Optimal PMU placement for power system observability using Taguchi binary bat algorithm

Accepted Manuscript Optimal PMU placement for power system observability using Taguchi binary bat algorithm Basetti Vedik, Ashwani K Chandel PII: DOI: Reference:

S0263-2241(16)30533-4 http://dx.doi.org/10.1016/j.measurement.2016.09.031 MEASUR 4348

To appear in:

Measurement

Received Date: Revised Date: Accepted Date:

1 December 2015 12 September 2016 16 September 2016

Please cite this article as: B. Vedik, A.K. Chandel, Optimal PMU placement for power system observability using Taguchi binary bat algorithm, Measurement (2016), doi: http://dx.doi.org/10.1016/j.measurement.2016.09.031

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1

Optimal PMU placement for power system observability using Taguchi binary bat algorithm Basetti Vedik1,* and Ashwani K Chandel2 1

Research Scholar, Electrical Engineering Department, NIT-Hamirpur, H.P, India 2 Professor, Electrical Engineering Department, NIT-Hamirpur, H.P, India *Email: [email protected]

Abstract Phasor measurement units (PMUs) are considered as a promising tool in wide area monitoring, protection, and control of power system networks. In this paper, a novel technique based on Taguchi binary bat algorithm (TBBA) is proposed to determine the optimal number and placement locations of PMUs such that power system is completely observable. The proposed TBBA combines the systematic reasoning ability of the Taguchi method with the traditional binary bat algorithm thereby enhances the initial population, and subsequently, improves the computational efficacy of the solution. The effectiveness of the proposed method is demonstrated on a number of power system test systems under different operating conditions. Results thus obtained are compared with the existing techniques published in the literature. From the results it has been found that the proposed technique not only yields multiple global optimal solutions in single experimental run but also provides solution with lesser computational time. Further, the proposed technique provides similar or better optimal solutions in comparison with the other techniques proposed in the literature.

Keywords: Binary bat algorithm, observability, optimal placement, phasor measurement unit, Taguchi method.

1.

Introduction Modern power industry has been witnessing a major transitional phase during the last decade. With the

integration of renewable energy resources and deregulation of power system network, the power flow pattern has become less predictable. Further, with the ever-increasing power demand the present day electrical grid is operated under intemperately stressed conditions. In such circumstances, so as to operate the power system in reliable, secure, and stable manner, the energy management system (EMS) has been equipped with a tool called state estimator (SE). The main function of SE is to provide the real time operating condition of the power system by utilising the network topology and the information obtained from various measurement devices and monitoring systems [1]. These measurement devices are judiciously dispersed across the power system such that

2 the power system network is completely observable [2]. Complete power system observability determines the ability of SE to provide unique solution from the available number of measurements. Phasor measurement units (PMUs) are intelligent electronic devices, which provide the coherent picture of the real-time operation of power system by utilising the synchronised voltage and current phasor measurements. In wide-area interconnected system, these measurements are synchronised by utilising the time stamp provided by the Global Positioning System (GPS) [3]. The distinctive ability of PMUs to provide synchronised voltage and current phasor measurements makes it to be one of the most essential measuring devices for monitoring and control of power systems. Once the PMU is installed at a bus, it provides voltage phasor of the bus and current phasors of some or all the branches incident to the installed bus. Subsequently, by employing Kirchhoff’s laws, voltage phasors of the neighbouring buses can be computed. Thus, a PMU with sufficient number of channel capacity installed at a bus can not only make the installed bus observable but also the adjacent buses which are incident to it. As a result, it is not essential to install the PMUs at all the buses for complete power system observability. Moreover, it is not economical to place PMUs at every bus of the power system due to their high installation cost and not possible due to lack of communication infrastructure [4]. Consequently, a suitable method is required to determine the optimal placement and locations of PMUs. Therefore, the optimal PMU placement (OPP) problem is formulated as an optimisation problem by the researchers. The main objective of OPP problem is to minimise the number of PMUs required to be deployed in the power system while maintaining complete system observability [5]. In order to make an N-bus system observable, there exists a solution space of 2 N possible combinations. Therefore, this OPP problem is considered as a NP-hard combinatorial optimisation problem and has been an area of research for many engineers and researchers [1-15]. As yet, several approaches have been employed in the technical literature to solve this NP-hard combinatorial optimisation problem [5-13]. These methods can be broadly divided into two main classes of algorithms, viz. numerical and topological. The numerical observability method is based on the measurement gain or Jacobian matrix. If the gain matrix is of full rank, then the power system network is said to be numerically observable. This method is iterative in nature and requires huge matrix handling and is computationally expensive [7]. On the other hand, topological observability method is the commonly used technique which is based on the concept of spanning tree. Some of the important contributions concerning the state-of-the-art optimisation techniques in the area of OPP problem can be found in [8]. In [9], a dual search algorithm is developed to obtain the minimal number of PMUs. This dual search algorithm utilises the modified bisecting search algorithm and the simulated annealing method to obtain the

3 minimal number and minimal placement locations of PMUs respectively. However, this method suffers from high computational burden. Further, it has been reported in [10] that the dual search technique may not yield optimal solutions for larger test systems. An evolutionary technique based on genetic algorithm is formulated in [6] for OPP by considering zero injection buses (ZIB). However, this approach [6] is studied on small test systems when compared to those exist in practice. Further, the implication aspect such as execution time has In [11], Tabu search algorithm is suggested to find the optimum number of PMUs by reducing the search space in obtaining the optimal solution. The shortcoming of this methodology is that it may not result in global optimal solution that can make the system completely observable [7]. In [7], a meta-heuristic optimisation technique based on immunity genetic algorithm (IGA) is presented to overcome the computational burden in obtaining the optimum solution. Although, the IGA approach provides optimal solution with minimal execution time, single line outage and loss of PMU contingencies are not considered. The authors in [4] formulated OPP problem using binary particle swarm optimisation (BPSO) technique by simultaneously considering multiple objectives, viz. minimising the installation cost of PMUs and maximising the measurement redundancy. However, this approach is highly sensitive to control parameters and requires high tuning of parameters in obtaining the optimal solution. In [12], a modified binary particle swarm optimisation (MPSO) technique is employed by considering both PMU loss and single loss of branch conditions. Further, a new rule based on observability analysis of ZIB is suggested to obtain the minimum number of PMUs. However, it has been observed that the execution time in obtaining the optimal solution using MPSO is quite high. The authors in [13] presented a flexible approach based on cellular genetic algorithm (CGA) in finding the optimal number of PMUs by considering channel limitations. However, the effect of unequal PMU cost is not incorporated in the problem formulation. A new technique based on binary imperialistic competition algorithm (BICA) is employed for OPP in [5]. Although the BICA method considers various contingency conditions, yet channel limitation and unequal PMU cost are not considered in the objective function. In [3], the multi-objective OPP is suggested using cellular learning automata (CLA) by simultaneously determining the minimum number of PMUs as well as maximum measurement redundancy. In [10], a two-step approach is developed to minimize the size of the PMU location set. In the first step, the best configuration of PMUs for complete system observability is obtained. Thereafter, in the second step a novel meta-heuristic technique known as improved fruit fly (IFF) optimisation algorithm is utilised to determine the minimum number of PMUs. However, it is stated in [1, 7, 14, 15] that the time taken to provide the solution using meta-heuristic techniques is high and the solution thus obtained may not always be

