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Transient stability and discontinuous solution in electric power system with dc transmission: A study with DAE system Yoshihiko Susuki , Takashi Hikihara , and Hsiao-Dong Chiang Department of Electrical Engineering, Kyoto University Katsura, Nishikyo-ku, Kyoto, 615-8510, Japan Department of Electrical and Computer Engineering, Cornell University Ithaca, NY, 14853, The United States Email: [email protected], [email protected], [email protected] Abstract—This paper focuses on transient stabil- ity of an electric power system with dc transmis- sion. When the transient stability is numerically evalu- ated based on a differential-algebraic equation (DAE) system, associated trajectories become discontinuous since constraint sets are different among correspond- ing pre-fault, fault-on, and post-fault DAE systems. In this paper, several discontinuous solutions are numer- ically and analytically discussed for the DAE system. 1. Introduction This paper addresses transient stability of an elec- tric power system with dc transmission. DC transmis- sion has been widely recognized as a novel technology for future power supply networks [1, 2, 3, 4]. Tran- sient stability of ac/dc power systems is of important concern with their ability to reach an acceptable op- erating condition following an event disturbance. The transient stability is mainly analyzed based on the two different approaches: time-domain (numerical) simula- tion [1, 2] and dynamical system theory [5, 6] involving energy function method [7, 1, 6]. By combing the two approaches, we can obtain sufficient and practical in- formation about the transient stability. The present paper investigates discontinuous solu- tions in a differential-algebraic equation (DAE) system using the numerical simulation. In [5, 6] we examined transient stability boundaries of the ac/dc power sys- tem based on the DAE system. Our previous stud- ies focused on geometric and topological structures of the stability boundaries. Unfortunately, the relation has not been clarified between the stability boundaries and possible system trajectories relative to accidental faults. The understanding of the relation is inevitable in order to apply the obtained results in [5, 6] to prac- tical situations and reveal the transient stability. Here, to clarify the relation, we consider particular discontin- uous solutions caused by accidental faults in the ac/dc system; the solutions have a potential to provide with us a clue about the relation. This paper shows several discontinuous solutions of the DAE system and evalu- ates them via the singular perturbation technique. γ dc transmission generator infinite bus dc(ref) I Figure 1 System model of electric power system with dc transmission 2. Differential-algebraic equation system Figure 1 shows the system model of an electric power system with dc transmission [8]. In [6] we derive the following DAE system as a mathematical model for the transient stability analysis: T 0 d0 L d - L 0 d dv 0 q dt = - U ac ∂v 0 q (v 0 q , δ, θ r ,V r ), dδ dt = Δω, 2H ω dt = -DΔω - U ac ∂δ (v 0 q , δ, θ r ,V r ), L dc dI dc dt = -R dc I dc +K V (e Vr cos α(I dc ) - V i cos γ ), 0 = - U ac ∂θ r (v 0 q , δ, θ r ,V r ) -K I e Vr I dc cos ϕ r , 0 = - U ac ∂V r (v 0 q , δ, θ r ,V r ) -K I e Vr I dc sin ϕ r , 0 = K I e Vr I dc cos ϕ r - K V e Vr cos α(I dc ) - 3 π X c I dc I dc , 2004 International Symposium on Nonlinear Theory and its Applications (NOLTA2004) Fukuoka, Japan, Nov. 29 - Dec. 3, 2004 423

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Page 1: Transient stability and discontinuous solution in electric ... · Transient stability and discontinuous solution in electric power system with dc transmission: A study with DAE system

Transient stability and discontinuous solutionin electric power system with dc transmission:

A study with DAE system

Yoshihiko Susuki†, Takashi Hikihara†, and Hsiao-Dong Chiang‡

†Department of Electrical Engineering, Kyoto UniversityKatsura, Nishikyo-ku, Kyoto, 615-8510, Japan

‡Department of Electrical and Computer Engineering, Cornell UniversityIthaca, NY, 14853, The United States

Email: [email protected], [email protected], [email protected]

Abstract—This paper focuses on transient stabil-ity of an electric power system with dc transmis-sion. When the transient stability is numerically evalu-ated based on a differential-algebraic equation (DAE)system, associated trajectories become discontinuoussince constraint sets are different among correspond-ing pre-fault, fault-on, and post-fault DAE systems. Inthis paper, several discontinuous solutions are numer-ically and analytically discussed for the DAE system.

