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Analysis and Detection of Forced Oscillation in Power System

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Analysis and Detection of Forced Oscillation in Power System IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 32, NO. 2, MARCH 2017 1149 Analysis and Detection of Forced Oscillation in Power System Hua Ye, Member, IEEE, Yutian Liu, Senior Member, IEEE, Peng Zhang, Senior Member, IEEE, and Zhengchun Du, Member, IEEE Abstract—To mitigate forced oscillation and avoid confusion due to malfunction of power system stabilizer (PSS) [2] and me- with modal oscillation, fundamentals of forced oscillation in multi- chanical oscillations of generator turbines [3], [9]. The causes machine power system are investigated. First, the explicit formula- of mechanical oscillations are complex and strongly related to tion of the oscillation is formulated in terms of forced disturbances and system mode shapes. Then, forced oscillation amplitude, thermal process. Specifically, the major contributors may components and envelope are intensively studied. Consequently, include unsteady combustion of boiler [10], turbo-pressure measures for mitigating the oscillation are obtained. Forced os- pulsations [11], undesirable steam turbine valve discharge char- cillation can also be effectively detected and discriminated from acteristics [12]–[14], etc. Besides, wind power may fluctuate modal oscillation by utilizing its uniqueness of components prop- periodically and becomes a potential forced oscillation source erties and envelope shapes. Study results of the 10-machine 39-bus New England test system and a real-life power system demonstrate because of wind shear and tower shadow effects [15], [16] the correctness of theoretical analyses and effectiveness of detection and vibration of floating offshore wind turbine [17]. There- methods for forced oscillation. fore, forced oscillation becomes not only an important issue of Index Terms—Beat frequency oscillation, eigen-analysis, enve- power system but also one concern of integrating wind energy lope, Forced oscillation, low frequency oscillation, resonance. into modern power grid. Since forced oscillation is significantly different from modal oscillation in nature, sources and control strategies, it is essen- I. INTRODUCTION tial to study its fundamentals and take unique countermeasures EYOND the well-known modal (or natural, free) oscil- to mitigate it. In addition, the rationale behind the similar ap- B lations that are excited by random load fluctuations and pearances of forced oscillation and undamped modal oscilla- sudden network switchings [1], sustained forced oscillations tion should be fully understood to avoid possible confusion. So can emerge when power system is perturbed by periodic dis- far, forced oscillation is still an open problem in power system turbances at frequencies close or equal to natural frequencies community and few literatures are established on its fundamen- of system modes [2], [3]. Such oscillations have been observed tals. Methods for detection and frequency estimation of forced from the western North American Power System, US West- oscillations are proposed in [18]–[21]. A spectral approach is ern Electricity Coordinating Council system, and the Nordic presented in [22] to distinguish forced and modal oscillations. In power system [4]. Compared with modal oscillation, forced os- [23], interactions between forced oscillation and system mode cillation exhibits extremely higher amplitude and may cause in the context of resonance are studied by numerical simulations catastrophic blackout, especially in the poorly damped operat- rather than theoretical analyses. ing condition. The origins of forced oscillation are sinusoidal To fill the gap in understanding forced oscillation, this paper in nature, including cyclic loads [5]–[8], electrical oscillations studies the fundamentals of forced oscillation in multi-machine power system. First, its explicit expression is derived by directly Manuscript received October 8, 2015; revised January 24, 2016 and May solving system state equation and formulated in terms of forced 7, 2016; accepted June 12, 2016. Date of publication June 14, 2016; date of disturbances and system mode shapes. Based on the expression, current version February 16, 2017. This work was supported in part by the Natural Science Foundation of China under Grant 51107073 and the Special- amplitude, components and envelope of forced oscillation are ized Research Fund for the Doctoral Program of Higher Education of China then intensively analyzed. Consequently, measures for mitigat- under Grant 11001311038, and Young Scholars Program of Shandong Uni- ing the oscillation are obtained. Besides, two methods based versity (YSPSDU). The work of P. Zhang was supported in part by the U.S. National Science Foundation under Grant CNS-1419076. Paper no. TPWRS- on uniqueness of components