State Estimation is the backbone of modern electric power system and is used by almost all Energy Management Systems (EMS) in the world to ensure the real-time monitoring and secure operation of a power system. Phasor Measurement Unit (PMU) is most popular meter in today’s electrical power industry because of its high refresh rates and measurement accuracy. Meanwhile, state estimation with only PMUs is not practical because of the very high initial installation cost. Consequently, the use of PMU meters along with conventional Supervisory Control and Data Acquisition (SCADA) meters can improve the performance of the state estimation. In this paper, phasor measurements (voltage and current phasors) are incorporated in two robust estimators: Weighted Least Absolute Value (WLAV) and Least Measurement Rejected (LMR). Further, we have investigated the importance of locating PMUs to save cost and improve the performance of state estimation. The performance of these two estimators after incorporating voltage and current phasors is investigated in terms of estimation accuracy of state variables and computational efficiency in the presence of different bad-data scenarios on IEEE-30 and IEEE-118 bus systems.
State estimator is an algorithm that process raw and redundant conventional meter readings and other information i.e. network topology, circuit breaker status etc. to estimates the state of a power system [1, 2, 3]. State estimation is one of the basic tools used to ensure that the system is operational in secure mode and all constraints are satisfied. The installed systems at control center of electrical utilities process different measurements from different types of sensors and meters to estimate the overall operating condition of a power system. The measurements which are recorded wrongly because of the large noise, aging of a meter or calibration issue etc. are referred as bad-data [4] that affects the estimation process and results in the wrong estimation of the system’s state variables (voltage magnitude and angle). Bad-data is mainly classified into two categories: (a) single bad-data and (b) multiple bad-data [5]. Multiple bad-data mostly occurs in very large systems and is strongly correlated to each other that poses a huge impact on the results of state estimation.
Weighted Least Square (WLS) is most popular state estimator deployed in electrical utilities and typically intakes reading from conventional meters including power flow meter, power injection meter, and voltage magnitude meter. The mathematical formulation of WLS is simple and has less computational burden, however, it is a non-robust state estimator because WLS fails to produce reliable estimation results even in the presence of single bad-data.
Weighted Least Absolute Value (WLAV) is a robust estimator, compared with WLS, however, it is susceptible to leverage points for certain configurations of meter distribution in a systems [6]. WLAV poses computational burden for large power systems that limits it utilization. For numerical stability and computational efficiency in linear programming (LP), the scaling technique is widely employed and proved an efficient tool [7] provided that scaling helps in reducing the effect of leverage points. In [8], a robust WLAV-T estimator is proposed to mitigate the effect of leverage points based on optimal transformation of associated rotation angles and scaling factor in systematic way, compared to heuristic approach. Further, the WLAV possesses auxiliary variables that reduce the convergence rate of the estimator, so Weighted Linear Least Square (WLLS) is proposed in [9] that possessed less number of variables than WLAV estimator and linear objective function.
Traditional state estimation is going through essential developments due to the innovation of Phasor Measurement Units (PMUs). State estimation problem is more easily formulated when there are adequate number of only PMUs installed in a power system because there is linear relationship between PMU’s phasor measurements and state variables [10]. Least Absolute Value (LAV) estimator is computationally efficient and competitive with WLS if only phasor measurements are provided for state estimation and the strategic scaling method may be used to avoid biasness of leverage points [11, 12]. In [10], the authors investigated the performance comparison between WLS and LAV when only PMUs were used in the presence of different bad-data scenarios and applied scaling technique to maintain the robustness of LAV. The accuracy, synchronization and redundancy of state estimation is improved by incorporating PMU phasor measurements with conventional SCADA measurement into the existing state estimator, however, there are many challenges in this implementation that are addressed with proposed solution [13]. In [14], the authors proposed a method to incorporate conventional SCADA having slow refresh rate of measurement and PMU phasor measurements with fast refresh rate in WLS estimator to keep track of the varying state of the power network. A hybrid state estimator [15] integrated with conventional Supervisory Control and Data Acquisition (SCADA) and PMU’s phasor measurements is utilized for observability analysis and state estimation of a power system. In this hybrid estimator, there is switching between WLS and LAV estimator depending upon the availability of measurements from conventional SCADA or PMU’s phasor measurements. The availability of measurements further depends upon the variation in refresh rate of SCADA and PMU’s phasor measurements. The basic idea of hybrid state estimator is presented in [16]. A two stage linear estimator with only PMUs is proposed in [17] that is not only robust estimator but also computationally efficient because of even distribution of processing burden among different areas.
