A Static State Estimation Scheme in Microgrid Utilizing <i>μ</i>PMU Measurements

Kumar, Kunal; Kumar, Prince; Kar, Susmita; Kumar Bohre, Aashish; Amde Gebereselassie, Samuel

doi:https://doi.org/10.1155/2023/1916299

International Transactions on Electrical Energy Systems

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Abstract Introduction Discussion Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 1916299 | https://doi.org/10.1155/2023/1916299

A Static State Estimation Scheme in Microgrid Utilizing μPMU Measurements

Kunal Kumar,¹Prince Kumar,²Susmita Kar,¹Aashish Kumar Bohre,³and Samuel Amde Gebereselassie⁴

Academic Editor: Kin Cheong Sou

Received30 Jan 2023

Revised14 Apr 2023

Accepted24 Apr 2023

Published09 May 2023

Abstract

The operation of power systems needs uninterrupted and precise wide-area real-time monitoring for secure and automated control of power networks, therefore, there is a requirement of robust state estimation (SE) by the smart control centre. An iterative free static state estimation (IFSSE) scheme is developed using the linear measurement model (LMM) in this paper for microgrids (MGs) integrating distributed generation energy resources (DGERs). The developed scheme of IFSSE utilizes M-estimator (ME) concept, which is combined with the linear static state estimation (SSE) approach in order to provide robust estimation. The developed IFSSE scheme is iteration-free due to the implementation LMM that is achieved from placing the microphasor measurement units (μPMUs) and separate bad data processing (BDP) is not required for this proposed scheme. The proposed scheme's performance has been verified by employing it on modified IEEE-14 bus test systems (BTS) integrating DGERs and modified IEEE 30 BTS integrating DGERs then comparison has been done by taking two cases, i.e., linear SSE without BDP and linear SSE with BDP.

1. Introduction

State estimation is an important tool for monitoring the power system networks for reliable and stable power system operations. Fred Schweppes first introduced the SE concept [1]. The main focus of SE is to extract the voltages and phases at each bus with the help of the measurements. The application of the SE technique and its research earlier focused on transmission networks. The transmission networks are equipped with measurement instruments that provide adequate measurement data for analysis. The development of electric distribution grids together with MGs demands monitoring and control for reliable operation [2, 3]. The significance of SE was appraised at the outset of SE and has continued in the industries [4].

The bad data (BD) effect and its suppression from the measurement scheme are discussed by Merrill et al. [5]. Koutiga and Vidyasagar [6] introduced the scheme which is able to detect and eliminate simultaneously the BD and a tracking SE, which is also discussed by Koutiga in [7]. Monticelli and Garcia [8] introduced a new scheme which is simple, reliable, and works on normalized residuals. The outliers can be present in the measurement of industrial and scientific data which is unavoidable, therefore, research is going on to design robust SE that will be insensitive to outliers [9, 10]. The ME concept is first discussed by Huber for robust SE which easily minimizes the measurement residual by suppress the BD [11]. The ME approach in SE is generally designed for ensuring robust SE by automatically rejecting the outliers. Milli et al. [12] proposed a fast and robust SE algorithm for identifying leverage points which is based on reweighted least squares. A PMU- and SCADA-based robust hybrid SE is proposed in [13] that is insensitive to BD. Zaho and Mili [14] proposed a robust GM unscented Kalman filter that cancels out the BD and noise effects in measurements obtained from PMU. A projection statistics-based SE in distribution system is proposed in [15] which is insensitive to BD, noise, and smart meter lost measurement.

