An Iterative Multiarea State Estimation Approach Using Area Slack ...

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An Iterative Multiarea State Estimation Approach Using Area Slack Bus Adjustment A. Sharma, Graduate Student Member, IEEE, S. C. Srivastava, Senior Member, IEEE, and S. Chakrabarti, Senior Member, IEEE

Abstract—This paper proposes a new multiarea state estimation (MASE) approach, which estimates the overall system states by utilizing the state estimation (SE) results of the subareas iteratively. In this approach, all the subareas run their SE sequentially in each iteration. The SE results of the boundary buses in a subarea are used as pseudomeasurements for running the SEs of the nearby subareas. An area slack bus angle adjustment approach has been utilized for estimating the bus angles of the overall system with reference to the global slack bus. It has been demonstrated that the use of the subarea SE results, as pseudomeasurements, provides better state estimates for the complete system. The effectiveness of the proposed method has been demonstrated on an IEEE 30-bus system and a 246-bus reduced Northern Regional Power Grid Indian system. Index Terms—Area slack bus angle adjustment, iterative approach, multiarea state estimation (MASE), weighted least square (WLS) approach.

I. I NTRODUCTION

S

TATE estimation (SE) is an important tool for online monitoring of the parameters (states) of various process plants and critical infrastructures, such as power system, water distribution system, and communication system. The SE utilizes field measurements containing noise and provides the best estimate of the system states based on certain statistical criteria. Due to increase in size and complexity of the systems, particularly the power system networks, performing SE has become a challenging and time-consuming task. It is required to divide the large system network into multiple subsystems and run the SE for the subsystems individually in an interactive manner. This paper proposes such a scheme to solve the SE problem iteratively for the power system network divided into smaller manageable subsystems (subareas). Generally, the SE for the power system network is solved using weighted least squares (WLS) algorithm [1]–[3]. The measurements considered to solve the SE problem are bus voltage magnitudes, real and reactive power flows, and real and reactive power injections.

Manuscript received August 27, 2013; revised November 28, 2013 and January 28, 2014; accepted April 4, 2014. This work was supported in part by the Department of Science and Technology under Project DST/EE/20100258. The authors are with the Department of Electrical Engineering, Indian Institute of Technology Kanpur, Kanpur-208016, India (e-mail: ankushar@ iitk.ac.in; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JSYST.2014.2316205

The subareas in a power system network can be formed using the boundaries of the utility companies controlling the operation in those areas. The utility companies run SE for their own subareas, and the central coordinator (CC) estimates the SE of the complete interconnected system utilizing the SE results and the measurements of the subareas. The role of a CC becomes critical in the SE of the complete interconnected system due to the hierarchical nature of the SE processing. With the advent of synchrophasor technology and associated power system monitoring and control tools [4]–[6], the conventional SE process is being replaced by the various hybrid SE techniques to incorporate the synchrophasor data in the SE formulation [7]–[9]. Some linear SE techniques, using only synchrophasor data, have been also proposed in the literature [10]. However, due to high cost of phasor measurement units (PMUs), the practical implementation of the hybrid SE or the linear SE using only PMUs has not been adopted in the power system network. Hence, there is still a need of SE with the angle referencing between different subareas [11] in the absence of the PMUs or with a limited number of PMUs in the system. Various procedures and algorithms have been proposed in the literature to solve the multiarea SE (MASE) problem. In [12], a distributed SE algorithm was proposed utilizing synchronized phasor measurements, and a sensitivity-analysis-based method was used to update the boundary buses at the aggregation level. In [13], a constrained optimization approach was used to solve the distributed SE problem. The overlapping subsystems with a common zero injection boundary bus were formed under this approach. Dantzig–Wolfe decomposition algorithm and linear programming were used in [14] to solve the MASE problem. A two-level SE approach was presented in [15], where the firstlevel results of the area SE and the tie-line flows are used to estimate the states of the complete network at the second level. A taxonomy of MASE methods was provided in [16]. In [17], MASE using the extended boundary of the individual areas was proposed using synchronized phasor measurements to consolidate state estimates at the coordination level. Although, in [17], the subarea slack bus angles are referred with respect to the system slack bus, the procedure to estimate the subarea slack bus angles with respect to the system slack bus was not discussed. In [18], the overall system was decomposed into nonoverlapping areas on a geographical basis. Iterative SE scheme was, then, run between areas and the CC to compute the system-wide states. In the preceding approaches, in addition to the state estimates, the subareas send a lot of information to the CC related to the boundary buses, such as boundary bus injection,

