An optimal Phasor Measurement Unit placement techniques for achieving complete perceptibility of a network even when PMU failure

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 IEEE-14 bus, IEEE-24 bus, IEEE-30 bus, IEEE-39 bus, and IEEE-57 bus are tested under normal operating conditions considering ZIB’s. The computational time for two techniques under test systems IEEE-14 bus, IEEE-24 bus, IEEE-30 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 three-stage 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).

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

• From the available bus data form the connection matrix and find the problem which is maximizing or minimizing function.
• Now form the corresponding linear constraints.
• Solving the constraints with relating to function, we are having the optimal set of PMUs’.

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.