The traditional PMUs are now replaced by μPMUs in electric distribution grids. μPMUs are more accurate, slowly less expensive as compared to traditional PMUs, and provide the available data in distribution grids [16, 17]. The SE in electric distribution grid is getting more recognition [18–20]. The DGERs integration in the electric distribution grids is increasing more and more to get environmental advantages. The concept of MG is introduced to manage the expansion of increased DGERs integration and reliable operation. Modern electric distribution grids are generally integrated by automated MGs [21]. The measurements can be erroneous due to noise and faults in sensors. Hence, SE ensures accurate estimations of measurements. Various researchers proposed SE in MGs for efficient and stable microgrids operation. Huang et al. [22] suggested electric grid estimator for electric distribution grids for MGs. Nguyen [23] discussed the thermal effect on SE in microgrids. Doorsamy and Cronje [24] discussed the SE on stand-alone DC microgrids using distributed intelligence. Mullen and Wade [25] discussed the SE of energy measurements for enhancing the trust for an application of distributed ledger on rural MGs. PMU-based SE in MGs are also introduced by some researchers. Ali et al. [26] proposed real-time SE in MG using PMU but they utilized weighted least squares (WLS) approach with PMU snapshots to estimate the state in an iterative manner. Lin et al. [27] proposed decentralized robust SE based on PMU measurements of electric distribution grids integrating MGs. However, the state is estimated in an iterative manner that leads to computational burden. Ahmed et al. [28] discussed the various methods of distribution system SE for enhancing the application of smart grid. Cintuglu and Ishchenko [29] introduced the secured distributed SE for networked MGs in which the BD identification in the main algorithm turns into misbehaving node identification. However, the scheme is not iterative free. Pandey [30] introduced a PMU-based linear SE for improving the quality of data and reliable applications but they utilized a separate BDP algorithm for detecting the BD that leads to computational burden. Feng et al. [31] proposed an error resilient SE for calculating the state of MGs. However, the weighted least squares approach is utilized in this paper and the state Jacobian matrix is calculated which is again a computational burden for the algorithm. Kumar et al. [32] proposed ME-based SE where complex PMU measurements have been taken for finding out the state but this scheme is not tested for MGs integrating DGERs. It is found after the literature survey, various researchers proposed different approaches for the SE algorithm to cancel out the effects of BD and noise in measurements. However, mainly the WLS approach and linear SE, i.e., PMU-based approaches are utilized by many of the researchers in their papers. The linear SE is more efficient approach than the WLS approach. Many researchers utilize separate BDP algorithm that makes the SE scheme more complex and computationally inefficient. Most of the SE scheme in the literature survey uses iterative step to find out the state that takes more time to estimate the state. Also, a robust scheme for SE has not been tested on MGs integrating DGERs. Therefore, there is motivation to develop a new scheme for robust static state estimation for MGs integrating DGERs which is iterative free, i.e., estimates the state in single iteration, computationally more efficient than the each proposed algorithm of SE in the literature survey and cancels out the effect of BD from measurement without the requirement of the extra algorithm of BDP.

In this paper, an iterative free static state estimation (IFSSE) scheme is developed by utilizing the linear measurement model for MGs integrating DGERs. The optimal number of μPMUs is placed on the buses of modified IEEE 14 BTS and modified IEEE 30 BTS with integrating DGERs. The optimally placed μPMUs are providing the complex branch current (CBC) and complex branch voltage (CBV) to make the measurement model in linear form. A constant observation matrix is utilized by the developed IFSSE scheme which makes the developed algorithm free from the calculation of the Jacobian matrix. The M-estimator concept is utilized in the proposed IFSSE scheme that automatically checks the existence of BD in measurement, and then cancels out of the measurement set. Therefore, no separate BDP algorithm is utilized in the proposed scheme of IFSSE. It is found after literature survey, the linear SSE is most efficient adapted scheme by the researchers. Therefore, the linear SSE scheme is adapted for comparison purposes in this paper. The developed scheme is examined on modified IEEE 14 BTS and modified IEEE 30 BTS by simulating it on the MATLAB R2021b platform and then the developed IFSSE scheme is compared with the linear SSE by considering two cases, i.e., linear SSE without BDP and linear SSE with BDP.

