OPEN ACCESS
For finding the power system state estimation, first we should know the observability of the system. In order to observe the system, we have to monitor the entire system for each and every second. For monitoring and control of an entire power system, number of techniques is available. Out of all phasor Measurement Unit (PMU) is one of most important technique for monitoring of system. Placement of PMU’s is the major trouble because they are little expensive. This problem can be overcome by finding minimum no of PMU’s and their optimal locations of PMU’s using network connectivity information. In this paper, we found optimal locations of PMU’s using two different techniques, they are Binary Integer Programming (BIP) and three stage algorithm. These methods are solved by considering a single PMU outage case along with Zero Injection buses (ZIB’s). The results are shown for different IEEE test systems like 14 bus,24 bus, 30 bus,39 bus,57 bus and 118 bus. The results of both techniques are shown along with their computational time.
state estimation, observability, optimization, phasor measurement unit (PMU), binary integer programming (BIP), PMU outage
As compared to past, power demand is increased now a day’s, crowded the existing network and as a result stability boundaries are decayed. To ensure appropriate and assured operation of power network, an accurate measurement set and network monitoring is essential. By using this measurement set we are finding state estimation of system (Yoon, 2005; Phadke et al., 1986; Jamuna and Swaup, 2011; Xu and Abur, 2005). In order to estimate the state of system we have to observe the network. For checking this observability, we are having mainly two methods (Sodhi et al., 2010).
(1) Numerical observability: In this we are finding rank of matrix by using measurement set, which involves number of combinations. Increases the computational burden of analysis explained in (Koutsoikis et al., 2013).
(2) Topological observability: Here we are considering topological graphs i.e. based on graph theory explained in (Nuqui and Phadke, 2005).
For finding this measurement set we are using SCADA which gives unsynchronized measurements. To get exact measurements we are using Wide Area Monitoring System’s (WAMS). This WAMS consists of PMU’s which gives synchronized voltages, currents and phase angles through the timing signals received from GPS.One of the advantage of PMU placement is it gives direct measurements of all system states without running any state estimator. As the PMU gives the synchronized phasors it will provide the system conditions like stability margins, maximum loading conditions, disturbance identification and dynamic response analysis (Phadke, 1993).
The techniques BIP and three stage algorithm finds the minimum number of PMU placement buses for complete of system. In three stage algorithm, the first two stages determine the imperative PMU placement bus locations. The third stage audits the more reduction of PMU’s from the final set. Binary Integer programming method is also solved using MATLAB. Two techniques are solved with and without ZIB’s and also considering the single PMU outage case. The several test systems like IEEE14 bus, IEEE24 bus, IEEE30 bus, IEEE39 bus, and IEEE57 bus are tested under normal operating conditions considering ZIB’s. The computational time for two techniques under test systems IEEE14 bus, IEEE24 bus, IEEE30 bus is compared. The intended method is examined only PMU measurements for complete perceptibility of the system. Optimal numbers of PMU’s obtained are equal but PMU placed bus locations may vary from one technique to another.
The main issue with PMU’s is placement. To find the minimal number of PMU’s needed for entire network observability, we are having number of Optimization methods. Some of the important methods are simulated annealing based graph theory approach, the practical development of PMU and its usage is explained by Phadke et al., (1986). Binary integer programming based method for optimal PMU placement (Nuqui and Phadke, 2005; Gou, 2008; Gou, 2008). For getting global optimal results for entire power system observability is explained in (Dua et al., 2008). A threestage optimal PMU placement technique is analysed (Chakrabarthi and Kyriakides, 2008). The optimal PMU placement by exhaustive search method (Saha et al., 2012) is explained. A two stage PMU placement technique for finding the optimality is discussed (Azizi et al., 2011).
For ascertaining system observability PMU’s are placed in two different ways:
(1) Allocating the PMU’s at all system buses and eliminate one by one by checking the observability of network.
(2) Allocating the PMU’s at preferred system buses by acquiring the observability of network.