A PMU measures voltage phasor at a bus and current phasor of all incoming and outgoing flows at substation where it has been installed [18]. A modified non-linear WLS estimator incorporating voltage and current phasors in rectangular and polar coordinates is presented in [19] and both approaches have been compared and evaluated on IEEE-14 bus system with different bad-data scenarios. In [20], a two-stage state estimator combining both SCADA and PMU is proposed and authors have claimed higher estimation accuracy over only conventional SCADA meters. An approach to integrate the PMU technology into the existing SCADA systems to improve the accuracy of state variable is proposed, however, the proposed approach has been tested on very small power system without providing solid mathematical formulation [21]. Another technique to combine both SCADA and \(\mu\)PMU meters in a distribution system state estimation is presented, however, the authors have proposed this approach for WLS, which is a non-robust state estimator [22]. In [23], the authors have proposed an approach to integrate both SCADA and \(\mu\)PMU meters together, however, the proposed approach is applied for network topology analysis and tested on IEEE-33 bus system only. Furthermore, the authors have not provided computational efficiency of the proposed approach.
Least Measurement Rejected (LMR) is a robust estimator which associate a tolerance range with each provided measurement. LMR is solved using mixed integer programming approach and it rejects unreliable, corrupted or wrongly recorded measurements during estimation process and it is not susceptible to leverage points [24]. LMR is simple and effective state estimation approach. In [25], the authors have proposed an iterative tuning approach to choose the appropriate tolerance value of LMR for a certain measurement configuration. The authors of [26] have proposed a novel approach to tune and select the best value for tolerance parameter of LMR. In literature, the different authors have integrated PMUs into existing SCADA system in different estimation algorithms e.g. WLS, WLAV, however, PMU’s phasor measurements have not been incorporated in LMR which is also the novelty of our work.
Though, the above authors were successful in implementing the integration of SCADA and PMUs, however, many of them have applied their proposed approaches on non-robust estimator or the chosen test case was small power system. Furthermore, many authors could not provide information about computational efficiency and number to iterations required to complete the estimation process. Our proposed technique has been applied on two robust state estimators: WLAV and LMR and evaluated on IEEE-30 and IEEE-118 bus systems in terms of state variables (voltage magnitude and angle), computational efficiency and number of iterations. Further, the objective of this paper is to improve the state estimation accuracy in the presence of different bad-data scenarios (single and multiple) by incorporating the voltage and current phasor from PMUs. The tolerance parameter of LMR estimator is chosen properly to reject large errors in conventional measurements. The final approach of achieving the objective is the proper selection of the PMUs locations to ensure best performance and accuracy.
The paper work is organized as follows. The section 2 covers the formulation and details about inclusion of voltage and current phasors in the state estimators and modified mathematical model of the state estimators. Performance validation of proposed technique is presented in section 3. Finally, section 4 concludes the paper with a brief summary.
Once a PMU is installed at a bus in a power system, the voltage phasor of the bus and current phasor of all the branches connected to the bus is measured accurately. In this paper, these PMU’s phasors are incorporated into the existing robust estimators and it was expected to achieve higher state estimation accuracy, compared with conventional measurements. When a phasor measurement is included in a state estimator, the weight of the phasor measurement must be increased because the PMU’s measurement is highly accurate [10]. In WLAV estimator, there is a weight matrix where any measurement is assigned a specific weight corresponds to its accuracy, however, there is no weight matrix or covariance matrix in LMR estimator. The simulation analysis of our paper reveals that reducing the tolerance parameter of LMR associated with a measurement is equivalent to increasing the weight of the measurement in WLAV.
