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Abstract - This paper addresses the problem of optimal placement of Phasor Measurement Units (PMUs) by comparing two different placement algorithms namely, Integer Linear Programming and Matrix Manipulation. The results are compared to check the suitability of these algorithms for networks of different sizes such as networks with higher number of buses such as utility networks and those with smaller number of buses such as small microgrids. Finally suggestions are made as to which algorithm should be used to calculate the optimal locations to render a given power network fully observable. The issue of installing PMUs in a system that has conventional measurement units is also discussed. This work forms a part of a joint project between Eskom, the national electric utility in South Africa and the University of Cape Town in creating indigenous PMU placement software for the utility to aid their asset management.

Index Terms – Phasor Measurement Unit, Optimal PMU placement, full system observability, matrix manipulation, integer linear programming

I. INTRODUCTION he state of a power network is determined by Power System State Estimation (PSSE) models which use bus voltage measurements in the network. These

measurements are currently made by SCADA (Supervisory Control and Data Acquisition system) measurement tools. However, the problem with SCADA is that measurements are not synchronised and in [5] it is shown that when dynamic events such as load changes occur, the asynchronous behaviour of SCADA measurements provide inaccurate state. Currently Phasor Measurement Units (PMUs) are more advanced commercially available measurement devices that are able to provide real-time, synchronised measurements of bus voltage and current phasors in the network using a GPS system in their architecture [2]. Therefore, when PMUs are placed in a power network, the PSSE models become more reliable, since the measurements are now synchronised and can be compared to each other in real-time. This improves the efficiency of the power network operation and the speed at

Abdul-Aziz Fish is with Electrical Engineering Department, University of

Cape Town, Cape Town 7701, South Africa. (e-mail: FSHABD001@uct.ac.za)

S.Chowdhury is with Electrical Engineering Department, University of Cape Town, Cape Town 7701, South Africa (e-mail: sunetra.chowdhury@uct.ac.za).

S.P.Chowdhury is with Electrical Engineering Department, University of Cape Town, Cape Town 7701, South Africa (e-mail: sp.chowdhury@uct.ac.za)

which it can react to disturbances such as faults, load changes, loss of generation etc. Since PMUs are costly devices to install, hence to make PMU deployment and utilization cost-effective, PMUs should be placed optimally at strategic locations such that full observability of the entire power network is achieved for the PSSE model with minimum number of PMUs. This issue provokes the study of Optimal PMU placement strategies and the development of various placement algorithms some of which are indicated to be well-established in the relevant research literature. Network structure differs widely in terms of the number of buses form large utility networks to small microgrids. Hence it is important to investigate the suitability of these algorithms with respect to the network size before developing a any PMU placement software tool for any specific network. In this paper, the authors investigate and compare the performances of the Matrix Manipulation algorithm and the Integer Linear Programming algorithm when applied to IEEE 14 bus and IEEE 30 bus systems. For all the case studies, the minimum number of PMUs required to achieve full system observability is calculated and the performance of the algorithm studied with respect to computation time and network size in terms of number of buses, as well as its ability to provide the same result consistently. Finally the results are used to provide a recommendation as to what the best procedure should be to select a PMU placement for a power network depending on network size. The issue of installing PMUs in the presence of conventional measurement units is also addressed in the paper. This work forms a part of a joint project between Eskom, the national electric utility in South Africa and the University of Cape Town in creating indigenous PMU placement software for the utility to help them in maximum utilization of the existing PMUs for state estimation of Eskom network and to decide on the procure and installation of PMUs in future.

II. PMU PLACEMENT PROCESS Several PMU placement algorithms have been developed where the central aim is to minimize the number of PMUs in the network due to cost and communication line availability. A good algorithm as defined in [13] must compensate for the issues such as (a) Outage of a PMU or communication line, (b) Acknowledging or identifying zero-injection buses and (c) Phasing the placements of PMUs. These issues are briefly discussed in this section. (a) Redundancy PMU placement algorithms must consider the possibilities of PMU or communication line outages [14]. This is achieved by observability reducndancy where more than one PMU

Optimal PMU Placement in a Power Network for Full System Observability

Abdul-Aziz Fish, Non-Member, S.Chowdhury, Member, IEEE, and S.P.Chowdhury, Member, IEEE

