Voltage Sensor Validation for Decentralized Power System Monitor ...

Voltage Sensor Validation for Decentralized Power System Monitor Using Polynomial Chaos Theory H. Li (*), A. Monti (**), F. Ponci (**) and G. D’Antona (***) (*) EE Department, University of South Carolina, Columbia, SC, USA (**) E.ON Energy Research Center, RWTH Aachen, Aachen, Germany (***) Energy Department, Politecnico di Milano, Milano, Italy

Abstract—Decentralized state estimation of power system is under investigation for the monitoring of modern power system. In this framework, the identification of sensor failure is a critical issue. In this work, a novel method is proposed to achieve this goal, yielding improved reliability of the decentralized power system monitoring. Sensor measurements are validated locally, before they are elaborated. The validation algorithm leverages on reasonable thresholds of the measurand, computed via Polynomial chaos theory (PCT) and determined based on to the effect of the uncertainty in the system, particularly that of loads. The application of this approach to decentralized state estimation is presented. Numerical results obtained from simulation of a shipboard DC zonal power system are presented.

Keywords- power system monitoring; estimation; sensor validation; PCT.

I.

distributed

state

INTRODUCTION

Online validation of sensor status is beneficial to the successful monitoring of power systems, usually realized through state estimation. A reliable monitoring strategy should also be equipped with advanced measurement features such as ability to propagate the measurement uncertainty to the state, the ability to detect, identify and isolate failed sensors, and the ability to distinguish measurement failures due to faulty sensors from system faults such as short or open circuit at loads. Considering the current trends in power system state estimation, the decentralized approach seems like a promising option. In particular, when applied to power systems with decentralized generation and control, it could improve the performance of the data communication and processing. In fact if data are processed locally, the amount of data that each node must handle is smaller that of the centralized case. Furthermore, the decentralized state estimation can be done in parallel, leveraging on node-to-node communication for the integration of the results. A fully decentralized algorithm is presented in [1], and has been used to perform state estimation of power system in [2], [3]. With such approach bad data rejection may be more challenging and the uncertainties in the knowledge of topology, the power system parameters and the measured data call for uncertainty analysis.. Particularly, the case of power

This work was supported by the US Office of Naval Research under the grant N00014-07-1-0603.

system state estimation is considered here. An analysis of some of these challenges in reported in [4]. For on line operation Monte Carlo methods do not represent a feasible option, and fast stochastic methods are to be considered instead. In this work in particular PCT is adopted. In this work, the authors leverage on the characteristics of PCT (detailed in [5], and applied in [6], [7]) to propagate uncertainty in decentralized power system monitoring. The proposed approach enables, in principle, the propagation of power system uncertainties in real time, given a known probability distribution function (PDF) of the main uncertainty sources. This feature, which has been shown to be a way to determine the boundaries of the dynamic behavior of the system [8], can be used to generate the thresholds for validating sensor behavior. Sensor validation is expected to improve the reliability of distributed state estimation. In this paper, a general method to construct decentralized PCT estimator (DPCTE) is presented in II; in III the algorithms for measurement validation are described; in IV the DC zonal power system of a ship is presented as a case study for voltage sensor validation; finally in V conclusions are presented together with future research directions. II.

FORMULATION OF THE GENERAL DPCTE

The concept of observing uncertain states using PCT has been introduced in [9] where the definition of Polynomial Chaos Observer (PCO) is proposed. The PCT state estimation consists in determining the PCT coefficients of the state variables and thus the probability density of these variables at every time instant. The DPCTE discussed in this paper is built upon the partition of the multi-sensor system. Consider a multisensory system modeled in PCT domain. Suppose that the system is the composition of local subsystems, each with its own estimator. Suppose also that each sensor provides its data only to one estimator. The operation scheme of decentralized estimation is illustrated in Figure 1Error! Reference source not found.Error! Reference source not found.

