Measurement placement algorithm for power system state estimation using PageRank - Case Studies Micha¨el Hurtgen∗, Pavel Praks†, Petr Zajac‡, Jean-Claude Maun§ Universit´e Libre de Bruxelles ˇ - Technical University of Ostrava, Universit´e Libre de Bruxelles VSB Sun Microsystems Czech, s.r.o. Universit´e Libre de Bruxelles

The aim is to determine a measurement distribution to make the system observable for power system state estimation. The power system is observable for state estimation when the measurements can be used to describe the system. The measurements should be distributed throughout the network in order to estimate the states of all nodes. The rank of the system matrix is used to determine the observability. For observability, nodes of the graph are selected by scrolling down the list of nodes sorted by importance. The PageRank algorithm is used as a tool for computation of the nodes’ importance. Our approach is illustrated on three IEEE test networks. Results are compared to those obtained using the Graph Theoretic Procedure.

1 Introduction Over the past decade, dynamic phenomena in power systems have made it ever more urgent to have a reliable tool to monitor the systems. The main goal is to maintain the system in a normal secure state as the operating conditions vary during the daily operation. Accomplishing this goal requires continuous monitoring of the system conditions, identification of the operating state and determination of the necessary

preventive actions in case the system state is found to be insecure(Abur and Esposito 2004). Based on measurements taken throughout the network, State Estimation allows estimation of the state of the power system while checking that these estimates are consistent with the measurements set. Traditionally, input measurements have been provided by the SCADA system (Supervisory Control and Data Acquisition) which is designed to capture only quasi-steady state operating conditions, preventing the monitoring of transient phenomena (Baldwin, Mili, Boisen, and Adapa 1993). With the advent of real-time Phasor Measurement Units (PMU’s), fast transients can be tracked at high sampling rates. What is more, the measurements are synchronised unlike the SCADA measurements, which allows monitoring of dynamic phenomena in the power system. The type of measurements needed by the state estimator will depend on the numerical method implemented. While classical state estimation uses active and reactive power flow or injected power measurements as well as voltage measurements, state estimation using PMUs requires voltage and current phasor measurements.

∗

Micha¨el Hurtgen, BEAMS dept., Universit´e libre de Bruxelles CP165/52 (energy), Facult´e des Sciences Appliqu´ees, Av. Fr. Roosevelt 50, 1050 Bruxelles, Belgium, http://beams.ulb.ac.be/beams/staff/all/view/169.html, email: [email protected] † Ing. Pavel Praks, Ph.D., Dept. of Mathematics and Descriptive Geometry, Dept. of Applied Mathematics, Faculty of ˇ - TU OsElectrical Engineering and Computer Science, VSB trava, 17. listopadu 15, CZ 708 33 Ostrava, the Czech Republic, http://praks.am.vsb.cz, tel.: +420 597 324 181, e-mail: [email protected], Service de M´etrologie Nucl´eaire (SMN), Universit´e Libre de Bruxelles, Facult´e des Sciences Appliqu´ees, Av. F.D. Roosevelt 50, 1050 Bruxelles, Belgium, tel +32-2650.20.90, fax +32-2-650.45.34 ‡ Ing. Petr Zajac, Sun Microsystems Czech, s.r.o., Evropsk´a 33E, CZ 148 00 Praha 6, http://www.sun.cz, tel.: +420 233 009 311, [email protected] § Prof. Jean-Claude Maun, BEAMS dept., Universit´e libre de Bruxelles CP165/52 (energy), Facult´e des Sciences Appliqu´ees, Av. Fr. Roosevelt 50, 1050 Bruxelles, Belgium, http://beams.ulb.ac.be/beams/staff/all/view/149.html, e-mail: [email protected]

A fundamental question with regard to the measurements on the network is whether it is possible to determine all bus voltage magnitudes and angles. 1