T

978-1-4577-1002-5/11/$26.00 ©2011 IEEE

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observes a specific bus. Thus even when a PMU is out, the observability of the bus measured by that PMU must be maintained by means of redundant measurements of surrounding PMUs.. (b) Acknowledgement of Zero-injection buses A PMU is able to observe the voltage phasors of the bus it is located at as well as those buses directly linked to that bus. Also it can observe the currents in the lines linking to those buses. Acknowledging or identifying zero-injection buses is important in optimal PMU placement. In case of a zero-injection, no load is connected to this bus and so the current passing through is the same as that in the lines linking the buses to the zero-injection bus. When a PMU is located at a bus directly linked to a zero-injection bus and since it is able to observe the current through the lines, the unobserved buses directly linked to the zero injection buses become observable [13]. However, it is important to note that a PMU need not be placed at the zero-injection bus for this to occur. The zero-injection bus presents a fundamental advantage in providing for requirement of less PMUs to make a system observable, if used correctly. If a PMU is placed at the zero-injection bus, then its advantage would be nullified. (c) Phasing the placements of PMUs Phasing of PMUs refers to installing PMUs in stages and not all at once. This approach is acceptable to utilities for phasing out the high expense linked to PMU installation. During phasing, sometimes more PMUs are installed than the optimal lowest amount that is needed. This happens when the first batch of PMUs are installed in locations with the highest level of observability. Then this is repeated for the next phase where the next batch of PMUs is installed at the next highest observability locations. This is repeated until the entire network becomes observable. Reference [13] suggests a phasing method where the total number of PMUs at the end of phasing is not more than the optimal amount of PMUs required for full observability. This is achieved by installing the PMUs with the highest level of observability in the initial stages chosen from the locations of the optimal solution and phased with installing the next PMU at the location for the following highest observability level and so on.

III. REVIEW OF PMU PLACEMENT ALGORITHMS

A.1) Binary search PMU placement algorithm This algorithm considers all the possible combinations of locations of PMUs and narrows down to the optimal solution by undertaking a binary search formulation. The first step is to generate all the possible combinations of solutions. From [14] the total number of candidate buses for a PMU placement is given as P; and the (initial) number of PMUs NPMU is given by

NPMU = [(N + s/2)/3] Where N is the total number of buses in the system and s is the number of unknown power injections. Then the number of combinations Nsol is given as:

( ))!(!

!PMUPMU

PPMUsol

NPNPNN−

==

A flow chart of the search method is given in Fig.1. Compensation for single line outages is achieved in this algorithm by an initial placement followed by rechecking the

network for observability once an outage occurs. It checks for each line outage separately and if the solution makes the system fully observable for all of the line outage instances, then it is considered a final solution. When the algorithm comes up with multiple solutions for the least number of PMUs then the one with highest redundancy is chosen as the optimal solution. This is done by defining a integer variable ri=[r1,r2,….rq] where q is the number of candidate solutions. Now one does a check and sums the number of times each bus of solution 1 to q is made observable by a PMU and sum each redundancy to each r value respectively. The solution with the largest r value is the optimal solution.

Fig.1 Flow chart for Binary search algorithm A.2) Binary Particle Swarm optimisation (BPSO) This Algoritm is based on the natural phenomena of individuals co-operating with each other in a swarm to find the optimal location where the swarm needs to be at. Each trial position and velocity is continously updated and compared to its past condition and also if it is better than the conditions adopted presently by the entire swarm. If it is better, then this is the new swarm position and velocity. This process is continued until the ideal location (optimal PMU placement solution) is found [1]. BPSO regulates and updates the particle position and velocity according to:

ti

ti

ti

ti dcdcvv ,2,1

1 21 ⊕+⊕+=+ 11 ++ += t

iti

ti vxx

where, tii

ti xpbestd ⊕=,1 ; t

it

i xgbestd ⊕=,2 ; 1+tidv is the next

velocity; 1+tix is the next position; c1 and c2 are two random

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binary vectors; pbest is the best position of the individual particle so far; gbest is the best position found by the swarm so far. The parameter vi

t is limited to vmax which sets the search domain for the particles. Too large vmax might cause that the optimal solution is never reached and too small vmax might cause that the best solution is not found due to it not lying in that domain. If vi

t reaches vmax, then a new vit is obtained by

reducing it to below vmax by randomly decreasing the number of ones in the binary velocity vector for vi

t<vmax. For PMU placement vmax is chosen as a fraction of the number of buses in the system.