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boundary line flows, and topology information to compute system-wide states. This requires a significant communication bandwidth. In [19], a simple multiarea decentralized SE is proposed. In this approach, the power system network is divided into nonoverlapping subareas, and only the state variables, estimated at the boundary buses, are exchanged iteratively. In [20], a fully distributed SE algorithm is proposed for widearea monitoring in the power system using iterative information exchange among the subareas utilizing dc SE. There have been attempts to solve the MASE problem using factorized SE algorithm [21], [22]. The SE problem is solved at multiple levels by introducing intermediate variables. The subareas send intermediate states along with the topology information to the CC to estimate the system-wide states. Bilinear formulation of the WLS SE problem is proposed in [23] to solve equality-constrained SE problem. In [24], a distributed robust power system SE algorithm is proposed, which is based on the alternating direction method of multipliers. Using this method, the SE problem converges even in the absence of local observability. In [25], master–slave splitting iterative method is developed for calculating largescale and mixed SE problem containing transmission and distribution network. A novel coordinated algorithm for distributed SE is proposed in [26], where the distributed SE with linear coordination is used at the subarea level and the linear SE is run at the coordination level. This approach needs PMU measurements at all the buses at the coordination level to make the coordination-level SE a linear SE. This paper proposes a new iterative approach to solve the MASE for those cases where either no PMU is installed in the subareas or sufficient PMU measurements for the area slack bus angle referencing are not available. This situation arises when sufficient PMUs are not installed in the power system network divided into various subareas, such as the Northern Regional Power Grid (NRPG) Indian system, due to large investment required in deployment of PMUs. In such scenario, by using the proposed approach, the additional cost required for installation of PMUs for the area slack bus angle referencing can be avoided. In the proposed approach, unlike [11], the SE process is run only at the subarea level without using the hierarchical level processing for coordination of the SE results of the complete network. The overall system is decomposed into overlapping subareas, i.e., creating extended subareas that include boundary buses of the nearby subareas. Each subarea will run its own SE and will report the estimated states, along with the corresponding state covariance matrix data of the boundary buses, to the nearby subareas. These estimated states of the boundary buses will be used as pseudomeasurements, while running the SEs for the nearby subareas. In case the PMU measurements are available for a subarea, the same will be utilized for the area slack bus angle referencing. In the subareas, where PMU measurement is not available, the available subarea pseudomeasurements will be utilized for the area slack bus angle referencing. The SE process is carried out iteratively for all the subareas until solution for the complete network converges. Using this approach, there will be no requirement of topology and/or measurement data exchange among the subareas. This

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Fig. 1.

Sample network divided into three subareas.

way, the sensitive internal information of the subareas can be preserved.

II. M ULTIAREA S TATE E STIMATOR A. Network Decomposition To run MASE, a complete power system network is divided into various subareas, separated by the tie-lines. Under the current electricity market scenario, boundaries of the utility companies or control centers divide the complete power system network into several nonoverlapping subareas or subsystems. For example, the network under the Northern Regional Load Dispatch Center (NRLDC) of India has eight State Load Dispatch Centers (SLDCs), which divide the complete NRLDC network into eight subareas. In this work, for MASE formulation, overlapping subarea boundaries are considered to create extended subareas. The extended subareas include the boundary buses of the nearby subareas. The formation of the extended subarea helps in performing area slack bus angle referencing. As shown in Fig. 1, a sample network is divided into three subareas. The buses of each extended subarea are divided into three categories [11]: 1) internal bus: a bus of an area that is not connected to any external boundary bus (e.g., bus 1-1 in the extended subarea 1); 2) internal boundary bus: a boundary bus of an area that is connected to at least one external boundary bus (e.g., bus 1-4 in the extended subarea 1); 3) external boundary bus: a boundary bus that belongs to another area and connected to an internal boundary bus of an area (e.g., bus 3-4 in the extended subarea 1).