4 optimum [1, 15] for higher order systems, i.e. IEEE 57-bus, IEEE 118-bus, etc. To overcome the shortcomings of the individual algorithms, several researchers are now paying attention to hybridise different optimisation techniques [16-20]. The Taguchi method (TM) is an optimisation technique proposed by Taguchi [20] based on the concept of statistical experimental design. It is used to obtain the robust solutions for evaluating and implementing enhancements in manufacturing design problems [20]. TM provides a systematic and efficient approach with reduced number of experiments to obtain optimal values of complex systems. Thus, this method has been applied to various optimisation problems in the electric power system [16-21]. The important tools used in the Taguchi method for searching the optimal solution are i) orthogonal arrays (OAs) and ii) signal-to-noise ratio (SNR). OA is used to analyse the decision variables concomitantly, on the other hand SNR is used to measure the quality of the solution [16]. The distinctive feature of the Taguchi method is that it has the ability to select optimal number of design experiments for testing from all the possible number of combinations [20]. Hence, TM requires less number of testing scenarios to provide optimum solution for NP-hard problems. Due to this reason, TM provides optimal solution with shorter computational time. Although, the Taguchi method is sufficient to deal with single-objective NP-hard problems, it can be hybridised with another optimisation method in order to obtain multiple solutions and further to enhance the initial population [19]. This paper proposes a novel hybrid Taguchi binary bat algorithm (TBBA) for OPP problem. Its silent feature is to incorporate the Taguchi method through generation of initial population for binary bat algorithm (BBA). The main aim of this hybrid technique is to obtain multiple optimal solutions in single experimental run and to improve the computational time by decreasing the feasible search locations (solution space). In real-time application many factors such as installation cost, communication infrastructure, zero-injection buses, etc., may affect the optimal number of PMUs. Consequently, a good PMU placement algorithm must not only find the minimum number of PMUs under normal conditions but also during the loss of PMUs, lack of communication infrastructure at substations, channel limitations of PMUs, unequal PMU cost, in the presence of conventional measurements, and by considering the effect of ZIBs so as to make the power system completely observable. The major contribution of this paper is threefold. i) To propose a novel Taguchi binary bat algorithm to solve OPP problem. ii) To obtain multiple optimal solutions in single experimental run. iii) To improve the computational speed and reliability in obtaining optimum number of PMUs under various constraints. The proposed TBBA has been implemented on IEEE 14-bus, IEEE 30-bus, IEEE 57-bus, IEEE 118-bus, and Polish 2383-bus test systems. The results thus obtained using the proposed technique is compared with

5 other evolutionary and conventional techniques reported in the literature. The obtained results show the efficiency of the proposed technique both in computational speed and accuracy. The paper is organised as follows. Section 2 briefly describes the optimal PMU placement problem formulation of each model. Taguchi method is introduced in Section 3. The proposed hybrid Taguchi binary bat algorithm approach is presented in Section 4. Section 5 briefly outlines the implementation of the proposed TBBA for OPP problem with flow chart. Simulation results are discussed in Section 6. Finally, conclusions are provided in Section 7.

2.

Problem formulation of the optimal PMU placement problem The main objective of the PMU placement problem is to minimise the number of PMUs required to make

the power system completely observable with minimal cost [7, 22]. For a N bus system, the optimal PMU problem is formulated as: min J ( x) = W t X x

s.t. f ( x)  1ˆ

(1)

where, J ( x) represents the objective function, f ( x) denotes the vector function, W signifies the installation cost of PMUs at different buses given by W  [w1 , w2 ,..., wN ]t , X indicates PMU location vector given by X  [ x1 , x2 ,...., xN ]t , xi denotes a binary variable, either 0 or 1, which represents the absent or present of a PMU

at bus i , and 1ˆ indicates unit vector. Installation costs of PMUs in (1) at all the buses are assumed to have equal cost. Hence, the installation cost vector W is a unit vector. The observability constraint f i for the i th bus is given by N

fi   Aij x j  1

(2)

j 1

where, A represents the connectivity matrix of the power system network and is defined by 1, if i  j or i is adjacent to j Aij   otherwise 0,

(3)

This constraint optimisation problem has been subsequently transferred to an unconstrained problem by adding the penalty factors in the objective function. As a result, the modified objective function of the OPP problem is defined according to (4) J ( x)  W t X   NUO

(4)

where, NUO indicates the number of unobservable buses associated with individual i and  represents the penalty factor.

6 2.1 Effect of PMU loss Optimal number of PMUs obtained using (1) and (2) guarantees that the power system is observable under normal operating condition. However, the constraint in (2) does not ensure complete system observability during the loss of single PMU or loss of any communication link [2]. Therefore, it is necessary to enhance the system reliability by installing the additional PMUs in order to protect the system against the loss of single PMU or any communication link [23-25]. These additional PMUs should be installed in such a way that each bus is observed by at-least two PMUs. This can be accomplished by multiplying the right hand side of the inequality given in (2) by 2. N

fi   Aij x j  2

(5)

j 1

2.2 Effect of unequal cost The number of current phasors that could be measured depends on the channel capacity of the PMUs. Moreover, as reported in the literature the cost of these PMUs varies from $30K to $40K depending upon the number of measuring channels [23]. Therefore, in order to determine the effect of unequal cost on the number and position of PMUs, the installation cost given in (1) has to be altered. In the present paper, the cost of a PMU that can measure one current phasor measurement in the incident branch connected to the installed bus is set to 1 p.u. Further, for each additional channel, the cost of PMU has been increased with a decimal fraction of 0.1 p.u. as suggested in [23]. 2.3 Effect of channel limitation In this subsection, the effect of channel limitation on the number and location of PMUs required to make the power system completely observable has been considered. As stated in Section 2.2, each PMU comes with channel limitation and the cost of each PMU mainly depends on the number of measurement channels. In practice, there are numerous types of PMUs that can be employed with different measurement channels. Consequently, in this work a more realistic OPP problem has been formulated according to the procedure given in [26] by considering the PMUs with limited channel capacity. Thus, the PMU installed at a bus can measure only the limited number of current phasors. Thus, a PMU can only make a limited number of buses observable among all the available adjacent buses [13, 27]. The number of ways to make this limited number of buses observable increases with increased number of branches connected to the installed bus. For a N bus system, the optimal PMU problem with channel limitation is formulated as: min J ( y ) = W t Y   NUO x

s. t. f (Y )  BT Y  1ˆ

(6)

7 where, J ( y ) represents the objective function, f (Y ) denotes the vector function, W signifies the installation cost of PMUs at different buses W  [w1 , w2 ,..., wN ]t , and Y indicates ( s 1) PMU location binary integer vector. Matrix B is formed according to connectivity matrix A according to (3) and number of rows in B for each bus is obtained according to (7).  n    if n  m rk   m  1 if n  m 

(7)

where, n indicates the number of buses adjacent to bus i , m denotes the number of measurement channels. For n  m the number of rows for bus i in matrix B is the number of m combinations of n that makes the adjacent buses observable. However, for n  m the number of available measurement channels can make all the adjacent buses observable. Hence, the corresponding row in connectivity matrix A is unaltered. 2.4 Effect of lack of communication infrastructure In wide area monitoring system, communication infrastructure (i.e. communication link) is required to transmit the data collected by the measurement devices to the control centre. The installation of these communication links and its upgrade are the two main factors affecting the installation cost of PMUs [28]. It is specified in [28] that owing to the absence of sufficient existing communication link and thereby up-gradation of the communication infrastructure increase the installation cost of PMUs by a factor of seven. Further, it is also stated in [28], the communications cost for installing additional PMUs is comparatively low once a communication network is installed. Hence, lack of communication infrastructure at substations should be considered to determine its effect on number and location of PMUs to make the system fully observable. In the present study, this is considered by imposing a higher cost at substations without communication infrastructure. Thus, the installation cost W should be modified in the objective function which is to be minimised according to (8). min J( x )  W t X   NUO x

