improved network parameter error identification using ... - PSCC

IMPROVED NETWORK PARAMETER ERROR IDENTIFICATION USING MULTIPLE MEASUREMENT SCANS Liuxi Zhang and Ali Abur Department of Electrical and Computer Engineering Northeastern University Boston, MA, USA [email protected] and [email protected] Abstract – This paper investigates the problem of detection and identification of parameter errors in large scale power systems. Maintaining error free parameter databases of power systems is a major challenge due to daily changes in network topology and network parameters. This challenge was recently addressed by a method which was based on calculation of Lagrange multipliers using a single measurement scan. This paper presents a simple yet effective improvement by using multiple measurement scans. It improves redundancy and allows observability of parameter errors, which could otherwise not be detected, without additional cost. In addition, the method is easy to implement since it uses the state estimation results from different scans without requiring changes in the existing state estimation code. Simulations on IEEE 14- and 30-bus systems are provided to illustrate the performance of the method.

Keywords: State estimation, parameter errors, bad data processing, optimization, multiple measurement scans. 1 INTRODUCTION State estimation plays a significant role among all the energy management system (EMS) applications, because it provides the network model for others. Network parameters, such as transmission line resistances, reactances, transformer reactances, taps and shunt capacitor/reactor parameters are all stored in the parameter database to build the network model. Traditionally, state estimation is implemented assuming perfect knowledge of the network model, thus suspecting no errors in the network parameters. However, this assumption may not always be true. Maintaining error free data bases for large scale power systems is an essential challenge because of the frequent changes in network topology and network parameters. Transformer taps may be moved but not recorded in the database, circuit additions to existing right-of-ways, upgrades to existing conductor configurations may not be reflected in the line models and their parameters used by the control center application functions. There may also be changes in parameters of equipment models due to aging, environmental effects and various nonlinearities not accounted for at the time of their installation. Detecting changes in network parameters represents a challenge due to the very large dimension of the parameter space for an interconnected power system.

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As a result, parameter error detection and identification in power system state estimation remains an issue that has not been fully resolved. Traditional approaches took advantage of the state estimation function and included a set of suspected parameters in the state vector, and estimated the augmented state vector thereby estimating the suspected parameter values [1-12]. In addition to steady state models, some of the methods relied on dynamic modeling [2, 8] or use of Kalman filter theory [9-13]. These approaches worked well as long as there were few parameters to be checked, however they became impractical when the entire network parameter errors were required to be tested for errors. Recently, a new method which used a fundamentally different approach to parameter error detection has been presented [14]. This approach revised the state estimation formulation by inserting a set of trivial constraints which stated the assumption that all parameters were error free. The problem would then be solved as an optimization problem where the sensitivity of the objective function to these trivial constraints would be used to identify those parameters which were in error. The method was shown to successfully detect and identify parameter errors in large systems without the need to specify a suspect set of parameters in advance. It was also shown that the effectiveness of the method depended on the existing measurement redundancy and in certain cases even with highly redundant measurement sets the method’s ability to identify certain types of parameters would be limited. This paper addresses this last issue by using multiple measurement scans instead of a single snap shot. This is a technique which was employed earlier for the traditional augmented state estimation method in order to improve redundancy and allow observability of parameters [15, 16]. In this paper, multiple scans are introduced to the new formulation. It is shown that some of the limitations of the technique which were reported in [14] will be lifted as a result of this reformulation. The proposed revision is particularly suitable for very large scale systems where certain regions may not have the required redundancy yet network parameters in those regions may be most critical to monitor. In addition, this method is easy to integrate because it can be applied after obtaining the different scans of estimation results separately, which indicates there is no need to change the existing estimation program.

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Considering multiple measurement scans increases the number of available measurement equations as well as the number of state variables which correspond to these different scans. However, parameter errors are assumed to remain the same since they are typically not a function of system state. This is the main advantage that produces redundancy and leads to identification of parameter errors which may otherwise be missed. After presenting the detailed derivation of the revised formulation, simulation examples on IEEE 14- and 30bus systems will be provided to illustrate the technical contributions. The method is simple to integrate, yet provides a powerful solution to improve network database accuracy at no extra investment.

