Application of Distributed Compressive Sensing to ... - IEEE Xplore

Application of Distributed Compressive Sensing to Power System State Estimation R. Jalilzadeh Hamidi, Student Member, IEEE, H. Khodabandehlou, H. Livani, Member, IEEE, and M. Sami Fadali, Senior Member, IEEE Abstract— This paper presents an application of distributed Compressive Sensing (CS) for data recovery/reconstruction in Power System State Estimation (PSSE). Transmitted measurements to power system control centers may disappear due to congestion or disconnection in communication links, sensor failures, and cyber-attacks. Consequently, the state estimator may encounter problems. In the proposed method, the identified (Phasor Measurement Unit) PMU bad/missing measurement(s) are reconstructed using CS. Data reconstruction exploits the correlation in both time and space among the PMU measurements using a random projection matrix and a wavelet dictionary. The linear state estimation is then carried out using the available and reconstructed PMU measurements. The proposed method is evaluated on the IEEE 57-Bus transmission system. The capabilities and limitations of the proposed method are also discussed. Keywords—Compressive sensing, data recovery, PMU data, state estimation.

I.

INTRODUCTION

Accurate power system state estimation is a critical task for secure and economic operations in power transmission networks. The state estimation is performed at monitoring and control centers, which are usually part of a Balancing Authority (BA) or an Independent System Operator (ISO). State estimation requires measurements from meters at different locations in a transmission network. These measurements include small random errors and bad data due to meter bias and noise. Measurements may also be lost as the results of communication links congestion or disconnection, sensor failures, and cyber-attacks. Thus state estimation results may become inaccurate and the system may become unobservable in case of missing critical measurements. Therefore, removal of bad data and reconstruction of missing data are essential for having reliable state estimation results. There are several bad data identification and detection methods for Power System State Estimation (PSSE). The simplest but, also, the least accurate is the Chi-squares method [1]. The Largest Normalized Residuals test is more accurate in bad data detection, but it fails in case of multiple conforming interacting bad data [2]. In [3], the Hypothesis Testing Identification technique is introduced for bad data identification in power systems. This method is based on the calculated estimates of measurement errors [1], [3]. In [4], an efficient algorithm is introduced to detect bad data resulting from unobservable/irreducible attacks. However, this algorithm is only applicable if exactly two injection meters are available. In [5], a topological-based un-detectability index is introduced This paper is based upon work partially supported by the National Science Foundation under Grant No. IIA-1301726. R. J. Hamidi, H. Khodabandehlou, H. Livani, and M. S. Fadali are with the Electrical and Biomedical Engineering Department; University of Nevada, Reno; Reno, NV. 89557, USA.

which quantitatively assesses the error in a measurement. In [6], a method is presented for identifying bad data in nearlycritical sets (i. e., critical tuples, for 2). In some works, bad data identification and state estimation are jointly accomplished [7], [8]. As mentioned in the existing literature, power system bad data are often detectable. However, only a limited number of methods have addressed the removal of the identified bad measurements and/or the reconstruction of missing measurements. In [1], bad data is removed when the estimated error is subtracted from the bad measurement, provided that there exists only one bad measurement. In [9], an approach is presented for recovering bad measurement using the estimated measurement residuals and the un-detectability index described in [5]. Both [1] and [7] fail to remove large errors in the bad measurements and also fail to reconstruct a missing measurement. Guo et al. applied ‘online sparsity-based missing data recovery methods’ in PSSE [10]. While there are several data compression techniques such as distributed source coding, matching pursuit, and other greedy basis pursuit algorithms, Compressive Sensing (CS) has received a great deal of attention recently. It is shown in [11] if a signal is sparse on some known basis, it is possible to recover the signal from a small number of samples using CS. In [12], a distributed CS approach is presented to process signals with inter and intra correlation. Applications of CS in power systems are studied in [13], [14]. Regarding PSSE, in [15], the power system measurements are compressed and then decompressed using CS, and it is shown that the PSSE solutions based on the decompressed data are reliable enough for PSSE applications. In [16], an iterative algorithm is introduced for sparse recovery from power system measurements based on the system ). In [17], a regression-based technique admittance matrix ( is proposed for power system state forecasting utilizing (Phasor Measurement Unit) PMU along with SCADA measurements. Given the rapid increase in the number of PMUs installed in substations in recent years, it is not unrealistic to assume that most transmission systems will be fully observable by only PMU measurements in the future. We assume that the system is fully observable with a minimal number of installed PMUs. A large number of data measured by PMUs increases the probability of data loss in communication systems. If any PMU data is lost, the system becomes partially unobservable. Referring to Fig. 1, once the bad/missing measurements are identified, Distributed CS is utilized to reconstruct the bad/missing data. Thus, the power system becomes fully observable and PSSE is performed using the available and the reconstructed measurement sets. The main advantages of this method are as follows: 1) The procedure to find correlation among the system states is not affected by a change in system