When enough measurements are available so that the entire state vector of bus voltage magnitudes and angles throughout the system can be estimated, the system is said to be observable. Because of telemetry failures and modelling constraints, it is possible that the power system is not observable using only real-time measurements (Clements, Kumpholz, and Davis 1983). Observability depends on the number of measurements available and their geographic distribution (Monticelli and Wu 1985). Regardless which state estimation formulation is used, the accuracy of state estimation solutions is dependent on the quality of data as well as measurement configuration and redundancy(Celik and Liu 1995). This measurement configuration needs to be well designed to ensure robust and accurate performance of the state estimator. The locations of the measurements should allow the state variables of the entire network to be calculated uniquely, therefore should render the network observable.

while allowing full network observability and enough redundancy, the Tabu search algorithm (Peng, Sun, and Wang 2006) can be used. In (Peng, Sun, and Wang 2006), a numerical formulation of the optimal PMU placement problem is presented. The installation cost is minimised but certain constraints are added to the system. In (Nuqui 2001), a tree search algorithm is used in order to successively determine the logical places for the PMUs. In addition, the PMUs are placed for an incomplete observability, to minimise the number of PMUs and therefore the installation cost. After this preliminary stage, a Simulated Annealing technique is used, which minimises an objective function calculated for each configuration. In this paper, an incremental measurement placement algorithm for PMUs, referred to as the PageRank placement algorithm is proposed. The chosen approach is to use the relative importance of the node in the network. A node’s relative importance can be determined using the PageRank algorithm. While scrolling down the list of nodes sorted by decreasing level of importance, nodes will be considered for placement of a PMU. After placement of a PMU, the algorithm checks if the measurement set allows observability. When the system is made observable by the selected measurement set, the selection process is stopped. The algorithm will determine a set of measurements that can then be further optimised. The algorithm is compared with a similar approach known as the Graph Theoretic Procedure. A brief review of power system state estimation is given in section II. The PageRank algorithm used by the proposed algorithm is described in section III. The proposed meter placement algorithm is presented in section IV. The proposed algorithm is applied to IEEE test networks with 14, 57 and 118 nodes.

Many measurement placement methods have been proposed in the literature. The methods can be classified in two groups. Either the technique redesigns the measurement configuration completely by choosing the ideal locations for measurement devices or it takes into account the existing measurement configuration and tries to improve it. In (Celik and Liu 1995), the authors propose an incremental algorithm to enhance the quality of state estimation solutions based on the topology of the studied network. The objective is to incrementally improve the state estimation when only a limited number of measurements can be added. In (Abur and Gou 2001), a method is proposed for adding measurements to an unobservable system using the gain matrix G in order to make it observable. Other methods try to redesign the measurement set completely using the topology of the network, like in (Huang, Lei, and Abur 2003). First a measurement configuration with a minimum number of measurements that satisfies the observability constraints is determined, then the measurement configuration is further optimised to reduce the number of measurement devices to be placed based on the network topology. In the decomposition technique (Rakpenthai, Premrudeepreechacharn, Uatrongjit, and Neville 2004), the entire network of the power system is first decomposed into smaller subsystems, in which the optimal positions for measurement placement are determined by using the minimum condition number criteria.

2 Background The output of a state estimator is the value of the state variables at a given time, deduced from a set of measurements taken on the network. If z is the measurement vector then the model used by power system state estimation is: z = h(x) + e,

(1)

where x is the current state vector, h is the vector of nonlinear measurement functions between x and z and e is the measurement noise vector. The global redundancy is defined as r=

With the advent of PMUs, another class of algorithms has been developed specifically for placing PMUs. When the Optimal PMU Placement problem is formulated so as to minimise the number of PMUs

m , n

(2)

where m is the number of measurements and n is the number of state variables to be estimated. The 2

Weighted Least Squares (WLS) estimator minimises the performance index or sum of the squared residuals J(x) = (z − h(x))T R−1 (z − h(x)),

T_1

(3)

where z − h(x) is the measurement residual and R is the covariance matrix which associates a variance to each measurement. In other words, the method ensures that the error between the measurements and the estimations of these measurements when using the value of the state variables, is minimised.

