control compen­ sator transfer function is designed to be

H(s) In

=

0.2 + 0.2/ s.

(6)

both of two cases, the inverterPLL compensator is

Zdc

(7) In the Bode plots of Fig. 7, Zeqv includes the large dc smoothing reactor but still shows capacitive at the low frequency. This capacitive behavior is due to integral gain of the current controller. From the Fig. 7, it can be observed that in the unstable case I the impedance Zinv and Zeqv intersect at 30 Hz, with a phase

=

sLd+

sLd 2

1 +s LdCd

(8)

It can be found that in the low frequency range where the resonance happens, the impedance of the rectifier greatly changed the dc impedance seen by the inverter.

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IV. vse HVDe

SYSTEMS

A 300 MW, 300 kV point-to-point vse HYDe system is shown in Fig. 8. Similar to the Lee system, the inverter terminal regulates the dc link voltage and the rectifier terminal controls the power flow on the dc link. Moreover, the rectifier terminal is designed to operate stably when supplying power to an ideal voltage source and the inverter terminal is designed to operate stably when being fed by an ideal current source. Following subsections develop terminal impedance models and apply them to the dc link stability analysis.

Modulation Signals

VSC Rectifier Output Impedance Modeling

The circuit diagram of vse based rectifier along with control implementation is shown in Fig. 9. The vse rectifier controls the active power flow on HVDe link by outer power control loop with 1 Hz bandwidth. The inner dq-current control is designed with 200 Hz bandwidth (10 % of the switching frequency). To model the rectifier output impedance, a small perturbation voltage signal, vs' is injected in the dc-link voltage. Based on power balance between the input and output of the vse, we have

Fig. 9. VSC rectifier circuit with active power control.

where •

•

(9)

•

•

The modeling process is simplified by assuming the ideal grid and neglecting the perturbations in the grid voltage. The assumption makes ac-side current perturbations to depend only on the rectifier bridge input voltages. It also ignores the effect ofPLL dynamics. The ac-side current perturbations are mapped back to the dc-side current using (9). The ratio of injected voltage perturbation to the resulting dc current perturbation gives the output impedance. The resulting rectifier output impedance is found to be

•

•

is the nominal output power;

OJ

is the dq-current control bandwidth;

e

Vp is the nominal phase voltage amplitude; HpCs) is the active power loop compensator and Zph

is the phase reactor impedance.

300 kV HVDC Transmission

vr,grid

400 kV/ 150 kV

400 kV Bus

Po

The simplified expression of impedance is obtained by assuming that thePI current control compensator has zero placed to cancel the pole introduced by the phase inductor impedance and its gain is designed to give bandwidth of OJe- Fig. 10 shows the comparison of the model prediction and the impedance obtained by point-by-point simulations for different grid conditions. As can be seen that the model is accurate for strong grid, but weak grid introduces resonant peaks not captured by the model. For large dc-link capacitor at the front of vse rectifier, these resonant peaks will not be seen by the cable and the HVDe inverter; hence model can be used for dc link stability analysis.

(10)

Zr,grid

Vs is the nominal dc-link voltage;

Rectifier

vi,grid Inverter

vr

vi

Zr  s  Z eqv  s  Z inv  s 

Zi  s 

Fig. 8. Single-line diagram of a point-to-point VSC-HVDC system.

971

150 kV/ 400 kV

400 kV Bus

A.

Ae Filter + Transformer + Grid

Zi,grid

The 2014 International Power Electronics Conference

,-.., � '-'

40

I I

fi.


.�

"0

�cI

0

(\l

�

.�!=;

o� 00

10 0

:::s


\

i.../\.

��

I

\rl

10Hz

Frequency (Hz)

lOO f"'i" 0 � 0 '-'

r---


�

\

0

� o '-'

':\

-50 - lOO

lOHz

·0

\ ...

i: -50 1kHz

Fig. 13. Comparison of VSC inverter input impedance responses (including ter­ minal capacitor of 1 mF): predicted by analytical model (solid lines) and point­ by-point simulation results for SCR of 2.5 (circles).

\

�r--

1Hz

Frequency (Hz)

c.