Ni: System order of the ith subsystem; Mi: Number of sensors in ith subsystem; INi and IMi: Unit matrices with dimension Ni and Mi; s_m and s_F : Spectral density of the measurement and plant noise; Apct and Bpct: The state and input matrices of the global PCT system model. The operation of each DPCTE nodal observer is described by three stages shown in equations (7) to (18): First stage, prediction:

Figure 1: Scheme of DPCTE with Sensor Validation for Power System

The two most important steps for decentralization are the decomposition of the state matrix, and the decomposition of the observation model. The decomposition of the state matrix requires a state nodal transformation matrix (Tipct). For each subsystem, Tipct makes the local states (states of the decentralized subsystem) a subset of the global states (states of the centralized model). One possible way to find Tipct is by examining the coupling relationship among all the uncertain states [10]. For instance, if a state is a variable of a component which is included in a subsystem (such as inductor current), it is a direct state of this subsystem. Concurrently, if another state variable is necessary to calculate the direct states of this subsystem, it is an indirect state of this subsystem. The decomposition of the observation model is accomplished via the observation nodal transformation matrix (Hipct). Hipct reduces the dimension of the observation model and is chosen so that only local sensors are included. Basically, it represents a map of the sensor location. With Tipct and Hipct, equations used to calculate the matrices for each local subsystem i are given as follows: + Aipct = Tipct ⋅ A pct ⋅ Tipct

(1)

Bipct = Tipct ⋅ B pct

(2)

+ C ipct = H ipct ⋅ C pct ⋅ Tipct

(3)

Dipct = H ipct ⋅ D pct

(4)

Qipct = s _ F 2 ⋅ I Ni

(5)

Ripct = s _ m 2 ⋅ I Mi

(6)

(7)

T PFipct ( k + 1 | k ) = Aipct ⋅ PFipct (k | k ) ⋅ Aipct + Qipct

(8)

Second stage, updating: T S ipct = Cipct ⋅ PFipct (k + 1 | k )Cipct + Ripct

T −1 K ipct = PFipct (k + 1 | k )Cipct ⋅ S ipct

(9) (10)

~ xipct (k + 1 | k + 1) = xipct (k + 1 | k ) + K ipct ⋅ (mi (k + 1)

(11)

− Cim ⋅ xipct _ Mean (k + 1 | k ) − Dim ⋅ ui (k + 1)) ~

T PF ipct (k + 1 | k + 1) = PFipct (k | k ) − K ipct ⋅ S ipct ⋅ K ipct

(12)

At the end of the updating step, three pieces of information of each local subsystem need to be communicated to all nodes: the local covariance error matrix, the state error vector, and the breaker status if breakers are involved. The former two matrices are calculated by equation (13) and (14) respectively. ~

cipct (k + 1) = PF ipct (k + 1 | k + 1) −1 − PFipct (k + 1 | k ) −1

(13)

~

Where, + : The Moor-Penrose generalized inverse of the nodal Tipct

transformation matrix Tipct;

xipct (k + 1 | k ) = Aipct ⋅ xipct (k | k ) + Bipct ⋅ ui (k )

s ipct (k + 1) = PF ipct (k + 1 | k + 1) −1 ⋅ ~ x ipct (k + 1 | k + 1) − PFipct (k + 1 | k ) −1 ⋅ x ipct (k + 1 | k )

(14)

Due to the varied dimension of each local state vector, these two quantities have to be mapped from one node to another using equation (15) (for covariance error) and (16) (for state error) before data assimilation. T T ci − jpct (k + 1) = T jpct ⋅ [Tipct ⋅ cipct (k + 1) ⋅ Tipct ] ⋅ T jpct

(15)

T si − jpct (k + 1) = T jpct ⋅ [Tipct ⋅ sipct ( k + 1)]

(16)

The final stage is the measurement assimilation described by equation (17) and (18) which guarantees the minimization of the estimation errors.

m

−1 PFipct (k + 1 | k + 1) = [ PFipct ( K + 1 | k ) + ∑ ci − jpct (k + 1)]−1

(17)

i =1

−1 xipct (k + 1 | k + 1) = PFipct (k + 1 | k + 1) ⋅ [ PFipct (k + 1 | k ) ⋅ m

xipct (k + 1 | k ) + ∑ s i − jpct (k + 1)] i =1

III.