Fig. 2 Flow chart for Binary Particle Swarm Optimization When the search space is considered, the zero-injection buses must be removed from the group of candidate PMU locations. This is because if a PMU is located at such as a bus, the advantage that a zero-injection bus presents to the PMU by allowing it to observe buses one bus beyond the zero-injection bus, would be lost. Moreover, if a PMU is placed at a radial bus, it would be wasting a PMU since at that location the PMU can only observe a maximum of 2 buses. Therefore a better location must be found that will make the radial bus observable without a PMU at that bus. The fitness function used to determine the fitness of all the individual particles is:

unobsPMU NcNcf .2.1 += where NPMU and Nunobs is the number of PMUs presently in the system and the number of unobserved buses, respectively. A flow chart of the aforesaid formulation steps is shown in Fig.2. A.3) Integer Linear programming (ILP) In ILP approach to PMU placement, an integer based formulation is used to arrive at the optimal solution. The objective function is given as: [15]

Minimise: x1 + x2 + x3 + x4 +…….+ xNbus where Nbus is the total number of buses in the system. A vector is defined as xi=[x1 x2 x3…..xNbus]T which represents the vector of the buses. The constraints are thus given as:

Ax ≥ e where e is a vector: [1 1 1…..1]T which generates the constraint that each bus must be observed by at least one PMU. An optimisation toolbox such as Tomlab can be used to solve this formulation by using the “cplex” function. The output is the number and locations for optimal PMUs. With regard to ILP, reference [13] presents a linear approach to solving the PMU placement problem including zero-injection buses. The proposed method [13] works with identifying the zero-injection buses and modifying their constraint equations and that of the buses directly linked to them. If a system has zero-injection buses at buses 1, 2 and 4, the following process is carried out: The method states that one of the buses in this inequality can remain unobservable. Now a vector is defined as u: [u1 u2 u3…..uNbus]T where u depicts the confirmation of observability of the bus. Parameter u is equal to 1 when the bus is observable and equal to 0 when it is not. Then the right hand side of the inequality Ax ≥ u must change from corresponding 1 to u1 and the same for buses 2 and 4. This is done by replacing the corresponding 1’s with u2 and u4 respectively. Then a new inequality constraint is added given by:

u1+u2+u4 ≥ 2 where the number “2” represents the advantage of one of the three buses being unobservable. This is due to a PMU at a bus directly linked to the zero-injection bus being able to observe other buses from the zero-injection bus without a PMU actually at the zero-injection bus. For ILP approach also redundancy criteria is used to select the optimal solution more than one solution arrives at the same minimum number of PMUs to provide full obsevability. In [13] a Bus Observability Index (BOI) is defined as the total amount of PMUs that can observe the bus in question. SORI or System Observability Redundancy Index is the sum of all the BOI’s of all the buses. The solution with the highest SORI is branded the optimal solution. Sometimes it is desireble to have at least 2 PMUs observing a single bus or 2 PMUs for all the buses. This is dealt with in ILP by changing the constraint equalities; as can be seen with the example bus 1:

Bus1: x1+x2+x4 ≥ 2 A.4) Spanning Tree Approach This algorithm uses a spanning tree method to obtain a solution for full observability of the power network. This approach uses the concept of depth of observability where the level of unobservability is explained as the number of directly linked unobservable buses connected to an observed bus which does not have a PMU situated at that bus. The spanning tree method works by starting off at the root node of a network (chosen by the user) and branches off to the terminal nodes at the end of the branch. There are two categories: parent terminal node, which is at the end of the main branch; and the spanning tree terminal node, which is an intermediate branch from the path. After reaching the terminal branch, it backtracks to the splitting point of the branch and explores the other directions in the same manner. Eventually it

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will backtrack to the root node after all terminal nodes have been reached. The first PMU is placed at the first bus after the root node for observability of the root node and of that particular bus. The next PMU is placed according to the following rule [16]

dp = u + 3 where dp is the number of buses away from the current PMU placement the next one will be; u is the desired level of unobservability.

