Annonce prochain congrès ESCAP

2015 IEEE Power, Communication and Information Technology Conference (PCITC) Siksha „O‟ Anusandhan University, Bhubaneswar, India.

Implementation of Integer Linear Programming and Exhaustive Search Algorithms for Optimal PMU Placement under Various Conditions K.SivaRamaKrishna Reddy1, D.Anantha Koteswara Rao2, A. Kumarraja3, B. Ravi Kumar Varma4 M.E Scholar, SRKR Engineering College, Bhimavaram. Assistant Professor, SRKR Engineering College, Bhimavaram. Assistant Professor, SRKR Engineering College, Bhimavaram. Professor, SRKR Engineering College, Bhimavaram. Abstract—PMU utilization is an emerging technology in the field of power systems. Optimal PMU placement (OPP) is the main concern due to the cost factor and non-existence of communication facilities. This paper gives the implementation of Integer Linear Programming (ILP) and Exhaustive Search (ES) algorithms for optimal PMU placement for full power system observability. Modeling of zero injection constraints in ILP frame work has given. A method has been proposed to the systems having zero injection busses in which we use binary connectivity matrix modification and the modified matrix can be used in Integer Linear programming (ILP) and Exhaustive Search (ES) for optimal PMU placement. ILP approach has also been given for the systems considering single PMU outage. The results specify that: 1) optimal PMU placement for full power system observability can be computed effectively; 2) connectivity matrix modification based approach for systems having zero injection buses is computationally efficient and easy to execute; 3) number of PMUs has to increase for systems considering single PMU outage. The proposed algorithms have been tested for IEEE – 9 bus, IEEE – 14 bus, IEEE – 24 bus test systems on MATLAB environment. Keywords - Integer linear programming (ILP); phasor measurment unit (PMU); optimal PMU placement (OPP); exhaustive search (ES); zero injection bus

I.

INTRODUCTION

Supervisory Control and Data Acquisition (SCADA) system is widely using in power systems to collect measurements for state estimation but it is slow and not accurate technique for real time power system monitoring. The traditional SCADA systems are not time synchronized, not update at faster rate and provide the measurements of the system state at relatively low sample rate. To run-over the draw backs of SCADA systems, PMU technology has been emerged and it is more and more attractive to power engineers from past decade. The technology of synchrophasors or synchronized phasor measurements was first developed in 1982 at Virginia tech. PMUs can be used in many applications like power system monitoring, protection, control and in capturing power system dynamic behavior. In SCADA it was not possible to determine the phasors of voltage and currents due to the technical problems in synchronizing the measurements from various places. This can be overcome by PMUs with all phasors precisely referenced to a common time frame with the aid of global positioning system (GPS) 978-1-4799-7455-9/15/$31.00 ©2015 IEEE

clock [1]. PMUs can not only measure the phasors of voltage and current but also the frequency and rate of change of frequency of measured signals. Recently there was a remarkable research activity on the problem of obtaining the minimum number of PMUs for full power system observability. The recent optimal PMU placement methods are given in [2] as a taxonomy. Here two types of algorithms are there one is mathematical algorithms and another one is heuristic algorithms. ILP and ES have come under mathematical algorithms and Immunity Genetic Algorithm (IGA), Particle Swarm Optimization (PSO), Tabu Search (TS), Simulated Annealing (SA), Greedy Algorithm (GA), Differential Evolution (DE) etc. are come under heuristic algorithms [2]. Reference [3] gives the ILP approach considering zero injection buses and outage of single PMU. Reference [4] and [5] gives the necessary information regarding ES algorithm. PSO, SA, DE, IGA algorithms for optimal PMU placement have been discussed in [6], [7], [8] and [9]. Another way of including zero injection busses (ZIB) into ILP and ES is by the modification of connectivity matrix which shows the connection of buses from one another. Reference [11] gives the adjustment of binary connectivity matrix for zero injection busses the algorithms can consume less CPU memory and also computational time can be minimized. This paper contains the following sections. Section II deals with the block diagram and working of a PMU. Here a brief description of PMU has been given. Section III describes the Integer Linear Programming (ILP) approach. Section IV covers the Exhaustive Search (ES) approach. In section V we report the simulation results for IEEE – 9, IEEE – 14, IEEE – 24 bus test systems. Section VI concludes the paper. II.

PHASOR MEASUREMENT UNIT (PMU)

An equivalent representation of a pure sinusoidal signal precisely characterized by the magnitude, phase angle, and frequency. In general a Phasor can be represented as

Here is magnitude at time t, is the phase angle measured w.r.t a common time reference.