2. Problem Statement

2.1. Conventional Nonlinear SSE

The mathematical formulation in which the measurement associates with true state vector “x” and error vector “e” is given as follows:where m = 1, 2, …, n. is measurement vector of size n; is function that associates state variables with measurements; x is state vector of size l; n denotes the number (no.) of measurements; and denotes measurement error. The error vector e possesses the following statistical properties:where “F” denotes the error covariance matrix of measurement, and denotes transpose matrix of e and Exp denotes the expectation operator. The variable of state can be calculated using WLS estimation scheme by minimizing the scheme function “S” as depicted in the following equation and it can be given as sum of square of square of weighted error “e.”

The scheme function S is minimizing by taking the differentiation with respect to x and then equated to zero.where Q(x) = , Q(x) represents the Jacobian matrix and after minimization of “S” provides iterative estimated state as follows:where G denotes Gain matrix (GM) and L denotes the measurement set.

2.2. Linear SSE

The Linear SSE is adapted in this paper for comparison purposes. Two cases have been considered in this paper, the first case is Linear SSE without BDP and second case is Linear SSE with BDP. The Linear SSE scheme utilizing the measurement vector provided by optimally placed μPMUs which comprises of branch current and branch voltage in complex form. A Linear measurement model can be designed as follows:where

denotes the branch currents in complex form of size where denotes no. of branch and denotes the voltage of bus in complex form of size where denotes the no. of buses. The is the size of observation matrix “” and it consisting of and matrix. The submatrix have an entries of branch admittances of order (). The () is the size of submatrix in which the diagonal and off diagonal entries are 1 and 0 respectively. is the size of an error vector “” and possess the given below statistical properties. i.e.,

denotes the covariance for branch voltage in complex form and and denotes the real and imaginary part of error covariance for measurement of branch voltage in complex form respectively. The covariance matrix for F can be formulated as follows:and estimated voltage “V” can be formulated as follows:

denotes the measurement vector comprises of “” and “.” “G” denotes the GM which can be calculated using equation (13). The flowchart of Linear SSE is shown in Figure 1.

2.3. Linear SSE with BDP

The linear SSE with BDP is the second case that is adapted for comparison purposes. An individual BDP algorithm is integrated with linear SSE algorithm in this scheme. The BDP algorithm utilizes largest normalized residual (LNR) test which check and identify the BD in measurement vector and then delete from it. The filter measurement vector is obtained just after the BDP, and then this filter measurement is utilized to estimate the state. The flowchart for this scheme is shown in Figure 2.

2.4. Modified IEEE 14 BTS

A modified IEEE 14 BTS [33] is represented in Figure 3 that is considered an MG integrating DGERs. This modified IEEE 14 test case is adapted for verifying the proposed scheme. The DGERs are integrated at buses such as wind power of 40 MW rated capacity at bus 2 and solar power of 60 MW rated capacity at Bus 3. The details of integrated DGERs are given in Table 1.

The details of line data for modified IEEE 14 BTS are provided in Table 2. A new line is added in between Bus 1 and Bus 3. The details of Bus data for modified IEEE 14 BTS are provided in Table 3.

2.5. Modified IEEE 30 BTS

Another test case, i.e., modified IEEE 30 BTS [34] is also adapted to test the efficacy of the proposed scheme for larger bus system. The modified IEEE 30 BTS is represented in Figure 4 that is considered as MG integrating DGERs. The DGERs are integrated at buses such as the wind power of 40 MW rated capacity at both Bus 13 and Bus 23, respectively. A solar power of 60 MW rated capacity is integrated at all three buses, i.e., Bus 2, Bus 22, and Bus 27, respectively. The details of integrated DGERs are available in Table 1. The details of bus data for modified IEEE 30 BTS are provided in Table 4. The details of line data for the modified IEEE 30 bus test system are provided in Table 5.

2.6. BDP

The largest normalized residual test is utilized for BDP to identify the existence of BD and subsequently eliminate it from the measurement set. Its mathematical expression is given as follows:where

obtain in that manner is largest among each of , k = 1,2, …, m. And then comparing with supposed threshold value “C.” If > C, then measurement as BD otherwise no BD is identified.