Method (1) is more expensive due to the placement of PMU’s in entire network. Method (2) is preferable, but the major problem is finding the peerless set of PMUs’ for obtaining complete Observability.
An optimal PMU placement problem is formed for a considering an Mbus system is as follows
Min $\sum _ { i = 1 } ^ { M } C _ { i } n _ { i }$Subject to G(N)>A
Where N is the binary selection variable for placement of PMU,
Whose n_{i}= 1 considering the PMU at i^{th} bus
= 0 no PMU for i=1, 2, 3 …M
A is a vector of unit length M i.e.
A= [1, 1, 1…M]
Is PMU cost installed at i^{th} bus
G (N) is the perceptibility constraint function
= 1 if buses are observable by considering the measurement set
= 0 not observable
The constrained vector function (G(N)) is formed using the bus incidence matrix (B) of network. The elements of B are defined as
b_{p,q}= 1 if p, q are connected each other and are also equal
= 0 p, q are not connected
The constraint function for any (i^{th}) bus in the considered test system is
G (N) = BN ≥ a
$\mathrm { f } _ { 1 } = \mathrm { a } _ { \mathrm { i } , 1 } \mathrm { n } _ { 1 } + \cdots + \mathrm { a } _ { \mathrm { i } , \mathrm { i } } \mathrm { n } _ { \mathrm { i } } + \cdots + \mathrm { a } _ { \mathrm { i } , \mathrm { M } } \mathrm { n } _ { \mathrm { M } }$
ZIB: ZIB’s are nothing but the buses which don’t any generation and load i.e. no electrical power or electrical current is supplied to the system. This can be used as pseudo measurements to get the system perceptible with minimum number of PMU’s.
3.1. BIP
In integer linear programming problem all of the variables or a set of variables are to be restricted to either ‘1’ or ‘0’. So, this method is considered to be a mathematical optimization problem.
ILP uses following steps to solve the problem
In mathematical optimization techniques, Binary Integer Programming (BIP) is one which solves the linear objective function by taking linear constraints corresponding to bus impedance matrix. By using the steps described above formation of problem is as follows:
$\min _ { n } G ^ { T } n$ Such that $\left\{ \begin{array} { c } { B \cdot n \leq a } \\ { B e q \cdot n = a e q } \\ { n \text { binary } } \end{array} \right.$
Now the PMU placement problem in the form of linear integer programming as follows
Min $\sum _ { \mathrm { k } = 1 } ^ { \mathrm { M } } \mathrm { n } _ { \mathrm { k } }$ Subject to $\mathrm { G } _ { \mathrm { pmu } } \mathrm { N } \geq \mathrm { A } _ { \mathrm { pmu } }$
Where $\mathrm { N } = \left[ \mathrm { N } _ { 1 } , \mathrm { N } _ { 2 } , \ldots \ldots . \mathrm { N } _ { \mathrm { m } } \right] ^ { \mathrm { T } }$ and $\mathrm { A } _ { \mathrm { PMU } } = [ 1,1,1 , \ldots 1 ] _ { \mathrm { Mx } 1 } ^ { \mathrm { T } }$
In above equation we can see function is minimization and constraints are maximization. So we have to convert the maximization to minimization constraints by using ILP solver in MATLAB.