SENSOR FAILURE DIAGNOSIS ALGORITHM

A. Measurement Validation Accounting for the uncertainties in the system will strengthen a failure detection algorithm. Thanks to the PCT analysis, it is possible to estimate the boundaries of the dispersion of values of each variable in the system due to parametric uncertainty. The knowledge of these boundaries can be exploited to set thresholds to detect sensor failure. The sensor diagnosis (SD) algorithm proposed in this paper diagnoses failures when on measurement results exceeds the thresholds therefore is outside the reasonable boundaries that the system variable can take. The upper and the lower thresholds of the voltage sensors can be obtained from the uncertain states estimated by each local DPCTE. The operation logic of SD and bad data reconstruction (BDR) can be summarized as follows: If measurement is within the uncertain thresholds, Sensor is healthy; Else, sensor is failed. SD identifies and isolates the failed sensors; BDR rebuilds the bad data. B. Failure Isolation and Missing Data Reconstruction Failure isolation is realized by permanently locking the sensors’ status at “Failed” if and only if a sensor has been diagnosed to be “Failed” by SD. This way data from such nodes will be blocked. The simplified logic of reconstructing missing data is portrayed in Figure 2. The sensor status diagnosed by SD acts as the trigger of the rebuilding action of BDR:

(18)

each DPCTE is built assuming a reliable system model. When the system has faults, the model used to build DPCT is no longer reliable. Thus the sensor validation algorithm should be disabled while the faulty conditions are in place (indicated by change of breaker status), and resumed after the faults is cleared, and the system has reached steady state. Since breakers may be involved in different subsystems, breaker status need to be communicated to all DPCTE Basically, the system condition is examined by checking every breaker status while the measurements are diagnosed. The determination of whether there is a system fault or not is based on the following signal index: N −1

CTR = ∑ ( Bi + Bk ) k =1 k ≠i

(19)

Where, Bk: Logic signal of breaker status received from other subsystems; Bi: Logic signal of breaker status of local subsystem; N: total number of the breakers in the whole system (i and k satisfy i + k = N); B =1 (faulty condition) or 0 (no fault). So, basically, transition of CTR from zero to one indicates the occurrence of a system faulty condition in one or more subsystems. Vice versa, a clearance of the faults is indicated. IV. CASE STUDY In this section, the DC zonal power system with the topology outlined in [2] is adopted as a study case. The main sources of uncertainty of this power network are assumed to be the resistive component R3 and R4 of the loads. The PDF of R3 and R4 used here are uniform distributions with an interval of 20% and 30% of the central value respectively. The equivalent circuit of the zonal system is shown in Figure 3:

Figure 2: Missing Data Reconstruction Algorithm

C. Distinguish System Faults from Sensor Failures We assume that the breakers are automatically operated. The purpose for retrieving breaker status is to avoid misunderstanding of system faults as sensor failures. In fact

Figure 3: Equivalent Circuit for the zonal power system

The sensor location can be mapped by matrix H in equation (20) and the observation nodal transformation matrix for subsystem i is given by equation (21). ⎡1 ⎢0 H =⎢ ⎢0 ⎢ ⎣0

0 1 0 0

0 0 1 0

0⎤ 0⎥⎥ 0⎥ ⎥ 1⎦

Hipct=H(i,:); i=1,…m

(20)

(21)

The number of rows of H equals the number of sensors; the number of columns equals the number of zonal subsystem. The coupling relationship between the states can be obtained from the state matrix. Let us exemplify the process with subsystem 2. This subsystem has only one direct state variable x3 (Figure 3). State x3 is uncertain because of the load uncertainty in the system. So the state variables for PCT expanded model of subsystem 2 is the triplet (x30, x31, x32). By inspection of matrix Apct, it can be seen that these three PCT states are coupled to other nine PCT states (x10, x11, x12, x40, x41, x42, x50, x51, x52), which are the indirect states of subsystem 2. So, T2pct, the state nodal transformation matrix for uncertain subsystem 2, is a 12 by 15 matrix with “1” for the eighteen relevant states (direct and indirect states) and “0”.for all other elements. Using Hipct and Tipct, the uncertain system obtained from subsection A can be decentralized into four uncertain subsystems using equations (1) to (6), and DPCTE can be built for each local system with equations (7) to (18) presented in II.

