This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRS.2017.2701649, IEEE Transactions on Power Systems IEEE TRANSACTIONS ON POWER SYSTEMS
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Tracking the Damping Contribution of a Power System Component Under Ambient Conditions Ruichao Xie, Student Member, IEEE, and Daniel J. Trudnowski, Fellow, IEEE
Abstract—This letter presents a method for tracking the damping contribution of a power system component in near realtime based upon oscillation energy dissipation. By transforming the energy-flow into the frequency domain, the new derivation demonstrates a new theoretical basis and a wider applicability of using the energy-flow theory to analyze power system damping problems. The concept is demonstrated via a non-linear simulation and an actual-system data set. Index Terms—Small-signal stability, energy-flow, damping torque, ambient noise, PMU measurements.
I. I NTRODUCTION OWER system small-signal stability is dictated by the electromechanical eigenvalues (or system modes). Poorlydamped modes threaten system reliability and limit power delivery. This letter presents an approach to estimate a component’s damping contribution to a given mode. The approach is based upon the energy-flow theory established in [1]-[3]. The energy-flow equations are transformed to the frequency domain resulting in a damping-torque coefficient (DTC) or dissipation energy-flow (DEF) spectra which allows one to monitor the component’s damping over a range of frequencies with the system in an ambient condition.
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II. R EVIEW OF E NERGY-F LOW M ETHOD It is shown R in [4] that R the energy out of a generation bus is: Wij = Pij dθi + Qij dVi , where Pij is the injection active power from bus i to j, Qij is the reactive power injection, θi is the voltage angle at bus i, and Vi = ln Ui is the natural logarithm of the bus voltage amplitude. The energy representation is satisfied for any system trajectory. The deterministic and random oscillations are considered in [1] and [2], respectively, where the branch energy Wij is termed the energy-flow and is found to be an indicator of the damping of a component that’s connected via the branch. This method was detailed for a generator’s damping evaluation using a 4th order generator model in [3]. The key conclusion is that for a particular modal (frequency) oscillation, the corresponding DEF thru a branch Lij can beR calculated byR using PMU measureable variables as: WijD = ∆Pij d∆θi + ∆Qij d∆Vi . For a generating component, a negative WijD indicates the component is dissipating oscillation energy and is adding damping to the oscillation. A positive WijD is vise-versa. It is proved in [1] that the DEF is consistent with the damping Manuscript received December 29, 2016; revised January 05, 2017 and March 20, 2017; accepted April 19, 2017. Paper no. PESL-00248-2016. This work was supported by the U.S. Department of Energy under grant DE-AC0205CH11231. Authors are with Montana Tech of The University of Montana, Butte, MT, 59701, USA. (e-mails: {rxie,dtrudnowski}@mtech.edu).
torque of a single-machine system. Moreover, as shown below, this is a common relation. III. F REQUENCY D OMAIN DEF AND DAMPING T ORQUE Consider a generation unit including the generator and its controls connected to bus i as a single rotating mass with respect to the system synchronous reference (e.g., 60Hz). For a particular oscillation, we decompose the imbalance torque at bus i as: ∆Ti (t) = KD,i ∆ωi (t) + KS,i ∆θi (t), KD,i is the well-known damping-torque coefficient, KS,i is the synchronizing-torque coefficient, ∆ωi is the deviation of the angular frequency at bus i, and t is time. Claim: KD,i is a measure of the unit’s damping performance, KD,i < 0 indicates good damping. Interpretation: Consider the system variables are oscillating around an equilibrium. For instance, consider a motion (say ∆ωi > 0) for two consecutive time points, if KD,i < 0, the imbalance torque ∆Ti tends to reduce, consequently the rate of change of ωi tends to be reduced due to the imbalance torque effect. A similar description can be given for any time points on the trajectory. It implies that a negative KD,i pulls the angular frequency back to its equilibrium. Such that KD,i can be a characterization of the unit’s local damping performance. As a rotating mass, from the definition of the energy, power, and torque, we have ˙ = P = T ω. Therefore, the energy can be represented W as the integral of power or the integral of Rthe product of R torque and speed, denoted as W = P dt = T ωdt. During an oscillation, the generation unit’s energy output can be represented as Z Z Wij = (Pij,s + ∆Pij )d(θi,s + ∆θi ) + (Qij,s + ∆Qij ) Z d(Vi,s + ∆Vi ) = (Ti,s + ∆Ti )(ωi,s + ∆ωi )dt, (1) where s denotes the steady-state value. Expanding (1) and cancellingR the DC and R transient energyR terms, we have WijD = ∆Pij d∆θi + ∆Qij d∆Vi = ∆Ti ∆ωi dt, then ˙ i , and using the consider d∆θi /dt = ∆ωi , d∆Vi /dt = ∆V R torque decomposition, we have K = ( ∆Pij ∆ωi dt + D,i R R R ˙ i dt − KS,i ∆θi ∆ωi dt)/( ∆ωi ∆ωi dt), which is ∆Qij ∆V hard to be further analyzed without an analytic form presumed for the deviations. Now consider any two real signals x1 (t) and x2 (t), using Parseval’s theorem [5], we have +∞ +∞ +∞ Z Z Z ∗ x1 (t)x2 (t)dt = X1 (f )X2 (f )df = X1 (f )X2∗ −∞
0
0
+∞ +∞ Z Z ∗ (f )df + X1 (f )X2 (f )df = 2 Re[Sx1 ,x2 (f )]df , (2) 0
0
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRS.2017.2701649, IEEE Transactions on Power Systems IEEE TRANSACTIONS ON POWER SYSTEMS
TABLE I E XAMPLE 1 SYSTEM MODES Gen Scenario 1 G1, G2 vs G3, G4 6.9%; 0.60Hz G1 vs G2 25.4%; 0.96Hz G3 vs G4 25.8%; 0.94Hz
Mode Inter-area Local 1 Local 2
Scenario 1 Scenario 2
0
-2 0.2
0.4
0.6
0.8 Freq. (Hz)
1
1.2
1.4
Fig. 1. DTC spectra for example 1. × 10-7
0
P.U.
