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Minimizing the number of PMUs and their optimal placement in power systems
Electric Power Systems Research 83 (2012) 66–72
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Minimizing the number of PMUs and their optimal placement in power systems S. Mehdi Mahaei ∗ , M. Tarafdar Hagh Department of Electrical Engineering, Ahar Branch, Islamic Azad University, Ahar, Iran
a r t i c l e
i n f o
Article history: Received 1 March 2011 Received in revised form 2 September 2011 Accepted 13 September 2011 Available online 21 October 2011 Keywords: PMU Number minimizing Optimal placement State estimation
a b s t r a c t This paper presents a new method for minimizing the number of PMUs and their optimal placement in power systems. The proposed method provides suitable constraints for power systems with two adjacent injection measurements (IMs). In addition, suitable constraints for considering the connection of two buses to each other and to an injection bus are proposed. The proposed constraints result in a reduction in the number of PMUs even though the system topological observability is complete. Existing conventional measurements are also considered. First, the number of PMUs is minimized in such a way that the system topological observability is complete. Then the optimal placement is done to maximize the measurements redundancy. The resulting phased to be installed in multiple stages. The optimal number of PMUs that ensure system topological observability under failure of a PMU or a line is also simulated. Simulations are performed on IEEE 30, 57 and 118 bus test systems by binary integer programming. The results show that the number of PMUs is equal to or less than the corresponding results of recently published papers, while the system topological observability is complete, and measurement redundancy is increased. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Current energy management systems (EMSs) require accurate monitoring of power system state variables, such as the voltage phasors at all buses. On the other hand, power systems may operate in contingency conditions, which require real-time monitoring for analysis and protection purposes. Phasor measurement units (PMUs) provide time-synchronized (real-time) phasor measurements in power systems [1]. This is available with the Global Positioning System (GPS) [2]. In addition, the PMUs are accurate devices for state estimation. Using PMUs results in linear state-estimation equations, and this makes them easier to solve than general nonlinear-state estimation equations. However, PMUs are expensive devices. Therefore, a suitable methodology is necessary to minimize the number of PMUs while maintaining complete observability of the power system (In this paper, observability means topological observability). A power system is considered completely observable when all of its states can be uniquely determined [3,4]. After establishing complete system observability, it is necessary to determine the optimal places of the PMUs to maximize measurement redundancy. A power system has measurement redundancy when its buses are observed by more than one PMU or the
∗ Corresponding author. Tel.: +98 9143075824/4115292986; fax: +98 4115261624. E-mail addresses: [email protected] (S.M. Mahaei), [email protected] (M.T. Hagh). 0378-7796/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2011.09.010
number of observable buses is maximized. In other words, some of the PMUs can be removed from the measurement system while all of the buses remain observable. System topology affects the minimize number of PMUs. For example, zero injection measurements (Zero IMs) reduce the number of PMUs. On the other hand, the conventional state estimators cannot provide a real-time picture of the power system, but they can be subjected to increased measurement redundancy. Therefore, it is necessary to consider zero injection. However, consideration of conventional measurements is optional in the optimal determination of the number and location of PMUs. Therefore, an effective PMU placement algorithm has to consider the following additional items: 1. Keeping complete system observability. 2. Maximizing measurement redundancy. 3. Modeling of zero injection buses to consider the topology conditions. These items result in a more complex optimization problem compared with some related published articles in this field. Refs. [5–8] have determined only the minimum number of PMUs required for complete observability. This approach may increase the risk of not having complete observability of the system in the contingency lines. The number and places of PMUs is determined without considering zero IMs in [9]. In [10], conventional measurements are considered, but zero IMs are not considered, which increases the number of PMUs. In [11,12], zero IMs are considered, but use of the islanding method leads to complex relations. Also
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in [13], the immunity genetic algorithm is used, but the system is not completely observable. Ref. [14] presents a simple method, but using this method in large systems may increase the number of PMUs because of the complex relations created by zero IMs buses. In this paper, a method is presented to implement suitable constraints in power systems having zero injection and/or conventional measurements. In some cases, this method reduces the number of PMUs to less than the corresponding results of published papers while maintaining complete system observability [13,15–21]. In addition, optimal placement is performed after minimizing the number of PMUs such that it increases measurement redundancy. Then the placements of the PMUs are phased for install in multiple stages. The number of PMUs that ensure system observability under PMU or line outages is also simulated. IEEE 30, 57 and 118 test bus systems are used to prove the simulation results.
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Fig. 1. Six-bus test system.
where x1 –x6 are elements of the x matrix that are generated by binary integer programming. Therefore, Eq. (7) represents the minimum number of PMUs. The bus observability constraints are defined as follows with Eq. (2): Bus 1 : x1 + x2 + x4 ≥ 1
(8)
2. Objective function to minimize the number of PMUs
Bus 2 : x1 + x2 + x3 + x5 ≥ 1
(9)
To define an objective function to minimize the number of PMUs, the following simple rules are considered:
Bus 3 : x2 + x3 + x6 ≥ 1
(10)
Bus 4 : x1 + x4 + x5 ≥ 1
(11)
Bus 5 : x2 + x4 + x5 + x6 ≥ 1
(12)
Bus 6 : x3 + x5 + x6 ≥ 1
(13)
1. A PMU placed at a specific bus can measure the voltage phasor of that bus and all the currents leaving it. 2. If the voltage phasor and current phasor at one end of a branch are known, the voltage phasor at the other end of the branch can be calculated using line impedance. 3. If voltage phasors at both ends of a branch are known, the branch current can be calculated using line impedance. 4. Each bus must be observed at least once by PMUs.
The unequal constraints (8)–(13) represent the number of observable buses for buses 1–6, respectively. For example, if at least one PMU is installed on bus 1, 2 or 4, constraint (8) is confirmed, and bus 1 will be observed. 3. Objective function for optimal placement of PMUs
Therefore, the objective function can be defined as follows [9–12,14,15]:
n
min
wk x
(1)
k=1
y = Ax ≥ b
(2)
where w, x, A and b are the weight (or cost [19]) of the PMUs, state matrix, topology and observability matrixes, respectively. In addition, n and k are the number of buses and the kth row of the w matrix, respectively. The w, x, A and b matrixes are defined as follows:
Wi,j =
xi =
0<≤1
1 if PMU installed in bus i 0 otherwise
Ai,j =
i=j 0 otherwise
1 i=j 1 if ith bus is connected to bus j 0 otherwise
b = [111 . . . 11]T
(3) max
n
(w−1 Ax)
(14)
k=1
(4)
y = Ax ≥ b
(15)
n
(5)
(6)
where w is a diagonal matrix whose iith element represents the weight (or cost) associated with placement of a PMU at the ith bus, x is a column matrix whose ith element represents the installation of PMUs in the ith bus, A is an n × n matrix whose ijth element represents the connection of the ith bus to the jth bus and b is a column matrix whose ith element represents the number of ith bus observables. For example, with Eq. (1), minimizing the number of PMUs for a 6-bus test system (Fig. 1) by assuming that w is a unit matrix can be formulated as follows: min{x1 + x2 + x3 + x4 + x5 + x6 }
Unequal constraint (2) is used for complete observability. Minimizing the number of PMUs may lead to similar sets of PMUs in which the number of PMUs is equal in each of them and system observability is complete. For example, in Fig. 1, installation of two PMUs in {1, 6}, {3, 4} and {2, 5} leads to complete observability. The question is, which of these sets is the most suitable for the installation of PMUs? The proposed method in this paper is as follows: after minimizing the number of PMUs by satisfying Eq. (1), placement is performed by maximizing observability. A suitable set is selected for the installation of PMUs subject to unequal constraint (2). The following function is used:
(7)
xk = num
(16)
k=1
where (w−1 A)k is the kth row of the (w−1 A) matrix and num is the minimum number of PMUs that is determined by minimization of Eq. (1). Therefore, for the six-bus test system shown in Fig. 1, set {2, 5} is more suitable because buses 2 and 5 are observed twice by PMUs. 4. Considering the existing measurements when minimizing the number and placement of PMUs Active, reactive and voltage measurements are used in power systems. The constraints from Eq. (2) have to be modified when these conventional measurements are considered. Two methods are used for considering both zero injection and conventional measurements:
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this paper proposes modification of constraint (18) to Eqs. (20) and (21), as follows: y1 + y2 + y3 + y4 ≥ 3
(20)
y5 ≥ 1
(21)
In fact, in this method, constraint (18) is considered for the bus that has IMs with more lines. Obviously, constraint (2) or (15) is usable for an adjacent bus. Considering the following cases is necessary. (a) Bus 4 can be eliminated from the network when placing and determining the number of PMU procedures if the number of connected lines to bus 4 is two such that its elimination may reduce the optimal number of PMUs because in this case, bus 4 will be observable if buses 3 and 5 are observed. (b) Inequality (21) should be eliminated and y5 added to inequality (20) if bus 5 is radial. (c) If two IMs are connected to each other by another bus that has two connected lines, as in Fig. 2e, constraint (18) is considered for the mentioned IMs buses, and their sum is considered a single constraint instead two constraints. 4.5. Connection of two buses to each other and an IM In some power systems, it is not possible to use constraint (18). In other words, the use of mentioned equation leads to system unobservablity. For example, in Fig. 2f, there is one IM on bus 5. Therefore, constraint (18) for buses 3, 4, 5, 6, 8 and 10 and constraint (2) for buses 1, 2, 7, 9, 11 and 12 will be as follows: Fig. 2. Conventional measurements in power systems.
