title application of hybrid system theory to power system

RIGHT:

URL:

CITATION:

AUTHOR(S):

ISSUE DATE:

TITLE:

Application of hybrid system theoryto power system stability analysis

Susuki, Yoshihiko; Ebina, Hiroaki; Hikihara, Takashi

Susuki, Yoshihiko ...[et al]. Application of hybrid system theory to power system stabilityanalysis. Proceedings of the 2005 International Symposium on Nonlinear Theory and itsApplications 2005: 202-205

2005-10

http://hdl.handle.net/2433/73435

Copyright(c)2005 IEICE. The definitive publisher-authenticated versionis available online at: http://www.oishi.info.waseda.ac.jp/NOLTA2005/

Application of hybrid system theory to power systemstability analysis

Yoshihiko Susuki, Hiroaki Ebina, and Takashi Hikihara

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

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

Abstract—This paper shows a basic framework forapplying hybrid system theory to power system sta-bility analysis. Hybrid dynamical systems and controlare nowadays developed in the intersection of com-puter science and control engineering. In this pa-per we discuss an application of hybrid system theoryto stability analysis of power systems, and propose anovel approach to the stability estimation based onthe revolution of reachable sets. This paper appliesthe proposed approach to transient stability analysisof a simple electric power system, thereby showing theeffecitiveness of the approach.

1. Introduction

Nowadays electric power systems become compli-cated in terms of their size, configuration, dynamics,operation, and control: see e.g., [1, 2, 3, 4]. In a techni-cal trend, various power apparatuses including HVDCsystems [5] and FACTS [6] are installed into conven-tional ac power systems. Their apparatuses are basedon switching operation of power conversion devices,and they are expected to contribute the operation ofpower systems. On the other hand, as a non-technicalissue, regulatory reforms of power markets require asubstantial modification of conventional power sys-tems. These technical and non-technical trends ob-viously cause the dynamics of power systems to becomplicated and therefore make it much difficult toanalyze and control the dynamics. A comprehensiveapproach to the analysis and control has been thusstrongly required [2].

Hybrid dynamical systems and their control are at-tracting a lot of interests in the fields of computer sci-ence and control engineering: see e.g., [7, 8, 9]. Hybridautomata are of central concern with hybrid systemsand control. The mathematical formulation is applica-ble to the analysis of various complicated systems thatinvolve the interaction of continuous and discrete dy-namics. Reachability analysis [7] of hybrid automata ishere of paramount importance for safety specificationsof engineering systems: for examples, steam boiler andflight management systems [7, 9].

The objective of this paper is to apply hybrid sys-tem theory to power system stability analysis. Powersystem stability is one of the fundamental concerns insystem planning, operation, and control [10, 11]. Re-cently several researchers have worked on the intersec-tion of power system analysis and hybrid system the-ory: Hiskens and Pai [12] propose a hybrid modeling ofpower systems including transformer tap change and

relay operation; DeMarco [13] proposes a phase transi-tion model for cascading failure via hybrid dynamicalsystems; and Fourlas et. al [14] investigate dynamicresponse of power transmission system via a hybridautomaton model. The present paper discusses a ba-sic framework for stability analysis of complex powersystems based on hybrid system theory. We here pro-pose a novel approach to the stability estimation basedon the revolution of reachable sets. This paper alsoapplies the proposed approach to transient stabilityanalysis of single machine-infinite bus (SMIB) system.Some of the results in this paper are preliminary pre-sented in [15, 16].

2. Basic framework

This section discusses a basic framework for apply-ing hybrid system theory to power system stabilityanalysis based on hybrid system theory [7, 17, 9].

2.1. Hybrid automaton as power system model

A hybrid automaton H is defined to be the tupleH = (Q × X, U × D,Σu × Σd, f, E, Inv, I,Ω) with

• Q × X is the state space, with Q ,q1, q1, · · · , qm a finite set of discrete states andX a n-dimensional manifold; a state of the systemis a pair (qi, x) ∈ Q × X;

• U × D ⊂ Ru × Rd is the product of the set ofcontinuous control inputs and the set of contin-uous disturbances; the space of acceptable con-trol and disturbance trajectories are denoted byU , u(·) ∈ PC0 | u(τ) ∈ U ∀τ ∈ R andD , d(·) ∈ PC0 | d(τ) ∈ D ∀τ ∈ R. PC0 de-notes the space of piecewise continuous functionsover R;

• Σu × Σd is the product of the finite set of dis-crete control actions and the finite set of discretedisturbance actions;

• f : Q×X×U ×D → TX is the vector field whichassociates a control system f(q, x, u, d) with eachdiscrete state q ∈ Q;

• E : Q × X × Σu × Σd → 2Q×X is the discretetransition function;

• Inv ⊆ Q×X is the invariant associated with eachdiscrete state, meaning that the system evolvesaccording to x = f(q, x, u, d) only if (q, x) ∈ Inv;

• I ⊆ Q × X is the set of initial states;• Ω is the trajectory acceptance condition (here

Ω = 2F for F ⊆ Q × X. 2 denotes a map,

Bruges, Belgium, October 18-21, 2005Theory and its Applications (NOLTA2005)

2005 International Symposium on Nonlinear

202

A Self-archived copy inKyoto University Research Information Repository

https://repository.kulib.kyoto-u.ac.jp

called property, from the set of all executions ofH to True,False[7].).