B. Subarea State Estimator Formulation The state estimator for each extended subarea is run by considering its own measurement set. For the external boundary buses of the extended subarea, only the line flows are

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considered in the measurement set. The subarea SE problem is formulated as [1]–[3] Minimize J = (z − h(x))T R−1 z (z − h(x)) subject to z = h(x) + e

(1) (2)

where z is the measurement vector, x is the state vector, h is a nonlinear function vector, which relates the measurement vector to the state vector, and e is the measurement error vector. Rz is the measurement error covariance matrix. It is assumed that the noise in the measurements is Gaussian distributed with zero mean. Hence E(e) = 0 E(eeT ) = Rz .

(3) (4)

The state estimates are found by solving the following equation iteratively:   −1 −1 z − h(xk ) (5) xk+1 = xk + (HT R−1 z H) Rz where xk is the state vector at the kth iteration, and H is the Jacobian of the measurement function h(x). The SE solution of an extended subarea will give state vector x, which is composed of three subvectors, i.e.,   T T extT (6) , x [x] = xint , xint b b T

where xint is a vector consisting of the voltage magnitudes and T is a vector consisting phase angles of the internal buses, xint b of the voltage magnitudes and phase angles of the internal T is a vector consisting of the voltage boundary buses, and xext b magnitudes and phase angles of the external boundary buses. The results provided by the subarea SE will include not only the states of the internal buses of the subarea but also the states of those external boundary buses of the nearby subareas, which are directly connected to the subarea. The state estimates of the boundary buses will be used as pseudomeasurements for the next nearby subareas. C. Existing MASE Approach [11] The methodology proposed in this paper is compared with the MASE approach presented in [11]. In [11], a CC receives the subarea SE results of the boundary buses along with the state covariance matrices from all the subareas. The CC, then, processes these results by running linear SE for boundary buses at the CC level to estimate the final states of the complete network. In the existing MASE approach, all the subareas run SE independently in parallel by considering extended subarea boundary and send results directly to the CC. Since the results provided by the subarea SE include the states of the external boundary buses of the nearby subareas, the angle referencing can be done by utilizing the state estimates of the boundary bus provided by the local and neighboring subareas. The angle referencing is done at the CC level by utilizing the following equation [11]: G L − θ(a−j) Δθ(a−j) = θ(a−j)

(7)

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where G is the global reference, a is the subarea index, j is G is the angle value of boundary bus j of the bus number, θ(a−j) subarea a estimated by the subarea where global slack bus lies, L is the angle of boundary bus j estimated by local subarea θ(a−j) a, and Δθ(a−j) is the area slack bus adjustment value used to adjust bus angle values of subarea a with respect to global slack. The updated estimated angles of a subarea, adjusted with respect to the global slack bus, are used further to adjust the other subarea bus angles connected to that subarea. After the subarea angle adjustment is done at the CC level using the preceding methodology, the CC, then, estimates the states of the boundary buses due to their sufficient data redundancy. Each boundary bus has at least two reported sets of states: the one provided by its own SE and the other provided by the neighboring subarea. At the CC level, the SE is linear in nature because the measurement vector is a linear function of the state vector, which is expressed as z = Ax + e