(7) (8) (9) where (10) (11) (12)

2 PARAMETER ERROR IDENTIFICATION BASED ON LAGRANGE MULTIPLIERS

(13)

2.1 Problem Formulation Incorporating parameter errors into the conventional weighted least squares (WLS) state estimation problem, yields the following optimization problem: Min:

and using Taylor approximation and equations (4) through (9), the following compact form will be obtained:

(1)

(14)

s.t. (2)

where (15)

where diagonal weight matrix; measurement vector; measurement residual vector; nonlinear function of system state and parameters related to measurements; nonlinear function of system state and parameters related to zero injections; vector containing parameter errors; system state vector One way to solve the optimization problem of Eq.(2) is by forming the Lagrangian which is given as:

(3) where , and are the Lagrange multipliers corresponding to the equality constraints in Eq.(2). Writing the first order optimality conditions: (4) (5) (6)

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(16) is the initial guess of the system state, and denote and respectively, in which is replaced by 0 using the equality constraint. This equation, which is nothing but the conventional WLS iteration, can be iteratively solved. Following convergence, the measurement residual vector and Lagrange multiplier vector for zero injections will be computed. Those Lagrange multipliers corresponding to the parameter errors can then be retrieved by the following equation [14]: (17) which is derived from Eq.(5) and Eq.(6). Finally, the normalized Lagrange multipliers can be calculated to identify the network parameter errors [14]. Normalized measurement residuals can also be computed simultaneously. This allows measurement errors to be identified as well. The entire identification procedure is summarized below: Step1: Run WLS state estimation, calculate r and µ and store results. Step2: Calculate and . The computation details can be found in [14].

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Step3: Detect and identify parameter errors and bad data. The largest one among all the normalized Lagrange multipliers and residuals computed in Step2 is selected if it is greater than the threshold (commonly chosen as 3.0). The corresponding parameter or measurement will be declared erroneous. Otherwise, it will be concluded that no parameter error or bad measurement exists in the system. n

2.2 Limitations The effectiveness of this method is closely related to the measurement redundancy. It has been shown in [14] that in certain cases, even with highly redundant measurement sets, the method may fail to identify certain types of errors. In this paper, this limitation is removed by integrating multiple measurement scans and reformulating the identification problem. In the following sections, the derivation of this algorithm will be introduced in detail and some simulation examples, involving those limitations presented in [14], will be presented to illustrate the benefits provided by this method. 3

nonlinear function of system state and parameter errors related to measurements in each scan; nonlinear function of system state and parameter errors related to zero injections in each scan; vector containing parameter errors; system state vector of each scan: voltage magnitudes and angles; number of scans.

Forming the Lagrangian:

(20) , and are Lagrange multipliers of the equality constraints in Eq.(19). The first order optimality conditions can then be applied to yield: (21)

ALGORITHM INTEGRATING MULTIPLE MEASUREMENT SCANS

(22) (23)

3.1 Proposed method Incorporating multiple measurement scans into state estimation is a technique which was applied before for conventional augmented state estimation method. In this section, the approach will be used to improve the method outlined in section 2. The basic assumption is that the network parameters, including parameter errors, remain unchanged for all the multiple scans. Therefore, the method takes advantage of repeated state estimation solutions without investing in new meters. This results in some parameters becoming observable even though they are unobservable in case of single scan estimation. Furthermore, different scan state estimation solutions can be obtained independently, without the need to modify the state estimation code that is currently in use. The detailed derivation of the method is given next. Similar to Eq.(1), the WLS problem with multiple measurement scans can be formulated as below: Min:

(24)

(25) (26)

where (27) (28) (29)

(18)

(30)

s.t. (19) where diagonal weight matrix of each scan; measurement vector of each scan; measurement residual vector of each scan;

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Then, Lagrange multiplier Eq.(22) and Eq.(23) as:

can be expressed using

(31)

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where (38) (32) (39) (33)

(34)

(35)

(36)

Eq.(31) can be used to recover the Lagrange multiplier vector corresponding to the parameter errors, after obtaining all the estimation results of multiple scans. Comparing Eq.(31) to Eq.(17), the total number of Lagrange multipliers does not change, because of the assumption that the number of the erroneous parameters remains the same. However, with multiple measurement scans, the value of each multiplier will be different, and they will be more sensitive to errors because of the improved redundancy. While not mentioned explicitly, the above formulation does not require or exclude any kind of measurement for it to work. Therefore, looking into the future, it is expected that any existing synchronized phasor measurements will be incorporated into the measurement set and corresponding equations will be augmented to the measurement equations. The above presented formulation will not be affected by this and added redundancy will likely to further improve the error identification capability. However, such measurements were not used in this particular study. Note that estimation of system states of different scans will still be carried out independently. The existing estimation program based on Lagrange multipliers can still be applied but run at different times to archive the results of different scans. As a result, Eq.(14) can still be applied to solve the WLS problem after reformulation to incorporate multiple measurement scans as follows