topology, and it is able to find the correlation again after a short while, 2) the proposed method is computationally costeffective and fast, 3) the CS basis functions are updated according to the last system state to find the best correlation among the measurements, 4) Wavelets are used for feature detecting since wavelets have sharp edges that enable them to handle data with sharp changes. This paper is organized as follows: Section II describes the proposed method, as well as the CS theory and its application to PSSE. Section III presents and discusses the simulation results for the IEEE 57-bus. The conclusion is given in Section IV. II.

METHODOLOGY

Referring to Fig. 2, in the proposed method, all the measurements are collected at a control center. If the control center identifies bad/missing measurements, they are reconstructed based on the correlation among the missing and available measurements using Distributed CS. Consequently, a state estimator obtains the PSSE solution. Then, CS basis function, , is updated by the last estimate of the system measurements which leads to accurate reconstruction of the bad or missing measurements. In the following sections, the CS theory and its application to PSSE are described. A. Compressive Sensing Theory Although there are several signal compression techniques like transform coding, distributed source coding, and greedy basis pursuit algorithms, most of these methods are computationally complex . CS is one of the well-known compression techniques that allows us to compress a signal using a small number of linear projections. Consider a realvalued measured signal of length . Assume that measured signal is sparse on basis Ψ, then (1) In the matrix form, the equations can be written as (2)

Fig. 2. The proposed method flowchart.

Most traditional methods, such as transform coding and greedy basis pursuit algorithms, require all the coefficients in (1) to determine the largest coefficients. CS uses a finite number of linear projections to obtain those coefficients. This can be done by calculating the projections of data on a second basis set as follows [12] ,

,

1,2, … ,

(3) If Φ and Ψ are incoherent, meaning that none of the elements of first basis have sparse representation on the terms of the second basis, and when the is sufficiently large, it is possible to recover those largest coefficients using projections [12], . .

(4) while it is enough to have at least 1 measurements to solve this minimization problem, solving this problem is time consuming. In fact, it is much easier to solve the equivalent minimization problem [12], . .

(5) This is equivalent to a much simpler linear programming problem. However, changing the norm minimization to norm minimization results to an oversampling factor. This means that we need more than 1 measurements to be able to recover the most significant coefficients from , 1,2, . . . , [12]. Theorem: Set with 0 1. Then there exists an oversampling factor 1/ , 1 , such that, for a -parse signal in basis, , the probability of recovering via Basis Pursuit from random projections, 0, converges to 1 as ∞ .In contrast, the probability of recovering via Basis Pursuit from ∞ [12]. random projections converges to 0 as Fig. 1. The overall concept of the proposed method.