Fig. 3 Flow chart for spanning tree approach When the spanning tree reaches the main terminal bus, it will not place the PMU at that bus even if is less than u + 3. Instead it places the PMU one bus away, which still makes the terminal bus observable. This is to avoid the issue of placing a PMU at the radial bus/ terminal bus, because as mentioned above, if a PMU is placed at a radial bus, the maximum number of buses it can observe is two (the bus it is located at and the one bus linked to it). Then it backtracks to the previous split and searches this area until the entire network is searched and the root node is reached. Thereafter a new root node is selected and the process is repeated a few times with a different root node each time in order to ascertain the minimum number of PMUs for system observability. The procedure is shown in Fig.3 as a flow chart. A.5) Immunity Genetic Algorithm (IGA) IGA is a uniquely adapted version of the Genetic Algorithm for Optimal PMU placement. The genetic algorithm (GA) is based on the concept of the evolution of species in the changing environments they are exposed to. The characteristics are stored on the chromosomes and are updated when breeding as well as mutation of the species takes place.

The immunity algorithm (IA) is based on the idea of how an immune system that protects the body against viruses, diseases and bacterial bodies. This provides for a good optimization procedure of choosing a vaccine and immunity selection. A vaccination implies amending some genes to raise the fitness of the population. Immune selection is to prevent degenerative aspects in the evolution of the population by selecting the fittest individuals in the population [17]. IGA is made up of combined characteristics of the GA and of the IA for harnessing the advantages of both. IGA is governed by two main rules as shown in Table-1, pertaining to the candidate PMU locations. Other rules fall under these two rules where they are applied as per circumstances.

TABLE 1 IGA RULES

Rule 1 When a PMU is situated at a bus, it can observe the voltage and current phasors of that bus as well as the buses directly linked to that bus.

Rule 2

If there is at least one PMU directly linked to a zero-injection bus, all the buses directly linked to the zero-injection bus become observable by use of Kirchhoff’s Current Law (KCL) making use of the current through the zero-injection bus.

The vaccines listed in Table-2 serve as limiters for the placement algorithm.

TABLE 2 IGA VACCINES

Vaccine 1 A PMU must not be placed at a radial bus. Placement at a radial bus is seen as a waste since it will only render the bus in question and the one bus directly linked to it observable.

Vaccine 2

The bus directly linked to a radial bus must be made a candidate PMU location if it is not a zero-injection bus.

Vaccine 3 PMUs should not be placed at zero-injection buses unless in the case that it will provide the optimal solution if it is placed at that specific bus.

Problem formulation for PMU placement as a minimization problem is explained below [18]: Step 1: Form the connectivity matrix to show which buses are linked in the system. Step2: Generate the initial population as matrix of dimension (Npop x N), where Npop is the number of populations and N the number of buses in the system. Step3: Calculate the fitness of each individual of the population. The fitness is given by the inverse of the following objective function [17]:

unobsPMUi NwNwxC .2.1)( += where NPMU is the number of PMUs in the network; Nunobs is the number of unobservable buses in the network; w1 and w2 are tunable constants. The objective function must be minimised because it is desired that there be a minimal number of PMUs in the system with no unobservable buses. Step4: Select the fittest individuals Step5: Apply the crossover and mutation processes [19]. Crossover is the process of reproduction between two parents where two offspring are produced. Relating this to the PMU placement problem, each individual parent is represented as a possible placement solution. The offsprings are new possible solutions determined by the genes of the parent. Mutation occurs randomly in nature. It is modelled with a mutation

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probability factor defined as 1/N. A random number between 0 and 1 is generated and compared to the mutation probability factor. If it is greater, then the individual’s genes are inverted, if it not, then nothing is done. Step6: Inject vaccinations as per vaccination conditions of Table-2 are applied to the network. Step7: Perform immune selection Step 8: Close off. This is the final stage where this generation is now completed and the next generation will be examined. The flow chart of IGA is given A.6) Evaluation of algorithms All the PMU placement algorithms discussed in this section have their own respective strengths and weaknesses. One decision criteria for selecting the most suitable algorithm for a particular power network is their ability to handle the network size. BPSO is suitable for all networks including the larger ones where it gives the least number of required PMUs (for larger systems) as shown in [1]. The ILP algorithm was modeled by non-linear equations, although in [13] it is linearized, which simplified the computation and enabled ILP to have low computation time. The IGA as well as BPSO use statistical methods with discrete variables since they do not use differentials of the cost function [1]. The IGA is an improvement of the GA algorithm due to application vaccinations for issues pertaining to degeneration which speeds up the converging time. Binary search is an exhaustive method where it assures that the proper solution is found, but will take a while to reach there for larger systems. Despite their differences, they all have similar results for the optimal placement problem. Therefore when an algorithm is to be chosen, the size, nature and method of installation of PMUs must be collectively considered.