(8)

Here, the lack of communication infrastructure at a particular bus i is indicated with high installation cost ( wi  27 104 ) and remaining buses are assigned with an installation cost of wi  1 as suggested in [28]. 2.5 Effect of flow measurements In this subsection, the OPP problem is discussed in the presence of few existing conventional flow measurements. This is because PMU technology is comparatively new and most of the existing power systems already consist of considerable number of traditional measurements to monitor the power system. Present work

8 also takes into account the situation where the power system consists of a few observable and unobservable islands [22, 25]. This situation may arise in practise due to the decision of replacing some of the aging or malfunctioning of existing conventional measurement units [22, 25]. In the present paper, a topological method proposed in [5, 22] has been considered for OPP problem in the presence of conventional measurements. With the existing conventional measurements, the power system can be made completely observable with fewer numbers of PMUs. This is because, the presence of power flow measurement on a branch can measure both active and reactive power flows of the corresponding branch say i  j . Hence, if the voltage phasor of bus i is known either directly or indirectly, the voltage phasor at other end of the branch, i.e. bus j can be calculated using the branch parameters. Thus, the constraints allied with the bus i and bus j (terminal buses of the measured branch) can be merged into one constraint [4]. The OPP problem in the presence of conventional flow measurements is to obtain the minimal number of PMUs to make the power system completely observable as a single island during normal operating conditions [25]. 2.6 Effect of zero injection buses In the present work, a method suggested in [29] has been employed for modelling the zero injection buses that exists in the power system network. A bus is said to be zero injection bus if the bus has neither a load nor a generator, i.e. net power injection at this bus is zero. These ZIBs can be used as pseudo measurements to make the power system network observable with less number of PMUs due to the following conditions: a)

If the ZIBs together with all its incident buses are observable except one, then the unobservable bus can also be made observable by applying Kirchhoff’s current law (KCL) at ZIB.

b) If all the adjacent buses to the ZIB are observable then the ZIB can be made observable by performing the nodal analysis at ZIB. c)

If there exist a group of adjacent ZIBs which are unobservable and all the buses adjacent to ZIBs are observable then these ZIBs can also be made observable by executing the nodal analysis at these buses.

As a result of above three conditions, the effect of ZIBs can be modelled as: N

k

j 1

l 1

fi   Aij x j   Ai , zl si , zl  1

(9)

where, si , zl represents the only unobservable bus among the ZIB along with its incident buses with value one. It is to be noted that similar procedure can be implemented for OPP problem by considering power injection measurements.

9

3.

Taguchi method The Taguchi method is an optimisation technique based on the concept of statistical experimental design.

This method has an ability to optimise quickly by changing the factors (variables) to obtain a desired outcome. The important tools used in the Taguchi method for searching the optimal solution are i) orthogonal arrays (OAs) and ii) signal-to-noise ratio (SNR). OA is used to analyse the decision variables concomitantly whereas SNR is used to measure the quality of the solution [16]. 3.1 Orthogonal array In exhaustive search (ES) technique, experiments are performed using all factorial combinations to achieve the optimal solution. In the Taguchi method, OA is utilised to obtain the desired outcome by performing a subset of experiments of all possible combinations of the values of all the factors (variables). An OA is a fractional factorial matrix that guarantees an equal comparison of levels of any variable or interaction of variables. This fractional matrix consists of numbers arranged in rows and columns. In this matrix each row indicates the level of the variable and each column indicates a particular variable that can be changed in each experimental run [30]. The experiment combinations are selected such that adequate information is provided to determine the effects of parameter values of each variable on the solution. Hence, this decreases the computational time compared to the ES technique in attaining the optimal solution [21]. This is explained with an example. Let us consider an optimisation problem that consists of seven factors, say, Ui ,i  1, 2,...,6,7 with each factor consist of two levels, say, A and B. In this example, the ES technique requires a total of 27  128 experiments to obtain the optimal combination of these factors. When there are F number of factors (variables) with Q number of levels for each factor, then a total number of Q F experiments (possible combinations) are required to be performed by ES technique in order to obtain optimal combinations of the factors [30]. Hence, it is very difficult to perform ES technique when these values are large. On the other hand, for the above example an orthogonal array requires only 32 experiments to obtain the optimal solution. An OA with F factors with each having Q levels is symbolised as LM  Q F  , where, L indicates a Latin square,

M represents the number of experiments. In general, M is much smaller than the total number of combinations, i.e. Q F [21]. For example, an orthogonal array with seven factors and two levels is symbolised as L32 (27 ). This L32 (27 ) orthogonal array is given in Appendix. The way the OAs are constructed and some of the

readily available OAs can be found in [31].

10 3.2 Signal-to-Noise ratio The second important tool in the Taguchi method is the Signal-to-Noise Ratio. The main function of SNR is to measure the quality of the solution and to find the suitable level for each variable that improves the quality of the solution. Depending on the types of characteristics, various types of SNRs are available, namely, i) continuous or discrete, ii) smaller-the-better (STB) or larger-the-better (LTB), and iii) nominal-is-best. For both STB and LTB characteristics, the SNR is calculated using

n

1 / J i 1

i

and

n

J i 1

i

respectively. Here, J i is the

fitness value of the i th level in an experiment run and n indicates the number of variables. The value with large SNR is the most favourable solution [18].

4.

Proposed Taguchi binary bat algorithm The binary bat algorithm (BBA) is a new heuristic optimisation technique based on the echolocation

behaviour of bats [32, 33]. This echolocation works on the principle of natural sonar which is used by the bats for navigating and hunting purposes. The three important characteristics of bats when detecting prey have been employed in designing the basic structure of bat algorithm [34, 35]. These characteristics are listed as follows. i) All the bats utilises the echolocation behaviour to sense the distance and to differentiate between the prey and the background barriers. ii) All the bats fly randomly with velocity vi at position xi . These bats emit pulses with a fixed frequency f min , varying wavelength  , and loudness A0 . The wavelength of the emitted pulses and the rate of pulse

emission r  0,1 are adjusted automatically depending upon the proximity of their target. iii) The loudness A0 is assumed to be varied from a large positive value to a minimum constant value Amin . In the present paper, a novel optimisation technique is proposed by applying Taguchi method to improve the initial population of BBA and to obtain the multiple optimal solutions. The application of proposed TBBA technique to OPP problem is discussed in the following subsections. 4.1 Generation of initial population Alike all other meta-heuristic techniques, BBA also begins by randomly generating an initial population

P with a certain number of individuals called position vectors n p with dimension size D .  x11  P x  np 1

x1D    xn p D 

(10)

11 In the present paper, number of buses represents the dimension size of the problem, i.e. for an N bus system the size of each position vector will be N . Therefore, the dimension size of P is n p  N . 4.2 Enhancement of initial population using Taguchi method After initialising, Taguchi method is performed on the initial population to exploit the area of potential solutions. Consequently, the positional vector of BBA has an optimal solution or a near optimal solution. Thus, the proposed TBBA method enhances the efficacy (in terms of convergence speed, reliability, and robustness) of the conventional BBA technique. After obtaining the initial population, Taguchi method is performed using two level OA and SNR. To perform two-level OA, for each positional vector another positional vector is randomly selected from the population. Let these two chromosomes be indicated as ulevel1 and ulevel 2 respectively [18, 36]. As stated previously, the number of buses represents the number of factors (variables) of OA. Hence, number of factors for an N bus system is N .