1. Introduction

This paper addresses transient stability of an elec-tric power system with dc transmission. DC transmis-sion has been widely recognized as a novel technologyfor future power supply networks [1, 2, 3, 4]. Tran-sient stability of ac/dc power systems is of importantconcern with their ability to reach an acceptable op-erating condition following an event disturbance. Thetransient stability is mainly analyzed based on the twodifferent approaches: time-domain (numerical) simula-tion [1, 2] and dynamical system theory [5, 6] involvingenergy function method [7, 1, 6]. By combing the twoapproaches, we can obtain sufficient and practical in-formation about the transient stability.

The present paper investigates discontinuous solu-tions in a differential-algebraic equation (DAE) systemusing the numerical simulation. In [5, 6] we examinedtransient stability boundaries of the ac/dc power sys-tem based on the DAE system. Our previous stud-ies focused on geometric and topological structures ofthe stability boundaries. Unfortunately, the relationhas not been clarified between the stability boundariesand possible system trajectories relative to accidentalfaults. The understanding of the relation is inevitablein order to apply the obtained results in [5, 6] to prac-tical situations and reveal the transient stability. Here,to clarify the relation, we consider particular discontin-uous solutions caused by accidental faults in the ac/dcsystem; the solutions have a potential to provide withus a clue about the relation. This paper shows severaldiscontinuous solutions of the DAE system and evalu-ates them via the singular perturbation technique.

γ

dc transmission

generator

infinitebus

dc(ref)I

Figure 1 System model of electric power system withdc transmission

2. Differential-algebraic equation system

Figure 1 shows the system model of an electric powersystem with dc transmission [8]. In [6] we derive thefollowing DAE system as a mathematical model forthe transient stability analysis:

T ′d0

Ld − L′d

dv′q

dt= −

∂Uac

∂v′q

(v′q, δ, θr, Vr),

dt= ∆ω,

2Hd∆ω

dt= −D∆ω −

∂Uac

∂δ(v′

q, δ, θr, Vr),

LdcdIdc

dt= −RdcIdc

+KV (eVr cos α(Idc) − Vi cos γ),

0 = −∂Uac

∂θr(v′

q, δ, θr, Vr)

−KIeVrIdc cos ϕr,

0 = −∂Uac

∂Vr(v′

q, δ, θr, Vr)

−KIeVrIdc sin ϕr,

0 = KIeVrIdc cosϕr

(

KV eVr cos α(Idc) −3

πXcIdc

)

Idc,

2004 International Symposium on NonlinearTheory and its Applications (NOLTA2004)

Fukuoka, Japan, Nov. 29 - Dec. 3, 2004

423

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where

α(Idc) = Gα(Idc(ref) − Idc),

Uac(v′q, δ, θr, Vr) = −pmδ

−(L′

d − Lq) cos 2(δ − θr) − (L′d + Lq)

2L′dLq

e2Vr

2

−v′

qeVr

L′d

cos(δ − θr) −eVrV∞

L∞

cos θr +e2Vr

2L∞

−V0

Ld − L′d

v′q +

Ld

L′d(Ld − L′

d)

v′2q

2,

or

Mdx

dt= f(x,y), 0 = g(x,y). (1)

x , (v′q, δ,∆ω, Idc)

T ∈ X and y , (θr, Vr, ϕr)T ∈

Y , and M denotes the positive-definite matrix: M ,diag(T ′

d0/(Ld − L′d), 1, 2H,Ldc). In (1) variables and

parameters are normalized with the well known perunit system. Their physical meaning is given in [6].

3. Singularly perturbed and boundary-layersystems

Let us consider the associated singularly perturbed(SP) system as follows:

Mdx

dt= f(x,y), ε

dy

dt= g(x,y), (2)

where ε is a small positive parameter. Let L ,{(x,y) ∈ X ×Y ; g(x,y) = 0} be the constraint set ofthe DAE system. The dynamics of the SP system (2)have the similarity to those of the DAE system (1) ina stable component Γs ⊂ L: Γs is a connected set suchthat for any point in Γs the Jacobian Dyg has no eigen-value with positive and zero real parts. In addition,applying the variable transformation s , ε/t to theSP system (2) and setting ε = 0 freeze the variables atx = x∗, and derive the following boundary-layer (BL)system:

dy

ds= g(x∗,y). (3)

The set of equilibrium points (EPs) in the BL systemcorresponds to L, and the set of asymptotically stableEPs also the union of stable components. This factis directly used to justify the discontinuous solutionstheoretically as discussed below.