(a) The islanding method [11,12]. (b) The topology-based method [14]. In this paper, a simple method is proposed as follows. 4.1. Power flow measurement By considering the power flow measurement on a line between buses i and j (Fig. 2a), both buses will be observable. Therefore, Eq. (17) is: y1 + y2 ≥ 1
(17)
In other words, the ith and jth rows of the A matrix are removed, and the above equation is replaced [14]. 4.2. Injection measurement If an IM exists on a bus, similar to Fig. 2b, then one of the constraints of Eq. (7) has to be changed [14]. Of course, the zero IM can be assumed as a pseudomeasurement. In this case, we have: y1 + y2 + y3 + y4 ≥ 3
(18)
4.3. Combining power flow and IMs
Buses 3, 4, 5, 6, 8 and 10 : y3 + y4 + y5 + y6 + y8 + y10 ≥ 5
(22)
Bus 1 : x1 + x2 + x3 ≥ 1
(23)
Bus 2 : x1 + x2 + x11 ≥ 1
(24)
Bus 7 : x6 + x7 + x11 ≥ 1
(25)
Bus 9 : x8 + x9 ≥ 1
(26)
Bus 11 : x2 + x3 + x7 + x12 ≥ 1
(27)
Bus 12 : x11 + x12 ≥ 1
(28)
Installation of three PMUs in buses 3, 8 and 11 satisfies constraints (22)–(28), but buses 4 and 6 are unobservable. In these cases, this paper proposes the following procedure: (a) Determine all adjacent buses of the IM bus. (b) Select those adjacent buses that are connected directly and indirectly to the same IM bus. “Indirectly” means that the selected bus should be connected to the IM bus via another adjacent IM bus. For example, in Fig. 2e, bus 4 has the aforementioned feature because it is connected directly to bus 5 (IM bus) and via bus 10 to bus 5, as well. (c) Use constraint (2) for the selected bus, in addition to using constraint (18) for the IM bus. The aforementioned procedure is used for the power system shown in Fig. 2e, and this results in the following constraints:
Fig. 2c shows the part of a power system that includes power flow and an IM. In this case, we have [14]:
Buses 3, 5, 6, 8 and 10 : y3 + y5 + y6 + y8 + y10 ≥ 4
(29)
y1 + y4 ≥ 1
Bus 1 : x1 + x2 + x3 ≥ 1
(30)
Bus 2 : x1 + x2 + x11 ≥ 1
(31)
Bus 4 : x4 + x5 + x10 ≥ 1
(32)
Bus 7 : x6 + x7 + x11 ≥ 1
(33)
Bus 9 : x8 + x9 ≥ 1
(34)
(19)
4.4. Two adjacent IMs Some large power systems may have several IMs in adjacent buses, such as in Fig. 2d. In this case, using unequal constraint (18) for each IM may increase the optimal number of PMUs. In this case,
S.M. Mahaei, M.T. Hagh / Electric Power Systems Research 83 (2012) 66–72
Bus 11 : x2 + x3 + x7 + x10 + x11 + x12 ≥ 1
(35)
Bus 12 : x11 + x12 ≥ 1
(36)
By considering the constraints (29)–(36), four PMUs are required for complete observation of the system shown in Fig. 2e. 4.6. Voltage measurement Installation of a voltage measurement on the ith bus results in its observation. Therefore, the iith element of A matrix will be zero. Therefore, before minimizing the number of PMUs, the constraints are first written the same as Eqs. (2) and (17)–(36). Then the corresponding elements of the A matrix are modified by considering the existing voltage measurements. In addition, before placement of PMUs, first, the constraints areplacements of PMUs, first, the constraints are written the same as Eqs. (14), (15) and (17)–(36); then the corresponding elements of the A matrix are modified by considering the existing voltage measurements. 5. Outage of a PMU or a line To ensure system observability under failure of a PMU, the righthand side of all of the constraints is multiplied by 2. If two PMUs are observing a bus, then a related line outage will not affect the node observability. Hence, the problem of ascertaining observability under a single line outage is a subset of the problem of the single PMU loss considered above [15]. 6. Phasing of PMUs Most utilities may not be able to install all the PMUs in one stage. Therefore, it is necessary for installation of the PMUs to be phased in multiple stages. It is assumed that the number of PMUs has resulted from optimizing objective function (1) for a system and that the places of them were obtained by optimizing objective function (14). At this point, the num PMUs should be inserted in specific places in t phases. In each stage, the objective function can be the same function (14) with different constraints, as follows: max
n
(w−1 Ax)
(37)
k=1 n
xk = numt
(38)
k=1
/S xk = 0|k ∈
k = 1, 2, . . . , n
(39)
xk = 1|k ∈ / S
k = 1, 2, . . . , n
(40)
numt , S and S are defined as follows. numt is the number of PMUs that are installed in the tth stage. S is the numbers of buses in which PMUs should be installed. In other words, these are the places of the PMUs that ensure complete system observability. S is the numbers of buses in which PMUs have been installed before the tth stage. Utilities may use other objectives for the multistage PMU placement, such as maximizing tie line observability or placing them at sparse locations [22,23]. In these cases, the objective should be optimized, subject to relative constraints. 7. Case study The case study was conducted in the following stages. In all of the stages, zero IMs are considered.