It is required to build a relationship between branch current flows in transmission lines and state variables to incorporate the current phasor in the state estimators. The proposed model will include all transmission lines and transformers between buses. An ampere flow in a branch is expressed as given below [27]:
where \(I_{ij,real}\) and \(I_{ij, imag}\) are rectangular components of the branch current flowing between bus \(i\) and \(j\), \(V_{i}\), \(V_{j}\), \(\theta_{i}\) and \(\theta_{j}\) are voltage magnitude and phase angle of bus \(i\) and \(j\) respectively, \(g_{ij}\) and \(b_{ij}\) are conductance and susceptance between bus \(i\) and \(j\) respectively, \(g_{si}\) and \(b_{si}\) are shunt conductance and susceptance respectively.
Practically, a PMU provides ampere measurements in polar coordinates rather than rectangular coordinates and can referred as direct measurements. However, it is better to use current phasors in rectangular coordinates, because the power flow measurements and power injection measurements are already in rectangular coordinates. In this regard, the ampere measurements should be converted to rectangular coordinates and utilized for state estimation. If the direct measurements are converted to rectangular coordinates, then error covariance values must be translated for rectangular coordinates. The relation between direct and indirect measurement is given by following equations:
In this section, the mathematical formulation of modified WLAV incorporated with voltage and current phasors is presented. Usually, LP solving approach like simplex method or interior point method is used for WLAV. The performance of WLAV is very good for rejection of bad-data, however, it fails to provide reliable estimation results in the presence of leverage points [4]. There are numerous methods to identify leverage measurements [29] which is not the scope of this work.
A power system consists of \(n\) buses with a specific network topology where \(m\) meters are placed at different bus and branch locations to provide measurements from meters to state estimators through Remote Terminal Unit (RTU). The measurement vector \(z\) of size \((m\) x \(1)\) is fed into a state estimator to obtain the state variable vector \(x\) of size \(n=(2n\) x \(1)\). The non-linear function relating measurements to system state variables is given below:
Measurement type | Description |
---|---|
\(PF_{i-j}\) | Real power flow from bus \(i\) to \(j\) |
\(PF_{j-i}\) | Real power flow from bus \(j\) to \(i\) |
\(PG_{inj}\) | Real power injection at bus \(i\) |
\(QF_{i-j}\) | Reactive power flow from bus \(i\) to \(j\) |
\(QF_{j-i}\) | Reactive power flow from bus \(j\) to \(i\) |
\(QG_{inj}\) | Reactive power injection at bus \(i\) |
\(Vm_{i}\) | Voltage magnitude at bus \(i\) |
Measurement Type | Description |
---|---|
\(\theta_{i}\) | Voltage Angle |
\(I_{i-j,real}\) | Real current flow from bus i to j |
\(I_{i-j,imag}\) | Imaginary current flow from bus i to j |
\(I_{j-i,real}\) | Real current flow from bus j to i |
\(I_{j-i,imag}\) | Imaginary current flow from bus j to i |
Measurement type | Standard deviation | |
---|---|---|
Without PMU | With PMU | |
\(PF_{i-j}, PF_{j-i}, PG_{inj}\) | 0.02 | 0.0002 |
\(QF_{i-j}, QF_{j-i}, QG_{inj}\) | 0.04 | 0.0004 |
\(Vm_{i}\) | 0.01 | 0.0001 |
Measurement type | Number of measurements |
---|---|
Real Power Flows | 41 |
Real Power Injection | 16 |
Reactive Power Flows | 40 |
Reactive Power Injection | 15 |
Voltage Magnitude | 14 |
State estimator | Convergence time (sec) | Number of iterations | |
---|---|---|---|
Min | Max | ||
WLAV | 0.31 | 0.36 | 4 |
LMR | 0.17 | 0.28 | 2 |
The results for absolute voltage magnitude (AVM) error with different bad-data scenarios are shown in Table 6. In the first column, `NO PMU’ means that there are only SCADA meters available in the test case. As already explained, one PMU is kept fixed at slack bus and another is relocated at different buses to get estimation results and highlight the importance of PMU placement. PMU-3 means that one PMU is fixed at slack bus and another is placed at bus 3.