Because multiple uncertain sources are considered in the system expansion, the mean and the variance of the sensor response could be reconstructed from the uncertain states with circuit laws as [11]: E[ f i ( x)] = f i ( x0 )

(22)

p

V [ f i ( x)] = ∑ f i ( xn ) 2

(23)

n =1

Where, i=1,…m; fi: vector of functions. D. Simulation Results and Analysis In the simulation presented here, the communication among all subsystems is realized by the broadcast function of UDP. Two cases are presented: in Case 1 one or more sensors fail; in Case 2 sensor failures are distinguished from short circuit fault at load branches. The test conditions for Case 1 are given in TABLE IError! Reference source not found.. Results are shown Figure 4 through Figure 6. The results demonstrate the effectiveness of the basic operation of the sensor validation algorithm discussed in the previous sections. TABLE I: Test Conditions for Case 1 Order

Operation Conditions

(1)

Voltage sensor in Zonal 3 fails (system tolerates single sensor failure)

(2)

Voltage sensor in Zonal 1 fails (system tolerates multiple sensor failures)

sensor 1, zonal 1 100

y1 (V)

B. Uncertain Model Decentralization The scheme can be decomposed into four sub-systems [2]: two generator branches and two load branches, as depicted in Figure 3. For each sub-system, the local bus voltage ym(k) with m=1,…4, are the inputs of the local DPCTE. These measurements are assumed to be obtained from various sensors located in the subsystems.

reconstruct the sensor response with the dynamic uncertain states referring to the given PDF.

95 1

2

3

4

100

y2 (V)

A. Model Expansion For this particular power system, the spectral expansion is applied to the two independent uncertain sources (nv=2) and using a first-order polynomial (np=1). This expansion is developed with the same procedure introduced in [9]

2

y3 (V)

7

8

4

6

8

9

10

Measured Threshold-U Threshold-L Reconstructed

10

sensor 3, zonal 3

90 88 1

y4 (V)

6

95

92

C. Validation Thresholds Generation Sensor failure detection with PCT is based on the analysis of the worst-case that is, the maximum possible variation with respect to the expected value. In other words, the occurrence of voltage values outside the thresholds in correspondence to PDF tail values of the parameters is interpreted as a symptom of sensor malfunction. This verification can be performed at every time step. So, the most reasonable way to find the thresholds of a sensor output is to

5

sensor 2, zonal 2

2

3

4

5

6

7

sensor 4, zonal 4

8

9

10

9

10

90 89.6

2

3

4

5

6

7

Time (s)

Figure 4: Voltage Sensor Validation

8

2

6

8

Time (s) sensor 3, zonal 3

2

4

6

System short circuit fault at load in zonal 2 (cleared after 0.15 seconds)

0

2

4

6

8

(3)

Voltage sensor in Zonal 1 fails (system tolerates multiple sensor failures)

10

Time (s)

(1)

0

(2)

1: Healthy; 0: Failed sensor 4, zonal 4 2

1

0

0

10

(1)

8

10

1

0

0

2

4

Time (s)

6

8

10

Time (s)

Figure 5: Voltage Sensor Status Monitoring

(3)

0

5

12

10

2

4

6

8

10

4

6

8

5

1

10

0 0

15

5

10

15

Time (s)

Figure 7: Voltage Sensor Status Monitoring

10

Time (s) state x5, zonal 4

sensor 1, zonal 1 y1 (V)

Current (Amp)

Current (Amp)

15

5

2

Measured Estimated 10

20

15

sensor 4, zonal 4

2

Time (s) 4

10

10

Time (s)