ulevel1  u1 ,...,uN 

(11)

ulevel 2  u1 ,...,uN 

(12)

In order to execute various set of experiments a two-level OA with N factors has been selected to obtain the best level or near best level for each factor. This can be better explained with an example. Consider a seven bus system [4] to explain the application of Taguchi method to OPP problem. For which, a two-level OA (L32) with seven factors has been selected which is given in Table A.1. Now, let us consider the following two positional vectors are selected randomly from the initial population, each consisting of seven factors which correspond to ui (i =1, 2,..., 6, 7) for performing the various matrix experiments of an OA. The fitness of these positional vectors has been found to be 4 and 3 respectively.

ulevel1  1, 1, 0, 0, 1, 1, 0

(13)

ulevel 2  0, 1, 0, 0, 1, 0, 1

(14)

Subsequently, Table 1 is developed by allocating the level 1 values with the values given in (13) and the level 2 values with the values given in (14), i.e. A and B in Table A.1 are replaced with the corresponding value of ulevel1 and ulevel 2 for each factor respectively. The values of the experiment #4 in Table A.1, i.e. [A A A B B B B] are replaced with the values [1 1 0 0 1 0 1] respectively for each factor shown in Table 1. Then, for each experiment in Table 1, the fitness is calculated according to (4) and SNR is calculated as the inverse of the objective function value [18, 36]. In experiment #4 the objective function value and SNR is found to be 4 and 0.2500 respectively and shown

12 in Table 1. Once all the experiments are performed, the effect of each factor f on the objective function for each level

l is computed in the following way:

S fl   SNRilf exp eriment i i

where, l signifies the level number, f represents the factor name, i denotes the experiment number, and S f 1 and S f 2 indicate the appropriateness of level-1 or level-2 for a factor respectively. For example, in order to

select suitable level for the factor u1 then the corresponding S11 and S12 are calculated as follows: S11  [0.2500  0.2000  0.3333  0.2500  0.2500  0.3333  0.2000  0.2500  0.2500  ... ...0.3333  0.2000  0.2500  0.2500  0.2000  0.3333  0.2500]  4.1333 S12  [0.3333  0.5000  0.2500  0.3333  0.3333  0.2500  0.5000  0.3333  0.3333  ... ...0.2500  0.5000  0.3333  0.3333  0.5000  0.2500  0.3333]  5.6667

For any factor, if S f 1 is larger than S f 2 , then level-1 is selected as the optimal level for that particular factor and vice-versa. For example, the optimal level for factor u1 is level-2, i.e. ‘B’ because S11  S12 . Similarly the optimal level for all the factors are computed and shown in Table 1. The obtained suitable level for each factor based on maximum SNR is [B A A A A B A]. Then corresponding best positional vector is [0 1 0 0 1 0 0]. This positional vector with objective function value 2 is the optimal solution for the corresponding positional vector. Here, it can be observed that the objective function value after performing the Taguchi method is better than the objective function values of original positional vectors. Thus, the Taguchi method is an efficient approach in obtaining the best or near best solution by executing less number of experiments. This process is repeated for each positional vector. The resultant population known as Taguchi population will act as initial population for binary bat algorithm [18, 36].

[Table 1 Here] 4.3 Evaluation The fitness of each positional vector is calculated using the objective function given in (4) on the current population. The current best solution is stored and updated [32, 33]. 4.4 Parameter updating As pointed in previous subsection, the frequency and velocity parameters of BBA are iteratively updated as Fi  Fmin  ( Fmax  Fmin )

(15)

Vi (t  1)  Vi (t )  Fi ( X i (t )  Gbest )

(16)

where, Fi and Vi (t  1) indicate the frequency and the velocity of bat i respectively, Fmin and Fmax denote the

13 minimum and maximum range of frequency with values 0 and 2 respectively,  signifies the uniform distributed random variable generated within in the interval [0, 1], X i symbolises the position vector of bat i , and Gbest indicates the best solution obtained so far that gives the optimal fitness value [32, 33]. 4.5 Generation of candidate solution In the BA, each artificial bat updates its position so as to find its prey/food by adding the velocities to its positions. However, in BBA updating the position means flipping the positions from “0” to “1” or vice-versa based on the velocity of the positional vectors (bats). Hence, the authors in [32] have suggested a v-shaped transfer function to change the position of the bats based on the probability of their velocity. This v-shaped transfer function offers a high probability of changing the position for a large absolute velocity and vice-versa [33]. This is mathematically formulated as

V (vik (t ))  2 / .tan 1 ( / 2.vik (t )) k 1 k  ( xi (t )) if rand  V (vi (t  1)) xik (t  1)   k rand  V (vik (t  1))   xi (t )

(17) (18)

where, xik (t ) and vik (t ) indicate the position and velocity of i th particle at iteration t in k th dimension, ( xik (t ))1 denotes the complement of xik (t ) , and rand signifies a uniform distributed random variable

generated within the interval [0, 1]. 4.6 Population updating After updating the position of each bat, the fitness of each position vector is evaluated. Now, the fitness of each position vector is compared with the fitness of position vector previously generated [32, 33]. The most fitted position vectors will act as population for next generation based on the loudness of the bat. This is represented mathematically as:   x ,if fiti  fiti and rand  A xi   i   xi , otherwise

(19)

where, A is loudness of the bat. This control variable varies from 0 to 1 and controls the acceptability of the improved position vectors. This process is iteratively repeated until the number of iterations reaches the maximum number of generations.

5.

Implementation of the proposed TBBA for optimal placement of PMUs The complete flow of the proposed TBBA technique is given in Fig. 1. The various steps in the procedure

are enumerated below.

[Figure 1 Here]

14 Step 1: Initialise the control parameters of bat algorithm such as initial population, pulse rate, and loudness. Step 2: Select suitable two-level orthogonal array and SNR ratio for enhancing the initial population using Taguchi optimisation method. Step 3: Select two position vectors from the initial population and calculate the fitness for various set of experiments in orthogonal array. Step 4: Perform SNR ratio on objective function value obtained using OA. Step 5: Now the effect of each factor on objective function for each level is calculated and a new position vector is obtained. Step 6: Repeat steps 3 to 5 is repeated until number of positional vectors in Taguchi population is equal to the population size. Step 7: In this step, frequency is adjusted and the velocity of each individual in the population is updated. Step 8: Calculate the transfer function value and update the position of each individual. Step 9: Select global best solution among the best solutions if the random number generated is greater than the pulse rate. Further, obtain the local solution around the global best solution. Step 10: Generate the new solution by flying randomly. Step 11: Evaluate fitness of each individual. Step 12: In this step, new solutions are accepted. Pulse rate and loudness are updated if the random number generated is less than the loudness and fitness of new solutions is less than the global best solution. Step 13: Ranking of individual solutions based upon their fitness value is carried out in this step and the global best solution is selected accordingly. Step 14: Steps 7 to 13 are repeated until number of iterations reaches maximum value.

6.