T_2

A fundamental question with regard to the measurements on the network is whether it is possible to determine the bus voltage magnitudes and angles at buses throughout the entire network from them. When enough measurements are available so that the entire state vector can be estimated, the system is said to be observable. Because of telemetry failures and modelling constraints, it is possible that the power system is not observable using only real-time measurements (Clements, Kumpholz, and Davis 1983). Observability depends on the number of measurements available and their geographic distribution (Monticelli and Wu 1985). The determination of network observability is equivalent to deciding whether the jacobian matrix H, which relates measurements to bus voltage magnitudes and angles in this linear model is of full rank (Kumpholz, Clements, and Davis 1980). A phasor measurement set allows observability of the network if rank(H) = N . It is supposed that the PMUs placed at node n are capable of measuring the current phasor on all incident branches as well as the voltage phasor at the node. The number of measurements mn made by a PMU placed at node n is thus

T_4

mn = bn + 1,

T_3

A

B

T_5

Figure 1: An example of links between internet pages for n = 5 and C(A) = 1. citations. If the selected paper is cited by other papers, it seems that the selected paper is important. So, we can measure the importance of the selected paper by counting the number of times it is cited (GTP approach). But let us assume that there are two papers with the same number of citations. Is the importance of these two papers really the same? Of course, it depends on the importance of cited papers. So, we can improve our model by assuming not only the count of the citations but also the ”quality” i.e. the importance papers citing the selected paper. And this is the idea behind the PageRank algorithm, which is used for instance in the Google search engine. The description of PageRank calculation is the following (Brin and Page 1998): Let m denote the total number of indexed pages1 . We assume page A has pages T1 , . . . , Tn which point to it (i.e., are citations). Also C(A) is defined as the number of links going out of page A, see Figure 1. The PageRank of a page A is denoted by the symbol P R(A) given as follows:

(4)

where bn is the number of branches incident to node n. Because a PMU measures the voltage phasor and the current phasors on all branches incident to a node, the voltage at an adjacent node can be easily determined. The Observability algorithm can then be deduced (Nuqui 2001): If a node v has a PMU, then all the buses incident to v are observed.

P R(A) =

1−d P R(T1 ) P R(Tn ) + d( +...+ ) (5) m C(T1 ) C(Tn )

The parameter d is a damping factor which can be set between 0 and 1. Note that the PageRanks form a probability distribution over web pages, so the sum of all web pages’ PageRanks will be one. The value of P R(Ti ), i = 1, . . . , n is obtained from the previous iteration of Eq. 5. For detailed description of the PageRank algorithm see (Brin and Page 1998) and (Zajac and Briˇs 2007).

3 Modelling the Importance by the PageRank algorithm When considering a network, a fundamental question is the importance of the nodes which make up the network. Using a recursive method, the PageRank algorithm associates a numerical value which describes the relative importance of each node. Let us begin with a simple example of the importance modelling related to research papers and their

1 In the original paper (Brin and Page 1998), the symbol m is missing because of a typing error.

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The PageRank algorithm, which was originally used for analyzing of importance of internet pages, is also well applicable for analysis of electrical networks. The results of the PageRank algorithm, which was successfully applied to a graph-model of real 110 kV overhead lines network located in the Czech Republic, correspond to the practical experiences of human experts (Praks, Zajac, Goˇno, and Briˇs 2007).

