Multi-time Interval Power System State Estimation Incorporating Phasor Measurements Ye Guo, Boming Zhang, Wenchuan Wu, and Hongbin Sun
As a crucial basic function in energy management system, the main task of power system state estimation is to provide the basic power flow model that obeys Kirchhoff’s laws according to the real-time measurements. Note that, Kirchhoff’s laws are describing electrical variables at exactly the same moment. While the measurements, on contrary, are measured at different moments. Especially for measurements from remote terminal units (RTU), their time difference can exceed several seconds. If the system has significant fluctuation in the meantime, the accuracy of state estimation may be deteriorated by the asynchronism of measurements. Traditionally, power systems were assumed to keep steady, and it is adequate to estimate their states with time interval around one minute. However, with the integrations of wind farms, states in modern power systems are becoming more and more fluctuant. For example, phasor measurements of the power injection and voltage magnitude at the integration node of a real wind farm are given in Fig.1 and Fig.2. We can see that within 3 seconds, the maximum fluctuations of power injection and voltage magnitudes exceed, respectively, 2MW and 4kV. With such significant fluctuations, the accuracy of state estimation may be deteriorated. To monitor the system states more accurately and promptly, it is necessary to make use of phasor measurements and estimate the system states with very short time intervals. For large scale practical power systems, however, this means huge increase in computation burden. In fact, only a portion of the today’s power system is fluctuant, while the rest part remains steady. Take Fig.2 as example, although the state for the integration node of the wind farm is fluctuant, the other voltage curve remains steady. The authors are with the Department of Electrical Engineering, Tsinghua University, Beijing, China. (Email:
[email protected]). This work is supported by National Key Basic Research Program of China (2013CB228203), China Postdoctoral Science Foundation (2014M550727), and National Science Fund for Distinguished Young Scholars (51025725)
978-1-4673-8040-9/15/$31.00 ©2015 IEEE
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I. I NTRODUCTION
Hence to monitor fast fluctuations in power systems while keep computation efficiency, an alternative is to estimate only the fluctuant states with shorter time interval and the rest states with longer time interval, then combine their results as state estimates for the entire system.
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Fig. 1. Phasor measurements of power injection from real wind farm
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Abstract—Although much of today’s power system remains steady, the integrations of wind power make a portion of the system become fluctuant. To monitor these fluctuations, we need to utilize phasor measurements and shorten the time interval for state estimation. On the other hand, however, this means huge increase in computation burden. As a tradeoff between accuracy and computation efficiency, a multi-time interval state estimation approach for power systems is proposed in this paper, in which fluctuant states are estimated with shorter interval and the rest steady states are estimated with longer interval. Their fusion forms the state estimates for entire power system.
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Fig. 2. Phasor measurements of voltage magnitudes at the integration
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Various papers have studied PSSE method with phasor measurements, while very few of them involves time interval problem. With pure phasor measurements, the measurement functions are linear and the solution of PSSE is non-iterative [1]. However, due to the limited number of phasor measurement units (PMU), a more practical way is to perform hybrid estimation with measurements from both RTUs and PMUs [2–4]. An important problem for hybrid estimation is how to deal with the ill-condition problem caused by phasor current measurements [5]. Another alternative is to perform PSSE with RTU measurements only, deem its state estimates as pseudomeasurements, and estimate the final states with these pseudomeasurements along with phasor measurements via linear esti-
mation [6]. Incorporating phasor measurements, other relative studies include mesurement uncertainty assessment [7], bad data processing [8], observability analysis [9] and multi-area state estimation [10, 11]. To monitor system fluctuations while reduce computation burden, an alternative is event triggered state estimation [12, 13]. Phasor measurements are used for event detection [14], and PSSE is performed only when event happens. However, the frequency of events is extremely high in large scale power systems as there are so many components, and it is still unaffordable to perform PSSE for the entire system whenever there is an event. In this paper, a multi-time interval PSSE framework is proposed. According to the positions of fluctuation sources and sensitivity factors, the system is partitioned to steady area, quasi-steady area, and fluctuant area. Fluctuant area has significant fluctuations within only several seconds, and we have to estimate their states with extremely short time intervals. Steady area, on the contrary, has little fluctuation and it is adequate to estimate their states with conventional time interval. Quasi-steady area is the intermediate area, and we can estimate their states with the time interval for RTU samplings. At any moment, the fusion of PSSE results with different time intervals provides the state estimates for the entire system. II. S TATE ESTIMATION BACKGROUND Due to the asynchronism of measurements, to estimate the states at time t, we should only use the measurements that measures at exactly time t or a different time but its states are very close to the states at time t. Fig.3 illustrates typical steady and fluctuant states. Assuming there are two RTU measurements in the same cycle but measure at different times, as the dash lines in Fig.3. If the states are steady, these measurements can be used for state estimation. Otherwise, if the states are fluctuant, inclusion of these measurements simultaneously will deteriorate the estimation accuracy.