The denominator of (3) is always a positive real number, the sign of KD,i (f ) is determined by the DEF only. Note that the damping torque defined here is not in conflict with the one in [6] where the negative electrical torque on the rotor is defined as the imbalance torque and then decomposed. If the oscillations are governed by the system modes such as the natural oscillations and the random modal oscillations, KD,i (f ) thereby indicates the damping contribution to the mode. As explained in [3], the damping from a generator is composed of two parts, one is the inherent positive damping from the windings, and the other is from its controls which is mainly from its excitation system. Therefore, a positive or a small negative KD,i (f ) indicates the PSS parameter might be inadequate for the underlying system condition. In the ambient conditions, one may use either (3) which is a DTC spectra or its numerator which is a DEF spectra to monitor the component’s damping over a continuous range of frequencies. For the spectral calculation of (3), multiple periods of the oscillation should be used for analysis in order to achieve a reasonable estimation of the Fourier transforms. For the ambient case, it is typical to use a several-minute data set to capture enough signal energy over the desired frequencies via periodogram averaging [5]. IV. M ETHOD VALIDATION Example1: We consider the Kundur’s two-area system in [6]. The generators are modelled using a detailed subtransient generator model and are equipped with exciters and PSS units; governors are installed on generators 1 and 3. To mimic ambient conditions, a small percentage of the loads are modelled as random. Two scenarios are considered: 1) all PSS parameters are well tuned; and, 2) the gain of the PSS installed on generator 3 is tuned to be negative. The modal parameters for scenario 1 and 2 are listed in Table I. Note that generator 3 has the largest and second largest participation factors for the inter-area mode and local mode 2, respectively. It shows that the modes damping are reduced. A 20-min ambient data set of the scenarios are analyzed via periodogram averaging. Fig.1 shows the estimated DTC spectra for generator 3. It clearly shows that for scenario 2, generator 3 has bad damping contribution over the electromechanical mode frequency range thus degrading all the modes damping. Note that the damping property of generator 3 has almost no influence on local mode 1 since it has trivial participation in that mode. Example2: Recently, a damping controller has been installed which modulates the Pacific DC Intertie (PDCI) power using remote PMU feedback. Reference [7] describes the control strategy and
Scenario 2 6.3%; 0.62Hz 25.3%; 0.96Hz 4.3%; 1.16Hz
2
P.U.
where ∗ denotes the conjugate, Re denotes the real part operator, and Xk (f ) is the Fourier transform of xk (t) at frequency f in Hz. If x1 and x2 are random as in the ambient case, we take the expectation of (2) without loss of generality [5]. Sx1 ,x2 (f ) is the well-known cross-energy spectral density of x1 (t) and x2 (t) [5]. Using (2) and recognizing that the integral collapses due to the evaluation at a single frequency, the damping-torque coefficient becomes # " S∆Pij ,∆ωi (f ) + S∆Qij ,∆V ˙ i (f ) . (3) KD,i (f ) = Re S∆ωi ,∆ωi (f )
2
-5 -10 -15 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Freq. (Hz)
Fig. 2. DEF spectra for example 2.
initial results of the installation showing increased damping to key interarea modes between 0.2 and 0.5 Hz. Fig. 2 shows the DEF spectra estimated from actual-system data at the northern inter-connection bus of the PDCI with the controller on-line under an ambient condition. The plot clearly shows that the controller is adding damping. Many other actual-system and simulation cases show similar results for the PDCI damping controller. V. C ONCLUSIONS This letter shows that the DEF and/or the DTC of an interconnecting component can be estimated in the frequency domain under ambient conditions. This can form the basis for future on-line system monitoring applications. One example demonstrates its performance for a generator via simulation and the second for an actual-system HVDC damping controller. Future work will further test the concept on more simulation and actual system data. R EFERENCES [1] L. Chen, Y. Min, and W. Hu, "An energy-based method for location of power system oscillation source," IEEE Trans. on Power Systems, vol. 28, no. 2, pp. 828-836, May. 2013. [2] L. Chen, M. Sun, and Y. Min, et al., "On line monitoring of generator damping using dissipation energy flow computed from ambient data," IET Generation, Transmission and Distribution, (Accepted), 2017. [3] L. Chen, Y. Min, Y. P. Chen, and W. Hu, "Evaluation of generator damping using oscillation energy dissipation and the connection with modal analysis," IEEE Trans. on Power Systems, vol. 29, no. 3, pp. 1393-1402, May. 2014. [4] Y. H. Moon, et al., "Derivation of energy conservation law by complex line integral for the direct energy method of power system stability," in Proc. 38th Conf. Decision & Control, Phoenix, AZ, 1999. [5] J. Bendat, and A. Presol, Engineering Applications of Correlation and Spectral Analysis, 2nd ed. John Wiley & Sons, Inc, 1993. [6] P. Kundur, Power System Stability and Control, New York: McGraw-Hill, 1994. [7] D. Trudnowski, et al., "Initial closed-loop testing results for the Pacific DC Intertie wide area damping controller," IEEE PES General Meeting, (To be published), July. 2017.
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