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1) Minimizing the number of PMUs with objective function (1). 2) Placement of PMUs by maximizing the measurements redundancies and objective function (14). 3) Comparing results with other references. 4) Considering conventional measurements and comparing with recently published papers. 5) Phasing of PMUs under normal conditions. 6) Outage of a PMU or a line and comparing it with references. Therefore, the IEEE 30, 57 and 118 bus systems are selected. These systems have 6, 15 and 10 zero IMs, respectively, which are: IEEE 30 bus system {6, 9, 22, 25, 27 and 28}. IEEE 57 bus system {4, 7, 11, 21, 22, 24, 26, 34, 36, 37, 39, 40, 45, 46 and 48}. IEEE 118 bus system{5, 9, 30, 37, 38, 63, 64, 68, 71 and 81}. 7.1. Constraints for the IEEE 30 bus system In the IEEE 30 bus system, buses 6 and 27 are adjacent with other buses that have small lines. Buses 11 and 26 are radially connected to IM buses. Therefore, constraint (20) for buses 6 and 27 and constraint (18) for bus 22 are considered as follows: y2 + y4 + y6 + y7 + y8 + y9 + y10 + y11 + y28 ≥ 7
(41)
y25 + y26 + y27 + y28 + y29 + y30 ≥ 5
(42)
y10 + y21 + y24 ≥ 3
(43)
Buses 29 and 30 have conditions, such as bus 4 in Fig. 2f. Therefore, the constraints are presented as follows for other buses that are not considered in unequal constraints (41)–(43): yi ≥ 1
(44)
7.2. Constraints for the IEEE 57 bus system In the IEEE 57 bus system, buses 21, 26, 34, 39, 40, 45 and 46 are IM buses with two connected lines. Therefore, these buses are eliminated from the network during the optimization procedures described in part (a), Section 4.4. Buses 36 and 37 are adjacent and have the same connected lines, so constraint (20) is used only for one of them. Buses 22 and 24 are two IM buses that are connected to each other by bus 23, which has two connected lines according to part (c), Section 4.4, so constraint (18) is used for buses 22 and 24; then the sides of the mentioned constraints are summed. Moreover, constraint (18) is used for buses 4, 7, 11 and 48. y3 + y4 + y6 + y18 ≥ 3
(45)
y6 + y7 + y8 + y29 ≥ 3
(46)
y9 + y11 + y13 + y41 + y43 ≥ 4
(47)
y20 + y22 + y23 + y24 + y25 + y27 + y38 ≥ 6
(48)
y35 + y36 + y37 + y56 ≥ 3
(49)
y38 + y47 + y48 + y49 ≥ 3
(50)
Bus 5 has characteristics of bus 4, according to bus 4 in Fig. 2f, so constraints are presented as constraint (44) for other buses that are not considered in unequal constraints (45)–(50). 7.3. Constraints for the IEEE 118 bus system In the IEEE 118 bus system, buses 9, 63 and 81 have the same characteristics as bus 4, which is mentioned in Section 4.5. Bus 38 is adjacent to other buses that have the most lines. Therefore, constraint (20) for buses 30 and 37 and constraint (18) for buses 5, 64, 68 and 71 are considered.
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Table 1 RESULTS of minimizing the number of PMUs with the proposed method considering zero IMs in IEEE 30, 57 and 118 bus systems. System
Objective function
No. of PMUs
Places of PMUs
Measurement redundancies
IEEE 30 bus IEEE 57 bus IEEE 118 bus
Without maximizing observation
7 11 28
1 5 10 12 15 19 27 1 4 13 20 25 29 32 38 51 54 56 3 8 11 12 17 21 27 31 32 34 40 45 49 53 56 62 65 72 75 77 80 85 86 90 94 101 105 110
6 4 38
Table 2 Results of placement of PMUs with the proposed method considering zero IMs in IEEE 30, 57 and 118 bus systems. System
Objective function
No. of PMUs
Places of PMUs
Measurement redundancies
IEEE 30 bus IEEE 57 bus IEEE 118 bus
With maximizing observation
7 11 28
2 4 10 12 15 20 27 1 4 13 20 25 29 32 38 51 54 56 2 9 11 12 17 21 27 31 32 34 40 45 49 52 56 62 65 72 75 77 80 85 87 90 94 101 105 110
11 4 41
Buses 4 and 39 that are adjacent with buses 5 and 37 have the same conditions as bus 4 in Fig. 2f. Therefore, the constraints are as follows. Of course, elimination of buses 9, 63 and 81 does not affect the number of PMUs in this system, unlike the IEEE 57 bus system.
7.5. Placement of PMUs by maximizing the measurement redundancies and objective function (14) For the maximizing observation, the objective function (14) is optimized, subject to the number of PMUs as an equality constraint, such as in Eq. (16). These results are given in Table 2. Table 2 shows that observation of the IEEE 30, 57 and 118 bus systems are complete, and observations are maximized by 7, 11 and 28 PMUs, respectively. These resulted were obtained by changing the allocation of PMUs using the proposed method compared with Table 1.
y3 + y8 + y11 + y5 + y6 ≥ 4
(51)
y8 + y9 + y10 ≥ 2
(52)
y8 + y17 + y26 + y38 + y30 ≥ 4
(53)
y33 + y35 + y34 + y38 + y40 ≥ 5
(54)
y64 + y63 + y61 ≥ 3
(55)
y81 + y116 + y68 + y65 + y69 ≥ 4
(56)
7.6. Comparing results with other references
y71 + y70 + y72 ≥ 3
(57)
Table 3 shows that in [7,15,16], the optimal number of PMUs is 7 for the IEEE 30 bus test system, but the places of the PMUs were not presented. In [13,17,18], the optimal number of PMUs is also 7, and their places ensures complete system observability, but the measurement redundancies for these places is less than the measurements redundancies of the resulting places in this paper. Table 4 shows that in [15,16,19–21], the optimal number of PMUs is more than 11 for the IEEE 57 bus test system. In [7,24], the number of PMUs for the same test system is reduced to 11, but their optimal places are not reported. In [17], the number of PMUs is
Therefore, the constraints are presented as constraint (44) for other buses that are not considered in unequal constraints (51)–(57). 7.4. Minimizing the number of PMUs by objective function (1) Appling constraints on objective function (1) leads to the results shown in Table 1. The determined PMU places result in a completely observable power system.
Table 3 Reported results of previously published papers by considering zero IMs in the IEEE 30 bus system. References
No. of PMUs
[7] [15] [16] [13] [17] [18]
7 7 7 7 7 7
Places of PMUs
Observation
Measurement redundancies
Without Placement
Unknown Unknown Unknown Observable Observable Observable
– – – 2 7 4
1 5 10 12 18 24 30 2 3 10 12 18 24 27 1 2 10 12 18 24 29
Table 4 Reported results of previously published papers by considering zero IMs in the IEEE 57 bus system. References [19] [15] [20] [16] [21] [24] [7] [13] [17] [18]
No. of PMUs 14 13 13 12 12 11 11 11 11 11
Places of PMUs
Observation
Measurement redundancies
The places are not reported because the number of PMUs is high.
It is not important because the number of PMUs is high.
Without Placement Without Placement 1 6 13 19 25 29 32 38 51 54 56 1 5 13 19 25 29 32 38 41 51 54 1 6 13 19 25 29 32 38 51 54 56
Unknown Unknown Observable Observable Observable
– – – – – – – 4 2 4
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Table 5 Reported results of previously published papers by considering zero IMs in the IEEE 118 bus system. Reference
No. of PMUs
[19] [25] [26] [16] [21] [22] [13]
29 29 29 29 29 28 28
[17]
28
[18]
28
Places of PMUs
Observation
Measurement redundancies
The places are not reported because the number of PMUs is high (29).
It is not important because the number of PMUs is high.
Without Placement 2 8 11 12 17 21 25 28 34 35 40 45 49 53 56 62 72 75 77 80 85 86 90 94 102 105 110 114 2 8 11 12 17 21 25 2833 34 40 45 49 52 56 62 72 75 77 80 85 86 90 94 101 105 110 114 3 8 11 12 17 21 25 28 34 35 40 45 49 53 56 62 72 75 77 80 85 86 90 94 102 105 110 114
Unknown Unobservable (buses 63 and 64)
– – – – – –
Unobservable (buses 63 and 64) Unobservable (buses 63 and 64)
Table 6 Conventional measurements for the IEEE 30, 57 and 118 bus systems. IEEE 30 bus Ms.