An additional condition is added before selection of the node to minimise the number of PMUs. A PMU is not placed at a node that is already observed. The algorithm proceeds as follows

4 Measurement placement algorithms Two measurement placement algorithms will be described. The first algorithm is the Graph Theoretic Procedure, which is generally used as a step to generate an initial configuration of measurement devices before further optimisation. The second algorithm is the new measurement placement algorithm proposed in this paper, the PageRank Placement Algorithm. The approach is somewhat different in that it selects the nodes by their relative importance after having made sure that end nodes with only one adjacent node, are observed. However, both algorithms consider the importance of nodes. But the graph theoretic procedure classifies the nodes with respect to the number of incident branches of a particular node while the PageRank placement algorithm uses the relative importance of nodes generated by the PageRank algorithm which was described in the previous section.

• Place a PMU at the node adjacent to the end nodes.

• Run the PageRank algorithm for the network which is considered. The output is the list of nodes sorted by decreasing importance.

• Successively place PMUs at the most important nodes after checking that the node is not already observed by another PMU. 5 Results and Discussion The measurement placement algorithm was applied to IEEE test networks with 14, 57 and 118 nodes (Christie 2003). The IEEE 14 test network is shown in Figure 2. Results obtained using the Graph Theoretic Procedure and the PageRank Placement Algorithm are shown in Table 1.

4.1 Graph Theoretic Procedure(GTP) The graph theoretic procedure is based strictly on the system’s topology. The procedure expands a subgraph from the part of the system so far made observable into the remaining unobservable parts (Baldwin, Mili, Boisen, and Adapa 1993). The subgraph is built according to the following steps: • A PMU is placed at the bus with the highest number of incident lines in the unobservable region • The region observed by the current PMU set is determined. • If the observed region does not cover the whole system, the process is repeated. The unobservable region is smaller because of the placement of one extra PMU. 4.2 PageRank Placement algorithm(PPA) Measurements are successively placed at the nodes sorted by decreasing importance. First observability of end nodes is ensured by placing a measurement device at the adjacent node. The next stage is to select the nodes at which measurement devices are to be placed. Let S be the list of nodes sorted by decreasing importance. The selection is stopped once the measurement set allows full observability of the network.

Figure 2: Test network IEEE14

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case 1 2 3 N 14 57 118 nen 1 1 6 nppa 5 20 41 ngtp 6 21 41 rppa 1.428 1.403 1.635 rgtp 1.571 1.456 1.585 Table 1: Comparison of the two algorithms for PMU placement

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Proportion of observed nodes

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In Table 1, the number of PMUs that have been placed using the GTP and PPA algorithms are respectively shown for the three test networks of IEEE by rows ngtp and nppa . The symbol N describes the total number of nodes for each case. Differences between the two methods are the total number of PMUs ngtp and nppa placed for observability of the entire network. For the IEEE test networks with respectively 14 and 57 buses, the number of PMUs which make the network observable using PPA is inferior to the number of PMUs placed using GTP. For the network with 118 nodes, there is no difference in the number of PMUs placed. However the locations chosen for the PMUs are not identical as shown by rows rgtp and rppa of Table 1, which represent the redundancy of the measurement set using PPA and GTP respectively, see equation 2. Although the number of PMUs placed are identical for the IEEE 118 bus test network, a better redundancy coefficient is observed when the PageRank placement algorithm is used. This means that the chosen locations by PPA are more important than those selected by GTP. It is also interesting to look at how the number of observed nodes evolves with the number of PMUs placed in the network by each algorithm. Results are shown in figures 3 to 5. It should be noted that in these figures, the number of nodes made observable by the first placed PMUs are smaller when PPA is used than when GTP is used. That observation is more marked in the third case when the number of end nodes nen is six instead of one in the two first cases. This might be due to the extra constraint used in the case of PPA, where the first PMUs are placed at a node adjacent to an end node.