4. Transient stability and discontinuous solu-tions

4.1. Faults setting

Two fault cases that we now adopt are as follows:one is a three-phase fault in the ac transmission line,and the generator operates as a result onto the dc link

Table 1 Parameters setting

Ld 1.79 Lq 1.77 L′d 0.34

T ′d0/(120π s−1) 6.3 s V0 1.7 pm 0.5

H/(120π s−1) 0.89 s D 0.05 L∞ 0.883

V∞ 1.0 Ldc 4.2 Rdc 0.014

Vi 1.0 KV 1.19 KI 1.19

Xc 0.12 Gα 30.0 Idc(ref) 1.0

only during the sustained fault. The other is a three-phase fault at the infinite bus, and thereby the infinitebus voltage is fixed at zero in the fault duration. Sup-pose that the system is at a known asymptotically sta-ble EP at t = 0−, the fault duration is confined to thetime [0+, t−cl ], and the fault is cleared at a time t = tcl.It is assumed for simplicity that the post-fault DAEsystem is identical to the pre-fault DAE one. Theabove two cases are now mathematically formulatedin the DAE system (1) for t ∈ [0+, t−cl ]: 1/L∞ = 0(case-1) and V∞ = 0 (case-2), respectively.

Two constraint sets are apparently different betweenthe pre-fault (or post-fault) and fault-on DAE systems.The difference generates discontinuous solutions of theDAE system (1) at t = 0 and tcl; they are called ex-ternal jumps [9, 10]. Jump behaviors or discontinuoussolutions have been discussed for constrained dynami-cal systems in [11, 12, 9]. The previous works [11, 12]define general discontinuous solutions based on associ-ated BL systems. In particular, when the BL systemsare gradient [13], the discontinuous solutions are sim-ply characterized based on the orbit structures of theBL systems. In the following, we use some previousresults in [9, 10] to validate numerical discontinuoussolutions (external jumps) in the DAE system (1).

The discontinuous solutions have bean reportedfor power system transient analysis using structure-preserving models [9, 10]. Practical power systems donot hold such discontinuous states, and they thereforeoriginate from modeling over-abstraction. However,since abstraction is inevitable for analyzing massivelycomplex power networks, understanding how the solu-tions affect the transient stability is essential, e.g., forthe development of controlling UEP method for thepower system analysis [10].

4.2. Numerical simulations

Numerical simulations are performed for the DAEsystem (1). The parameters setting are identical toTab. 1. They are obtained for the practical system[8]. We here adopt the 3rd-stage Radau-IIA implicitRunge-Kutta method [14] to integrate the DAE sys-tem numerically. Since the numerical scheme is one-step type, we possibly obtain numerical discontinuoussolutions although they do not generally hold unique-ness properties.

Figure 2 shows the transient behavior of ∆ω, Vr,and active power with setting the case-1 fault attcl/(120π s−1) = 140 ms. In the figures, 2 denotes

424

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-0.015

-0.010

-0.005

0

0.005

0.010

0.015

0 0.2 0.4 0.6 0.8 1

∆ω

TIME / s

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

0 0.2 0.4 0.6 0.8 1

Vr

TIME / s

-0.2

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1

AC

TIV

E P

OW

ER

TIME / s

generator outputinfinite bus output

dc input

Figure 2 Discontinuous solution of the DAE systemwith case-1 fault setting. The fault-clearingtime tcl/(120π s−1) is fixed at 140 ms.

the initial condition, which coincides with a pre-faultsteady state, and • the state at each t = 0+, t−cl ,

or t+cl. The solution in the figure converges to apost-fault asymptotically stable EP. In Fig. 2, ∆ωin x is continuous, while Vr in y is discontinuous att/(120π s−1) = 0 s and 140 ms. This property is dis-cussed in the next subsection. Fig. 2 also shows theactive power swing in the system. During the faultduration, the active power output of the infinite bus iszero, and the generator output is therefore identical tothe dc power input. Since the generator output dur-ing the fault duration is greater than the mechanicalpower input pm = 0.5, the generator is de-acceleratedas shown in Fig. 2. After the fault is cleared at t = tcl,both generator and infinite bus output show oscilla-tory motions and converge to the values at the steadystate.