1) Distributed Compressed Sensing While the basic compressed sensing theory deals with just one signal, many real world applications involve more than one signal. In such cases, one may apply compressed sensing to each signal separately. When there is no correlation between signals, this solution is simple and efficient, but if the ensemble of signals are mutually correlated, this solution will be inefficient. Distributed Compressive Sensing is an expansion of traditional compressed sensing theory to deal with ensembles of correlated signals. Distributed CS is particularly useful and will provide a better compression rate if the ensemble of signals show strong mutual correlation. 1,2, … , , . Assume that Given signals , on which all the signals can be there exists a basis Ψ for sparsely represented. Also assume that all signals have a common sparse component plus an innovation component . The signals can thus be written as ,

1,2, … ,

(6)

and their components can be written as , ,

,

1,2, … ,

(7) (8)

The main advantage of this approach to distributed compressed sensing is that it requires a single linear program. To be able to jointly recover the ensemble of signals using a single linear program we define the following matrices,

,

,

(9)

B. State Estimation Method Different methods have been introduced for PMU-only linear PSSE. In [20], a non-iterative method uses voltage, current, and line admittances in phasor form. In [21], a fast decoupled state estimator utilizes real and imaginary parts of voltage, current, and line admittances. In this paper, we use the same approach as [20]. When a PMU is installed on a bus, the bus voltage phasors and all the branch current phasors connected to that bus are measured. To utilize the phasor measurements for PSSE, the SE is formulated as follows [20] (14) is the measurement vector consisting of where is the state vector voltage and current phasors, consisting of bus voltage phasors, is the identity matrix with the rows corresponding to the buses equipped with a PMU, and is defined as follows [20] (15) , is the series admittance of the where branch on which the mth current measurement installed, is the current measurement-bus incidence matrix, and is a current-measurement-by-bus matrix where is the shunt admittance of the branch that measured by the mth current measurement and connected to bus . Let

, then the state estimation problem becomes

where is the weighting matrix, and . transposition. III.

Φ

Ψ

Φ ,Φ ,…,Φ Ψ Ψ Ψ

Ψ 0 0

0 Ψ … …

… … 0

(10) 0 0 Ψ

(11)

Using the measurement matrix, , compressed measurements are calculated and using both Φ and Ψ the sparse coefficient vector, is calculated, using . .

(12)

∑ nonzero elements and the where Contains complete signal can be recovered using (13) If the measured signals have a strong correlation, this method allows us to do more compression. In fact, in this method, distributed compressed sensing is performed by stacking measurements to form a single vector and running a single linear program. This makes the method easy to implement but increases the computational cost. The computational complexity of a linear program is cubic and the computational cost increases dramatically with an increase in the size of the vector [12].

(16) denotes matrix

SIMULATION RESULTS

In this paper, the simulation results are presented for the IEEE 57-Bus which includes 57 buses, 7 generators, and 80 transmission lines [22]. It is assumed that PMUs are utilized as the measurement devices and the buses equipped with PMUs are shown in Fig. 3 (in green) and also stated in Table I. The considered transmission system requires 21 PMUs to become fully observable. It is assumed that the PMUs send voltage and current phasors, and (if applicable) power injections to the control center every second. As shown in Fig. 2, if the control center identifies bad and/or missing measurements, the bad measurements are reconstructed based on the correlation among the missing and available measurements using distributed CS. Also, CS basis function, , is updated by the last available system state. The simulations are performed in MATLAB-2015 environment on a computer with Intel Xeon E5420 CPU and 16 [GB] of RAM. Drastic changes in power system state reduce correlation among PMU measurements and make it harder for CS to reconstruct the bad/missing measurements. In order to evaluate the performance of our approach, total load is increased by 7% in 5 [min] as a drastic change in power system operation. We also assume that the control center receives measurements and runs state estimator every second. In order to evaluate the performance of distributed CS technique on reconstructing the power system measurements, no noise is added into the

measurements (i.e., identity weighting matrix). Therefore, the errors in PSSE originate from measurement reconstruction. Three scenarios are simulated. In all three scenarios, an increase in the total load initiates at 0 [s] and some of the PMUs become unavailable at 30 [s]. Table II shows unavailable PMUs, missing measurements, and the resulting unobservable islands. We consider the effect of losing PMUs on buses 15 and 38 that are connected to five branches. Therefore, their loss has a significant impact on PSSE results. First, to provide an insight into the reconstructed measurements by the proposed method, the actual and the reconstructed measurements are presented. Fig. 4 and Fig. 5 show the reconstructed voltage and current phasors and their corresponding errors. Fig. 4 shows voltage phasors at bus 15 in the first scenario. The PMU data at bus 15 becomes unavailable at 30 [s]. Thus, the missing measurements are reconstructed using distributed CS. As it can be observed in Fig. 4, the reconstructed voltage magnitudes closely pursue the real values. However, the reconstructed voltage angle drifts after 10 [s]. The absolute errors are used to compare the actual and reconstructed values. The absolute errors are calculated by |

|

where |. | denotes absolute value, and reconstructed values of a measurement.