0.8 0.7 0.6 0.5 0.4 0.3 0.2 Pagerank placement algorithm Graph Theoretic Procedure

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Figure 3: Relationship between number of PMUs and observed part of the network for IEEE 14

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6 Conclusions and future work The aim was to determine a measurement distribution to make the system observable for power system state estimation. We used two approaches for placement of PMUs: the classical Graph Theoretic Procedure and the PageRank Placement Algorithm proposed in this paper. Both approaches are illustrated on three IEEE test networks. Results indicate that the classical Graph Theoretic Procedure allows a bigger observable area for the first placed PMUs. However

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Figure 4: Relationship between number of PMUs and observed part of the network for IEEE 57

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Abur, A. and B. Gou (2001). An improved measurement placement algorithm for network observability. Transactions on Power Systems 16.

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Baldwin, T., L. Mili, M. Boisen, and R. Adapa (1993). Power system observability with minimal phasor measurement placement. IEEE Transactions on Power Systems.

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Brin, S. and L. Page (1998). The anatomy of a large-scale hypertextual Web search engine. Computer Networks and ISDN Systems 30(1– 7), 107–117.

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Christie, R. (2003). Power systems test case archives. Technical report, Power Systems Test Case Archives. [Online]. Available: http://www.ee.washington.edu/research/pstca/.

Figure 5: Relationship between number of PMUs and observed part of the network for IEEE 118 PageRank provides a better final measurement configuration in all cases: Test results for the 118 bus IEEE network show that the same number of PMUs were required but the redundancy coefficient rppa is higher than rgtp . The next step of our research will be based on preference modelling by a specialised evaluation of power-lines and customers. It is clear that in practise importance of nodes depends for instance on the voltage level and injected power. We believe that the so called PageRank personalised vector could be used for these purposes. In the future we would like to analyse more complex electrical networks.

Clements, K., G. Kumpholz, and P. Davis (1983). Power system state estimation with measurement deficiency: an observability/measurement placement algorithm. IEEE Transactions on Power Apparatus and Systems 102. Huang, G., J. Lei, and A. Abur (2003). A heuristic approach for power system measurement placement design. Proceedings of the International Symposium on Circuits and Systems. Kumpholz, G. R., K. A. Clements, and P. W. Davis (1980). Power system observability: A practical algorithm using network topology. IEEE Transactions on Power Apparatus and Systems 99.

7 Acknowledgment We would like to thank Prof. Pierre-Etienne Labeau and Prof. Radim Briˇs. The support of the Universit´e Libre de Bruxelles, Belgium related to the ARC Project - ”Advanced supervision and dependability of complex processes: application to power systems” is graciously acknowledged. The research has been also supported by the Ministry of Education, Youth and Sports of the Czech Republic under the Research project CEZ MSM6198910007 ”Research of the reliability of power systems in connection with nontraditional ecological sources of energy and valuation of unsupplied energy” and the F.R.I.A which funds the project ”Wide-area static and dynamic state estimation using synchronized phasor measurements” and other belgian research programs in industry and agriculture.

Monticelli, A. and F. F. Wu (1985). Network observability: Theory. Transactions on Power Apparatus and Systems 104. Nuqui, R. (2001). State Estimation and Voltage Security Monitoring Using Synchronized Phasor Measurements. Ph. D. thesis, Virginia Polytechnic Institute and State University. Peng, J., Y. Sun, and H. Wang (2006). Optimal pmu placement for full network observability using tabu search algorithm. Electrical Power and Energy Systems 28, 223–231. Praks, P., P. Zajac, R. Goˇno, and R. Briˇs (2007). Finding the most important electricity substations using pagerank - a case study. In Electric ˇ - TechPower Engineering (EPE 2007). VSB nical University of Ostrava.

REFERENCES Abur, A. and A. Esposito (2004). Power System State Estimation: Theory and Implementation. Marcel Dekker.

Rakpenthai, C., S. Premrudeepreechacharn, S. Uatrongjit, and W. Neville (2004). Measurement placement for power system state estimation 6

using decomposition technique. In 11th International Conference on Harmonics and Quality of Power. Zajac, P. and R. Briˇs (2007). Critical software defects are located in important source code. In ˇ Risk, Quality and Reliability - RQR 2007. VSB - Technical University of Ostrava.

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