(8)

where A is a constant matrix having elements either zero or one, z is the measurement vector, x is the state vector, and e is the measurement error vector; Rcc is the measurement error covariance matrix at the CC level. The WLS solution of (8) is obtained from  −1 T −1 x = AT R−1 A Rcc z. (9) cc A The solution procedure of the preceding SE problem is noniterative in nature. The measurement error covariance matrix Rcc is obtained from the state covariance matrices, reported by the subareas to the CC, and the standard uncertainty of the bus angles used to calculate area slack bus angle adjustment value [11], [27]. III. P ROPOSED M ETHODOLOGY The methodology proposed in this paper is different from [11] in terms of the MASE formulation and the execution procedure. In the existing MASE formulation [11], it is required to have hierarchical structure of the power system monitoring and control at the control center level. If the CC is not available, the hierarchical type of MASE is difficult to implement. Instead, an iterative approach of MASE, proposed in this paper, is needed among the subareas for the angle referencing and finalizing the system states. The measurements from the existing supervisory control and data acquisition system are regularly received at a discrete time instant, typically in the interval of 1–5 s, for the execution of the power system SE. Hence, the proposed approach is formulated to address only the solution of the static SE problem at a particular operating point. The power system dynamical components, such as load variations and rate of change of frequency, are not considered while formulating the MASE approach. In the proposed approach, each subarea is responsible to run SE for its own extended subarea and exchanges the state estimates and the corresponding state covariance matrix values of the boundary buses to the neighboring subareas, which are

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Fig. 2. Process flow of the proposed iterative MASE.

directly connected to it. These state estimates will be used by the neighboring subareas as pseudomeasurements for running their own SEs and performing area slack bus angle referencing. If the PMUs are installed at the desired neighboring boundary buses or at the slack buses of the nearby subareas, the state estimates of those boundary/slack buses will be used in preforming the area slack bus angle referencing because the PMU measurements are generally more accurate than the conventional measurements. The SE process starts from that subarea where global slack bus lies. Generally, the generator bus having a PMU installed on it is considered as a global slack bus. The SE results related to the internal and external boundary buses of this subarea are used as pseudomeasurements by the neighboring subareas to run their own SEs. The slack bus angle referencing is done by the neighboring subareas using (7), therefore setting the angle values of the buses of the neighboring subareas with respect to the global slack bus. This process is repeated for all the subareas iteratively until the overall solution converges (see Fig. 2). In the subarea SE formulation, each boundary bus will have at least two reported or estimated sets of states: the one provided by its own SE and the other provided by the neighboring subarea. This will provide adequate data redundancy at the subarea level for the SE processing. The objective function of the iterative MASE process is a set of individual subarea objective functions, which can be defined as follows: ⎞ ⎛ nbi mi li  2 2   (z − μ ) (z − μ ) j j k k ⎠. + Min Ji = min ⎝ 2 2 σ σ j k j=1 b=1 k=1

(10) In matrix form, it can be defined as   Min Ji = (zi − hi (x))T R−1 (z − h (x)) i i zi subject to zi = hi (x) + ei

(11) (12)