2015 IEEE Power, Communication and Information Technology Conference (PCITC) Siksha „O‟ Anusandhan University, Bhubaneswar, India. represents a deviation from nominal power system frequency f0 at time t. Generally we use Discrete Fourier Transform (DFT) and least square estimation based methods for phasor estimation. The process of calculating the phasors of a commercial PMU with DFT based phasor estimation is shown in Fig. 1.



Its connectivity information, captured usually in single line diagram (SLD), is fully available.



All bus voltage phasors can be measured or estimated with high level accuracy.

To make the system fully observable we have two options, 

Placing of PMUs at all busses in the system.



Placing of PMUs at selected busses in the system by using the placement algorithms which are available.

Usually we will go for the placing of PMUs at selected buses in the system because of the PMUs are more expensive devices and lack of communication facilities [7].

Fig. 1. Model of a commercial PMU with DFT.

The input voltage and current signals are passed through internal potential and current transformers (PTs and CTs). PT and CT can step down the signals to a level appropriate for an analog-to-digital converter (ADC). For elimination of high frequency interference signals before the sampling, the signals from CT and PT are passed through the frontend antialiasing filter. The ADC which has fixed frequency sampling is synchronized to a GPS clock. The errors occurred due to internal instrument transformers can be corrected by using proprietary algorithms, which are tuned through a calibration exercise. By the use of DFT phasor estimation we can calculate the real and imaginary parts of the phasor. The back end performance class filter is used to get the accurate output measurements. The group delays occurring due to the presence of back-end filters and other PMU processing delays are compensated with the aid of proper time stamps. The derived quantities such as phasors, frequency and ROCOF are given to particular application through proper communication interface [1]. III.

INTEGER LINEAR PROGRAMMING (ILP)

Through the whole of the paper we can assume that the PMU placed on a bus measures the following: 1) Voltage phasor (voltage magnitude and phase angle) of the bus. 2) Branch current phasors of every branch connected to the bus. From the basics of electrical engineering which are Ohm‟s law and Kirchhoff‟s equations we can say that, if a PMU is installed at a bus, the adjacent busses can also maintain observability. So we can say that a system will be made observable with lesser PMUs than number of buses. This work on optimal PMU placement for full observability using ILP was first started by Abur [10]. A. Observability criteria A power system is said to be observable if,

B. ILP without considering zero injections Consider an IEEE – 14 bus test system as given in Fig 2. Let xi is the binary decision variable corresponding to bus i. The binary variable xi is set to zero if a PMU is not installed at bus i and it‟s one if a PMU is installed at the bus i. The optimal PMU placement problem for IEEE – 14 bus system is given as : OPP: Subject to the bus observability constraints defined as follows: Bus-1: Bus-2: Bus-3: Bus-4: Bus-5: Bus-6: Bus-7: Bus-8: Bus-9: Bus-10: Bus-11: Bus-12: Bus-13: Bus-14: The objective function defined in (2) is the minimization function, which has to be minimized to get total number of PMUs required for complete power system observability [3]. The equations given [(3)-(16)] describe the constraint equations. The first constraint (3) implies that at least one PMU must be placed at either one of busses 1, 2, or 5 in order to make bus 1 observable. The remaining constraints are also modeled depending on the above observability criteria. After the formulation we can say that, minimum of four PMUs required at busses 2, 6, 7 and 9. A general ILP

2015 IEEE Power, Communication and Information Technology Conference (PCITC) Siksha „O‟ Anusandhan University, Bhubaneswar, India. formulation for minimum PMU placement problem, referred as OPP, is summarized in Appendix A.

Bus-2: Bus-3: Bus-4: Bus-5: Bus-6: Bus-7: Bus-8: Bus-9: Bus-10: Bus-11: Bus-12:

Fig. 2. IEEE – 14 bus test system (7th bus as ZIB)

The same formulation can be applied to IEEE – 9 bus and IEEE – 24 bus systems to get minimum PMU placement for full power system observability. C. ILP considering zero injection busses The zero injection buses (ZIB) are the buses through which no external current is being injected nor extracted. The number of PMUs to be placed in the system can be further reduced if zero injection buses are modeled. 7th bus is zero injection bus in the IEEE – 14 bus system shown in fig 2. Total concept of observability for the systems having zero injection buses is given in [3]. We impose following additional constraints for the system to be fully observable: 1) If any unobservable buses are there after taditional single stage formulation of ILP, they all must belong to the cluster of zero injection buses and the buses connected to the ZIBs. 2) For a zero injection bus i, Let Ai be the set of busses adjacent to bus i. Let Bi be the union of Ai and the bus i itself. So the number of unobsrvable busses in each and every cluster defined by Bi is at most one. For IEEE – 14 bus system with 7th bus as zero injection bus. A7 = {4,8,9} and set B7 = {4,7,8,9}. Then the additional constraint to be modeled in the ILP formulation will become The above equation states that out of four buses 4,7,8 and 9 we need to have atleast three busses observable. Then the modified ILP formulation considering zero injection bus constraints is given by OPP – Z Subject to the following bus observability and zero injection constraints: Bus-1:

Bus-13: Bus-14: Zero injection: Solving the above problem in ILP we identify that we only need three PMUs to be placed in the system for full power system observability and the placement is at the busses 2,6 and 9. The general formulation of modeling zero injection constraints in ILP is given in Appendix B. D. ILP considering zero injection busses by connectivity matrix modification The other way of associating zero injection busses in integer linear programming (ILP) is the modification of binary connectivity matrix. The general binary connectivity matrix shows the connection between directly connected busses. To incorporate zero injection busses we need to consider the busses which are connected through ZIB. The modified binary connectivity matrix elements can be formed as follows:

For a 14 bus system shown above, the general binary connectivity matrix is given as follows

2015 IEEE Power, Communication and Information Technology Conference (PCITC) Siksha „O‟ Anusandhan University, Bhubaneswar, India. For the same IEEE – 14 bus test system with 7th bus as zero injection bus the adjusted binary connectivity matrix is given as follows:

time to execute for large power systems like IEEE – 24, IEEE – 57 bus test systems. An evident approach to get OPP solution is to use every different valid PMU placement combinations. The power system with N busses, by considering two possible states of placing or not placing a PMU at each bus, the search space has 2N different combinations of PMU placement [4]. If search space is completely checked thoroughly by counting all placement combinations which makes network fully observable are determined. OPP solution is the placement combination requires the minimum number of PMUs. Then the PMU placement algorithm is as follows: As said earlier PMU placed at a specific bus, measures voltage phasor of that bus and currents of all the branches attached to it. Below given the observability of bus i as a function of PMUs location

Here by solving ILP considering zero injection busses by making slight adjustments in binary connectivity matrix we can identify those busses 2, 6 and 9 for PMU placement. So, we can say that when zero injection constraints are modeled, we will achieve complete system observability with just three PMUs, i.e., minimum number of PMUs required for system observability was reduced by one. Utilization of modified binary connectivity matrix for zero injection busses can reduce the memory requirement and CPU running time. E. ILP with single PMU outage To increase the reliability of the system observability, every bus must be observed by minimum of two PMUs. This leads to full power system observsbility even though an outage of PMU occurs in the system. In ILP framework it is easier to achieve by slight modification of the right hand side of the observability constraints given in (3) – (16). The modification here is multiplying the right hand side of the inequalities by 2. In case of 14 bus system, bus – 2 will make observable by at least two PMUs by replacing inequality (4) in the formulation of OPP by the following inequality: Bus-2: The same modification is given to all the constraint inequalities given in (3) – (16). Solving ILP problem for outage will give PMU placement at busses 2, 4, 5, 6, 7, 8, 9, 11, and 13. So, we get full power system observability by placing the PMUs at the above given 9 busses even though a single PMU outage occurs. The OPP formulation for PMU outage has been given in Appendix - C IV.

EXHAUSTIVE SEARCH (ES)

This section proposes exhaustive search algorithm to find the minimum number and optimal placement of PMUs for state estimation to get secure power system operation. This method is searching for all possible binary combinations, so that it gives the global optimal solution. Proposed exhaustive search algorithm has been tested on IEEE – 9 and 14 bus test systems. Because of its exhaustive nature, this method exceeds its range and taking very long

Where Oi ≥ 1 representing that bus i is observable; ai, j is the i-j th entry of the incidence matrix which is defined as given below

is a binary parameter which is equals to 1 if a PMU is placed at bus j and 0 otherwise; N is meant for number of network buses. A. Proposed ES algorithm for OPP Step 1) Generate 1 to 2N binary combinations representing the cases of installing and not installing a PMU at each and every bus and let it is p1 (which is the pj in (33)). Step 2) for every combination, use (33) to know all observable busses by means of PMUs and the resulting matrix is taken as F. Step 3) Check the columns of F, if every element of the column is not equal to zero then save that combination as F1. Step 4) Find the sum of the elements of all the columns of F1 and the resulting matrix will be named as F2. Step 5) Find the minimum of all the columns of F2 and save it as F3. Step 6) The column which gives the minimum value will be taken as the optimal combination for the PMU placement. Step 7) Print the Optimal PMU placement combination

2015 IEEE Power, Communication and Information Technology Conference (PCITC) Siksha „O‟ Anusandhan University, Bhubaneswar, India. The above ES method has tested on IEEE – 9 bus and 14 bus test systems. We need to place three PMUs for 9 bus system and the placement is at busses 3, 4 and 7 for full system observability and for 14 bus system the number is four and the placement is at 2, 6, 7 and 9 busses. B. ES incorporating Zero Injection Busses ES algorithm given above can also be used when the system has zero injection busses. By slight modification of general binary connectivity matrix we can obtain OPP for the systems having ZIB. Earlier we have been discussed the procedure for modification of general binary connectivity matrix. So, by replacing the general binary connectivity matrix with the modified matrix in ES algorithm we can get the OPP for full system observability. V.