It can be noticed from the results in Table 6 that AVM error of ‘NO PMU’ is higher than all PMU placement cases i.e. PMU-3, PMU-5, PMU-10 and PMU-21 for both WLAV and LMR. However, any optimal location cannot be suggested as the optimal placement of PMUs is not the scope of this work. In this paper, the objective was to incorporate PMUs in existing state estimators and to compare the performance of WLAV and LMR in terms of computational efficiency and accuracy.
It can be observed from the results that inclusion of PMUs has improved the estimation accuracy of both WLAV and LMR estimators in all PMU placements which is reflected by the lower value of AVM error in all PMU placement cases than AVM of `No PMU’ case and LMR has better state estimation accuracy than WLAV.PMUlocation | White Noise | SBD aspower flowat \(PF_{2-5}\) | SBD as powerinjectionat \(PG_{2}\) | SBD as voltage magnitude \(Vm_{12}\) |
MNIBD at \(PF_{2-4}\), \(PG_{5}\), \(Vm_{12}\) |
MNIBD at \(PF_{6-9}\) , \( PG_{24}\), \(Vm_{12}\) |
||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
WLAV | LMR | WLAV | LMR | WLAV | LMR | WLAV | LMR | WLAV | LMR | WLAV | LMR | |
NO PMU | 0.0498 | 0.0200 | 0.0499 | 0.0200 | 0.0490 | 0.0212 | 0.0841 | 0.0664 | 0.0856 | 0.0559 | 0.0731 | 0.0584 |
PMU-3 | 0.0394 | 0.0100 | 0.0394 | 0.0098 | 0.0328 | 0.0093 | 0.0655 | 0.0187 | 0.0650 | 0.0160 | 0.0500 | 0.0126 |
PMU-5 | 0.0397 | 0.0104 | 0.0397 | 0.0104 | 0.0397 | 0.0198 | 0.0627 | 0.0201 | 0.0661 | 0.0200 | 0.0398 | 0.0139 |
PMU-10 | 0.0207 | 0.0108 | 0.0207 | 0.0112 | 0.0205 | 0.0121 | 0.0311 | 0.0215 | 0.0230 | 0.0212 | 0.0242 | 0.0207 |
PMU-21 | 0.0217 | 0.0169 | 0.0217 | 0.0169 | 0.0231 | 0.0163 | 0.0260 | 0.0219 | 0.0255 | 0.0247 | 0.0325 | 0.0197 |
PMUlocation | White Noise | SBD aspower flowat \(PF_{2-5}\) | SBD as powerinjectionat \(PG_{2}\) | SBD as voltage magnitude \(Vm_{12}\) |
MNIBD at \(PF_{2-4}\), \(PG_{5}\), \(Vm_{12}\) |
MNIBD at \(PF_{6-9}\) , \( PG_{24}\), \(Vm_{12}\) |
||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
WLAV | LMR | WLAV | LMR | WLAV | LMR | WLAV | LMR | WLAV | LMR | WLAV | LMR | |
NO PMU | 2.8236 | 0.9927 | 2.9621 | 0.9927 | 2.2643 | 1.6630 | 3.5216 | 1.9322 | 4.0834 | 3.5875 | 4.4249 | 2.4865 |