where mi is the number of measurements in subarea i, li is the number of subareas connected to subarea i, nbi is the number of measurements belonging to the neighboring subarea b of subarea i, μj is the expected value of measurement j, σj is the standard deviation of measurement j, zi is the measurement vector for subarea i, xi is the state vector for subarea i, hi is a nonlinear function vector, which relates the measurement vector to the state vector for subarea i, and ei is the measurement error vector for subarea i. Rzi is the measurement error covariance matrix for subarea i. To run the subarea SE, the conventional WLS process is modified to accommodate the pseudomeasurements related to the state estimates of the boundary buses reported by the neighboring subareas and the available PMU measurements in the subarea. In this scenario, the measurement function for the subarea is defined as follows: ⎤ ⎡ h (x ) ⎤ ⎡ ⎤ ⎡ eVi zVi Vi i hVj eˆ ⎥ ˆ (xi ) ⎥ ⎥ ⎢ ⎢ zVj ˆ ⎥ ⎢ ⎢ Vj ⎥ ⎥ ⎢ ⎢ ⎢ e ⎢ ⎥ ⎢ zVipmu ⎥ ⎢ hVipmu (xi ) ⎥ Vipmu ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ePi ⎥ (x ) h ⎥ ⎢ zPi ⎥ ⎢ Pi i ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ + ⎢ eQi ⎥ (x ) h zi = ⎢ zQi ⎥ = ⎢ ⎥ (13) Qi i ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ePFi ⎥ (x ) h ⎥ ⎢ zPFi ⎥ ⎢ PFi i ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ eQFi ⎥ (x ) h ⎥ ⎢ zQFi ⎥ ⎢ QFi i ⎥ ⎦ ⎢ ⎣ ⎣ h ˆ (xi ) ⎦ ⎣ e ˆ ⎦ zθj ˆ θj θj zθipmu eθipmu hθipmu (xi ) where subscript i denotes the values related to subarea i; subscript j denotes the values provided by the neighboring subarea j in the form of pseudomeasurements to run the SE for subarea i; subscript V denotes the values related to voltage measurement; subscript P denotes the values related to real power injection measurement; subscript Q denotes the values related to reactive power injection measurement; subscript PF denotes the values related to real power flow measurement; subscript QF denotes the values related to reactive power flow ˆ and θˆ denote the pseudomeasuremeasurement; subscripts V ment values related to bus voltage and angle provided by the neighboring subareas, respectively; and subscript ipmu denotes the measurements related to the PMUs of subarea i. The states of subarea i are estimated by solving the following equation iteratively: −1 −1    xk+1 = xki + HTi R−1 Rzi zi − hi (xki ) (14) zi Hi i where xki is the state vector of subarea i at the kth iteration, and Hi is the Jacobian of the measurement function hi (x) [28]. The measurement error covariance matrix for subarea i is also modified to add the state covariance submatrix related to the boundary buses reported by the neighboring subareas and PMU measurements, i.e., ⎤ ⎡ Rzir 0 0 (15) 0 ⎦ Rzipmu Rzi = ⎣ 0 0 0 Rzib where Rzir is the regular measurement error covariance matrix for subarea i, Rzipmu is the measurement error covariance matrix related to the PMU measurements of subarea i, and

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Rzib is the measurement error covariance matrix related to the boundary buses reported by the neighboring subareas. The elements of the Jacobian Hi are also modified to incorporate the pseudomeasurements and PMU measurements. The structure of Hi is defined in (16), shown at the bottom of the page. The modified WLS approach is run for all the subareas sequentially for minimization of their respective objective functions. The process is repeated for the subareas iteratively until the sum of the objective functions of all the subareas, which is defined in (17), is within the specified tolerance level, i.e., M=

N 

(Ji )

(17)

i=1

where N is the number of subareas. In summary, the following advantages are achieved over the existing MASE [11] by using the proposed methodology. 1) The proposed MASE does not require CC to aggregate the SE results. 2) The proposed MASE approach is more suitable for the nonhierarchical type of power system networks. 3) The proposed MASE is more accurate. 4) The proposed MASE is solved at the subarea level only. There is no need to run linear state estimator at the coordination level for the overall system. 5) The proposed MASE approach can utilize synchrophasor measurements, if available, in performing the area slack bus angle referencing. IV. S IMULATION R ESULTS To demonstrate the proposed methodology, an IEEE 30-bus system [30] and a 246-bus reduced NRPG Indian system [31] were used. A MATLAB application was developed to test and verify the proposed approach of the MASE. The SE results were obtained using the conventional integrated SE approach [1]–[3], the MASE methodology proposed in [11], and the methodology proposed in this paper. The results of all the aforementioned approaches are compared. To demonstrate the accuracy of the proposed methodology, the results are also compared with the actual values of the system states. The actual values of the system states are taken from the load flow results. A MATLAB function is used to add Gaussian noise into the actual values of the load flow results to consider them as a measurement data for the MASE input. The MATLAB function adds Gaussian noise into the actual values of a measurement by picking a random value between ±3σ, where σ is the standard deviation of the corresponding measurement type. Various measurement types and their standard deviations, used in this work, are given in Table I [8], [27], [32]. Utilizing Guide to the Expression of