(40)

(41)

(42) , and ( denote , and of each measurement scan. This expanded version can also be solved iteratively as done in conventional WLS state estimation problem. After the results including system states of each scan have been obtained, the Lagrange multipliers corresponding to all parameter errors can be recovered by Eq.(31). Normalized Lagrange multipliers , can be computed by the same method described in [14]. The entire identification procedure is summarized below. Step1: Run WLS state estimation with multiple measurement scans which can be executed separately. Step2: Recover using Eq.(31). Calculate normalized Lagrange multipliers and normalized residuals by the method presented in [14]. Step3: Identify the parameter errors and bad data via the same method explained before. The flow chart of this process considering multiple scans is shown as Figure 1. Comparing this approach to the single scan estimation, two main advantages of the proposed method will be noted. One is the implicit improvement in redundancy without actually adding any new meters. This allows certain types of parameter errors, which cannot be estimated before via single scan, become identifiable. The other is that it can be implemented without modifying the existing state estimation programs. These programs can be executed repeatedly at different times. Simulation results on IEEE standard test systems will be presented in the next section.

(37)

where

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Existing system with network parameters Multiple measurement scans State estimation

Calculation of system states (xi), residuals (ri) and Lagrange multipliers λ

Computation of normalized residuals rN

Computation of normalized Lagrange multipliers λN

Figure 3: IEEE 30-bus system Compare and identify parameter or measurement errors

Figure 1: Flowchart of the identification method integrating multiple measurement scans.

4

SIMULATION RESULTS

The proposed method has been implemented and tested on IEEE 14- and 30-bus systems. In order to illustrate the improvements brought out by this method, some examples are recalled from [14] and comparatively solved by both single and multiple measurement scan methods.

Two examples are recalled from [14]. Errors are introduced to shunt susceptances at bus 9 of IEEE 14-bus system and bus 24 of IEEE 30-bus system respectively. The normalized Lagrange multipliers and measurement residuals are computed. Test A uses the single measurement scan method, which is the same as before, and Test B applies 5 different measurement scans. The number of scans “n” is selected based on the desired increase in redundancy to ensure identification of each and every parameter and measurement error. Given the local nature of redundancy in sparsely connected power systems, the required number of scans will not grow with system size and is expected to remain a small percent of the number of buses. Full conventional measurement set (including power flow, injection and voltage magnitude measurements) is used in each scan. The results are presented in Table 1 and Table 2. Test A (Single Scan) Parameter/ Measurement

/

Test B (Multiple Scans) Parameter/ Measurement

/

7.6305 16.7925 7.6305 7.6349 2.3828 7.6305 Table 1: IEEE 14-bus system results of error identification Figure 2: IEEE 14-bus test system

4.1 Case 1: Shunt capacitor/reactor parameter errors

Test A (Single Scan) Parameter/ / Measurement 7.7012

As described in [14], the parameter error in shunt capacitor is detectable but not identifiable even with high measurement redundancy. The main reason is that having reactive power injection measurement is not enough to differentiate the source of error, i.e. whether it is due to errors in capacitor susceptance or in the reactive power injection measurement.

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Test B (Multiple Scans) Parameter/ / Measurement 15.9898

7.7012 7.8153 2.7821 7.7012 Table 2: IEEE 30-bus system results of error identification

As shown in Table 1, using the single scan method (Test A), the normalized residual corresponding to reactive injection at bus 9 ( ) is identical to the normalized

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Lagrange multiplier corresponding to the shunt susceptance at bus 9 ( ). Hence, the error on the injection measurement or the parameter cannot be identified when only single measurement scan is applied. As evident from Test B, implementing multiple measurement scans overcomes this limitation and yields the normalized Lagrange multiplier for the shunt susceptance ( ) as the largest. Similar results are obtained for other systems like IEEE 30-bus system, which are shown in Table 2. It should be added that the errors on the injection measurements have also been tested and correctly identified by the proposed method. 4.2 Case 2: Errors in areas lacking local redundancy Parameter error identification method that is based on Lagrange multipliers also relies on measurement redundancy and network topology. The ability to identify will be limited if the errors occur in an area which has relatively low local redundancy. An illustrative example for such a case is an error in the impedance of Branch 6 to 13 ( in IEEE 14-bus system. Figure 4 shows the local area where the error occurs. The measurements in this part of the system are provided in Table 3. The results of parameter and measurement error identification are shown in Table 4. Single measurement scan is applied in Test A, while 5 different measurement scans are implemented in Test B.