SIMULATION RESULTS

The ILP algorithm have been successfully tested on IEEE – 9, 14 and 24 bus systems with and without considering zero injection busses and for a single PMU outage. The ES approach as it is suffering from the draw backs like memory out of bounds and large CPU time taking, have been applied on IEEE – 9 and 14 bus test systems by both considering and ignoring the case of zero injection busses. All these systems are tested in MATLAB environment. The SLDs of test systems (IEEE – 9 and IEEE – 24) have given in Fig. 3 and Fig. 4.

The zero injection bus locations for IEEE – 9 bus system [13] are {4, 6, 8} and for IEEE – 24 bus system [12] it is {11, 12, 17, 24}. Table .1, Table .2 and Table .3 gives the comparison of IEEE – 9, 14 and 24 bus test systems with the existing results proposed in [12]. TABLE I.

Meth od used

COMPARISON OF RESULTS FOR IEEE – 9 BUS TEST SYSTEM Without Modelling ZIB

Modelling ZIB

With single PMU outage

Optimal number of PMUs

Location of PMUs

Optimal number of PMUs

Location of PMUs

Optimal number of PMUs

Locati on of PMUs

ILP

3

4,7,9

2

7,9

6

1,2,3, 4,7,9

ES

3

3,4,7

3

1,3,7

N/A*

N/A

[12]

N/A

N/A

N/A

N/A

N/A

N/A

TABLE II.

Meth od used

COMPARISON OF RESULTS FOR IEEE – 14 BUS TEST SYSTEM Without Modelling ZIB

Modelling ZIB

With single PMU outage

Optimal number of PMUs

Location of PMUs

Optimal number of PMUs

Location of PMUs

Optimal number of PMUs

Locati on of PMUs

ILP

4

2,6,7,9

3

2,6,9

9

2,4,5, 6,7,8, 9, 11,13

ES

4

2,6,7,9

3

2,6,9

N/A

N/A

[12]

4

2,6,7,9

3

2,6,9

N/A

N/A

TABLE III.

COMPARISON OF RESULTS FOR IEEE – 24 BUS TEST SYSTEM

Fig. 3. IEEE – 9 bus test system Met hod used

Without Modelling ZIB

Modelling ZIB

With single PMU outage

Optimal number of PMUs

Location of PMUs

Optima l number of PMUs

Location of PMUs

Optimal number of PMUs

Locati on of PMUs

ILP

7

3,4,8, 10,16, 21,23

6

2,8,10, 15,16,23

14

1,2,7, 8,9,1 0,11, 15,16 ,17,2 0,21, 23,24

ES

N/A

N/A

N/A

N/A

N/A

N/A

[12]

7

2,3,8, 10,16,2 1,23

6

1,2,8,16, 21,23

N/A

N/A

Fig. 4. IEEE – 24 bus test system

N/A – Not available

2015 IEEE Power, Communication and Information Technology Conference (PCITC) Siksha „O‟ Anusandhan University, Bhubaneswar, India. VI.

CONCLUSION

This paper presents an ILP and, ES based formulation and the associated solution to the problem of PMU placement in power systems. Numerical results are given for different size systems and the minimum number of PMUs to be placed and the locations where they have to be placed has been given with and without modeling ZIB. Also ILP considering PMU outage has been given.

APPENDIX A. OPP Formulation For n bus system, if the PMU placement vector x having elements xi gives possibility of PMUs on a bus, i.e.

and if wi is the cost associated with the placement of PMU at bus i. so, a minimum cost PMU placement problem can be defined as [5];

Or

Here Bi is the union of Ai and i. |Ai| indicates the cardinality of Ai. The condition that busses which are not adjacent to ZIB are made observable is given by the constraint (40). C. Formulation of ILP with PMU outage The ILP formulation for single PMU outage can be obtained by a simple modification to the constraint given in (37) as

REFERENCES [1]

[2]

[3]

[4]

Subject to the constraints: [5]

Where, b is a unit vector of length n, i.e.

[6]

b = [1 1 1………..1] T And

x=[x1 x2………xn] T

Here A is a binary connectivity matrix of the system defined earlier. It is a general ILP formulation. We have to set all weights wi to unity to get Optimal PMU placement problem. B. Formulation of ILP with Zero Injections In the ILP frame work the zero injection constraints are modelled as given below.

[7]

[8]

[9]

[10]

[11]

Subject to:

[12]

And and

[13]

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