IV. DEVELOPMENT OF PMU PLACEMENT ALGORITHMS FOR PERFORMANCE ANALYSIS WITH IEEE TEST SYSTEMS

The authors developed two PMU placement algorithms, viz., Matrix Manipulation algorithm and Integer Linear Programming (ILP) algorithm and tested and compared their performance on IEEE test systems. The Matrix Manipulation algorithm was developed as a basic exhaustive algorithm for determining optimal PMU locations by manipulating the connectivity matrix. Its convergence criterion is similar to BPSO where past and present conditions are compared. Moreover, it is relatively easy to understand and could be used as a good tool for personnel training in utilities who would handle the PMU placement tool. ILP was chosen as a reference of comparison to decide how far Matrix Manipulation could be considered as a PMU placement tool for utility usage. This is because ILP is highly developed in terms of its accuracy (of finding minimal amounts of PMUs required) and speed of convergence. Both the algorithms are tested on IEEE 14 bus and IEEE 30 bus test systems to study their performance on networks with different bus numbers. IEEE 14 bus configuration is shown in Fig.4 and IEEE 30 bus configuration is shown in Fig.5.

Fig. 4 IEEE 14 bus configuration [14]

Fig. 5 IEEE 30 bus configuration: [14] B.1) Development of ILP Algorithm To solve the optimal placement problem with the Integer Linear Programming formulation, Tomlab’s Optimisation toolbox was used for its CPLEX ILP solver. The standard Mixed Integer solver was used and the binary constraints of the connectivity-matrix were added. Below is an explanation of the variables used which was acquired from [20]. The objective function to be optimised is as follows:

xcxf T== )(minimisex

where x represents each bus and c is a transpose vector of size Nbus x 1 (where Nbus is the number of buses in the system) representing the system to be optimised. To illustrate this, let c be given as:

c = 1 1 1 1 1 1 . . . . . 1 (Nbus columns all ones) This means that the number of PMUs must be minimised in using as little of the buses as possible. The objective function is constrained by the following:

vL xxx ≤≤ which gives:

vbL bAx ≤≤ where of the vectors: xL, xv, bL, bv and A are made up of binary only values. A is the connectivity matrix. For the Optimal

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placement problem, xL and xv always remain the same and are given as:

xL = 0 0 0 0 0 0 0. . . 0 (Nbus columns all zeroes) and

xv = 1 1 1 1 1 1 . . . . . 1 (Nbus columns all ones) This sets the binary limitation that the specific entity is either present or not. The values of the bL vector determine the desired redundancy level of each bus with all of its constituents bigger or equal to one. If bL was made up of just ones then it would mean that all of the buses need at least one PMU. This is the basic criterion for full observability. If some or all of the columns of bL are greater than one, for example, 3, it means that at least 3 PMUs are required for that specific optimisation. The basic form of bL is therefore:

bL = 1 1 1 1 1 1 . . . . . 1 (Nbus columns all ones) and bv is the maximum number of PMUs in the system and should is not necessary to be limited since for the Optimal placement problem, the bL vector is the main constraint that is used. So bL is set to:

bv = inf inf inf inf inf …..(Nbus columns all infinity) The above variables: A, c, xL, xv, bL and bv are the basic inputs for the program to produce an optimal solution using the CPLEX function. This means of tackling the problem does not take into account zero-injection buses and is defined for a minimum redundancy level of one PMU observing a bus. It does however compensate for the issue with radial buses. B.2) Development of Matrix Manipulation Algorithm