The case-2 fault solution is shown in Fig. 3.The solution here converges to a singular surfaceS of the fault-on DAE system: S , {(x,y) ∈L; det(Dyg)(x,y) = 0}. The solution which reaches Sis also possibly discontinuous; the discontinuity hereis qualitatively different from external jumps [9, 10].This result implies an application limit of the DAE sys-tem (1) for the transient stability analysis. It is statedin [15] that the transient stability is determined by thedynamics of the post-fault DAE system. The transientbehavior in Fig. 3 therefore suggests that the presentDAE system (1) is not relevant to clarifying the tran-sient stability problem relative to the case-1 fault. To

-0.7-0.6-0.5-0.4-0.3-0.2-0.1

0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Vr

TIME / ms

-0.6-0.4-0.2

00.20.40.60.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

AC

TIV

E P

OW

ER

TIME / ms

generator outputinfinite bus output

dc input

Figure 3 Discontinuous solution of the DAE systemwith case-2 fault setting

rV

θr

t=t

t=0clt=t t=

cl

− +

+

0−

−0.01

−0.02

−0.03

−0.04

−0.05

−0.06

−0.07

−0.08−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2

DAESP

stable EP

Figure 4 Projected trajectories of the DAE and SPsystems with case-1 fault setting onto θr−Vr

plane. The perturbation parameter ε is setat 0.5. The solid line denotes the trajectoryof the DAE system (1) and the dotted linethat of the SP system (2).

overcome the limitation, we need to model the tran-sient dynamics in detail with taking some equipmentsof the ac/dc system: AVRs, shunt capacitors at con-verter stations, and so on.

4.3. Discontinuous solution and boundarylayer systems

Next we discuss the discontinuous solution of thecase-1 fault setting through the associated SP system(2). As discussed before, the SP system plays a keyrole in validating numerical simulations for the DAEsystem (1). Fig. 4 shows the projected trajectoriesof the DAE and associated SP systems onto θr − Vr

plane. The perturbation parameter ε is set at 0.5. Thesolid line denotes the trajectory of the DAE system(1) and the dotted line that of the SP system (2). Thefigure implies that the trajectory of the SP system (2)precisely traces the discontinuous solution of the DAEsystem (1).

The discontinuous solution of the case-1 fault is nowanalytically justified based on the BL system. Each

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t=

t=0

t=tcl

+

0

t=tcl−

+

rV

stable EPpre BL

post BLDAE

0.05

0.00

0.10

−0.10

−0.05

−0.15

−0.20−0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2

Figure 5 Projected discontinuous solution and tra-jectories of the pre- and post-BL systemswith case-1 fault setting onto θr −Vr plane.The solid line denotes the trajectory of theDAE system (1) and the two dotted linesthose of the pre- and post-fault BL systems.

point in a stable component Γs becomes an asymptot-ically stable EP of the following post-fault BL systemat t = tcl:

dθr

ds= −

∂Uac

∂θr(v′∗

q , δ∗, θr, Vr)

(

KV eVr cos α∗ −3

πXcI

∗dc

)

I∗dc,

dVr

ds= −

∂Uac

∂Vr(v′∗

q , δ∗, θr, Vr)

K2I e2VrI∗2dc −

(

KV eVr cosα∗ −3

πXcI∗dc

)2

I∗2dc ,

(4)

where x(t+cl) , (v′∗q , δ∗,∆ω∗, I∗dc)

T, and we assume

that I∗dc > 0 and KIeVrI∗dc sinϕr > 0; this is rele-

vant to the transient stability analysis. From [12, 9]we here state that if the DAE system (1) admits of thediscontinuous solution at t = tcl, then the trajectoryof the BL system (4) with the initial condition y(t−cl)

converges to the point y(t+cl) in the stable component

Γs satisfying x(t−cl) = x(t+cl); the coincident property

holds in Fig. 2. This follows that the point y(t−cl) is

on a stable manifold of the stable EP y(t+cl) in the BLsystem (4). Fig. 5 definitely describes the trajectoriesof the pre- and post-BL systems and the projected dis-continuous solution of the DAE system. In the figure,all trajectories of the BL systems converge to asymp-totically stable EPs of the BL systems, and these EPscoincide with the starting points of the discontinuoussolution at t = 0+ and t+cl. This implies that our nu-merical discontinuous solution is valid in the analyticalsense.Acknowledgments This research is partially sup-ported by the Ministry of Education, Culture, Sports,Sciences and Technology in Japan, The 21st CenturyCOE Program #14213201, and Grant-in-Aid for Ex-ploratory Research, #16656089, 2004.

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