as Fig.3. As RTU measurements do not have time labels, it is impossible to distinguish them. Accordingly, state estimation in fluctuant area should use phasor measurements only. Inclusion of RTU measurements in fluctuant area can deteriorate, rather than enhance, the accuracy of state estimation. In general, power system measurement model is given by: z = h(x) + e.
(1)
where z and x denote, respectively, the measurement values and state variables. Mapping h denotes measurement functions, which is nonlinear for RTU measurements. Vector e denotes measurement errors. Power system state estimation is to estimate the the weighted least square solution for given measurement set z: min f (x) = r(x)† W r(x), x
(2)
where W is the measurement weight matrix and r(x) = z − h(x) is the residual vector. The optimization model (2) is usually solved by GaussNewton iterations, with the correct equation as (H(x)† W H(x))Δx = H(x)† W r(x),
(3)
where H(x) is the measurement Jacobian matrix computed at point x. In particular, if a pure phasor measurement set is used, the measurement model will become linear: z = Hx + e.
(4)
And the weighted least square model (2) can be directly solved without iterations: (H † W H)x = H † W z.
(5)
For pure phasor measurement set, the measurement Jacobian H is constant and usually referred as measurement matrix. III. M ULTI - TIME INTERVAL STATE ESTIMATION A. Framework
Steady states
Fluctuant states
Fig. 3. Steady and fluctuant states
For fluctuant area like Fig.1, the states may have significant change within only several seconds. At this time, it is possible that some of the RTU measurements measure at moments whose states are significant different with other measurements,
The proposed multi-time interval state estimation framework is like Fig.4. Some typical time intervals are given in Fig.4 as examples. If necessary, these numbers can be adjusted within a reasonable range. In Fig.4, the entire system is partitioned into steady area, quasi-steady area and fluctuant area. In steady area, the states change slowly and it is adequate to use conventional estimation time interval such as 1 minute. In quasi-steady area, the states change faster, but remain steady during several seconds and it is proper to use RTU measurements in state estimation. In Fig.4, time interval for state estimation in quasi-steady area is set as 3 seconds. In fluctuant area, the states may have significant changes even within several seconds. Accordingly, we have to estimate these states with even shorter time interval, such as 300ms. At any time t, the state estimates for the entire system are the fusion of the newest state estimates, shown by dot lines. For steady and quasi-steady areas, their states change
SE with RTU and PMU measurements
SE with PMU measurements only
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Fig. 4. Framework for multi-time interval state estimation
slightly during the sampling interval of RTU measurements. Accordingly, the PSSE in steady and quasi-steady areas should use RTU and PMU hybrid measurements, denoted by in Fig.4. For fluctuant area, however, its states change extremely fast and the state estimation should use phasor measurements only, denoted by . Subsequent subsections will elaborate some important modules in framework Fig.4. B. System partition Fluctuations of power flow distribution can be attributed as fluctuation of power injections. Hence in this paper, fluctuation sources are modeled as injections. For fluctuation source i, we know its connecting node ki , standard error for fluctuation of active and reactive power injections ΔPi and ΔQi . The values for ΔPi and ΔQi can be assigned by operation experience. Especially, for wind farms, these values can be set according to the variance of real-time wind power prediction. Subsequently, the fluctuation level of power flow on branch l, denoted by ΔPl and ΔQl , can be computed by n ΔPl = |SP (ki , l)ΔPi |, i=1 (6) n |SQ (ki , l)ΔQi |, ΔQl = i=1