Binary connectivity matrix (B) and constraints corresponding to bus connectivity matrix are defined for 14 bus system as follows
_{$Gpmu=\left[ \begin{matrix} {1}&{1}&{0}&{0}&{1}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0} \\{1}&{1}&{1}&{1}&{1}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0} \\{0}&{1}&{1}&{1}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0} \\{0}&{1}&{1}&{1}&{1}&{0}&{1}&{0}&{1}&{0}&{0}&{0}&{0}&{0}\\ {1}&{1}&{0}&{1}&{1}&{1}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0} \\{0}&{0}&{0}&{0}&{1}&{1}&{0}&{0}&{0}&{0}&{1}&{1}&{1}&{0}\\ {0}&{0}&{0}&{1}&{0}&{0}&{1}&{1}&{1}&{0}&{0}&{0}&{0}&{0} \\ {0}&{0}&{0}&{0}&{0}&{0}&{1}&{1}&{0}&{0}&{0}&{0}&{0}&{0}\\ {0}&{0}&{0}&{1}&{0}&{0}&{1}&{0}&{1}&{1}&{0}&{0}&{0}&{1}\\ {0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{1}&{1}&{1}&{0}&{0}&{0}\\ {0}&{0}&{0}&{0}&{0}&{1}&{0}&{0}&{0}&{1}&{1}&{0}&{0}&{0}\\ {0}&{0}&{0}&{0}&{0}&{1}&{0}&{0}&{0}&{0}&{0}&{1}&{1}&{0}\\ {0}&{0}&{0}&{0}&{0}&{1}&{0}&{0}&{0}&{0}&{0}&{1}&{1}&{1}\\ {0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{1}&{0}&{0}&{0}&{1}&{1} \end{matrix}\right]$}
_{$Gpmu.N=\left( \begin{array} { c } { g _ { 1 } = n _ { 1 } + n _ { 2 } + n _ { 5 } \geq 1 } \\ { g _ { 2 } = n _ { 1 } + n _ { 2 } + n _ { 3 } + n _ { 4 } + n _ { 5 } \geq 1 } \\ { g _ { 3 } = n _ { 2 } + n _ { 3 } + n _ { 4 } \geq 1 } \\ { g _ { 4 } = n _ { 2 } + n _ { 4 } + n _ { 4 } + n _ { 5 } + n _ { 7 } + n _ { 9 } \geq 1 } \\ { g _ { 5 } = n _ { 1 } + n _ { 2 } + n _ { 4 } + n _ { 5 } + n _ { 6 }\geq 1 } \\ { g _ { 6 } = n _ { 5 } + n _ { 6 } + n _ { 11 } + n _ { 12 } + n _ { 13 } \geq 1 } \\ { g _ { 7 } = n _ { 4 } + n _ { 7 } + n _ { 8 } +n_9 \geq 1 } \\ { g _ { 8 } = n _ { 7} + n _ { 8 } \geq 1 } \\ { g _ { 9 } = n _ { 4 } + n _ { 7 } + n _ { 9 } +n_{10}+n_{14} \geq 1 } \\ { g _ { 10 } = n _ {9} + n _ { 10 } +n_{11} \geq 1 } \\ { g _ { 11 } = n _ { 6 } + n _ { 10 } + n _ { 11 } \geq 1 } \\ { g _ { 12 } = n _ { 6 } + n _ { 10 } + n _ { 13} \geq 1 } \\ { g _ { 13 } = n _ { 6 } + n _ { 12 } + n _ { 13 }+n_{14} \geq 1 }\\ { g _ { 14 } = n _ { 9 } + n _ { 13 } + n _ { 14 } \geq 1 } \end{array} \right.$}
By solving these constraints, we are having the optimal set for placing the PMU’s. Due to communication collapse or line interruption, PMU may not be able to give the measurements for total perceptibility of system. In that case, we are moving onto a PMU outage problem. The results are shown for different standard IEEE bus systems.
3.1.1. For single PMU outage case
If we consider the single loss case (20), (21) the test system modified as
Min $\sum _ { \mathrm { k } = 1 } ^ { \mathrm { M } } \mathrm { n } _ { \mathrm { k } }$ Subject to $\mathrm { G } _ { \mathrm { pmu } } \mathrm { N } \geq \mathrm { A } _ { \mathrm { pmu } }$
Where $N = \left[ N _ { 1 } , N _ { 2 } , \ldots \ldots N _ { m } \right] ^ { T }$ and $A _ { P M U } = [ 2,2,2 \ldots . .2 ] _ { M \times 1 } ^ { T }$
3.2. Three stage algorithm
This technique is an iterative procedure. A PMU placement problem in three stage technique is same as the BIP problem i.e.