 Hi =

∂Vi ∂θ i ∂Vi ∂Vi

∂Vj ∂θ i ∂Vj ∂Vi

∂Vipmu ∂θ i ∂Vipmu ∂Vi

∂Pi ∂θ i ∂Pi ∂Vi

5

TABLE I M EASUREMENT T YPES AND C ORRESPONDING S TANDARD D EVIATIONS

Uncertainty in Measurement specifications [33], the standard deviation of the PMU is derived from the maximum measurement uncertainty specified in the data sheet of a PMU [32], which is available in the authors’ laboratory, and assuming a uniform probability distribution of the measurement [27]. For the simulation, an Intel Core i7 3.4-GHz processor based computer having 4-GB random access memory was used. The convergence criterion used in the proposed approach has utilized the value of tolerance as 0.00001, which is lesser than the minimum standard deviation among all types of measurements (see Table I). The solution is said to be converged if the maximum change in sum of all the objective functions defined in (17), between two successive iterations, is less than 0.00001. A. IEEE 30-Bus System The IEEE 30-bus system is divided into two subareas in such a way that each subarea has 15 buses, as shown in Fig. 3. The extended subarea is created by containing the internal buses and the boundary buses of the nearby subareas. In the IEEE 30-bus system, the extended subarea 1 contains 23 buses, and the extended subarea 2 contains 20 buses, as shown in Table II. Bus 1 is considered as the global slack bus and the local slack bus for subarea 1. Bus 16 is considered as the local slack bus for subarea 2 and is highlighted in Table II. Bus 16 is a part of the extended subarea 1 and the extended subarea 2. In the IEEE 30-bus system, only one PMU is assumed to be installed at bus 1, which is not sufficient for the area slack bus angle referencing. Hence, pseudomeasurements are utilized to perform area slack bus angle referencing. Under the proposed MASE approach, only the updated boundary bus state estimates, reported by the neighboring subareas, are used as a pseudomeasurement in running the SE for the subarea. For the IEEE 30-bus system, the SE results for boundary buses 16, 17, 18, 20, 21, 22, 23, and 28, which are provided by the extended subarea 1, are used as pseudomeasurements in running the SE for subarea 2. The angle difference between the global slack bus, i.e., bus 1, and the local slack bus of subarea 2, i.e., bus 16, is used for the area slack bus angle adjustment. Similarly, the SE results for boundary buses 6, 8, 10, 12, and 15, which are provided by the extended subarea 2, are used as pseudomeasurements in running the SE for subarea 1. The process is repeated iteratively until the combined objective function of both the subareas reaches to its minimum value. The solution for the IEEE 30-bus system under this scenario converges in

∂Qi ∂θ i ∂Qi ∂Vi

∂PFi ∂θ i ∂PFi ∂Vi

∂QFi ∂θ i ∂QFi ∂Vi

∂θ j ∂θ i ∂θ j ∂Vi

∂θ ipmu ∂θ i ∂θ ipmu ∂Vi

T (16)

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Fig. 3. IEEE 30-bus system. TABLE II E XTENDED S UBAREAS OF THE IEEE 30- BUS S YSTEM

SE and the noniterative MASE methods. It is so because the proposed MASE approach solves the objective function for the local minimum for the subareas and the global minimum for the complete network iteratively. It is also observed that the estimated states using the proposed iterative MASE method are very close to their actual values.

B. Reduced NRPG System of 246 Buses three iterations. To demonstrate the relative accuracy, errors in the states estimated with the integrated SE, the noniterative MASE [11], and the proposed MASE, with respect to their actual values, are graphically compared in Figs. 4 and 5. Fig. 4 compares the error values in the voltage magnitude estimation, and Fig. 5 compares the error values in the angle estimation. To numerically compare the error values with the integrated SE, the noniterative MASE, and the proposed MASE with respect to the actual values, the maximum error value and the average sum of square error value for voltage magnitude and angle against these approaches were computed, as listed in Table III. Comparing the SE results, it is found that the errors with the proposed iterative MASE are lesser than with the integrated

Another system considered to demonstrate the effectiveness of the proposed iterative MASE methodology is the practical 246-bus reduced NRPG system under the NRLDC in India [31]. The schematic of the 246-bus reduced NRPG system displaying three major extended subareas is shown in Fig. 6. Under the NRLDC, there are eight SLDCs. Hence, the NRPG system is divided into eight subareas by considering the SLDC boundaries. The extended subareas of the SLDCs are created by including the boundary buses of the nearby SLDCs. For example, the extended subarea of the Uttar Pradesh (UP) SLDC is created by including native buses of the UP SLDC, along with boundary buses 2, 185, 208, 209, and 219 of the Uttarakhand (Ukh) SLDC; boundary buses 20, 21, 165, 179, 181, and 230 of the Haryana (HR) SLDC; and boundary buses 17, 70, 109,

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Fig. 4.