IEEE 57 bus Ms.
Power flow
Injection
15–23 12–14 19–18 6–8 – –
15 – – – – –
Power flow 41–43 34–35 42–41 40–56 – –
IEEE 118 bus Ms. [21] Injection 56 57 – – – –
Power flow
Injection
1–2 4–5 20–21 21–22 17–113 86–87
– – – – – –
Table 7 Places of PMUs in the presence of conventional measurements for IEEE 30, 57 and 118 bus systems. IEEE 30 bus
IEEE 57 bus
IEEE 118 bus
1 5 10 12 18 27 Nub. = 6 Ms. redundancies = 8
1 4 13 20 25 29 32 38 51 54 56 Nub. = 11 Ms. redundancies = 11
8 12 15 19 23 29 34 40 45 49 53 56 62 65 72 75 77 80 85 90 94 101 105 110 115 Nub. = 25 Ms. redundancies = 28
11, but the number of measurement redundancies for these places is 2, whereas the number of measurement redundancies under the resulting places in this paper is 4. Refs. [13,18] have results similar to this paper for the IEEE 57 bus system. Table 5 shows that in [16,19,21,25,26], the optimal number of PMUs is 29 for the IEEE 118 bus test system. In [24], the number of PMUs for the same test system is reduced to 28, but their optimal places were not reported. In [13,17,18], the number of PMUs is 28, but the resulting PMU places did not result in complete observation of the system. In the aforementioned paper, buses 63, 64 and 65 are not observable. By using the proposed method in this paper, the system is observable, and the number of PMUs is 28. The measurement redundancies for the resulting places for the same system in this paper are 41.
Table 7 shows that the number of PMUs reduces to 6 and 25 for the IEEE 30 and 118 bus systems, respectively, whereas the number of PMUs in [21] with the same conventional PMUs was 26. With the presence of conventional measurements, the number of PMUs for the IEEE 57 bus system is 11, but the measurement redundancies increase from 4 to 11.
7.8. Phasing of PMUs in normal conditions Phasing of PMUs is performed in three stages, according to Section 6. The number of PMUs is selected as optional according to Table 8. The results are presented in Table 8.
7.9. Outage of a PMU or a line compared with references 7.7. Considering conventional measurements and comparing with recently published papers Conventional measurements for IEEE 30, 57 and 118 bus systems are considered in Table 6. The results of the simulation containing them are presented in Table 7.
To ensure system observability under failure of a PMU or a line, the right-hand side of constraints (41)–(57) is multiplied by 2, and objective functions (1) and (14) are optimized, respectively, for the IEEE 30, 57 and 118 bus systems. The results are presented in Table 9.
Table 8 Phasing of PMUs with the proposed method considering zero IMs in the IEEE 30, 57 and 118 bus systems. System
Phase I
Phase II
Phase III
IEEE 30 bus
Nub. = 2 10 27
Nub. = 3 2 4 12
Nub. = 2 15 20
IEEE 57 bus
Nub. = 4 1 4 13 38
Nub. = 4 25 29 32 56
Nub. = 3 20 32 51 54
IEEE 118 bus
Nub. = 9 12 17 49 56 65 75 77 80 85
Nub. = 10 9 11 31 34 40 45 62 94 105 110
Nub. = 9 2 21 27 32 52 72 87 90 101
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Table 9 Outage of a PMU or a line considering zero IMs in the IEEE 30, 57 and 118 bus systems. System
Proposed method
[17]
[18]
[19]
IEEE 30 bus
Nub. = 15 1 2 3 5 10 12 13 15 17 19 20 23 25 27 30
Nub. = 15 2 3 4 8 10 12 13 15 16 18 20 22 24 27 30
– –
IEEE 57 bus
Nub. = 25 1 3 4 6 9 10 12 13 15 19 20 25 27 29 30 32 33 37 38 41 49 51 53 54 56 Nub. = 61 3 5 7 9 11 12 15 17 19 21 22 23 27 29 31 32 34 35 37 40 42 44 45 46 49 50 51 53 54 56 59 62 65 66 70 71 75 76 77 78 80 83 85 86 87 89 90 92 94 96 100 102 105 106 109 110 111 112 115 117
Nub. = 22 1 2 4 9 12 15 18 19 25 28 29 30 32 33 38 41 47 50 51 53 54 56 Nub. = 62 1 3 7 8 10 11 12 15 17 19 21 22 24 25 27 28 29 32 34 35 40 41 44 45 46 49 50 51 52 54 56 59 62 66 68 72 73 74 75 76 77 78 80 83 85 86 87 89 90 92 94 96 100 101 105 107 109 110 111 112 115 117
Nub. = 17 1 3 5 7 10 11 12 13 15 16 19 20 23 24 26 27 30 Nub. = 26 1 2 4 6 9 12 14 19 20 24 25 27 29 30 32 33 36 38 41 44 46 50 51 53 54 56 Nub. = 65 1 3 5 7 8 10 11 12 15 17 19 21 22 24 25 27 28 29 32 34 35 37 40 41 44 45 46 49 50 51 52 54 56 59 62 66 68 72 73 74 75 76 77 78 80 83 85 86 87 89 90 92 94 96 100 101 105 107 109 110 111 112 115 116 117
IEEE 118 bus
Table 9 shows that the number of PMUs under failure of a PMU or a line in the IEEE 30, 57 and 118 bus systems is less than the resulted reported in [17–19], except the number of PMUs for the IEEE 57 bus system in [18], which is 22. However, under this number of PMUs, the system is not ensured complete observability. For example, an outage of line 4–5 causes bus 5 to be unobservable. 8. Conclusion In large power systems, the buses that have injection measurements may be connected to each other. It is necessary to consider suitable unequal constraints for these buses; otherwise, the optimal number of PMUs may be increased. In addition, if two buses are connected to a bus that has injection measurements and also to each other, the unequal constraints should be changed to make the system completely observable. In this paper, a method is proposed for solving the aforementioned problems. A case study on IEEE 30, 57 and 118 bus systems resulted in a complete observable power system and an optimal number of PMUs that was equal or less than all published papers, whereas the measurement redundancies were higher. Consideration of conventional measurements and outages of a PMU or a line is also discussed. References [1] G. Phadke, Synchronized phasor measurements in power systems, IEEE Comput. Appl. Power 6 (April (2)) (1993) 10–15. [2] A.G. Phadke, J.S. Thorp, Synchronized Phasor Measurements and Their Applications, Springer Science Business Media, LLC, 2008. [3] Monticelli, State Estimation in Electric Power Systems. A Generalized Approach, vol. 88, Kluwer, Norwell, MA, 1999, pp. 262–282. [4] Abur, A.G. Exposito, Power System State Estimation: Theory and Implementation, Marcel Dekker, New York, 2004. [5] A.B. Antonio, J.R.A. Torreao, M.B. Do Coutto Filho, Meter placement for power system state estimation using simulated annealing, Proc. IEEE Porto Power Technol. Conf. 3 (September) (2001) 10–13. [6] H. Mori, Y. Sone, Tabu search based meter placement for topological observability in power system state estimation, Proc. IEEE Transm. Distrib. Conf. 1 (April) (1999) 172–177. [7] R.F. Nuqui, A.G. Phadke, Phasor measurement unit placement techniques for complete and incomplete observability, IEEE Trans. Power Deliv. 20 (October (4)) (2005) 2381–2388. [8] G.B. Denegri, et al., A security oriented approach to PMU positioning for advanced monitoring of a transmission grid, Proc. Power Syst. Technol., Power Conf. 2 (2002) 798–803.