Results and Discussion The proposed TBBA technique determines the optimal number of strategic PMU bus locations for

complete observability of the power system [2]. The proposed method has been applied on five different standard systems, viz. IEEE 14-bus, IEEE 30-bus, IEEE 57-bus, IEEE 118-bus, and Polish 2383-bus systems [37]. In the present work, for all the test systems the initial population, pulse rate, and loudness control parameters values are initialised as 60, 0.025, and 8 respectively. The performance of the proposed algorithm has been compared with the other techniques proposed in the literature. The comparison is performed with respect to the computational time and optimal number of PMUs required for complete observability. To validate the efficiency and accuracy of the proposed method, six different scenarios have been considered to solve the

15 OPP problem. Test Case 1. Normal operating condition In this case study, observability analysis has been performed under normal operating condition. The optimal number of PMUs and corresponding placement locations are obtained using the proposed method for different test systems. Results thus obtained are given in Table 2. From Table 2 it can be observed that the present method provides multiple placement locations with the same number of PMUs. Optimal placement location can be selected based upon the applications such as state estimation accuracy and monitoring of tielines. Table 3 provides the comparative analysis of the test results obtained using the proposed technique with other evolutionary and conventional methods suggested in [1, 2, 4, 22, 24, 25, 38]. It can be observed that all the methods provide the same minimal number of PMUs for complete observability. However, in comparison the proposed technique provides multiple solutions in a single experimental run. For instance, in case of IEEE 14bus system the proposed technique provides same number of PMUs, i.e. 4 in comparison with the methods proposed in [1], [2], [25], and [38] so as to obtain complete system observability. On the other hand, the main advantage of the proposed method compared to [1], [2], [25], and [38] is that the proposed approach provides multiple solutions with same number of PMUs, however, with different placement locations, i.e. [2, 6, 7, 9], [2, 6, 8, 9], and [2, 7, 10, 13]. This shows the higher reliability of the proposed technique.

[Table 2 Here] [Table 3 Here] Test Case 2. Loss of single PMU In this case study, the effect of loss of single PMU has been investigated. The loss of PMU measurements not only makes the bus unobservable at which PMU is installed, but also the neighbouring buses (the buses that are indirectly observable) unobservable. For example, consider a PMU placement location set {2, 7, 10, 13} of IEEE 14-bus system under normal operating condition given in Table 2. This PMU placement location set can measure 12 current phasor measurements in the branches connected to these buses. From this placement location set, let us assume an unexpected failure of a PMU has occurred on bus 7. The loss of PMU at bus 7 not only makes bus 7 to be unobservable, but also its adjacent buses {4, 8, 9} unobservable. However, in the present case only buses 7 and 8 are unobservable. This is because the buses 4 and 9 are observed using PMUs 2 and 10 respectively. In order to make these two buses, i.e. buses 7 and 8 to be observable an additional PMU is required to be placed at bus 8. Hence, in order to make the system completely observable under loss of any single PMU additional PMUs are needed to be placed in the power system. For the test systems under consideration

16 placement locations are given in Table 4. From Table 4, it can be seen that similar to case 1 the proposed technique provides multiple solutions with same number of PMUs. Table 5 shows the comparative results obtained by other techniques suggested in [22, 24, 25]. It can be observed that the proposed method provides similar or better results in comparison with the other techniques. For example, in case of IEEE 118-bus system the proposed technique along with [24] and [25] require 68 PMUs to make complete system observability as compared to 72 in case of ILP technique suggested in [22].

[Table 4 Here] [Table 5 Here] Test Case 3. Unequal cost of PMUs In this case study, the effect of unequal costs of PMUs on optimal location and number has been considered. As stated in Section 2, for each additional channel the cost of each PMU is increased by 0.1. Results for different test systems considering the effect of unequal PMU cost are given in Table 6. The comparative results for IEEE 14-bus, IEEE 30-bus, and IEEE 57-bus systems under equal and unequal cost of PMUs are shown in Table 7. From Table 7 it can be seen that the optimal number of PMUs for both equal and unequal PMU costs is identical, however, the placement locations are different. It can also be observed from Table 7 column 6 that due to the difference in number of measuring channels overall installation cost of PMUs is affected (decreased) by considering unequal costs. For example, by considering unequal cost of PMUs the overall installation cost in case of IEEE 57-bus system results in 20.1 p.u., in contrast the installation cost of PMUs by considering equal cost results in 20.5 p.u. This shows the effect of unequal cost of PMUs on optimal placement locations.

[Table 6 Here] [Table 7 Here] Test Case 4. Lack of communication infrastructure Optimal placement locations and minimal number of PMUs required for power system observability by considering the effect of lack of communication infrastructure at buses (substations) is shown in Table 8. There are four columns in Table 8. Second column of the table indicates the buses with lack of communication facility for different test systems under consideration. The optimal number of PMUs and their locations are given in columns 3 and 4 of Table 8 respectively. The results thus obtained by applying the proposed TBBA method illustrate that the optimal number of PMUs required to make the power system completely observable, is greater

17 or equal to the optimal number of PMUs achieved by ignoring the effect of lack of communication infrastructure [39]. In order to validate the efficacy of the proposed method the results thus obtained are compared with the results provided in [39]. Table 9 shows the comparative results of the proposed and SQP method. It can be observed from Table 8 and from [39] that both the proposed and sequential quadratic programming (SQP) methods provide multiple placement locations with same number of PMUs. However, the main advantage of the proposed method compared to SQP method is that the proposed method being the population based technique provides multiple placement locations in single experimental run while the SQP requires multiple experimental runs with different initial points. This shows the effectiveness of the proposed method compared to SQP method.

[Table 8 Here] [Table 9 Here] Test Case 5. Inclusion of conventional measurements The presence of traditional power flow measurements is considered in the power system network in this case. Table 10 presents the results obtained using the proposed technique for different test systems. Table 10 consists of four columns. Second column of the table indicates the placement locations of conventional measurements that already exist in the power system network. These conventional measurements are randomly distributed such that there exist some unobservable islands. Hence, the proposed method is executed to ensure complete system observability under normal operating condition. The optimal number of PMUs and their locations thus obtained using the proposed method are given in columns 3 and 4 of Table 10 respectively. It can be seen from Table 10 column 3 that in the presence of conventional power flow measurements the optimal number of PMUs required to make complete system observability decreases in comparison with the absence of conventional measurements.

[Table 10 Here] Test Case 6. Channel limitation In this case study, the OPP problem has been solved by considering the effect of channel limitation. Test case is solved by assuming predefined number of channel capacity of PMUs, i.e. 1, 2, 3, and 4. Given a PMU with the channel capacity one, the PMU installed at a bus can only measure one voltage phasor and one current phasor measurements in any one of the branches connected to the installed bus. For example, a PMU installed at bus 3 of IEEE 14-bus system can measure voltage phasor at bus 3 and current phasor either in branch {3-2} or in branch {3-4}. Thus, a PMU with channel capacity of one can only make two buses observable, i.e. either

18 buses 3 and 2 are observable or buses 3 and 4 are observable in this case. Hence, under single channel capacity condition the IEEE 14-bus, IEEE 30-bus, and IEEE 57-bus systems require 7, 15, and 29 PMUs in order to make the system completely observable. The results obtained for the IEEE 14-bus, IEEE 30-bus, and IEEE 57bus systems are shown in Table 11. It can be observed from Table 11 that as the channel capacity of PMUs increases the optimal number of PMUs required to be installed in the power system network decreases. Furthermore, the results thus obtained by the proposed technique have been compared with the results obtained in [26-27]. It can be observed from Table 12 that all the methods provide same minimal number of PMUs for complete system observability.