Min $\sum _ { i = 1 } ^ { M } C _ { i } n _ { i }$
Subject to G (N)>a
Where N is the decision variable for PMU placement having either 1 or 0,
Whose n_{i}= 1 considering the PMU at i^{th} bus
0 no PMU for i=1, 2, 3 …M
The Three stages are explained below with considering Zero Injection Buses (ZIB).
Stage (1): It founds the important buses where PMU’s are confined among distinct valency buses.
Stage (2): Some of the buses are unobservable due to not having any connection with already placed PMU’s or with ZIB’s. These unobservable buses are tested here.
Stage (3): This is an eliminating stage. This checks feasible ways for further minimization of PMU’s obtained from stages (1), (2).
If we can solve these three steps, a peerless group of PMU’s is obtained for complete perceptibility of system.
3.2.1. for single PMU outage case
After getting a peerless group of PMU’s, the algorithm for one PMU outage case is modified as below.
(1) Locale PMU at minimum valency buses along with optimal set obtained from above case.
(2) Finding the buses which are perceptible through one PMU with already placed PMU’s.
(3) Find highest valency from the buses which are perceptible through one PMU ignoring the placed PMU buses.
(4) Choose any bus from the list of maximum valency buses and remove doubly observable buses.
(5) In order to eliminate the surplus PMU’s other than the doubly perceptible, we are performing pruning and final optimal PMU’s are obtained.
If we can see the simulation results, for IEEE14bus system the optimal set is 2,6,7,9 without ZIB’s and with ZIB’s finalized optimal set is 2, 6, and 9. Due to PMU outage the optimal set is 1, 2, 4, 6, 9, 10, and 13.The optimal results for IEEE bus systems are shown in Table 2 without ZIB’s under normal operating conditions. The results with ZIB’s are also shown in table 3 and also a single PMU failure or outage is shown in table 4. The computational time for IEEE bus systems are also shown in table 1.
The comparison of computational time of different test systems is shown. BIP without ZIB’s will take more time to compare three stage algorithms with ZIB’s. BIP take 1.16 sec while three stage algorithms take 0.66 sec for IEEE14 bus system.
Table 1. Computational time of two techniques for obtaining optimal solution
Test System IEEE 
BIP 
Three Stage Algorithm 
14 bus 
1.16s 
0.66s 
24 bus 
1.34s 
0.76s 
30 bus 
1.24s 
0.83s 
Table 2. Optimal no of PMU’s and their positions under normal operating condition
IEEE Test system 
No of PMU’s 
BIP 
Three Stage Algorithm 
Without ZIB’s Optimal PMU Locations 

14 bus 
4 
2, 6, 7, 9 
2, 6, 7, 9 
24 bus 
7 
2, 3, 8, 10, 16, 21, 23 
2, 3, 8, 10, 16, 21, 23 
30 bus 
10 
1, 7, 9, 10, 12, 18, 24, 25, 27, 28 
2, 3, 6, 9, 10, 12, 15, 19, 25, 27 
39 bus 
13 
2, 6, 9, 10, 12, 14, 17, 19, 20, 22, 23, 25, 29 
2, 6, 9, 10, 12, 14, 17, 19, 20, 22, 23, 25, 29 
57 bus 
17 
1, 4, 9, 20, 24, 27, 29, 30, 32, 36, 38, 41, 45, 51, 54 
1, 4, 9, 20, 24, 27, 29, 30, 32, 36, 38, 41, 45, 51, 54 
118 bus 
32 