The hybrid automaton H is relevant for modelingan electric power system that includes relay opera-tions, switching operation of power conversion appa-ratuses, and possible disturbances caused by deregu-lation of power markets. In the continuous part in H,f describes the dynamics of rotor angles and voltages,for which we usually use swing equations [10, 11] anddifferential-algebraic equations [18, 10]; u(·) ∈ U is re-garded as the control input such as dc links, SVCs,and so on; and d(·) ∈ D is as a possible and irregulardisturbance caused by deregulation of power markets.On the other hand, in the discrete part in H, E repre-sents the discrete transition of system states (modes);(σu[·], σd[·]) ∈ Σu ×Σd also implies the controlled anduncontrolled line switching by relay operation, acci-dental faults, and so on. The hybrid automaton H isthus applicable to complex power systems which in-volve the interaction of continuous dynamics and dis-crete events. Note that the above description is rele-vant to any aspect of power system analysis: transientstability, voltage stability, multi-swing instability, andso on. In the next section we will deal with transientstability analysis of a simple power system.

2.2. Reachable sets for stability analysis

Here we introduce a new approach for stability es-timation via hybrid system theory. Let us define anunsafe set G ⊂ Q × X for the hybrid automaton H.The unsafe set is interpreted as a subset in which thepower system cannot be safely operated: large rotorswing, stepping-out, low voltage amplitude, and so on.A reachable set Rt(G) (t < 0) for the hybrid automa-ton H is roughly defined to be a subset of Q × Xin which any system state reaches to the boundary∂G of G until at least |t| time despite of the control(u(·), σu[·]): precise discussion of hybrid reachable setsis in [17]. Fig. 1 shows the concept of reachable setsfor continuous systems. An usable part in the figure,for continuous systems, is a part of the boundary ∂Gfor which there exists a disturbance d ∈ D such thatfor all inputs u ∈ U the vector field points into G. Theusable part is utilized to calculate the reachable set inSection 3. The concept of the reachable sets is muchimportant for validating the stability of power systems,because, if a system state exists in Rt(G), then we canevaluate that the system will reach to an unaccept-able operation for any control input. In particular thereachable sets have a great potential to contribute thestability analysis of hybrid power systems caused byline switching, etc. Note that an interesting methodfor calculating time-growth of reachable sets has beenproposed based on a Hamilton-Jacobi equation and alevel set method in [7, 17, 19, 9].

3. Transient stability estimation via reachablesets

This section analyzes transient stability of a singlemachine-infinite bus (SMIB) system, shown in Fig. 2,via reachable sets. The SMIB system consists of a

PartG

Usable

flow

(a) Usable part reachable set

flow

t

G PartUsable

(b) Reachable set

Figure 1 Concept of reachable sets for continuoussystems

eInfinite bus

E οοejj 0δ E

PM

Generator Transmission line

Figure 2 Single machine-infinite bus (SMIB) system

synchronous machine, an infinite bus, and two paralleltransmission lines.

3.1. Mathematical model

The electro-mechanical dynamics of synchronousmachine is described by the following swing equationsystem:

δ = ω,ω = PM − b sin δ − kω,

(1)

where δ is the rotor position with respect to syn-chronous reference axis, and ω the rotor speed devia-tion relative to system angular frequency. PM denotesthe mechanical input power to generator, b the crit-ical transmission power of SMIB system, and k thedamping coefficient in generator. The derivation ofthe system (1) is given in [11]. This section uses thefollowing parameters setting [20]:

b = 0.7, k = 0.05, PM = 0.2. (2)

3.2. Numerical simulation

3.2.1. Continuous case

First of all we examine reachable sets for continuousswing equation system (1). Let us define the unsafe setG with the boundary ∂G as follows:

G , (δ, ω) ∈ S1 × R1 | ω2

c − ω2 ≤ 0,∂G , (δ, ω) ∈ S1 × R1 | ω2

c − ω2 = 0, (3)

where ωc = π. Any state in G physically implies anunacceptable operation of the SMIB system becauseof the large value of ω. The usable part UP of ∂G isthen derived as follows:

UP = (δ, ω) ∈ ∂G |∓2π(PM − b sin δ ∓ πk) < 0 . (4)

203



-3

-2

-1

0

1

2

3

0-π π

ω

δ

(a) TR = 1.37 s

-3

-2

-1

0

1

2

3

0-π π

ω

δ

(b) TR = 1.65 s

-3

-2

-1

0

1

2

3

0-π π

ω

δ

(c) TR = 1.86 s

-3

-2

-1

0

1

2

3

0-π π

ω

δ

(d) TR = 2.06 s

Figure 3 Growth of reachable set in continuous swingequation system

-3

-2

-1

0

1

2

3

0-π π

ω

δ

Figure 4 Reachable set and stability region in contin-uous swing equation system. The red dot-ted closed loop is an analytical criterion forstability based on closest u.e.p. method.

The derivation of UP is based on [7, 17, 9]. Fig. 3shows the growth of reachable set of G for continuousswing equation system (1). TR in the figure denotesthe time until at least which any state in the reach-able set reaches to ∂G. As TR increases, the reachableset expands into the phase space; actually the comple-ment of the reachable set approximates the stabilityregion of an asymptotically stable equilibrium point(sin−1(PM/b), 0) as time goes to infinity. This is con-firmed in Fig. 4; the red closed loop is the sufficientcondition for transient stability based on closest u.e.p.method [10]. The complement of the obtained reach-able set in Fig. 3 hence corresponds to a necessarycondition for the transient stability of the SMIB sys-tem.

3.2.2. Hybrid case

This section discusses hybrid reachable sets relatedto re-closing operation of transmission lines. Fig. 5shows a switching sequence for the SMIB system. Thethree modes are as follows: (i) fault-on is the systemstate during a sustained fault on one line; (ii) 1 lineoperation is the state after clearing the fault line byrelay operation; and (iii) 2 line operation is the state

ω =P −Dδ ω=

M

δ ω.ω

=. δ ω.

=ω. MM

t=0 clearing

ω

.

.

ct <t<tr0<t<t c t <t<tr

1 line 2 lineoperationoperation

=P −(b/2) sin −Dδ ω =P −bsin −Dδ ω

re closing−

fault−on

Figure 5 Switching sequence governing SMIB systemwith clearing and re-closing operations

after re-closing the line. tc denotes the fault-clearingtime, and tr the re-closing time. In the figure, we canregard the fault clearing and circuit re-closing as thediscrete transitions, which drive depends on time vari-able only, in the hybrid automaton. In the swing equa-tion system (1), the fault-on, 1 line operation, and 2 lineoperation modes coincide with the parameter settings:b = 0.0, 0.35, 0.7, respectively.

We are in a position to investigate hybrid reachablesets of the unsafe set G at ωc = 2.0 for each mode.The transmission line is here re-closed after the re-closing period Trc , tr − tc. The derived reachable setis then decomposed into the two subsets Rbefore andRafter; Rbefore is the subset of S1 ×R1 from which anystate reaches to ∂G before the re-closing; and Rafter

is the one from which any state reaches to ∂G afterthe re-closing. Fig. 6 shows the two reachable subsetsRbefore and Rafter under the condition Trc = 0.5 s. Theproduct of the two subsets corresponds to the hybridreachable set of G.

-2

-1

0

1

2

0 π-π

ω

δ

(a) Rbefore

-2-1.5

-1

0

1

2

0 π-π

ω

δ

(b) Rafter

Figure 6 Reachable set in swing equation system un-der re-closing time Trc = 0.5 s

Figure 7 describes how the reachable set changes asthe re-closing period Trc increases. The red closed loopstands for the sufficient condition for transient stabil-ity of 1 line operation. In Fig. 7 we can observe thatsome of the system states can survive by the slow re-closing Tcr = 0.3 s rather than the fast one Tcr = 0.1 s:see the neighborhoods of (−π/2,−0.5) in Figs. 7 (a)and (b). This observation is not given by any classi-cal method of transient analysis. Fig. 7 thus suggeststhat the reachability analysis makes it possible for usto estimate the transient stability taking the switchingoperation into account.

4. Summary

This paper showed a basic framework for applyinghybrid system theory to power system stability anal-

204



-2

-1

0

1

2

0 π-π

ω

δ

stability limit

(a) Trc = 0.1 s

-2

-1

0

1

2

0 π-π

ω

δ

(b) Trc = 0.3 s

-2

-1

0

1

2

0 π-π

ω

δ

(c) Trc = 0.5 s

-2

-1

0

1

2

0 π-π

ω

δ

(d) Trc = 0.8 s

Figure 7 Hybrid reachable sets in swing equation sys-tem. The red closed loop stands for thesufficient condition based on closest u.e.p.method.

ysis. The mathematical model via hybrid automatais much relevant to the analysis and control of fu-ture power networks involving power conversion ap-paratuses, deregulation of power markets, and so on.This paper also performed the transient stability esti-mation based on the revolution of reachable sets. Ourfuture direction is to apply the present approach topractical system analysis [21].