[Table 11 Here] [Table 12 Here] Test Case 7. Effect of zero injection buses Optimal placement locations and minimal number of PMUs required for power system observability by considering the effect of zero injection buses has been tabulated in Table 13. There are three columns in Table 13. Second and third column of Table 13 indicates the optimal number of PMUs and their placement locations respectively. The results thus obtained by applying the proposed TBBA method illustrate that the optimal number of PMUs required to make the power system completely observable, is found to be less than the optimal number of PMUs obtained by ignoring the effect of ZIBs. Further, it can be observed from Table 13 that the proposed method provides multiple placement locations with same number of PMUs. In order to validate the efficacy of the proposed method the results thus obtained are compared with the results provided in the literature in Table 14. It can be seen from Table 14 column 4 that in case of IEEE 118 bus system the proposed method requires only 28 PMUs to make the system completely observable while other methods [3, 4, 7, 9, 23, 24, 39] require 29 PMUs. This shows the effectiveness of the proposed method in obtaining the optimal number of PMUs.

[Table 13 Here] [Table 14 Here] Further, computational time taken by the proposed method in obtaining the optimal solution under normal condition is given in Table 15 for different systems under consideration. From Table 15, it can be seen that the computational time taken to provide minimal number of PMUs by the proposed method (shown in bold in the table) is minimal in comparison with other evolutionary and conventional techniques [2, 7, 13, 40, 41, 42, 43]. For instance, in case of IEEE 57-bus system the proposed technique gives a computation time of 1.3103s as

19 compared to 15.462s, 10.705s, 30.41s, 345.04s, 128.388s, and 21.402s in case of [7], [13], [40], [41], [42], and [43] respectively. This shows the computational superiority of the proposed method compared to other techniques in obtaining the optimal number of PMUs.

[Table 15 Here] 7.

Conclusions In this paper, a new hybrid Taguchi binary bat algorithm has been proposed for optimal placement of

PMUs by ensuring complete observability of the power system. The proposed approach consists of two steps. In the first step, systematic reasoning capability of the Taguchi method has been utilised to enhance the initial population by providing optimal or near optimal solution. In the second step, binary bat algorithm has been employed to obtain multiple optimal solutions by utilising the enhanced population provided by the Taguchi method. The efficiency of the proposed TBBA approach has been implemented on different IEEE standard and Polish 2383-bus test systems. The performance of the proposed technique has been validated under different conditions, namely, normal condition, loss of single PMU, effect of unequal PMU cost, lack of communication infrastructure, presence of conventional power flow measurements, effect of channel limitation, and effect of zero injection buses. Comparison of results obtained using the proposed method has been performed with conventional and evolutionary techniques. Unlike conventional techniques, the proposed method works with set of solutions. This distinctive feature of the proposed approach alleviates the necessity of multiple executions of OPP with different starting points to obtain multiple global optimal solutions with same minimal number of PMUs. Another potential advantage of the proposed TBBA method is that it provides these multiple global optimal solutions with lower computational time in comparison with other evolutionary and conventional approaches. Therefore, the proposed TBBA approach can be utilised in various applications such as state estimation, identification of fault location, etc. to obtain minimum number of PMUs under various conditions. Further, investigations in this area can be accomplished by considering the effect of loss of multiple PMUs. This shall be the future work of the authors.

8.

Appendix A An orthogonal array with seven factors and each factor consisting of 2 levels is symbolised as L32 (27 ).

This L32 (27 ) orthogonal array is given in Table A.1.

[Table A.1 Here]

20

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21 [18] P. Subbaraj, R. Rengaraj, S. Salivahanan, Enhancement of self-adaptive real-coded genetic algorithm using Taguchi method for economic dispatch problem, Applied Soft Computing 11 (2011) 83–92. [19] H. M. Hasanien, Design optimization of PID controller in automatic voltage regulator system using Taguchi combined genetic algorithm method, IEEE Systems Journal 7, (2013) 825-831. [20] H. Yu, C. Y. Chung, K. P. Wong, Robust transmission network expansion planning method with Taguchi’s orthogonal array testing, IEEE Transactions on Power Systems 26 (2011) 1573-1580. [21] D. Liu, Y. Cai, Taguchi method for solving the economic dispatch problem with nonsmooth cost functions, IEEE Transactions on Power Systems 20 (2005) 2006-2014. [22] Optimal Placement of Phasor Measurement http://www.pserc.org, assessed May 2014.

Units

for

State

Estimation

Power

Systems,

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23

List of Tables: Table 1. Generation of better positional vector from two positional vectors using the Taguchi method. Table 2. Optimal placement of PMUs under normal condition Table 3. Comparison of optimal number of PMUs of the proposed method with other methods under normal condition Table 4. Optimal placement of PMUs for single PMU loss condition Table 5. Requirement of optimal number of PMUs in the proposed method as compared to the other methods for single PMU loss condition Table 6. Optimal placement of PMUs considering unequal cost Table 7. Comparison of optimal number and locations of PMUs of the proposed method under equal and unequal costs of PMUs Table 8. Optimal placement of PMUs considering lack of communication infrastructure Table 9. Comparison of optimal number of PMUs of the proposed method with other method for lack of communication infrastructure condition Table 10. Optimal placement of PMUs considering the effect of power flow measurements Table 11. Optimal placement of PMUs considering the channel limitation condition Table 12. Comparison of optimal number of PMUs of proposed method with other methods considering the channel limitation condition Table 13. Optimal placement of PMUs considering the effect of ZIBs Table 14. Comparison of optimal number and locations of PMUs by considering the effect of ZIBs Table 15. Comparison of computational time(s) for obtaining optimal number of PMUs using the proposed method with other methods under normal condition Table A.1. Orthogonal array L32(27)

24

List of Figures: Figure 1. Flow chart of optimal PMU placement problem using TBBA technique.

25 Table 1. Generation of better positional vector from two positional vectors using the Taguchi method. Factors U1

U2

U3

U4

U5

U6

U7

Objective Function Value

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1

0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0

4 5 3 4 4 3 5 4 4 3 5 4 4 5 3 4 3 2 4 3 3 4 2 3 3 4 2 3 3 2 4 3

Sf1

4.1333

4.9000

4.9000

4.9000

4.9000

4.1333

5.6667

Sf2

5.6667

4.9000

4.9000

4.9000

4.9000

5.6667

4.1333

Optimal Contribution Values

0

1

0

0

1

0

0

Exp No

Signal-to-Noise Ratio 0.2500 0.2000 0.3333 0.2500 0.2500 0.3333 0.2000 0.2500 0.2500 0.3333 0.2000 0.2500 0.2500 0.2000 0.3333 0.2500 0.3333 0.5000 0.2500 0.3333 0.3333 0.2500 0.5000 0.3333 0.3333 0.2500 0.5000 0.3333 0.3333 0.5000 0.2500 0.3333

2

Table 2. Optimal placement of PMUs under normal condition System

Optimal number of PMUs

IEEE 14-bus

4

2, 6, 7, 9 2, 6, 8, 9 2, 7, 10, 13

IEEE 30-bus

10

3, 6, 7, 10, 11, 12, 15, 20, 25, 27 2, 3 ,6, 9, 10, 12, 18, 23, 26, 30 1, 5, 10, 11, 12, 18, 24, 26, 27, 28 1, 6, 7, 10, 11, 12, 18, 23, 25, 29