Nub. = 29 – Nub. = 64 –
[9] S. Chakrabarti, E. Kyriakides, D.G. Eliades, Placement of synchronized measurements for power system observability, IEEE Trans. Power Deliv. 24 (January (1)) (2009). [10] M. Hurtgen, P. Praks, J.-C. Maun, Zajac, Measurement placement algorithms for state estimation when the number of phasor measurements by each PMU is limited, in: 43rd International Universities Power Engineering Conference, UPEC, 1–4 September, 2008. [11] Jian Chen, Ali Abur, Improved bad data processing via strategic placement of PMUs, Power Eng. Soc. Gen. Meet., IEEE 1 (June) (2005) 509–513, 12–16. [12] Jian. Chen, A. Abur, Placement of PMUs to enable bad data detection in state estimation, IEEE Trans. Power Syst. 21 (November (4)) (2006) 1608–1615. [13] F. Aminifar, C. Lucas, A. Khodaei, M. Fotuhi, Firuzabad, Optimal placement of phasor measurement units using immunity genetic algorithm, IEEE Trans. Power Deliv. 24 (July (3)) (2009) 1014–1020. [14] Bei Gou, Optimal placement of PMUs by integer linear programming, IEEE Trans. Power Syst. 23 (August (3)) (2008) 1525–1526. [15] K.S. Cho, J.R. Shin, S.H. Hyun, Optimal placement of phasor measurement units with GPS receiver, in: Proc. IEEE Power Engineering Society Winter Meeting, vol. 1, January–February, 2001, pp. 258–262. [16] F.J. Marın, F. Garcıa-Lagos, G. Joya, F. Sandoval, Genetic algorithms for optimal placement of phasor measurement units in electric networks, Electron. Lett. 39 (September (19)) (2003) 1403–1405. [17] Mahdi Hajian, Ali Mohammad Ranjbar, Turaj Amraee, Babak Mozafari, Optimal placement of PMUs to maintain network observability using a modified BPSO algorithm, Electron. Power Syst. Res. 33 (2011) 28–34. [18] F. Aminifar, A. Khodaei, M. Fotuhi-Firuzabad, M. Shahidehpour, Contingencyconstrained PMU placement in power networks, IEEE Trans. Power Syst. 25 (February (1)) (2010) 516–522. [19] D. Dua, S. Dambhare, R.K. Gajbhiye, S.A. Soman, Optimal multistage scheduling of PMU placement: an ILP approach, IEEE Trans. Power Deliv. 23 (October (4)) (2008) 1812–1820. [20] J. Peng, Y. Sun, H. Wang, Optimal PMU, placement for full network observability using tabu search algorithm, Int. J. Electron. Power Energy Syst. 28 (4) (2006) 223–231. [21] B. Xu, A. Abur, Observability analysis and measurement placement for system with PMUs, Proc. IEEE Power Syst. Conf. Expo. 2 (October) (2004) 943–946. [22] R. Sodhi, S.C. Srivastava, S.N. Singh, Multi-criteria decision-making approach for multistage optimal placement of phasor measurement units, IET Gen. Transm. Disturb. 5 (2) (2011) 181–190. [23] Wang Bo, Quanyuan Jiang, Yijia Cao, Transmission network fault location using sparse PMU measurements, in: International Conference on Sustainable Power Generation and Supply, Nanjing, China, 6–7 April, 2009. [24] Korkali, Mert, Ali Abur, Placement of PMUs with Channel Limits, in: Power and Energy Society General Meeting, PES’ 09, 26–30 July, 2009, pp. 1–4. [25] T.L. Baldwin, L. Mili, M.B. Boisen Jr., R. Adapa, Power system observability with minimal phasor measurement placement, IEEE Trans. Power Syst. 8 (May (2)) (1993) 707–715. [26] B. Milosevic, M. Begovic, Nondominated sorting genetic algorithm for optimal phasor measurement placement, IEEE Trans. Power Syst. 18 (February (1)) (2003) 69–75.
Contents lists available at SciVerse ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Minimizing the number of PMUs and their optimal placement in power systems S. Mehdi Mahaei ∗ , M. Tarafdar Hagh Department of Electrical Engineering, Ahar Branch, Islamic Azad University, Ahar, Iran
a r t i c l e
i n f o
Article history: Received 1 March 2011 Received in revised form 2 September 2011 Accepted 13 September 2011 Available online 21 October 2011 Keywords: PMU Number minimizing Optimal placement State estimation
a b s t r a c t This paper presents a new method for minimizing the number of PMUs and their optimal placement in power systems. The proposed method provides suitable constraints for power systems with two adjacent injection measurements (IMs). In addition, suitable constraints for considering the connection of two buses to each other and to an injection bus are proposed. The proposed constraints result in a reduction in the number of PMUs even though the system topological observability is complete. Existing conventional measurements are also considered. First, the number of PMUs is minimized in such a way that the system topological observability is complete. Then the optimal placement is done to maximize the measurements redundancy. The resulting phased to be installed in multiple stages. The optimal number of PMUs that ensure system topological observability under failure of a PMU or a line is also simulated. Simulations are performed on IEEE 30, 57 and 118 bus test systems by binary integer programming. The results show that the number of PMUs is equal to or less than the corresponding results of recently published papers, while the system topological observability is complete, and measurement redundancy is increased. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Current energy management systems (EMSs) require accurate monitoring of power system state variables, such as the voltage phasors at all buses. On the other hand, power systems may operate in contingency conditions, which require real-time monitoring for analysis and protection purposes. Phasor measurement units (PMUs) provide time-synchronized (real-time) phasor measurements in power systems [1]. This is available with the Global Positioning System (GPS) [2]. In addition, the PMUs are accurate devices for state estimation. Using PMUs results in linear state-estimation equations, and this makes them easier to solve than general nonlinear-state estimation equations. However, PMUs are expensive devices. Therefore, a suitable methodology is necessary to minimize the number of PMUs while maintaining complete observability of the power system (In this paper, observability means topological observability). A power system is considered completely observable when all of its states can be uniquely determined [3,4]. After establishing complete system observability, it is necessary to determine the optimal places of the PMUs to maximize measurement redundancy. A power system has measurement redundancy when its buses are observed by more than one PMU or the
∗ Corresponding author. Tel.: +98 9143075824/4115292986; fax: +98 4115261624. E-mail addresses: [email protected] (S.M. Mahaei), [email protected] (M.T. Hagh). 0378-7796/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2011.09.010
number of observable buses is maximized. In other words, some of the PMUs can be removed from the measurement system while all of the buses remain observable. System topology affects the minimize number of PMUs. For example, zero injection measurements (Zero IMs) reduce the number of PMUs. On the other hand, the conventional state estimators cannot provide a real-time picture of the power system, but they can be subjected to increased measurement redundancy. Therefore, it is necessary to consider zero injection. However, consideration of conventional measurements is optional in the optimal determination of the number and location of PMUs. Therefore, an effective PMU placement algorithm has to consider the following additional items: 1. Keeping complete system observability. 2. Maximizing measurement redundancy. 3. Modeling of zero injection buses to consider the topology conditions. These items result in a more complex optimization problem compared with some related published articles in this field. Refs. [5–8] have determined only the minimum number of PMUs required for complete observability. This approach may increase the risk of not having complete observability of the system in the contingency lines. The number and places of PMUs is determined without considering zero IMs in [9]. In [10], conventional measurements are considered, but zero IMs are not considered, which increases the number of PMUs. In [11,12], zero IMs are considered, but use of the islanding method leads to complex relations. Also
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in [13], the immunity genetic algorithm is used, but the system is not completely observable. Ref. [14] presents a simple method, but using this method in large systems may increase the number of PMUs because of the complex relations created by zero IMs buses. In this paper, a method is presented to implement suitable constraints in power systems having zero injection and/or conventional measurements. In some cases, this method reduces the number of PMUs to less than the corresponding results of published papers while maintaining complete system observability [13,15–21]. In addition, optimal placement is performed after minimizing the number of PMUs such that it increases measurement redundancy. Then the placements of the PMUs are phased for install in multiple stages. The number of PMUs that ensure system observability under PMU or line outages is also simulated. IEEE 30, 57 and 118 test bus systems are used to prove the simulation results.