properties and envelope shapes 01410-2015. respectively are presented for detection of forced oscillation. H. Ye and Y. Liu are with the Key Laboratory of Power System Intelligent The contributions of this paper are threefold. First, how sys- Dispatch and Control of Ministry of Education, Shandong University, Ji’nan 250061, China (e-mail: [email protected]; [email protected]). tem mode affects amplitude of forced oscillation is fully un- P. Zhang is with the Department of Electrical and Computer Engineer- derstood by formulating and analyzing its explicit expression. ing, University of Connecticut, Storrs, CT 06269 USA (e-mail: peng.zhang@ Second, a distinction between forced and modal oscillations is uconn.edu). Z. Du is with the Department of Electrical Engineering, Xi’an Jiaotong Uni- made by comparing their differences in component quantities, versity, Xi’an 710049, China (e-mail: [email protected]). frequencies and damping ratios. It enables one to detect forced Color versions of one or more of the figures in this paper are available online oscillation excited by multiple disturbance sources. Third, the at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPWRS.2016.2580710 envelope of forced oscillation is intensively investigated. The 0885-8950 © 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information. 1150 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 32, NO. 2, MARCH 2017 resultant envelope shapes are desirable for detection of forced Since only the force oscillation is of our interests in this oscillation in the case of single forced disturbance by visual paper, a zero initial state deviation ∆x(0) is assumed in (4). inspection. Accordingly, only ∆x2 (t) is presented in ∆x(t). The remainder of the paper is organized as follows. In the following Sections II-C and II-D, ∆x2 (t) is first ex- Section II derives the explicit formulation of forced oscillation. panded and then reduced by utilizing the eigen-analysis theory, Its amplitude, components and envelope are analyzed in leading to an explicit expression of forced oscillation. Sections III–V, respectively. Studies of forced oscillations in the 10-machine 39-bus New England test system and a practical C. Expansion of Forced Oscillation power system are given in Sections VI and VII, respectively. Section VIII draws conclusions of this paper. ∆x2 (t) can be reformulated as follows by substituting the eigen-decomposition A = ΦΛΨ into (4): II. EXPLICIT FORMULATION OF FORCED OSCILLATION t Φ Λ(τ t)Ψ In this section, the core of this paper, i.e., explicit formulation ∆x2 (t)= e− − Bu(τ)dτ (5) 0 of forced oscillation, is obtained by directly solving system state equation and reducing the resultant solution. where Λ is a diagonal matrix with the distinct conjugate complex λ λ eigenvalue pairs ( r , r∗ ), r =1, ..., q, as diagonal elements. λ λ A. Power System State Equation r and r∗ are also termed electromechanical oscillation modes of the system. Φ =[φ , ..., φ , φ∗ , ..., φ∗ ] and Ψ = The power system model for small signal stability analysis 1 q 1 q [ψT , ..., ψT , ψ T ,...,ψ T ]T are the corresponding right can be formulated by a set of homogeneous differential and 1 q 1∗ q∗ and left eigenvector matrices. algebraic equations linearized around an operating point [24]: Equation (5) is further expanded and expressed in terms of x˙ J 1 J 2 ∆x forced disturbances, system modes, and the associated left and = (1) 0 right eigenvectors. J 2 J 4 ∆z q Rn 1 m t where x × is the vector of system state variables, and z λr t λr τ ∈ ∆x2 (t)= ∆Pl φirψrle e− sin(ωl τ)dτ is the vector of algebraic variables. J 1 –J 4 are sparse Jacobian r=1 l=1 0 matrices. The system state equation can be obtained from (1) by t λ λ eliminating the vector of algebraic variables: r∗ t r∗ τ + φir∗ ψrl∗ e e− sin(ωl τ)dτ 1 0 ∆x˙ = J 1 J 2 J 4− J 3 ∆x = A∆x (2) − q m where A is the system state matrix. = ∆Pl r=1 l=1 B. Solution of System State Equation λ φ ψ [( λ sin ω t ω cos ω t)+ω e r t ] ir rl − r l − l l l Physically, periodic and forced disturbances mean there exists × λ2 + ω2 a certain amount of disturbed mechanical or electromagnetic r l λ λ r∗ t power imposed on generator shaft. Accordingly, the dynamics φ∗ ψ∗ [( ∗ sin ωl t ωl cos ωl t)+ωl e ] + ir rl − r − of the power system can be described by superimposing a non- λ 2 + ω2 r∗ l homogeneous term on (2). q m ∆P = l ∆x˙ (t)=A∆x(t)+Bu(t) (λ2 + ω2 )(λ 2 + ω2 ) (3) r=1 l=1 r l r∗ l ∆y(t)=C∆x(t) λ λ λ 2 2 r t λ2 2 r∗ t ( r∗ +ωl )φirψrle +( r +ωl )φir∗ ψrl∗ e ωl where B and u(t) are identity input matrix and forced dis- ×{ − λ 2 2 λ2 2 turbance vector, respectively. The forced disturbances on ( r∗ + ωl )φirψrl +( r + ωl )φir∗ ψrl∗ ωl cos ωl t m − generators have the form of ul =∆Pl sin(ωl t),l=1, ..., m. λ 2 2 λ λ2 2 λ ( r∗ + ωl )φirψrl r +( r + ωl )φir∗ ψrl∗ r∗ sin ωl t ∆Pl and ωl are magnitude and frequency (in rad/s) of the lth } disturbance, respectively.
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