3. Proposed IFSSE Algorithm

The proposed IFSSE algorithm utilizes ME approach which main work is to minimize the scheme function. The SF is given as a function of measurement residual.

Subject to z = h(x) + e

For minimization of ,

After applying first order Taylor Series approximation for q(x) ≈ q that yielding the following:

The GM of robust estimation can be formulated as follows:

The introduced IFSSE algorithm uses the same linear model for measurement as linear SSE. Hence, the state can be estimated directly due to its iterative free property. The proposed IFSSE algorithm incorporates the robust ME with a Linear SSE approach. Therefore, the robust bus voltage can be estimated as follows:

Measurement “” consists of “” and “.” “ρ” denotes a diagonal weight matrix, that element entries are prescribed as follows:

“” denotes the constant and its value has been chosen as 1.49 in this paper. The IFSSE algorithm flowchart is shown in Figure 5.

4. Simulation Study

The load time evolution at each bus of the modified IEEE-14 and modified IEEE-30 bus system was acquired over an interval of twenty-four time instances. The load variation has been done by considering the benchmark test system for networked MGs [35]. The load is varied according to the provided load data in the paper [35]. The weekly load data (WLD) in percentage of annual peak load (APL) which is provided by Table 6, and the daily load data (DLD) in percentage of weekly peak load (WLD) which is provided by Table 7.

The hourly load data (HLD) in percentage of daily peak load (DPL) is provided by Table 8. The simulation has been performed on MATLAB R2021b Simulink platform. The week (WLD in % APL is 77.5) and Wednesday as a day (DLD in % of WPL is 95) have been selected for load variation in simulation study for both the test case. The entire load variation was assigned among the generators. The power factor is supposed to be fixed so that “Q” counterpart the “.” At each time instances, the successive load flow analysis is performed for finding the true value of bus voltage in complex form. The error is Gaussian in nature, having a zero mean and 0.5% of the standard deviation. Multiple BD is simulated by adding an error magnitude of 15σ to the true value of complex measurements at various time instances of twenty-four time instances.

The details of simulated BD for both the test cases are presented in Table 9 above. The optimal location (OL) of μPMU and its no. are for both the test case is presented in Table 10.

4.1. Conduct Analysis (CA)

The robustness and performance of the proposed IFSSE scheme has been analyzed using various conduct analysis parameters which is discussed as follows:(i)The conduct index CI (k) can be mathematically calculated as in (22) where denotes the value obtained after estimation, denotes the true value, denotes measured value of measurement, denotes total no. of utilized measurement.(ii)The average absolute state error () and maximum absolute state error () can be mathematically formulated as follows:where denotes the value for estimated state and denotes true value of state.

4.2. Statistical Analysis (SA)

The parameters of SA have been calculated [36] for validating the ascendancy of the suggested IFSSE approach. The Maximum of Mean Square Error () for estimated state variables values can be formulated as follows:where denotes the estimated SV, denotes the true state value, and denotes total no. of time instances.

The maximum standard deviation error (), maximum of sum of squared error () and average of absolute error () of the estimated values of the state variables are formulated as follows:where denotes the estimated state SV, represents the true value of state, denotes total no. of time instances, and denotes the no. of buses.

5. Result and Discussion

The proposed IFSSE scheme potency has been tested on modified IEEE 14 BTS and modified IEEE 30 BTS in presence of multiple BD. Afterwards, the comparison has been done among obtained results of proposed IFSSE, Linear SSE, and Linear SSE with BDP by calculating various conduct analysis parameters (i.e., conduct index CI (k), , and ) and statistical analysis parameters (i.e. , , and , and ). The results for both conduct analysis parameter and statistical analysis parameter are obtained by simulating the algorithm of proposed IFSSE, linear SSE, and linear SSE with BDP on MATLAB R2021b Simulink Platform.