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Fig. 1. Six-bus test system.
where x1 –x6 are elements of the x matrix that are generated by binary integer programming. Therefore, Eq. (7) represents the minimum number of PMUs. The bus observability constraints are defined as follows with Eq. (2): Bus 1 : x1 + x2 + x4 ≥ 1
(8)
2. Objective function to minimize the number of PMUs
Bus 2 : x1 + x2 + x3 + x5 ≥ 1
(9)
To define an objective function to minimize the number of PMUs, the following simple rules are considered:
Bus 3 : x2 + x3 + x6 ≥ 1
(10)
Bus 4 : x1 + x4 + x5 ≥ 1
(11)
Bus 5 : x2 + x4 + x5 + x6 ≥ 1
(12)
Bus 6 : x3 + x5 + x6 ≥ 1
(13)
1. A PMU placed at a specific bus can measure the voltage phasor of that bus and all the currents leaving it. 2. If the voltage phasor and current phasor at one end of a branch are known, the voltage phasor at the other end of the branch can be calculated using line impedance. 3. If voltage phasors at both ends of a branch are known, the branch current can be calculated using line impedance. 4. Each bus must be observed at least once by PMUs.
The unequal constraints (8)–(13) represent the number of observable buses for buses 1–6, respectively. For example, if at least one PMU is installed on bus 1, 2 or 4, constraint (8) is confirmed, and bus 1 will be observed. 3. Objective function for optimal placement of PMUs
Therefore, the objective function can be defined as follows [9–12,14,15]:
n
min
wk x
(1)
k=1
y = Ax ≥ b
(2)
where w, x, A and b are the weight (or cost [19]) of the PMUs, state matrix, topology and observability matrixes, respectively. In addition, n and k are the number of buses and the kth row of the w matrix, respectively. The w, x, A and b matrixes are defined as follows:
Wi,j =
xi =
0<≤1
1 if PMU installed in bus i 0 otherwise
Ai,j =
i=j 0 otherwise
1 i=j 1 if ith bus is connected to bus j 0 otherwise
b = [111 . . . 11]T
(3) max
n
(w−1 Ax)
(14)
k=1
(4)
y = Ax ≥ b
(15)
n
(5)
(6)
where w is a diagonal matrix whose iith element represents the weight (or cost) associated with placement of a PMU at the ith bus, x is a column matrix whose ith element represents the installation of PMUs in the ith bus, A is an n × n matrix whose ijth element represents the connection of the ith bus to the jth bus and b is a column matrix whose ith element represents the number of ith bus observables. For example, with Eq. (1), minimizing the number of PMUs for a 6-bus test system (Fig. 1) by assuming that w is a unit matrix can be formulated as follows: min{x1 + x2 + x3 + x4 + x5 + x6 }
Unequal constraint (2) is used for complete observability. Minimizing the number of PMUs may lead to similar sets of PMUs in which the number of PMUs is equal in each of them and system observability is complete. For example, in Fig. 1, installation of two PMUs in {1, 6}, {3, 4} and {2, 5} leads to complete observability. The question is, which of these sets is the most suitable for the installation of PMUs? The proposed method in this paper is as follows: after minimizing the number of PMUs by satisfying Eq. (1), placement is performed by maximizing observability. A suitable set is selected for the installation of PMUs subject to unequal constraint (2). The following function is used:
(7)
xk = num
(16)
k=1
where (w−1 A)k is the kth row of the (w−1 A) matrix and num is the minimum number of PMUs that is determined by minimization of Eq. (1). Therefore, for the six-bus test system shown in Fig. 1, set {2, 5} is more suitable because buses 2 and 5 are observed twice by PMUs. 4. Considering the existing measurements when minimizing the number and placement of PMUs Active, reactive and voltage measurements are used in power systems. The constraints from Eq. (2) have to be modified when these conventional measurements are considered. Two methods are used for considering both zero injection and conventional measurements:
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this paper proposes modification of constraint (18) to Eqs. (20) and (21), as follows: y1 + y2 + y3 + y4 ≥ 3
(20)
y5 ≥ 1
(21)
In fact, in this method, constraint (18) is considered for the bus that has IMs with more lines. Obviously, constraint (2) or (15) is usable for an adjacent bus. Considering the following cases is necessary. (a) Bus 4 can be eliminated from the network when placing and determining the number of PMU procedures if the number of connected lines to bus 4 is two such that its elimination may reduce the optimal number of PMUs because in this case, bus 4 will be observable if buses 3 and 5 are observed. (b) Inequality (21) should be eliminated and y5 added to inequality (20) if bus 5 is radial. (c) If two IMs are connected to each other by another bus that has two connected lines, as in Fig. 2e, constraint (18) is considered for the mentioned IMs buses, and their sum is considered a single constraint instead two constraints. 4.5. Connection of two buses to each other and an IM In some power systems, it is not possible to use constraint (18). In other words, the use of mentioned equation leads to system unobservablity. For example, in Fig. 2f, there is one IM on bus 5. Therefore, constraint (18) for buses 3, 4, 5, 6, 8 and 10 and constraint (2) for buses 1, 2, 7, 9, 11 and 12 will be as follows: Fig. 2. Conventional measurements in power systems.