PMU-3 | 1.1620 | 0.5947 | 1.1620 | 0.6011 | 1.2285 | 0.5570 | 1.1042 | 0.5036 | 1.1199 | 0.4136 | 1.1445 | 0.6171 |
PMU-5 | 1.0387 | 0.4735 | 1.0387 | 0.4715 | 1.2676 | 1.0270 | 1.1646 | 0.3330 | 1.0109 | 0.3414 | 1.0387 | 0.9417 |
PMU-10 | 0.6354 | 0.3040 | 0.6354 | 0.3061 | 0.5994 | 0.4121 | 1.0486 | 0.4871 | 0.7861 | 0.5908 | 0.5848 | 0.5200 |
PMU-21 | 0.8452 | 0.2509 | 0.8452 | 0.2313 | 1.0931 | 0.9487 | 0.8626 | 0.3419 | 0.9876 | 0.4605 | 0.8827 | 0.4030 |
Measurement type | Number of measurements |
---|---|
Real Power Flows | 134 |
Real Power Injections | 55 |
Reactive Power Flows | 134 |
Reactive Power Injections | 56 |
Voltage Magnitude | 61 |
State estimator | Convergence time (sec) | No. of iterations | |
---|---|---|---|
Min | Max | ||
WLAV | 6.40 | 9.17 | 6 |
LMR | 3.73 | 4.26 | 6 |
PMUlocation | White Noise | SBD aspower flowat \(PF_{23-32}\) | SBD as powerinjectionat \(PG_{66}\) | SBD as voltage magnitude at \(Vm _{36}\) |
MNIBD at \(PF_{4-11}\), \(PG_{16}\), \(Vm_{3}\) |
MNIBD at \(PF_{34-36}\) , \( PG_{33}\), \(Vm_{18}\) |
||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
WLAV | LMR | WLAV | LMR | WLAV | LMR | WLAV | LMR | WLAV | LMR | WLAV | LMR | |
NO PMU | 0.2792 | 0.1976 | 0.2602 | 0.1775 | 0.2700 | 0.1816 | 0.2791 | 0.1976 | 0.2841 | 0.2090 | 0.2883 | 0.1816 |
PMU-5 | 0.2035 | 0.1622 | 0.1910 | 0.1189 | 0.1992 | 0.1689 | 0.2034 | 0.1621 | 0.2135 | 0.1702 | 0.2135 | 0.1157 |
PMU-12 | 0.2199 | 0.1786 | 0.2111 | 0.1715 | 0.2154 | 0.1811 | 0.2198 | 0.1856 | 0.2198 | 0.1872 | 0.2299 | 0.1514 |
PMU-23 | 0.1952 | 0.1770 | 0.1951 | 0.1709 | 0.1904 | 0.1802 | 0.1951 | 0.1770 | 0.2034 | 0.1794 | 0.2066 | 0.1703 |
PMU-30 | 0.1695 | 0.1197 | 0.1701 | 0.0901 | 0.1629 | 0.1017 | 0.1695 | 0.1196 | 0.1765 | 0.1462 | 0.1801 | 0.1261 |
PMU-37 | 0.1737 | 0.1244 | 0.1655 | 0.1122 | 0.1692 | 0.0911 | 0.1737 | 0.1315 | 0.1876 | 0.1801 | 0.1740 | 0.1242 |
PMU-49 | 0.1969 | 0.1048 | 0.1617 | 0.1176 | 0.1950 | 0.0482 | 0.1968 | 0.1048 | 0.2026 | 0.1706 | 0.2061 | 0.0847 |
PMU-56 | 0.2121 | 0.1797 | 0.1781 | 0.1558 | 0.2131 | 0.1692 | 0.2121 | 0.1797 | 0.2179 | 0.1894 | 0.2213 | 0.1407 |
PMU-77 | 0.2180 | 0.1446 | 0.1830 | 0.1328 | 0.2120 | 0.1365 | 0.2180 | 0.1445 | 0.2237 | 0.1577 | 0.2275 | 0.1032 |