IEEE 57-bus

17

1,5,9,15,19,22,26,29,30,32,36,38,41,47,51,53,57 1,6,12,15,19,22,25,27,32,36,41,44,47,50,52,55,57 1,4,6,13,20,22,25,27,29,32,36,39,41,45,47,51,54 1,4,9,20,23,25,26,29,32,36,39,41,45,46,48,50,54

IEEE 118-bus

32

1,6,10,11,12,17,21,23,28,30,34,37,42,45,49,53,56,62,64,71,75,77,80,85,87,90,94,101,105,110,114,116 1,5,10,12,15,17,21,23,28,30,36,40,44,46,50,51,54,62,64,71,75,77,80,85,87,91,94,101,105,110,114,116

Optimal locations of PMUs

Polish 746 2383-bus * Optimal PMU locations of Polish 2383-bus system is not provided due to large number

26 Table 3. Comparison of optimal number of PMUs of the proposed method with other methods under normal condition Method

IEEE 14-bus

IEEE 30-bus

IEEE 57-bus

IEEE 118-bus

[1]

4

10

-

-

[2]

4

10

17

32

[25]

4

10

17

32

[38]

4

10

17

32

Proposed

4

10

17

32

Table 4. Optimal placement of PMUs for single PMU loss condition System

Optimal number of PMUs

IEEE 14-bus

IEEE 30-bus

IEEE 57-bus

IEEE 118 bus

Optimal locations of PMUs

9

2, 4, 5, 6, 7, 8, 9, 11, 13 1, 2, 4, 6, 7, 8, 9, 10, 13 2, 3, 5, 6, 7, 8, 9, 11, 13 1, 2, 4, 6, 7, 8, 9, 11, 13

21

2,3,4,5,6,9,10,11,12,13,15,17,18,19,21,24,25,26,27,28,30 1,3,5,6,7,8,9,10,11,12,13,15,16,18,19,21,24,25,26,29,30 1,2,4,6,7,9,10,11,12,13,15,16,18,19,21,24,25,26,28,29,30 1,2,3,6,7,8,9,10,11,12,13,15,16,18,19,21,23,25,26,27,30

33

1,2,4,6,9,12,15,19,20,22,24,25,27,28,29,31,32,33,35,36,37,38,39,41,43,45,46,47,50,51,53,54,56 1,3,4,6,9,12,15,19,20,22,24,25,26,28,29,31,32,33,35,36,37,38,41,43,44,46,47,50,51,53,54,56,57 1,3,4,6,9,11,12,15,19,20,22,24,25,26,28,29,30,32,33,35,36,37,38,41,45,46,47,50,51,53,54,56,57 1,2,4,6,9,12,15,19,20,22,24,26,28,29,30,31,32,33,35,36,38,39,41,43,45,46,47,50,51,53,54,56,57

68

2,3,5,6,9,10,11,12,15,17,19,21,22,26,27,28,29,30,32,34,35,37,40,41,44,45,46,49,52,53,56,57,58 ,59,62,63,65,67,68,70,71,72,73,75,77,78,80,83,85,86,87,89,91,92,94,96,100,101,105,106,108,1 10,111,112,114,116,117,118 2,3,5,6,9,10,11,12,15,17,19,21,22,25,27,28,30,31,32,34,35,37,40,42,43,45,46,49,50,51,52,54,56 ,59,61,64,66,67,68,70,71,72,73,75,76,77,78,80,84,85,86,87,89,91,92,94,96,100,101,105,106,10 8,110,111,112,115,116,117 1,2,5,7,9,10,11,12,15,17,19,21,22,25,26,27,28,29,32,34,36,37,40,42,44,45,46,49,50,51,52,54,56 ,59,62,63,65,67,68,70,71,72,73,75,76,77,78,80,83,85,86,87,89,91,92,94,96,100,101,105,106,10 8,110,111,112,115,116,117

Polish 23831681 bus * Optimal PMU locations of Polish 2383-bus system is not provided due to large number

Table 5. Requirement of optimal number of PMUs in the proposed method as compared to the other methods for single PMU loss condition Method [22]

IEEE 14-bus 9

IEEE 30-bus 22

IEEE 57-bus 35

IEEE 118-bus 72

[24]

9

-

33

68

[25]

9

21

33

68

Proposed

9

21

33

68

Table 6. Optimal placement of PMUs considering unequal cost

IEEE 14-bus

Optimal number of PMUs 4

IEEE 30-bus

10

3, 5, 8, 10, 11, 12, 18, 23, 26, 30

IEEE 57-bus

17

1,4,7,9,13,20,23,27,30,32,36,39,41,44,47,51,53

IEEE 118 bus

32

1,5,9,12,15,17,20,23,28,30,36,40,43,46,50,51,54,62,64,68,71,75,77,80,85,86,90,94,102,105,110,114

System

Optimal locations of PMUs 2, 8, 10, 13

27

Table 7. Comparison of optimal number and locations of PMUs of the proposed method under equal and unequal costs of PMUs

IEEE 14-bus

Equal Cost Unequal Cost

Optimal number of PMUs 4 4

IEEE 30-bus

Equal Cost Unequal Cost

10 10

3, 6, 7, 10, 11, 12, 15, 20, 25, 27 3, 5, 8, 10, 11, 12, 18, 23, 26, 30

36 25

12.6 11.5

IEEE 57-bus

Equal Cost Unequal Cost

17 17

1,5,9,15,19,22,26,29,30,32,36,38,41,47,51,53,57 1,4,7,9,13,20,23,27,30,32,36,39,41,44,47,51,53

52 48

20.5 20.1

Equal Cost

32

117

40.5

Unequal Cost

32

114

40.2

System

IEEE 118 bus

Optimal location of PMUs 2, 6, 7, 9 2, 8, 10, 13

1,6,10,11,12,17,21,23,28,30,34,37,42,45,49,53,56,62,64, 71,75,77,80,85,87,90,94,101,105,110,114,116 1,5,9,12,15,17,20,23,28,30,36,40,43,46,50,51,54,62,64,6 8,71,75,77,80,85,86,90,94,102,105,110,114

Number of measuring channels 15 10

Installation cost (p.u.) 5.1 4.6

Table 8. Optimal placement of PMUs considering lack of communication infrastructure System IEEE 14bus

IEEE 30bus

IEEE 57bus

IEEE 118 bus

Lack of CI bus

Optimal number of PMUs

2, 9

10, 14, 30

1, 9, 25, 38, 54

3, 7, 9, 11, 12, 17

Optimal locations of PMUs

5

1, 3, 8, 10, 13 4, 5, 8, 11, 13 1, 4, 7, 10, 13 4, 5, 7, 11, 13

11

3,5,11,12,17,19,21,23,25,28,29 3,5,9,12,15,16,20,22,25,28,29 2,3,6,11,12,15,17,20,22,26,29 2,4,6,11,12,16,19,22,23,26,29

18

2,6,10,12,19,22,24,28,31,32,36,39,41,45,46,49,52,55 2,6,12,15,19,22,24,28,31,32,36,39,41,44,47,50,52,55 2,6,12,15,19,22,26,29,30,32,36,39,41,44,47,50,53,55 2,6,12,19,22,26,27,30,32,36,41,45,46,48,50,52,55,57