PMU place [bus index]

1 20

TABLE I. PMU Placement. 3 4 9 14 24 25 27 29

38

40

41

48

51

15 32

18 34

53

57

Totally, 21 PMUs are installed. TABLE II. Lost Measurements and the Unobservable Islands. Unobservable islands Lost phasor Lost PMU measurements bus [bus index] branch index Scenario 1

15

V15, I15-1 I15-3, I15-13 I15-14, I15-45

45

15-45 44-45

Scenario 2

38

V38, I38-22 I38-37, I38-44 I38-48, I38-49

22 37

21-22, 22-23 22-38, 36-37 37-38, 37-39 38-44, 44-45

15 & 38

V15, V38 I15-1, I15-3 I15-13, I15-14 I15-45, I38-22 I38-37, I38-44 I38-48, I38-49

22 37 45

15-45, 44-45 21-22, 22-23 22-38, 36-37 37-38, 37-39 38-44, 44-45

Scenario 3

(17) are actual and

According to the absolute errors shown in Fig. 4, although the reconstructed voltage angle drifts from the actual value, it does not diverge and remains within an acceptable range for PSSE (max absolute error is less than 0.05 [deg]). Fig. 5 shows the current magnitude [pu] and angle [deg] for branch 15-13 in the first scenario. Similar to the voltage angle, the reconstructed values closely follow the actual values for the first 10 [s] then start drifting. Although they lose their consistent pursuit of the actual value, the reconstructed current magnitude and angle are still reliable for PSSE application. The correlation between the missing measurements and available ones deteriorates as time passes. Therefore, the drifting trend in the reconstructed values is normal. Regardless of the drifts in the measurement reconstructions, the PSSE results need to be considered for assessing the proposed method.

Fig. 4. Voltage phasors and the absolute errors at Bus 15 in the first scenario.

Fig. 5. Current phasors and the absolute errors at branch 15-13 in the first scenario.

Fig. 3. The 57-bus IEEE test case, the buses equipped with a PMU are shown in green, listed as {1, 3, 6, 9, 14, 15, 18, 20, 24, 25, 27, 29, 32, 34, 38, 40, 41, 48, 51, 53, 57}.

Fig. 6 shows the estimated voltage magnitudes and their corresponding errors according to the actual and reconstructed data, scenario 1, at 50 [s]. As it is observed in Fig. 6(b), the largest absolute error of the voltage magnitudes is around 1.2 10 [pu] which is at buses 19, 50, and 54.

Fig. 8. Voltage magnitude and angle RMSEs in different scenarios over the time.

Fig. 6. a) Voltage magnitudes estimation using the actual and reconstructed measurements, scenario 1, t=50 [s]. b) Absolute error between the PSSE solutions based on the actual and reconstructed measurements.

shows that the bus angles are generally estimated more accurately than the bus voltages. It is noteworthy that bus angles are often more important to system operators. As a common practice, the Root-Mean-Squared Error (RMSE) is calculated in order to evaluate the performance of the proposed method on PSSE results [8], [17], [21]. 1

(18)

where is the number of buses, is the difference between state estimation results using the actual and reconstructed measurement for the ith bus at time [s].

Fig. 7. a) Voltage angles estimation using actual and reconstructed measurements, scenario 1, t=50 [s]. b) Absolute error between the PSSE solutions based on the actual and reconstructed measurements.