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Voltage magnitude error in SE for the IEEE 30-bus system. Fig. 6. Schematic of the 246-bus reduced NRPG system displaying three major extended subareas.

Fig. 5.

Angle error in SE for the IEEE 30-bus system.

TABLE III E RROR C OMPARISON OF VARIOUS SE A PPROACHES W ITH R ESPECT TO ACTUAL VALUES FOR THE IEEE 30- BUS S YSTEM

111, 115, 122, 162, 229, 233, and 236 of the Rajasthan (RA) SLDC. It may be noted that the bus numbers in the SLDCs are not in numerical order [31]. For the purpose of demonstration, it is assumed that the PMUs are installed at buses 1, 2, and 20, belonging to different SLDCs in the 246-bus reduced NRPG system. As per the proposed iterative MASE approach, the SE for the UP SLDC is run first by considering extended subarea boundary

because bus 1 in the UP state is considered as the global slack bus. The SE results of the UP SLDC provide state estimates for the boundary buses to Ukh, HR, and RA SLDCs. This helps in running SE for the Ukh, HR, and RA SLDCs by using the state estimates of the boundary buses reported by the UP SLDC as pseudomeasurements. The SE results provided by the UP SLDC will also help in estimating the slack bus angles of the Ukh, HR, and RA SLDCs with respect to the global slack bus. Similarly, the SE results of the extended subarea of the HR SLDC provide state estimates for the boundary buses of the Punjab (PB), Himachal Pradesh (HP), and Delhi (DL) SLDCs. The state estimates of the boundary buses of HR are used as pseudomeasurements in running the SE for the PB, HP, and DL SLDCs and estimating the slack bus angles of the PB, HP, and DL SLDCs with respect to the global slack bus. Finally, the state estimates of the boundary buses of PB are used as pseudomeasurements in running the SE for the Jammu and Kashmir (JK) SLDC and estimating the slack bus angles of JK with respect to the global slack bus. The SE results for the extended subarea of the UP SLDC give state estimates for the boundary buses 2, 185, 208, 209, and 219 of the Ukh SLDC; boundary buses 20, 21, 162, 165, 179, 181, and 230 of the RA SLDC; and boundary buses 17, 70, 109, 111, 115, 122, 229, 233, and 236 of the HR SLDC. The state estimates provided by the UP SLDC for the preceding boundary buses are used as pseudomeasurements in running the SEs for their respective SLDCs. The angle estimates for bus 2 provided by UP and Ukh SLDCs are used in calculating the area slack bus angle adjustment value for adjusting the angle estimates of the buses of the Ukh SLDC with respect to global slack bus because buses 1 and 2 are the PMU buses. Similarly, the angle estimates of buses 1 and 20 provided by UP and RA SLDCs are used in adjusting the angle estimates of the buses of the RA SLDC with respect to the global slack bus. Buses 1 and 20 get priority in providing area slack bus angle adjustment because these buses are the PMU buses. The angle estimates of bus 70 provided by UP and HR SLDCs are used in adjusting the angle estimates

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TABLE IV E RROR C OMPARISON OF VARIOUS SE A PPROACHES W ITH R ESPECT TO ACTUAL VALUES FOR THE 246- BUS NRPG S YSTEM

Fig. 7. Voltage error in SE for the 246-bus NRPG system.

compares the error values with the integrated SE, the noniterative MASE, and the proposed MASE with respect to the actual values by showing maximum error value and average sum of square error value for the voltage magnitude and the angle. On comparing the results, it can be observed that, for this system also, the error with the proposed iterative MASE is lesser than that with the integrated SE and the noniterative MASE methods. Furthermore, the estimated state values, computed using the proposed iterative MASE approach, are very close to their actual values.