Fig. 6 Flow diagram of Matrix Manipulation algorithm This method uses an exhaustive approach as in the binary search, but has some differences. The algorithm is developed

in Matlab. The main concept used in developing this is to allow for the basic constraint that at least one PMU must observe each bus for an optimal solution. It also does not compensate for zero-injection buses (but can be adapted for that purpose) but like above with the ILP, it compensates for radial buses. The basic concept of its operation is shown in Fig.6. The first step is to attain the connectivity matrix of the system that is being dealt with. The user then has to enter the locations of the zero-injection buses. The following step, generation of all the possible solutions, is the longest step in the algorithm. It works by taking the connectivity matrix and firstly calculates how many solutions are available. This is done by taking the sum of each specific row and multiplying it by the next row and the next, till all the rows are complete. To make all the buses observable, all the solutions are taken into account which means multiplying the sum of all the rows together. In the program this is done by using inner for-loops with the outer most one being the first row and the inner most one the last row of A. In each for-loop a Q vector is declared = [0 0 0 …0 Nbus times]. In each loop, Q is appended with a single solution for that specific bus and then for the following loop, Q is re-declared as [0 0 0 …0 Nbus times] and the next single solution is chosen. This is the reason for the inner for-loop strategy; where all the singular solutions would be used to build a complete candidate solution for the placement problem. The complete candidate solution is made in the inner-most loop and is defined as the sum of all the Q vectors for that loop iteration. The complete solution is given below:

busNQQQQZ ++++= .......321 The Z-vector is an Nbus x 1 vector and is likely to be made up of integer values instead of just binary values. This is due to adding of Q solutions with PMUs (a binary 1 value) at the same location. The Z-vector however must be a binary vector representing a complete candidate solution and therefore a check is made that if any of the values in Z is ≥ 1, then it is made equal to one. This means that a PMU must be placed at that location. A variable “x” is set as = . A new Z-vector is made for every complete iteration of the inner for-loop setup. If the sum of the elements of Z is equal to x, then that Z vector is appended to a solution pool matrix P, such that:

)(

)1(

.

.

.

.

lastsolution

solution

Z

Z

P =

where P is a matrix with Nbus columns and number of Z-solutions rows. If the sum of the elements of Z < x, then x is set to the sum of the elements of the newer Z. The matrix P is cleared and the Z vectors abiding to the new x will fill P. The

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Matrix Manipulation convergence criterion can therefore be simplified as to say that it compares past solution data to present data. This cycle is perpetuated until all the Z vectors or all solutions are generated. All the rows of P are solutions with the minimum (“x” amount) number of PMUs for full observability. A redundancy test is then performed on P to single out the solution with the highest level of redundancy. The idea behind the redundancy test is taken from the SORI concept of ILP in [13]. It is applied to this formulation by taking each solution at a time and then multiplying the columns of each Z-vector by every row in A. the sum of every row is then taken and added up. This final sum is the redundancy factor or SORI. The Z-vector which produces the highest redundancy is the optimal solution. This solution is the one that will be printed out when the program is run. For example if a Z-vector was given as: Z = 0 1 1 0 1 Then it means for this system (5 bus system), the optimal PMU placement locations are at Bus 2, 3 and 5 to give full observability. After the redundancy test, this final solution is put through the zero-injection bus check to see whether there is a possibility of further minimising the number of required PMUs. This is done by firstly checking if there is a PMU to be placed at a zero-injection bus (as indicated by the user), if so, then a check is carried out to see if a PMU is placed at a bus directly linked to that specific zero-injection bus. If so, then the PMU at the zero-injection bus can be removed (as explained above in acknowledgement of zero-injection buses). By now, the solution is the final and optimal solution presented by this algorithm.

V. RESULTS The results for the two IEEE test systems are summarized in Table 3 and Table 4 below:

TABLE 3 RESULTS FOR IEEE 14 BUS TEST SYSTEM IEEE 14 bus system ILP Matrix Manipulation Number of PMUs with consideration of zero-injection buses 3 3

Locations of PMU placement 2, 6, 9 2, 6, 9 Time taken 0.015625 sec 391.145657 sec

TABLE 4 RESULTS FOR IEEE 30 BUS TEST SYSTEM IEEE 30 bus system ILP Matrix Manipulation Number of PMUs with consideration of zero-injection buses 7 Unable to provide a solution Locations of PMU placement 1, 2, 10, 12, 15, 18, 25 -Do- Time taken 0.031250 sec -Do-

The results show that the main problem of Matrix Manipulation algorithm is its inability to handle large systems. This is because it takes extremely long to calculate and converge when the number of connection lines suddenly increase for larger systems. Further development of this method would therefore be aimed at cutting down on convergence time. Another sideline test would be to switch the order of the redundancy test with the zero-injection-bus test to see if for some systems if this switching further reduces the number of PMUs to be installed.