Fig. 7 shows voltage angles and their corresponding errors based on the actual and reconstructed measurements, scenario 1, at 50 [s]. The absolute errors of the voltage angles have a peak around 1 10 [deg] at buses 14, 39, and 50, which is acceptable for PSSE. Comparing Fig. 6(b) and Fig. 7(b)

TABLE III. RMSEs in Different Scenarios. Scenario 1 Scenario 2 Scenario 3 RMSE RMSE RMSE T[s] T[s] T[s] Vol. Ang. Vol. Ang. Vol. Ang. 10-6 10-6 10-6 10-6 10-6 10-5 30 3.281 3.310 30 2.957 1.551 30 9.152 4.429 31 1.807 2.944 31 8.064 2.911 31 9.293 2.059 32 3.277 3.284 32 3.295 6.654 32 8.469 1.659 33 2.357 3.087 33 5.451 5.040 33 16.15 4.324 34 2.189 3.002 34 1.081 1.692 34 8.088 1.939 35 2.728 3.129 35 6.609 3.490 35 4.253 4.874 36 3.383 3.279 36 6.173 3.890 36 2.439 2.279 37 1.983 2.962 37 7.527 5.586 37 4.215 1.833 38 2.846 3.177 38 4.807 2.391 38 3.061 1.482 39 2.955 3.210 39 5.361 4.501 39 3.872 4.017 40 3.284 3.299 40 3.745 3.659 40 5.021 3.857 41 4.542 3.698 41 2.791 1.343 41 5.526 6.146 42 5.072 3.845 42 3.878 8.931 42 6.272 3.060 43 5.767 4.114 43 5.580 1.266 43 4.940 8.654 44 3.058 3.281 44 3.733 3.580 44 8.612 8.060 45 2.816 3.240 45 5.167 2.455 45 7.160 4.706 46 3.923 3.443 46 3.779 4.510 46 8.010 1.400 47 3.173 3.337 47 4.756 3.963 47 5.953 8.399 48 4.270 3.601 48 7.489 7.761 48 6.808 5.768 49 8.143 4.809 49 5.971 5.908 49 4.633 6.509 50 7.709 4.648 50 5.530 2.093 50 8.721 2.137

Fig. 8 and Table III show RMSEs for voltage magnitudes [pu] and angles [deg] in all three scenarios from 30 [s] to 50 [s] in which the PMU(s) are unavailable. As expected, the error is generally larger when two PMU measurements are missing but the RMSEs are in an acceptable range.

[6]

As PMU-only State Estimation (PSE) is assumed, the solution method is linear non-iterative [20]. This method is far faster than iterative methods and most of the CPU time is due to the matrix inversion by Cholesky decomposition method. The CPU time comprised the times for data reconstruction and state estimation solution. The average CPU times for case studies are, negligibly different, approximately 630 [ms], 450 for data reconstruction and 180 for state estimator.

[8]

It is observed that, the accuracy of the power system state forecasting utilizing our proposed method is improved compared with the existing forecasting approaches in the literature. Furthermore, the proposed method provides reconstructed measurements for several seconds which are reliable for PSSE applications. However the proposed method requires available data from all PMUs before losing some of the measurements.

[11]

IV.

CONCLUSION

In this paper, we propose a method for reconstructing bad/ missing PMU measurements by distributed CS for PSSE (bad/missed data should be detected using any given method before data reconstruction). The available measurements at a control center are first tested for identifying the probable bad/missing measurement(s). If bad/missing measurement(s) are identified, then they are reconstructed by distributed CS. Then, state estimator obtains the PSSE solution using all available measurements, including received and reconstructed measurements. The CS basis set is updated every second using the last state estimation results to preserve its capability of accurately reconstructing bad/missing measurements. The proposed method is tested on the IEEE 57-bus test system and the results are discussed. The proposed algorithm will be tested on the IEEE 118-Bus and 390-Bus Nevada systems. We will also investigate the application of iteratively reweighted algorithms-based CS to PMU data reconstruction.

[7]

[9]

[10]

[12]

[13]

[14]

[15]

[16]

[17]

[18] [19] [20]

[21]

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