1, 5, 9, 12, 13, 17, 21, 23, 26, 28, 34, 37, 41, 45, 49, 53, 56, 62, 63, 68, 71, 75, 77, 80, 85, 86, 90, 94, 101, 105, 110, 114 
1, 5, 9, 12, 13, 17, 21, 23, 26, 28, 34, 37, 41, 45, 49,53, 56, 62, 63, 68, 71, 75, 77, 80, 85, 86, 90, 94,101, 105, 110, 114 
Table 3. Optimal no of PMU’s and their positions under normal operating condition considering ZIB’s
IEEE Test System 
No of PMU’s 
BIP 
Three Stage Algorithm 
With ZIB’s Optimal PMU Locations 

14 bus 
3 
2, 6, 9 
2, 6, 9 
24 bus 
6 
2, 8, 10, 15, 18, 20 
1, 2, 8, 16, 21, 23 
30 bus 
7 
2, 4, 10, 12, 15, 18, 27 
2, 3, 10, 12, 18, 24, 30 
39 bus 
8 
3, 8, 12, 16, 20, 23, 25, 29 
3, 8, 12, 16, 20, 23, 25, 29 
57 bus 
11 
1, 6, 9, 19, 25, 27, 32, 37, 38, 46, 51, 53, 56 
1, 6, 13, 19, 25, 29, 32, 38, 51, 54, 56 
118 bus 
28 
3, 8, 12, 15, 17, 21, 27, 31, 32, 34, 40, 45, 49, 53, 56, 62, 65, 70, 75, 77, 80, 85, 87, 90, 94, 102 105, 110 
1, 6, 8, 12, 15, 17, 21, 25, 29, 34, 40, 45, 49, 53, 56, 62, 72, 75, 77, 80, 85, 86, 90, 94, 101, 105, 110, 114 
Table 4. Optimal no of PMU’s and their positions under normal operating condition considering ZIB’s with single PMU outage
IEEE Test System 
No of PMU’s

BIP 
Three Stage Algorithm 
Optimal PMU Locations FOR Single PMU Outage With ZIB’s 

14 bus 
7 
1, 2, 4, 6, 9, 10, 13 
1, 2, 4, 6, 9, 10, 13 
24bus 
13 
1, 2, 7, 8, 9, 10, 11, 15, 16, 17, 20, 21, 23 
1, 2, 7, 8, 9, 10, 11, 15, 16, 17, 20, 21, 23 
30bus 
15 
1, 2, 3, 5, 6, 10, 12, 13, 15, 16, 18, 19, 24, 27, 30 
1, 2, 3, 5, 6, 10, 12, 13, 15, 16, 18, 19, 24, 27, 30 
39 bus 
18 
2,3,5,6,8,13,16,17,20,22,23,25,26,29,34,36,37,38 
2,3,5,6,8,13,16,17,20,22,23,25,26,29,34,36,37,38 
57bus

26 
1, 2, 5, 6, 8, 9, 11, 12, 15, 17, 19, 22, 24, 25, 27, 29, 30, 32, 33, 36, 38, 41, 47, 50, 51, 53, 54, 56 
1, 2, 5, 6, 8, 9, 11, 12, 15, 17, 19, 22, 24, 25, 27, 29, 30, 32, 33, 36, 38, 41, 47, 50, 51, 53, 54, 56 
118 bus 
64 
1, 2, 5, 6, 8, 9, 11, 12, 15, 17, 19, 20, 21, 23, 25, 27, 28, 29, 32, 34, 35, 37, 40, 41, 43, 45, 46, 49, 50, 51, 52, 53, 56, 59, 62, 66, 68, 70, 71, 72, 75, 76, 77, 78, 80, 83, 85, 86, 87, 89, 90, 92, 94, 96, 100, 101, 105, 106, 108, 110, 111, 112, 114, 117 
1, 2, 5, 6, 8, 9, 11, 12, 15, 17, 19, 20, 21, 23, 25, 27, 28, 29, 32, 34, 35, 37, 40, 41, 43, 45, 46, 49, 50, 51, 52, 53, 56, 59, 62, 66, 68, 70, 71, 72, 75, 76, 77, 78, 80, 83, 85, 86, 87, 89, 90, 92, 94, 96, 100, 101, 105, 106, 108, 110, 111, 112, 114, 117 
Two optimized techniques are discussed in this paper which gives complete observability of system. BIP is simple technique but we cannot achieve minimization and maximization at the same time. The three step method is quick and smooth to implement by using network connectivity information. The results for IEEE test systems sight the optimal set of PMUs’ for complete perceptibility of system. The intended method is examined only PMU measurements for complete perceptibility of the system. Optimal numbers of PMU’s obtained are equal but PMU placed bus locations may vary from one technique to another. Further work we have to mainly focus on an optimal PMU placement for incomplete perceptibility of system.