VI. CONCLUSION From the results it can be concluded that ILP converges to the optimal solution very quickly and can be used for big and small systems. This makes ILP a feasible option for use in locating the PMUs. On the other hand, the Matrix manipulation algorithm, which also converges to the optimal solution, takes much longer to get there. It also showed that it cannot handle large power systems well. The reason for this is that when lots of connection lines are introduced, more possible solutions are generated since the sum of each row of the connectivity matrices will increase. Therefore, this algorithm should be able to handle large power systems with few connection lines. The application of this method should therefore be used only on smaller power systems or large ones with little lines. The strength of this algorithm is that it is exhaustive and can thus determine the exact solution since it considers all possible solutions. Although, when a system is large with relatively few lines, more PMUs will be required for full observability. This is one of the reasons that brought about the study for optimal placement in the first place. Therefore there is a trade-off between the amount of connection lines and PMUs. If there are lots of connection lines, less PMUs will be needed and if there are lots of PMUs, less connection lines will be required. When calculating the optimal solution for the locations of PMU placements, it can be noted that this calculation need not be performed often. Therefore it is recommended that as many algorithms as possible are used to calculate the best solution in the event that one algorithm might provide a solution where even a single PMU less is required. From research done in [1] it shows that most of the algorithms give the same results, but the Binary Particle Swarm Optimisation Algorithm provides less PMUs for large systems. So the final recommendation is that if lots of algorithms are not able to be applied due to whatever reason, then at least let the ILP and Binary Particle Swarm Algorithms be used. When installing PMUs in a power system that already has SCADA measurement units, it was suggested that the minimum amount of PMUs first be calculated and then install those PMUs in phases, where in each phase a certain number of PMUs are installed in locations providing the highest possible level of obsevability chosen from the optimal locations as in [13].

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VII. ACKNOWLEDGEMENT The authors are grateful to the authorities of Electrical Engineering Department, University of Cape Town, South Africa for providing the necessary infrastructure for carrying out this research.

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IX. BIOGRAPHY

Abdul-Aziz Fish is currently a BSc(Engg.) student of Electrical Engineering Department of University of Cape Town, South Africa. Email : FSHABD001@uct.ac.za

S.Chowdhury received her BEE and PhD in 1991 and 1998 respectively from Jadavpur University, Kolkata, India. She was connected to M/S M.N.Dastur & Co. Ltd as Electrical Engineer from 1991 to 1996. She served Women’s Polytechnic, Kolkata, India as Senior Lecturer from 1998 to 2006. She is currently the Senior Research Officer in the Electrical Engineering Department of The University of Cape Town, South Africa. She became member of IEEE in 2003. She visited Brunel University, UK and The University of Manchester, UK several times on collaborative research programme. She has published two books and over 100 papers mainly in power systems. She is a Member of the IET (UK) and IE(I) and Member of IEEE(USA). Email: sunetra.chowdhury@uct.ac.za

S.P. Chowdhury received his BEE, MEE and PhD in 1987, 1989 and 1992 respectively from Jadavpur University, Kolkata, India. In 1993, he joined E.E.Deptt. of Jadavpur University, Kolkata, India as Lecturer and served till 2008 in the capacity of Professor. He is currently Associate Professor in Electrical Engineering Department of the University of Cape Town, South Africa. He became IEEE member in 2003. He visited Brunel University, UK and The University of Manchester, UK several times on collaborative research programme. He has published two books and over 150 papers mainly in power systems and renewable energy. He is a fellow of the IET (UK) with C.Eng. IE (I) and the IETE (I) and Member of IEEE (USA). He is a member of Knowledge management Board and Council of the IET (UK).Email: sp.chowdhury@uct.ac.za