Acknowledgments

The authors are grateful to Mr. Takuji Uemura,Mr. Takashi Ochi, and Mr. Kosaku Yokota in KansaiElectric Power Company Ltd. for valuable discussions.This research is partially supported by the Ministry ofEducation, Culture, Sports, Sciences and Technologyin Japan, The 21st Century COE Program (GrandNo. 14213201).

References

[1] S. Strogatz. Exploring complex networks. Nature,410:268–276, March 2001.

[2] P. Fairley. The unruly power grid. IEEE Spectrum,41(8):22–27, August 2004.

[3] I. Dobson, B. A. Carreras, V. E. Lynch, and D. E.Newman. Complex systems analysis of series ofblackouts: Cascading failure, criticality, and self-organization. In Proceedings of the Bulk Power SystemDynamics and Control–VI, 2004.

[4] C. W. Gellings and K. E. Yeager. Transforming theelectric infrastructure. Physics Today, 57(12):45–51,December 2004.

[5] K. R. Padiyar. HVDC Power Transmission Systems:Technology and System Interactions. Wiley Eastern,New Delhi, 1990.

[6] CIGRE and IEEE FACTS Working Groups. FACTSOverview, IEEE Catalog #95-TP-108, 1995.

[7] J. Lygeros, C. Tomlin, and S. Sastry. Controllers forreachability specifications for hybrid systems. Auto-matica, 35(3):349–370, March 1999.

[8] M. Domenica, D. Benedetto, and A. Sangiovanni-Vincentelli, editors. Hybrid Systems: Computationand Control, Lecture Notes in Computer Science 2034.Springer-Verlag, 2001.

[9] C. J. Tomlin, I. Mitchell, A. M. Bayen, and M. Oishi.Computational techniques for the verification of hy-brid systems. Proceedings of the IEEE, 91(7):986–1001, July 2003.

[10] H. -D. Chiang, C. -C. Chu, and G. Cauley. Directstability analysis of electric power systems using en-ergy functions: Theory, applications, and perspective.Proceedings of the IEEE, 83(11):1497–1529, November1995.

[11] J. Machowski, J. W. Bialek, and J. R. Bumby. PowerSystem Dynamics and Stability. John Wiley & Sons,England, 1997.

[12] I. A. Hiskens and M. A. Pai. Hybrid systems view ofpower system modeling. In Proceedings of the Inter-national Symposium on Circuits and Systems, pagesII–228–II–231, 2000.

[13] C. L. DeMarco. A phase transition model for cascad-ing network failure. IEEE Control Systems Magazine,21(6):40–51, December 2001.

[14] G. K. Fourlas, K. J. Kyriakopoulos, and C. D. Vour-nas. Hybrid systems modeling for power systems.IEEE Circuits and Systems Magazine, 4(3):16–23,Third Quarter 2004.

[15] T. Hikihara et. al. Application of hybrid system the-ory to power system analysis (I) and (II). In AnnualMeeting Record I.E.E.Japan, volume 6, pages 187–189, 2005. (in Japanese).

[16] H. Ebina et. al. A study on transient stability ofelectric power system based on reachable set analy-sis. In Proceedings of the 49th Annual Conference ofthe ISCIE, pages 25–26, 2005. (in Japanese).

[17] C. J. Tomlin. Hybrid control of air traffic managementsystems. PhD dissertation, University of California atBerkeley, 1998.

[18] A. R. Bergen and D. J. Hill. A structure preserv-ing model for power system stability analysis. IEEETransactions on Power Apparatus and Systems, PAS-100(1):25–35, January 1981.

[19] I. Mitchell and C. Tomlin. Level set methods forcomputation in hybrid systems. In B. Krogh andN. Lynch, editors, Hybrid Systems: Computation andControl, Lecture Notes in Computer Science 1790,pages 310–323. Springer-Verlag, 2000.

[20] Y. Susuki and T. Hikihara. An analytical criterionfor stability boundaries of non-autonomous systemsbased on melnikov’s method. Transactions of theISCIE, 15(11):586–592, November 2002.

[21] T. Uemura et. al. Application of hybrid system theoryto power system analysis (III). In Annual MeetingRecord I.E.E.Japan, volume 6, page 190, 2005. (inJapanese).

205



title application of hybrid system theory to power system

Documents