6

100

95 2

5

4

6

0

2

Time (s)

4

6

8

10

From the results, it can be noticed that the decentralized state estimation could survive even if some of the local systems lose their sensors. Real measurements are used by DPCTE when sensor is healthy (the blue and read lines are overlapped in Figure 4), and the DPCTE best case estimation is adopted instead to reconstruct the missing sensor measurements when sensor is diagnosed as failed (the blue line is out of the two thresholds in Figure 4). The failed sensors could be detected and reported (Figure 5), and each DPCTE could still operate normally with the constructed measurements (Figure 6). Case 2 shows that the proposed method can distinguish typical system faults (short circuit at load is considered here) from sensor failures. To simplify the problem, it is assumed in this example that all system faults could be cleared before the system becomes unstable. Test conditions and sequence for Case 2 are given in TABLE II. Simulation results are shown from Figure 7 through Figure 9

10

12

14

16

18

Measurement Threshold-U Threshold-L Reconstructed

98 96

Time (s)

Figure 6: DPCTE State Estimation

8

sensor 2, zonal 2

100

y2 (V)

8

0 0

8

(2)

y3 (V)

6

Time (s) state x4, zonal 3

5

1: Healthy; 0: Failed

10

5

10

15

sensor 3, zonal 3

90 88 2

4

6

8

10

12

14

16

18

12

14

16

18

sensor 4, zonal 4 y4 (V)

4

1

0 0

15

(1)

90.5 90 89.5

2

4

6

8

10

Time(s)

Figure 8: Voltage Sensor Validation States x1 (Amp)

2

10

Time (s)

1

25

x5 (Amp) x4 (Amp) x3 (Amp)

Current (Amp)

Current (Amp)

10

Status

state x3, zonal 2

state x1, zonal 1 25

data1

sensor 3, zonal 3

2

15

2

1

0

20

sensor 2, zonal 2

sensor 1, zonal 1 2

Status

4

Operation Conditions Voltage sensor in Zonal 3 fails (system tolerates single sensor failure)

Status

2

Order

1

Status

0

2

Status

Status

1

Status

Status

(2)

0

TABLE II Test Conditions for Case 2

sensor 2, zonal 2

sensor 1, zonal 1 2

11

20

6

2

4

6

8

10

12

14

16

18

2

4

6

8

10

12

14

16

18

2

4

6

8

10

12

14

16

18

2

4

6

8

10

12

10 5 2 0

Time (s)

14 Measured

Figure 9: DPCTE State Estimation

16

18 Estimated

For Case 2, the expected outcomes of the algorithm are as follows. In test condition (1), failure of sensor 3 should be reported by SV module in zonal 3, and other sensors should be reported as “Healthy”. In test condition (2), all sensor status should keep the same as those between test conditions (1) and (2). In test condition (3), another failure of sensor 1 should be reported by the corresponding SV module in zonal 1 and other sensors’ status should remain unchanged. Finally, in test condition (4), all sensor status should stay the same as those between test conditions (3) and (4). These expected results are all proved by status monitoring shown in Figure 7. Figure 8 presents the voltage validation thresholds for each sensor. Proofs of the state estimation performance during all test conditions are illustrated by Figure 9. Transients are obvious (circled) but all DPCTE could survive. V.

CONCLUSIONS AND FUTURE WORK

This work presents a novel method to perform sensor validation in decentralized state estimation of electric power system. Propagation of parametric uncertainty is used to generate the thresholds to bind the sensor output. The validation algorithms are implemented along with local estimators. For each local subsystem, validation algorithms can also sort out typical system faults, which may occur in the system, from the effect of failed local sensors. Numerical results of this method, applied to a DC zonal shipboard electric power system, demonstrate a prospective use of this approach. Further developments currently pursued focus on two aspects: improving the current algorithms with artificial intelligence techniques, and testing the proposed method in the real time environment. ACKNWLEDGMENT This work has been partly supported by the US Office of Naval research under grant N00014-08-1-0080.

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