PMU-85 | 0.2177 | 0.1225 | 0.1823 | 0.1206 | 0.2108 | 0.1156 | 0.2176 | 0.1225 | 0.2232 | 0.1822 | 0.2271 | 0.1408 |
PMU-94 | 0.2083 | 0.1055 | 0.1729 | 0.0879 | 0.2014 | 0.1018 | 0.2082 | 0.0869 | 0.2138 | 0.1250 | 0.2177 | 0.0896 |
PMU-105 | 0.2196 | 0.1115 | 0.1842 | 0.0988 | 0.2127 | 0.1043 | 0.2195 | 0.1150 | 0.2251 | 0.1829 | 0.2290 | 0.1264 |
PMU-110 | 0.2188 | 0.1370 | 0.1834 | 0.1281 | 0.2119 | 0.0946 | 0.2188 | 0.1369 | 0.2243 | 0.1580 | 0.2282 | 0.0827 |
PMUlocation | White Noise | SBD aspower flowat \(PF_{23-32}\) | SBD as powerinjectionat \(PG_{66}\) | SBD as voltage magnitude at \(Vm _{36}\) |
MNIBD at \(PF_{4-11}\), \(PG_{16}\), \(Vm_{3}\) |
MNIBD at \(PF_{34-36}\) , \( PG_{33}\), \(Vm_{18}\) |
||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
WLAV | LMR | WLAV | LMR | WLAV | LMR | WLAV | LMR | WLAV | LMR | WLAV | LMR | |
NO PMU | 6.5645 | 5.7658 | 8.4241 | 6.9760 | 6.9005 | 6.2859 | 6.5645 | 5.7658 | 8.7292 | 8.4180 | 8.8401 | 7.8542 |
PMU-5 | 4.8424 | 4.1135 | 5.5419 | 4.4315 | 4.1214 | 4.0842 | 4.8424 | 4.1136 | 7.1270 | 6.2995 | 6.6821 | 6.5527 |
PMU-12 | 5.6592 | 4.1274 | 4.9530 | 4.7770 | 5.0185 | 4.2916 | 5.6592 | 4.1614 | 7.6667 | 6.6985 | 7.7004 | 7.4777 |
PMU-23 | 5.1023 | 3.9799 | 5.1023 | 4.4299 | 4.4990 | 4.0794 | 5.1023 | 3.9799 | 7.4183 | 6.7431 | 8.1718 | 6.9019 |
PMU-37 | 4.9576 | 3.8861 | 6.7908 | 4.3050 | 4.5046 | 4.0361 | 4.9576 | 3.8754 | 6.4610 | 6.4533 | 4.2436 | 3.6609 |
PMU-49 | 4.9691 | 3.3357 | 4.8502 | 4.1246 | 4.6477 | 4.1873 | 4.9691 | 3.3368 | 7.8704 | 5.5164 | 6.6806 | 6.6307 |
PMU-56 | 4.7916 | 3.6311 | 6.8126 | 4.4770 | 5.0280 | 4.7945 | 4.7916 | 3.6311 | 7.6820 | 6.1747 | 7.4555 | 6.6350 |
PMU-77 | 5.1490 | 4.1643 | 7.6845 | 4.3710 | 4.8917 | 4.8848 | 5.1490 | 4.1636 | 7.8193 | 6.8693 | 7.9457 | 7.1038 |
PMU-85 | 5.2203 | 3.6269 | 7.6517 | 4.6830 | 4.8087 | 4.5386 | 5.2203 | 3.6269 | 7.8273 | 6.3858 | 7.5832 | 7.1190 |
PMU-94 | 5.4897 | 3.7719 | 7.9211 | 4.0410 | 5.0780 | 4.0052 | 5.4897 | 3.4912 | 8.0966 | 6.2631 | 7.3938 | 7.3884 |
PMU-105 | 5.0139 | 3.9057 | 7.4453 | 4.7130 | 4.6023 | 4.0723 | 5.0139 | 3.9050 | 7.6209 | 6.4835 | 7.5176 | 6.9126 |
PMU-110 | 4.9884 | 3.8172 | 7.4198 | 4.7730 | 4.5768 | 4.4168 | 4.9884 | 3.8172 | 7.5954 | 6.0187 | 7.6356 | 6.8871 |