5.1. Results for Modified IEEE-14 BTS and Modified IEEE-30 BTS

The obtained conduct analysis results for modified IEEE 14 BTS and modified IEEE 30 BTS are depicted in Figures (6–11). The value for conduct index should be less than one to get better robustness. The obtained plot for conduct index of modified IEEE-14 BTS is represented in Figure 6 and observed that the proposed IFSSE scheme efficiently rejects the BD because its yields lower value of conduct index as compared to linear SSE, and linear SSE with BDP. Similarly, the obtained plot for conduct index of modified IEEE-30 BTS is represented in Figure 9 and it can be observed that the proposed IFSSE efficiently rejects the BD as compared to Linear SSE, and Linear SSE with BDP due to the yielding lower value of conduct index. The obtained plot for and are represented in Figures 7 and 8, respectively, for modified IEEE 14 BTS. Similarly, the obtained plots for and of modified IEEE 30 BTS are represented in Figures 10 and 11 respectively. It can be clearly understood from the obtained plots for both test case that the proposed IFSSE scheme yields lower value of and as compared to Linear SSE, and Linear SSE with BDP. Each obtained plot of conduct analysis in Figures (6–8) for modified IEEE 14 BTS and Figures (9–11) for modified IEEE 30 BTS clearly reveals the superiority of proposed IFSSE scheme by comparing with the linear SSE, and linear SSE with BDP. Furthermore, the proposed IFSSE scheme effectiveness has been also verified by calculating the various statistical analysis parameters, i.e., , , , and for both test case.

The active power () and reactive power (Q) has been determined by utilizing the algorithm of proposed IFSSE, linear SSE, and linear SSE with BDP and then comparison has been performed with the true value of both active and reactive powers that are denoted by and . The obtained “” and “Q” plots at bus 9 for modified IEEE 14 BTS are shown in Figures 12 and 13 respectively. Similarly, the obtained and Q plots at bus 9 for modified IEEE 30 BTS are represented in Figures 14 and 15, respectively, and after analysing it can be clearly exhibits that the obtained value of the and Q of proposed IFSSE scheme came nearly to the both and as compared to the value of Linear SSE, and Linear SSE with BDP for both the IEEE test system. Therefore, the proposed IFSSE scheme is robust and works efficiently as compared to another adapted scheme, i.e., linear SSE and linear SSE with BDP.

The , , , and in SA has been calculated for modified IEEE-14 BTS in presence of multiple BD and their obtained results are provided in Table 11. It can be analyzed from the results provided in Table 11, the proposed IFSSE scheme is superior as compared to the linear SSE, and linear SSE with BDP due to lower yielding value of statistical errors.

Similarly, the , , , and in SA has been calculated for modified IEEE-30 BTS in presence of multiple BD and their obtained results are provided in Table 12. It can be analyzed from the results provided in Table 12 that the proposed IFSSE scheme is superior as compared to the Linear SSE, and Linear SSE with BDP due to lower yielding value of statistical errors.

6. Conclusion

A newly robust IFSEE scheme has been proposed in this paper for MGs integrating DGERs by utilizing the measurements which is obtained from optimally placed μPMU. In order to provide robust estimation, the proposed IFSSE scheme implements ME approach to cancel the BD presence in the measurements. A constant observation matrix is utilized by the proposed IFSSE scheme which exempts the calculation of Jacobian matrix and makes the proposed IFSSE algorithm efficacious in computation. The developed IFSSE scheme is iteration free by the employment of LMM and provides the explicit solution. The proposed IFSSE scheme robustness is tested and verified on modified IEEE-14 BTS and modified IEEE-30 BTS in presence of simulated multiple BD by varying the load for twenty-four hours according to the benchmark test system for networked MGs. The obtained test results depicted in this paper clearly indicate that the suggested IFSSE scheme is highly efficacious and robust than the linear SSE and linear SSE with BDP. Therefore, the proposed IFSSE scheme is best suited for state estimation in MG integrating DGERs.

Data Availability

All data generated or analyzed during this research are included in the manuscript.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2023 Kunal Kumar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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