and a measurement is suspected instead. The main reason for this failure can be related to the very low local redundancy. In test B, the parameter error can be identified by applying multiple measurement scans but without incorporating any new measurements. The problem of low local redundancy is resolved without additional cost. Several other examples have been found in larger systems with similar results. These results suggest that the proposed algorithm can be especially suitable for very large systems containing pockets of very low measurement redundancy. Incidentally, these may be the parameters that are most crucial to monitor. This revision is convenient to integrate and provide a powerful tool to improve the accuracy and reliability of the parameter database with no additional cost.

5

CONCLUSIONS

The paper first reviews the existing method of identifying network parameter errors based on Lagrange multipliers using single measurement scan. This method is then shown to have certain limitations which cannot be easily overcome by simple increase in measurement redundancy. Instead a simple extension to this method is proposed where several measurement scans are used in detecting and identifying parameter errors. The main advantage of this approach is that it makes certain types of parameter errors that are not identifiable by existing single scan method, readily identifiable. Thus an error free parameter data base can be maintained without investing in additional meters. Furthermore, this revision is easy to implement without requiring changes to the existing state estimation programs. REFERENCES

Figure 4: Part of IEEE 14-bus system

Measurement Type Power Injection Power flow

Location Bus 6 ,

, ,

, ,

, ,

Table 3: Measurements of the part of IEEE 14-bus system

Test A (Single Scan) Parameter/ / Measurement 5.9619 5.6402 5.5904

Test B (Multiple Scans) Parameter/ / Measurement 16.4211 16.1147 14.9704

Table 4: IEEE 14-bus system results of error identification

As shown in Table 4, in Test A when only single measurement scan is used, parameter error is missed

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[6] O. Alsac, N. Vempati, B. Stott and A. Monticelli, “Generalized state estimation,” IEEE Trans. Power Syst., vol. 13, no.3, pp. 1069-1075, Aug. 1998. [7] P. Zarco and A. Gomez, “Off-line determination of network parameters in state estimation,” Proceedings 12th Power System computation conference, pp. 1207-1213, Dresden, Germany, Aug. 1996. [8] E. Handschin and E. Kliokys. “Transformer tap position estimation and bad data detection using dynamic signal modeling,” IEEE Trans. on Power Systems, vol. 10, no. 2, pp. 810-817, May 1995. [9] S. Arafeh and R. Schinzinger, “Estimation algorithms for large-scale power systems,” IEEE Transactions on Power Apparatus and Systems, vol. pas98, no.6, pp. 1968-1977, Nov. /Dec. 1979. [10]I. Slutsker, S. Mokhtari and K. Clements, “On-line parameter estimation in energy management systems,” American Power Conference., pp. 169, Chicago, Illinois, Apr. 1995. [11]I. Slutsker and S. Mokhtari, “Comprehensive estimation in power systems: state, topology and parameter estimation,” American Power Conference., pp. 170, Chicago, Illinois, Apr. 1995. [12]K. Clements and R. Ringlee, “Treatment of parameter uncertainty in power system state estimation,” IEEE Transactions on Power Apparatus and Systems, paper C74 311-7, Anaheim, Cal., July 1974. [13]I. Slutsker and K. Clements, “Real time recursive parameter estimation in energy management systems,” IEEE Trans. Power Syst., vol. 11, no.3, pp. 1393-1399, Aug. 1996. [14]J. Zhu and A. Abur, “Identification of network parameter errors,” IEEE Trans. Power Syst., vol. 21, no. 2, pp. 586–592, May 2006. [15]A. Abur and A. Gómez-Expósito, Power System State Estimation: Theory and Implementation, NewYork: Marcel Dekker, 2004. [16]S. Zhong and A. Abur, “Combined State Estimation and Measurement Calibration,” IEEE Trans. Power Syst., vol. 20, no. 1, pp. 458–465, Feb. 2005.

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