where SP (ki , l) is the entry corresponding to node ki and branch l in active power shift factor matrix, and SQ (ki , l) is the same entry in reactive power shift factor matrix. Scalar n denotes the number of fluctuation sources. The fluctuation level of node j is Δj defined by ΔPj
=
max ΔPl ,
ΔQj
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Δj
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max ΔQl , l∈Aj ΔPj2 + ΔQ2j ,
l∈Aj
(7)
where Aj denotes the adjacent branch set for node j. With the fluctuation level of each node, the system node set N is partitioned into steady area node set NS , quasi-steady area node set NQ and fluctuant area node set NF : NS NQ NF
= {j ∈ N |Δj ≤ q }, = {j ∈ N |q < Δj ≤ f }, = {j ∈ N |Δj > f },
(8)
where q and f are artificial thresholds. Similarly, the system measurement set M is also partitioned into steady area measurement set MS , quasi-steady area measurement set MQ and fluctuant area measurement set MF . The steady area measurement set MS includes measurements for node in NS and measurements for head (tail) side for branches whose head (tail) nodes are in NS . Note that the system partition can change with system operations. Hence it is necessary to update the system partition after system operations are scheduled. C. State estimation method Take steady area as example, its states are denoted by vector xS , and the states for nodes outside but adjacent to steady area are denoted by xBS . Then the measurement functions for steady area are zS = hS (xS , xBS ) + eS ,
(9)
where zS , hS and eS denote, respectively, measurement values, measurement functions and errors for measurement set MS . The state estimates for steady area are obtained by solving following weighted least square problem: min (zS − hS (xS , xBS ))† WS (zS − hS (xS , xBS )), xS
(10)
where WS is the weight matrix for measurement set MS . Model (10) estimates the states in steady area with states in quasi-steady and fluctuant areas fixed. Similarly, the state estimation in quasi-steady and fluctuant areas will only update their own states as well. At any moment t, the state estimates for the entire power system is given by ˆQ , x ˆF ], xˆ = [ˆ xS , x
(11)
where x ˆS , xˆQ and xˆF are newest state estimates for steady area, quasi-steady area and fluctuant area, respectively. There are several effective hybrid PSSE methods, in this paper we use the method in [6] for hybrid PSSE. The PSSE method with pure PMU measurements are given in section ??. With the state estimate for the entire system in (11) and the measurement values, we are able to identify the potential bad data. The conventional bad data identification procedure is based on maximum normalized residual principle [15, 16]. The normalized residual for measurement i can be computed as (12) rN i = ri / Ωii , where Ω is the residual covariance matrix, which can be computed conveniently [15]. Consequently, the state estimator will pick the measurement with largest normalized residual out of the normal measurement set. According to our practical experience, however, largest normalized residual based approach sometimes may fail to identify bad data, especially when conforming bad data exists. An alternative is to employ robust estimators [17, 18]. While new approach for bad data identification is beyond the scope of this paper.