Leqv


�ooo

lOOHz

0

� .L. I ·

Z , inv

50

6'

"\

1Hz

-

-

50

1kHz

100Hz

10Hz

'\

I--

1kHz

100Hz

Frequency (Hz) Fig. 14.Jmpedance-based analysis of VSC system; inverter input impedance (solid lines), equivalent impedance seen by inverter terminal (dashed lines).

VSC HVDC System Stability

The dc-link stability in vse systems is analyzed using impedance models from (10) and (11). Same as Lee based systems, the dc-link stability depends on the ratio of inverter input impedance (Zinv) to the impedance of the remaining network seen by inverter (Zeqv) as shown in Fig. 8. The Bode plots in Fig. 14 compare Zinv and Zeqv and the impedance-based analysis predicts a damped resonance at the natural frequency of 110 Hz with 30 degrees of phase margin. Fig. 15 shows the simulated response of the dc-link voltage when at t 1 s, the rectifier output power is increased from 20 MW to 300 MW, confirming the predicted resonance.

:>

320

:::, 300 1 +----;-,
� �

280 260 l.0

V.

1.1

l.05

1.15

t (s)

=

Fig. 15. Time domain simulation of the damped resonance.

SUMMARY

REFERENCES

This paper showed the applicability of impedance-based approach to the stability analysis of point-to-point HVDe systems. Both Lee and vse are considered and impedance models required for stability analysis are developed. Impedance-based approach to stability can provide an analytical tool for the control design of HVDe terminal to potential unstable modes and ensure dc link stability for different operation conditions.

[I]

S. Shah, R. Hassan and J. Sun, "HVDC transmission system architectures and control - a review," in Proc. of2013 IEEE COMPEL Workshop, June 2013.

[2]

[3]

[4]

973

D. Jovcic, N. Pahalawaththa, and M. Zvahir, "Stability analysis of HVDC control loops," in lEE Proceedings - Generation. Transmission and Distribution, vol. 146, no. 2, pp. 143-148, March 1999. A. E. Hammad, "Analysis of second harmonic instability for the Chateauguay HVDC/SVC scheme," IEEE Trans. on Power DelivelY, vol. 7, pp. 410-415, Jan. 1992. R. S. Whitehouse, 'Technical challenges of realizing multi-terminal networking with VSC," in Proceedings of EPE 2011, Sep. 2011.

The 2014 International Power Electronics Conference [5]

G. O. Kalcon, G. P. Adam, O. Anaya-Lara, S. Lo and K. Uhlen, "Small­ signal stability analysis of multi-tenninal YSC-based dc transmission system," IEEE Trans. Power Sys. , vol. 27, no. 4, pp. 1818-1830, Nov.

[9]

[10]

2012. [6]

[7]

[8]

R. D. Middlebrook, "Input filter considerations in design and application of switching regulators," in Proc. IEEE Ind. Appl. Soc. Annu. Meeting, 1976, pp. 366-382. X. Feng, J. Liu, and F. C. Lee, "Impedance specification for stable dc distributed power systems," IEEE Trans. Power Electron. , vol. 17, no. 2, pp. 157-162, Mar. 2002. F. Luo, J. Li, Z. Xu, Y. Li, J. Zhang and S. Liu, "Study on impedance frequency characteristics of HYDC filter commutate converter," in Proc.

[11]

[12]

[13]

of 2008 Electric Utility Deregulation and Restructuring and Power Technologies Conference,

pp. 1652-1656, April 2008.

974

A. R. Wood and 1. Arrillaga, "Composite resonance: a circuit approach to the waveform distortion dynamics of an HYDC converter," IEEE Trans. Power DelivelY, vol. 10, pp. 1882-1888, Oct. 1995. A. R. Wood and J. Arrillaga, 'The frequency dependent impedance of an HYDC converter," IEEE Trans. Power Delivry, vol. 10, pp. 1635-1641, July 1995. J. Sun, "Impedance-based stability criterion for grid-connected inverters," IEEE Trans. Power Electronics, vol. 27, no. 11, pp. 30753078, November 2011. H. Liu and 1. Sun, "Modeling and analysis of dc-link harmonic instability in LCC HYDC systems," in Proc. of 2013 IEEE COMPEL Workshop, June 2013. H. Liu and 1. Sun, "DC terminal impedance modeling of LCC HYDC converters," in Proc. of2013 JEEE COMPEL Workshop, June 2013.