(a) The islanding method [11,12]. (b) The topology-based method [14]. In this paper, a simple method is proposed as follows. 4.1. Power flow measurement By considering the power flow measurement on a line between buses i and j (Fig. 2a), both buses will be observable. Therefore, Eq. (17) is: y1 + y2 ≥ 1
(17)
In other words, the ith and jth rows of the A matrix are removed, and the above equation is replaced [14]. 4.2. Injection measurement If an IM exists on a bus, similar to Fig. 2b, then one of the constraints of Eq. (7) has to be changed [14]. Of course, the zero IM can be assumed as a pseudomeasurement. In this case, we have: y1 + y2 + y3 + y4 ≥ 3
(18)
4.3. Combining power flow and IMs
Buses 3, 4, 5, 6, 8 and 10 : y3 + y4 + y5 + y6 + y8 + y10 ≥ 5
(22)
Bus 1 : x1 + x2 + x3 ≥ 1
(23)
Bus 2 : x1 + x2 + x11 ≥ 1
(24)
Bus 7 : x6 + x7 + x11 ≥ 1
(25)
Bus 9 : x8 + x9 ≥ 1
(26)
Bus 11 : x2 + x3 + x7 + x12 ≥ 1
(27)
Bus 12 : x11 + x12 ≥ 1
(28)
Installation of three PMUs in buses 3, 8 and 11 satisfies constraints (22)–(28), but buses 4 and 6 are unobservable. In these cases, this paper proposes the following procedure: (a) Determine all adjacent buses of the IM bus. (b) Select those adjacent buses that are connected directly and indirectly to the same IM bus. “Indirectly” means that the selected bus should be connected to the IM bus via another adjacent IM bus. For example, in Fig. 2e, bus 4 has the aforementioned feature because it is connected directly to bus 5 (IM bus) and via bus 10 to bus 5, as well. (c) Use constraint (2) for the selected bus, in addition to using constraint (18) for the IM bus. The aforementioned procedure is used for the power system shown in Fig. 2e, and this results in the following constraints:
Fig. 2c shows the part of a power system that includes power flow and an IM. In this case, we have [14]:
Buses 3, 5, 6, 8 and 10 : y3 + y5 + y6 + y8 + y10 ≥ 4
(29)
y1 + y4 ≥ 1
Bus 1 : x1 + x2 + x3 ≥ 1
(30)
Bus 2 : x1 + x2 + x11 ≥ 1
(31)
Bus 4 : x4 + x5 + x10 ≥ 1
(32)
Bus 7 : x6 + x7 + x11 ≥ 1
(33)
Bus 9 : x8 + x9 ≥ 1
(34)
(19)
4.4. Two adjacent IMs Some large power systems may have several IMs in adjacent buses, such as in Fig. 2d. In this case, using unequal constraint (18) for each IM may increase the optimal number of PMUs. In this case,
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Bus 11 : x2 + x3 + x7 + x10 + x11 + x12 ≥ 1
(35)
Bus 12 : x11 + x12 ≥ 1
(36)
By considering the constraints (29)–(36), four PMUs are required for complete observation of the system shown in Fig. 2e. 4.6. Voltage measurement Installation of a voltage measurement on the ith bus results in its observation. Therefore, the iith element of A matrix will be zero. Therefore, before minimizing the number of PMUs, the constraints are first written the same as Eqs. (2) and (17)–(36). Then the corresponding elements of the A matrix are modified by considering the existing voltage measurements. In addition, before placement of PMUs, first, the constraints areplacements of PMUs, first, the constraints are written the same as Eqs. (14), (15) and (17)–(36); then the corresponding elements of the A matrix are modified by considering the existing voltage measurements. 5. Outage of a PMU or a line To ensure system observability under failure of a PMU, the righthand side of all of the constraints is multiplied by 2. If two PMUs are observing a bus, then a related line outage will not affect the node observability. Hence, the problem of ascertaining observability under a single line outage is a subset of the problem of the single PMU loss considered above [15]. 6. Phasing of PMUs Most utilities may not be able to install all the PMUs in one stage. Therefore, it is necessary for installation of the PMUs to be phased in multiple stages. It is assumed that the number of PMUs has resulted from optimizing objective function (1) for a system and that the places of them were obtained by optimizing objective function (14). At this point, the num PMUs should be inserted in specific places in t phases. In each stage, the objective function can be the same function (14) with different constraints, as follows: max
n
(w−1 Ax)
(37)
k=1 n
xk = numt
(38)
k=1
/S xk = 0|k ∈
k = 1, 2, . . . , n
(39)
xk = 1|k ∈ / S
k = 1, 2, . . . , n
(40)
numt , S and S are defined as follows. numt is the number of PMUs that are installed in the tth stage. S is the numbers of buses in which PMUs should be installed. In other words, these are the places of the PMUs that ensure complete system observability. S is the numbers of buses in which PMUs have been installed before the tth stage. Utilities may use other objectives for the multistage PMU placement, such as maximizing tie line observability or placing them at sparse locations [22,23]. In these cases, the objective should be optimized, subject to relative constraints. 7. Case study The case study was conducted in the following stages. In all of the stages, zero IMs are considered.
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1) Minimizing the number of PMUs with objective function (1). 2) Placement of PMUs by maximizing the measurements redundancies and objective function (14). 3) Comparing results with other references. 4) Considering conventional measurements and comparing with recently published papers. 5) Phasing of PMUs under normal conditions. 6) Outage of a PMU or a line and comparing it with references. Therefore, the IEEE 30, 57 and 118 bus systems are selected. These systems have 6, 15 and 10 zero IMs, respectively, which are: IEEE 30 bus system {6, 9, 22, 25, 27 and 28}. IEEE 57 bus system {4, 7, 11, 21, 22, 24, 26, 34, 36, 37, 39, 40, 45, 46 and 48}. IEEE 118 bus system{5, 9, 30, 37, 38, 63, 64, 68, 71 and 81}. 7.1. Constraints for the IEEE 30 bus system In the IEEE 30 bus system, buses 6 and 27 are adjacent with other buses that have small lines. Buses 11 and 26 are radially connected to IM buses. Therefore, constraint (20) for buses 6 and 27 and constraint (18) for bus 22 are considered as follows: y2 + y4 + y6 + y7 + y8 + y9 + y10 + y11 + y28 ≥ 7
(41)
y25 + y26 + y27 + y28 + y29 + y30 ≥ 5
(42)
y10 + y21 + y24 ≥ 3
(43)
Buses 29 and 30 have conditions, such as bus 4 in Fig. 2f. Therefore, the constraints are presented as follows for other buses that are not considered in unequal constraints (41)–(43): yi ≥ 1
(44)
7.2. Constraints for the IEEE 57 bus system In the IEEE 57 bus system, buses 21, 26, 34, 39, 40, 45 and 46 are IM buses with two connected lines. Therefore, these buses are eliminated from the network during the optimization procedures described in part (a), Section 4.4. Buses 36 and 37 are adjacent and have the same connected lines, so constraint (20) is used only for one of them. Buses 22 and 24 are two IM buses that are connected to each other by bus 23, which has two connected lines according to part (c), Section 4.4, so constraint (18) is used for buses 22 and 24; then the sides of the mentioned constraints are summed. Moreover, constraint (18) is used for buses 4, 7, 11 and 48. y3 + y4 + y6 + y18 ≥ 3
(45)
y6 + y7 + y8 + y29 ≥ 3
(46)
y9 + y11 + y13 + y41 + y43 ≥ 4
(47)
y20 + y22 + y23 + y24 + y25 + y27 + y38 ≥ 6
(48)
y35 + y36 + y37 + y56 ≥ 3
(49)
y38 + y47 + y48 + y49 ≥ 3
(50)
Bus 5 has characteristics of bus 4, according to bus 4 in Fig. 2f, so constraints are presented as constraint (44) for other buses that are not considered in unequal constraints (45)–(50). 7.3. Constraints for the IEEE 118 bus system In the IEEE 118 bus system, buses 9, 63 and 81 have the same characteristics as bus 4, which is mentioned in Section 4.5. Bus 38 is adjacent to other buses that have the most lines. Therefore, constraint (20) for buses 30 and 37 and constraint (18) for buses 5, 64, 68 and 71 are considered.
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Table 1 RESULTS of minimizing the number of PMUs with the proposed method considering zero IMs in IEEE 30, 57 and 118 bus systems. System
Objective function
No. of PMUs
Places of PMUs
Measurement redundancies
IEEE 30 bus IEEE 57 bus IEEE 118 bus
Without maximizing observation
7 11 28
1 5 10 12 15 19 27 1 4 13 20 25 29 32 38 51 54 56 3 8 11 12 17 21 27 31 32 34 40 45 49 53 56 62 65 72 75 77 80 85 86 90 94 101 105 110
6 4 38
Table 2 Results of placement of PMUs with the proposed method considering zero IMs in IEEE 30, 57 and 118 bus systems. System
Objective function
No. of PMUs
Places of PMUs
Measurement redundancies
IEEE 30 bus IEEE 57 bus IEEE 118 bus
With maximizing observation
7 11 28
2 4 10 12 15 20 27 1 4 13 20 25 29 32 38 51 54 56 2 9 11 12 17 21 27 31 32 34 40 45 49 52 56 62 65 72 75 77 80 85 87 90 94 101 105 110
11 4 41
Buses 4 and 39 that are adjacent with buses 5 and 37 have the same conditions as bus 4 in Fig. 2f. Therefore, the constraints are as follows. Of course, elimination of buses 9, 63 and 81 does not affect the number of PMUs in this system, unlike the IEEE 57 bus system.