V. C ONCLUSION

Fig. 8. Angle error in SE for the 246-bus NRPG system.

of the buses of the HR SLDC with respect to the global slack bus. Bus 70, a non-PMU bus, of the HR SLDC is chosen for the area slack bus angle adjustment because the state covariance matrix element corresponding to bus 70 is minimum, i.e., bus 70 state estimate is the most accurate among the other boundary buses [11]. The boundary bus state estimates provided by Ukh, RA, and HR SLDCs are further used as pseudomeasurements in running SEs and performing area slack bus angle adjustment for the other remaining neighboring SLDCs. This process is repeated iteratively by covering all the SLDCs until the overall solution converges. The iterative MASE solution for the 246-bus NRPG system converges in two iterations. To demonstrate the relative accuracy, errors in the states estimated with the integrated SE, the noniterative MASE [11], and the proposed MASE, with respect to their actual values, are graphically compared in Figs. 7 and 8. Fig. 7 compares the error values in the voltage magnitude estimation, and Fig. 8 compares the error values in the angle estimation for the 246-bus reduced NRPG system. Table IV numerically

This paper has proposed an iterative MASE for a power system divided into multiple subareas. The iterative MASE process starts from an extended subarea, where global slack bus lies. The state estimates of the boundary buses of this subarea have been used as pseudomeasurements in running SEs for the neighboring subareas. The iterative process is repeated to estimate the final states of the complete network. In case the synchrophasor measurements are available at the desired location in the subarea, the same are utilized for the area slack bus angle adjustment process. For the subareas without any PMU, the pseudomeasurements provided by the nearby subareas have been utilized in estimating the subarea slack bus angle estimates with reference to the global slack bus. The effectiveness of the proposed approach has been demonstrated on an IEEE 30-bus test system and on a practical 246-bus power system network of northern India. The state estimates obtained using the conventional integrated SE [1]–[3] and the noniterative MASE [11] are compared with those obtained with the proposed MASE. The state estimates obtained with the proposed MASE are found to be more accurate than the conventional integrated SE and the noniterative MASE and are quite close to the actual values. In the proposed MASE approach, only the state estimates of the boundary buses are used as pseudomeasurements, and the data related to only the state estimates of the boundary buses and the corresponding covariance matrix values are shared among the subareas. There is no data exchange related to the topology and measurements among the subareas. Thus, the subareas can preserve their internal information. The proposed MASE approach is well suited for the nonhierarchical types of power system networks as there is no requirement for the CC to aggregate the SE results.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. SHARMA et al.: ITERATIVE MULTIAREA STATE ESTIMATION APPROACH USING AREA SLACK BUS ADJUSTMENT

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A. Sharma (M’01–S’10) received the M. Tech. degree in electrical engineering in 2001 from the Indian Institute of Technology Kanpur, Kanpur, India, where he is currently working toward Ph.D. degree. He has worked in the energy domain of Tata Consultancy Services for 11 years. His research interests include state estimation, energy market, smart grid technology, and information technology applications in power system.

S. C. Srivastava (SM’91) received the Ph.D. degree in electrical engineering from the Indian Institute of Technology (IIT) Delhi, New Delhi, India. He is currently a Professor with the Department of Electrical Engineering, IIT Kanpur, Kanpur, India. His research interests include energy management systems, synchrophasor applications, power system security, stability, and technical issues in electricity markets. Prof. Srivastava is a Fellow of the Indian National Academy of Engineering, the Institution of Engineers (India), and and the Institution of Electronics and Telecommunication Engineers (India).

S. Chakrabarti (S’06–M’07–SM’11) received the Ph.D. degree in electrical engineering from Memorial University of Newfoundland, St John’s, NF, Canada, in 2006. He is currently an Assistant Professor with the Indian Institute of Technology Kanpur, Kanpur, India. His research interests include power system dynamics and stability, state estimation, and synchrophasor applications.