the three fluctuation sources, Gaussian distributed random fluctuations with zero mean and standard deviation ΔP , ΔQ are added to their power injections. Step.2 Construct the true states by performing power flow calculation with corresponding power injections. Step.3 For all generator nodes, PMUs are installed to measure their complex voltages and currents for adjacent branches. The standard deviations for phasor measurements are set as 0.0001p.u. for voltage measurements and 0.001p.u. for current measurements. Every 3 seconds, we construct RTU measurements for all nodes and branches. The standard deviations for voltage and power measurements are set as 0.001p.u. and 0.01p.u., respectively. Five different PSSE approaches are tested: i) Estimate the states for entire system every minute. ii) Estimate the states for entire system every 3 seconds. iii) Estimate the states for entire system every 300 milliseconds. iv) Estimate the states for entire system every 300 milliseconds with removal of RTU measurements in fluctuation area. v) Multi-time interval PSSE. In which method 4 tries to provide the most accurate state estimates for this problem. In practise, however, we can never know which RTU measurements should be removed without the system partition. Set q = 0.2 and f = 0.4, the node sets for steady area, quasi-steady area and fluctuant area in this test are given in TABLE II. TABLE II
S YSTEM PARTITION RESULTS Area Fluctuant area Quasi-steady area Steady area
IV. N UMERICAL T ESTS In this section we illustrate the proposed multi-time interval power system state estimation approach via numerical tests on IEEE 118-bus system. A time period of 2 minutes is considered, and it is assumed that the system load is linearly increasing, active and reactive power injections increase 0.4MW/0.2MVar per minute for all nodes except three specified fluctuation sources. The position and power fluctuations for fluctuation sources is given in TABLE I. Our simulation is based on a commercial grade state estimation software which is developed by C++, and is executed on a laptop with CPU Inter i7 2.40GHz.
nodes 49,52-56,59,89,90,91 45-48,60-64,66,67,69, other nodes
Every 300 milliseconds, we compare the state estimates and the true states and get the state estimation accuracy index α defined as α = max(|V˙ SE − V˙ T RUE |), (13) where V˙ SE and V˙ T RUE denote estimates and true values for node complex voltages, respectively. The values for index α during the 2 minutes time period is given by Fig.5. 1 min
3s
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300ms remove improper meas.
multiple time interval
TABLE I
F LUCTUATION SOURCE SETTING Index 1 2 3
Connected node 54 55 90
ΔP (MW) 2.0 1.0 0.8
ΔQ (MVar) 1.0 0.5 0.4
Every 300 milliseconds, we construct the true states and phasor measurement values with following steps: Step.1 For all loads and generators, update their power injections with the constant changing rate. Furthermore, for
Maximum error for state estimates
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Fig. 5. Accuracy for different approaches
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From Fig.5 we can see that approaches i and ii cannot monitor the fluctuations between their estimation intervals accurately. Comparing with iv and v, the estimation accuracies for i-iii are affected by the asynchronous RTU measurements. And the mean values for α indices of the five approaches are given in TABLE III.
[6]
[7] TABLE III
E STIMATION ERRORS AND TIME COSTS Approach i (1 min interval) ii(3s interval) iii(300ms interval) iv(optimal) v(multi-time interval)
Mean errors (p.u.) 1.727 × 10−3 1.607 × 10−3 1.499 × 10−3 0.50 × 10−3 0.57 × 10−3
Total computation time (s) 0.27 3.79 31.57 30.95 1.52
The first three rows in TABLE III shows that although estimation accuracy can be improved by shorten time interval, huge cost of computation efficiency is needed. Furthermore, the asynchronous RTU measurements can deteriorate estimation accuracy in fluctuant area and such problem cannot be overcome by simply shorten the estimation time interval. The proposed multi-time interval SE (v) is very close to the optimal estimation (iv) on estimation error, while its computation is much more efficient. V. C ONCLUSION A multi-time interval state estimation method is proposed in this paper. To monitor fast fluctuations while keep computation efficiency, only the states from fluctuant area are estimated with extremely high frequency, and the other states are estimated with lower frequency. The state estimates for entire system are given by the fusion of estimation results with different time intervals. The proposed method can achieve real-time monitoring for system fluctuations with minimum cost of computation efficiency.