35

2,5,6,10,15,16,19,22,26,27,31,32,34,37,42,45,49,52,56,62,63,68,70,71,76 ,79,84,86,89,92,96,100,105,110,117 1,5,6,10,15,16,19,22,23,26,27,28,32,34,37,41,45,49,53,56,62,64,68,71,75 ,77,80,85,86,91,94,102,105,110,117 1,5,6,10,15,16,18,21,27,29,30,32,34,36,40,45,49,52,56,62,63,68,70,71,76 ,79,85,87,89,92,96,100,105,110,117 1,4,6,10,15,16,19,22,27,30,31,32,36,40,43,46,50,51,54,62,64,68,70,71,78 ,84,86,89,92,96,100,105,110,117,118

6, 16, 90, 195, 568, 978, 1017, 1564, 1756, 1939, 752 2167, 2296, 2374, 2380 * Optimal PMU locations of Polish 2383-bus system is not provided due to large number Polish 2383-bus

Table 9. Comparison of optimal number of PMUs of the proposed method with other method for lack of communication infrastructure condition Method [39]

IEEE 14-bus 5

IEEE 30-bus -

IEEE 57-bus -

IEEE 118-bus 35

Proposed

5

11

18

35

28

Table 10. Optimal placement of PMUs considering the effect of power flow measurements System

Optimal number of PMUs 2

Flow Measurements

Optimal locations of PMUs 4, 6

IEEE 14-bus

1-2, 7-8, 9-10, 9-14

IEEE 30-bus

1-3, 6-7, 6-8, 9-11, 12-13, 16-17, 18-19, 24-25, 25-26

4

2, 10, 15, 27

IEEE 57-bus

6-7, 15-45, 22-38, 24-25, 30-31, 34-35, 36-40, 37-38, 3844, 39-57, 41-43, 50-51, 53-54.

11

1, 4, 9, 20, 24, 27, 32, 46, 49, 52, 56

IEEE 118-bus

1-3, 5-6, 11-13, 16-17, 20-21, 22-23, 23-25, 27-28, 2931, 34-43, 35-36, 41-42, 44-45, 46-48, 50-57, 51-52, 5354, 56-58, 60-62, 65-66, 66-67, 68-81, 71-73, 75-118, 76-77, 77-82, 78-79, 86-87, 90-91, 95-96, 100-101, 114115.

18

9, 10, 11, 19, 26, 32, 37, 49, 59, 61, 70, 71, 80, 85, 92, 105, 110, 116

Table 11. Optimal placement of PMUs considering the channel limitation condition Channel limit 1

IEEE 14-bus 7

IEEE 30-bus 15

IEEE 57-bus 29

2

5

11

19

3

4

10

17

4

4

10

17

Table 12. Comparison of optimal number of PMUs of proposed method with other methods considering the channel limitation condition Channel limit 1 2 3 4

IEEE 14-bus Proposed [26] Method 7 7 5 5 4 4 4 4

[27] 7 5 4 4

IEEE 30-bus Proposed [26] [27] Method 15 15 15 11 11 11 10 10 10 10 10 10

IEEE 57-bus Proposed [26] [27] Method 29 29 29 19 19 19 17 17 17 17 17 17

Table 13. Optimal placement of PMUs considering the effect of ZIBs System IEEE 14-bus

Optimal number of PMUs 3

Optimal locations of PMUs 2, 6, 9

IEEE 30-bus

7

1, 2, 10, 12, 15, 20, 27 2, 4, 10, 12, 18, 24, 29 2, 4, 10, 12, 19, 23, 27 3, 7, 10, 12, 18, 24, 29

IEEE 118bus

28

3, 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 1, 8, 11, 12, 17, 21, 25, 28, 34, 37, 41, 45, 49, 53, 56, 62, 72, 75, 77, 80, 86, 87, 91, 94,102,105, 110, 114 2, 10, 11, 12, 17, 21, 25, 28, 34, 36, 40, 45, 49, 52, 56, 62, 72, 75, 77, 80, 85, 86, 90, 94,101,105, 110, 114 2, 6, 9, 12, 15, 17, 21, 25, 29, 34, 40, 45, 49, 53, 56, 62, 72, 75, 77, 80, 85, 87, 91, 94,101,105, 110, 114

Polish 2383553 bus *Optimal PMU locations of Polish 2383-bus system is not provided due to large number

29 Table 14. Comparison of optimal number and locations of PMUs by considering the effect of ZIBs Method Proposed

IEEE 14-bus 3

IEEE 30-bus 7

IEEE 118-bus 28

[3]

3

7

29

[4]

3

7

29

[7]

3

7

29

[9]

3

-

29

[22]

3

8

28

[23]

-

7

29

[24]

3

-

29

[39]

3

-

29

Table 15. Comparison of computational time(s) for obtaining optimal number of PMUs using the proposed method with other methods under normal condition Technique

IEEE 14-bus

IEEE 30-bus

IEEE 57-bus

IEEE 118-bus

[2]

1.3s

2.6 hr

-

-

[7]

0.444s

1.483s

15.462s

75.219s

[12]

1min

6 min

43 min

85 min

[13]

0.162s

1.121s

10.705s

53.122s

[40]

0.05s

-

30.41s

1625.30s

[41]

0.298s

18.43s

345.04s

-

[42]

1.114s

11.366s

128.388s

572.610s

[43]

0.205s

2.205s

21.402s

187.144s

Proposed

0.032s

0.0706s

1.3103s

20.5334s

Table A.1. Orthogonal array L32(27) Factors Exp No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

U1

U2

U3

U4

U5

U6

U7

A A A A A A A A A A A A A A A A B B B B B B B B B B B

A A A A A A A A B B B B B B B B A A A A A A A A B B B

A A A A B B B B A A A A B B B B A A A A B B B B A A A

A A B B A A B B A A B B A A B B A A B B A A B B A A B

A B A B A B A B A B A B A B A B A B A B A B A B A B A

A A B B B B A A B B A A A A B B B B A A A A B B A A B

A B A B B A B A B A B A A B A B B A B A A B A B A B A

Objective Function Value

Signal-to-Noise Ratio

J1

1/J1

J2 J3 J4 J5 J6 J7 J8 J9 J10 J11 J12 J13 J14 J15 J16 J17 J18 J19 J20 J21 J22 J23 J24 J25 J26 J27

1/J2 1/J3 1/J4 1/J5 1/J6 1/J7 1/J8 1/J9 1/J10 1/J11 1/J12 1/J13 1/J14 1/J15 1/J16 1/J17 1/J18 1/J19 1/J20 1/J21 1/J22 1/J23 1/J24 1/J25 1/J26 1/J27

30 28 29 30 31 32

B B B B B

B B B B B

A B B B B

B A A B B

B A B A B

B B B A A

B B A B A

Sf1

V1(A)

V2(A)

V3(A)

V4(A)

V5(A)

V6(A)

V7(A)

Sf2

V1(B)

V2(B)

V3(B)

V4(B)

V5(B)

V6(B)

V7(B)

Optimal Contribution Values

B

A

A

B

A

B

A

J28 J29 J30 J31 J32

J

Figure 1. Flow chart of optimal PMU placement problem using TBBA technique.

1/J28 1/J29 1/J30 1/J31 1/J32

Highlights



To find the optimal placement locations of phasor measurement units in power system network.



A novel Taguchi binary bat algorithm is proposed to solve optimal phasor measurement unit placement problem.



The proposed method provides multiple optimal placement locations (solutions) in single experimental run.



Results validate the computational efficacy and reliability in obtaining optimum number of PMUs under various constraints.