Azizi S., Dobakhshari A. S., Nezam Sarmadi S. A. (2011). An optimal PMU placement by an Equivalent Linear formulation for Exhaustive search. IEEE Transactions on smart grid, Vol. 3, No. 1, pp. 174182. https://doi.org/10.1109/tsg.2011.2167163
Chakrabarthi S., Kyriakides E. (2008). Optimal placement of phaor measurement units for power system observability. IEEE Transactions on Power Systems, Vol. 23, No. 3, pp. 143340. https://doi.org/10.1109/TPWRS.2008.922621
Dua D., Dambhare S., Gajbhiye R. K., Soman S. A. (2008). Optimal multistage scheduling of PMU Placement: an ILP approach. IEEE Transactions on Power Systems. https://doi.org/10.1109/TPWRD.2008.919046
Gou B. (2008). Generalized integer linear programming formulation for optimal PMU placement. IEEE Transactions on Power Systems, Vol. 23, No. 3, pp. 10991104. https://doi.org/10.1109/TPWRS.2008.926475
Gou B. (2008). Optimal placement of PMUs by integer linear programming. IEEE transactions on Power Systems, Vol. 23, No. 3, pp. 15251526. https://doi.org/10.1109/TPWRS.2008.926723
Jamuna K., Swarup K. S. (2011). Optimal placement of PMU and SCADA measurements for security constrained state estimation. International Journal of Electrical Power Energy Systems, Vol. 33 No. 10, pp. 165865. https://doi.org/10.1016/j.ijepes.2011.08.002
Koutsoikis N. C., Manousakis N. M., Georgilakis P. S., Korres G. N. (2013). Numerical observability method for optimal phasor measurement units placement using recursive Tabu search method. IET Generation, Transmission and. Distribution. Vol. 7, No. 4, pp. 347356. https://doi.org/10.1049/ietgtd.2012.0377
Nuqui R. F., Phadke A. G. (2005). Phasor measurement unit placement for complete and incomplete observability. IEEE Trans Power Delivery, Vol. 20, No. 4, pp. 23812388. https://doi.org/10.1109/TPWRD.2005.855457
Nuqui R. F., Phadke A. G. (2005). Complete topological observablility for optimal PMU placement. IEEE Transactions on Power Delivery, Vol. 20, No. 4, pp. 23812388.
Phadke A. G. (1993). Synchronized phasor measurements in power systems. IEEE Computer Applications in Power, Vol. 6, No. 2, pp. 1015.
Phadke A. G., Thorp J. S., Karimi K. J. (1986). State estimation with phasor measurements. IEEE Transactions on Power Systems, Vol. 1, No. 1, pp. 233 241. https://doi.org/10.1109/MPER.1986.5528179
Saha Roy B. K., Sinha A. K., Pradhan A. K. (2012). An optimal PMU placement technique for power system observability. IEEE Transactions on Power Systems, Vol. 42, No. 1, pp. 7177. https://doi.org/10.1016/j.ijepes.2012.03.011
Sodhi R., Srivastava S. C., Singh S. N. (2010). Optimal PMU placement method for complete topological and numerical observability of power system. International Journal of Electrical Power Energy Systems, Vol. 80, No. 9, pp. 11541159. https://doi.org/10.1016/j.epsr.2010.03.005
Xu B., Abur A. (2005). Optimal placement of phase measurement units for state estimation. Final Project Report.
Yoon Y. J. (2005). Study of utilization and benefits of phasor measurement units for large scale power system state estimation. Master of Science.