[8]
[9]
[10] [11] [12]
[13]
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R EFERENCES [1] A. Phake, J. Thorp, and K. Karimi, “State estimation with phaser measurements,” IEEE Trans. on Power Syst., vol. 1, no. 1, pp. 233–237, Feb. 1986. [2] T. Bi, X. Qin, and Q. Yang, “A novel hybrid state estimator for including synchronized phasor measurements,” Elect. Power Syst. Res., vol. 78, pp. 1343–1352, Feb. 2008. [3] A. Costa, A. Albuquerque, and D. Bez, “An estimation fusion method for including phasor measurements into power system real-time modeling,” IEEE Trans. on Power Syst., vol. 28, no. 2, pp. 1910–1920, May 2013. [4] G. Valverde, S. Chakrabarti, E. Kyriakides, and V. Terzija, “A constrained formulation for hybrid state estimation,” IEEE Trans. on Power Syst., vol. 26, no. 3, pp. 1102–1109, Aug. 2011. [5] S. Chakrabarti, E. Kyriakides, G. Ledwich, , and A. Ghosh, “A comparative study of the methods of inclusion of PMU current phasor measurements in a
[15] [16]
[17]
[18]
hybrid state estimator,” in IEEE PES General Meeting, Minneapolis, MN, July 2010. M. Zhou, V. Centeno, J. Thorp, and A. Phadke, “An alternative for including phasor measurements in state estimators,” IEEE Trans. on Power Syst., vol. 21, no. 4, pp. 1930–1937, Nov. 2006. S. Chakrabarti and E. Kyriakides, “Pmu measurement uncertainty considerations in wls state estimation,” IEEE Trans. on Power Syst., vol. 24, no. 2, pp. 1062–1071, May 2009. G. Korres and N. Manousakis, “State estimation and bad data processing for systems including PMU and SCADA measurements,” Elect. Power Syst. Res., vol. 81, pp. 1514–1524, April 2011. ——, “State estimation and observability analysis for phasor measurement unit measured systems,” Proc. IET Gen., Transm., Distrib., vol. 6, no. 9, pp. 902–913, Sep. 2012. W. Jiang, V. Vittal, and G. Heydt, “Diakoptic state estimation using phasor measurement units,” IEEE Trans. Power Syst., vol. 23, no. 5, pp. 1580–1589, Nov. 2008. L. Zhao and A. Abur, “Multiarea state estimation using synchronized phasor measurements,” IEEE Trans. Power Syst., vol. 20, no. 2, pp. 611–617, May 2005. R. Francy, A. Farid, , and K. Youcef-Toumi, “An event triggered tracking state estimator for power systems with integrated wind generation,” in PowerTech, Grenoble, France, June 2013. L. Vanfretti, M. Baudette, I. Al-Khatib, M. Almas, and J. Gjerde, “Testing and validation of a fast real-time oscillation detection pmu-based application for wind-farm monitoring,” in First International Black Sea Conference on Communications and Networking, Batumi, Georgia, July 2013. L. Xie, Y. Chen, and P. Kumar, “Dimensionality reduction of synchrophasor data for early event detection: Linearized analysis,” IEEE Trans. Power Syst., vol. 29, no. 6, pp. 2784–2794, April 2014. A. Abur and A. Expsito, Power System State Estimation: Theory and Implementation. New York: Marcel Dekker, 2007, pp. 27–36. B. Zhang, S. Wang, and N. Xiang, “A linear recursive bad data identification method with real-time application to power system state estimation,” IEEE Trans. Power Syst., vol. 7, no. 3, pp. 1378–1385, Aug 1992. L. Mili, M. Cheniae, N. Vichare, and P. Rousseeuw, “Robust state estimation based on projection statistics [of power systems],” IEEE Trans. Power Syst., vol. 11, no. 2, pp. 1118–1127, May 1996. W. Wu, Y. Guo, B. Zhang, A. Bose, and H. Sun, “Robust state estimation method based on maximum exponential square,” Generation, Transmission Distribution, IET, vol. 5, no. 11, pp. 1165–1172, Nov 2011.