7.5. Placement of PMUs by maximizing the measurement redundancies and objective function (14) For the maximizing observation, the objective function (14) is optimized, subject to the number of PMUs as an equality constraint, such as in Eq. (16). These results are given in Table 2. Table 2 shows that observation of the IEEE 30, 57 and 118 bus systems are complete, and observations are maximized by 7, 11 and 28 PMUs, respectively. These resulted were obtained by changing the allocation of PMUs using the proposed method compared with Table 1.
y3 + y8 + y11 + y5 + y6 ≥ 4
(51)
y8 + y9 + y10 ≥ 2
(52)
y8 + y17 + y26 + y38 + y30 ≥ 4
(53)
y33 + y35 + y34 + y38 + y40 ≥ 5
(54)
y64 + y63 + y61 ≥ 3
(55)
y81 + y116 + y68 + y65 + y69 ≥ 4
(56)
7.6. Comparing results with other references
y71 + y70 + y72 ≥ 3
(57)
Table 3 shows that in [7,15,16], the optimal number of PMUs is 7 for the IEEE 30 bus test system, but the places of the PMUs were not presented. In [13,17,18], the optimal number of PMUs is also 7, and their places ensures complete system observability, but the measurement redundancies for these places is less than the measurements redundancies of the resulting places in this paper. Table 4 shows that in [15,16,19–21], the optimal number of PMUs is more than 11 for the IEEE 57 bus test system. In [7,24], the number of PMUs for the same test system is reduced to 11, but their optimal places are not reported. In [17], the number of PMUs is
Therefore, the constraints are presented as constraint (44) for other buses that are not considered in unequal constraints (51)–(57). 7.4. Minimizing the number of PMUs by objective function (1) Appling constraints on objective function (1) leads to the results shown in Table 1. The determined PMU places result in a completely observable power system.
Table 3 Reported results of previously published papers by considering zero IMs in the IEEE 30 bus system. References
No. of PMUs
[7] [15] [16] [13] [17] [18]
7 7 7 7 7 7
Places of PMUs
Observation
Measurement redundancies
Without Placement
Unknown Unknown Unknown Observable Observable Observable
– – – 2 7 4
1 5 10 12 18 24 30 2 3 10 12 18 24 27 1 2 10 12 18 24 29
Table 4 Reported results of previously published papers by considering zero IMs in the IEEE 57 bus system. References [19] [15] [20] [16] [21] [24] [7] [13] [17] [18]
No. of PMUs 14 13 13 12 12 11 11 11 11 11
Places of PMUs
Observation
Measurement redundancies
The places are not reported because the number of PMUs is high.
It is not important because the number of PMUs is high.
Without Placement Without Placement 1 6 13 19 25 29 32 38 51 54 56 1 5 13 19 25 29 32 38 41 51 54 1 6 13 19 25 29 32 38 51 54 56
Unknown Unknown Observable Observable Observable
– – – – – – – 4 2 4
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Table 5 Reported results of previously published papers by considering zero IMs in the IEEE 118 bus system. Reference
No. of PMUs
[19] [25] [26] [16] [21] [22] [13]
29 29 29 29 29 28 28
[17]
28
[18]
28
Places of PMUs
Observation
Measurement redundancies
The places are not reported because the number of PMUs is high (29).
It is not important because the number of PMUs is high.
Without Placement 2 8 11 12 17 21 25 28 34 35 40 45 49 53 56 62 72 75 77 80 85 86 90 94 102 105 110 114 2 8 11 12 17 21 25 2833 34 40 45 49 52 56 62 72 75 77 80 85 86 90 94 101 105 110 114 3 8 11 12 17 21 25 28 34 35 40 45 49 53 56 62 72 75 77 80 85 86 90 94 102 105 110 114
Unknown Unobservable (buses 63 and 64)
– – – – – –
Unobservable (buses 63 and 64) Unobservable (buses 63 and 64)
Table 6 Conventional measurements for the IEEE 30, 57 and 118 bus systems. IEEE 30 bus Ms.
IEEE 57 bus Ms.
Power flow
Injection
15–23 12–14 19–18 6–8 – –
15 – – – – –
Power flow 41–43 34–35 42–41 40–56 – –
IEEE 118 bus Ms. [21] Injection 56 57 – – – –
Power flow
Injection
1–2 4–5 20–21 21–22 17–113 86–87
– – – – – –
Table 7 Places of PMUs in the presence of conventional measurements for IEEE 30, 57 and 118 bus systems. IEEE 30 bus
IEEE 57 bus
IEEE 118 bus
1 5 10 12 18 27 Nub. = 6 Ms. redundancies = 8
1 4 13 20 25 29 32 38 51 54 56 Nub. = 11 Ms. redundancies = 11
8 12 15 19 23 29 34 40 45 49 53 56 62 65 72 75 77 80 85 90 94 101 105 110 115 Nub. = 25 Ms. redundancies = 28
11, but the number of measurement redundancies for these places is 2, whereas the number of measurement redundancies under the resulting places in this paper is 4. Refs. [13,18] have results similar to this paper for the IEEE 57 bus system. Table 5 shows that in [16,19,21,25,26], the optimal number of PMUs is 29 for the IEEE 118 bus test system. In [24], the number of PMUs for the same test system is reduced to 28, but their optimal places were not reported. In [13,17,18], the number of PMUs is 28, but the resulting PMU places did not result in complete observation of the system. In the aforementioned paper, buses 63, 64 and 65 are not observable. By using the proposed method in this paper, the system is observable, and the number of PMUs is 28. The measurement redundancies for the resulting places for the same system in this paper are 41.
Table 7 shows that the number of PMUs reduces to 6 and 25 for the IEEE 30 and 118 bus systems, respectively, whereas the number of PMUs in [21] with the same conventional PMUs was 26. With the presence of conventional measurements, the number of PMUs for the IEEE 57 bus system is 11, but the measurement redundancies increase from 4 to 11.
7.8. Phasing of PMUs in normal conditions Phasing of PMUs is performed in three stages, according to Section 6. The number of PMUs is selected as optional according to Table 8. The results are presented in Table 8.
7.9. Outage of a PMU or a line compared with references 7.7. Considering conventional measurements and comparing with recently published papers Conventional measurements for IEEE 30, 57 and 118 bus systems are considered in Table 6. The results of the simulation containing them are presented in Table 7.
To ensure system observability under failure of a PMU or a line, the right-hand side of constraints (41)–(57) is multiplied by 2, and objective functions (1) and (14) are optimized, respectively, for the IEEE 30, 57 and 118 bus systems. The results are presented in Table 9.
Table 8 Phasing of PMUs with the proposed method considering zero IMs in the IEEE 30, 57 and 118 bus systems. System
Phase I
Phase II
Phase III
IEEE 30 bus
Nub. = 2 10 27
Nub. = 3 2 4 12
Nub. = 2 15 20
IEEE 57 bus
Nub. = 4 1 4 13 38
Nub. = 4 25 29 32 56
Nub. = 3 20 32 51 54
IEEE 118 bus
Nub. = 9 12 17 49 56 65 75 77 80 85
Nub. = 10 9 11 31 34 40 45 62 94 105 110
Nub. = 9 2 21 27 32 52 72 87 90 101
72
S.M. Mahaei, M.T. Hagh / Electric Power Systems Research 83 (2012) 66–72
Table 9 Outage of a PMU or a line considering zero IMs in the IEEE 30, 57 and 118 bus systems. System
Proposed method
[17]
[18]
[19]
IEEE 30 bus
Nub. = 15 1 2 3 5 10 12 13 15 17 19 20 23 25 27 30
Nub. = 15 2 3 4 8 10 12 13 15 16 18 20 22 24 27 30
– –
IEEE 57 bus
Nub. = 25 1 3 4 6 9 10 12 13 15 19 20 25 27 29 30 32 33 37 38 41 49 51 53 54 56 Nub. = 61 3 5 7 9 11 12 15 17 19 21 22 23 27 29 31 32 34 35 37 40 42 44 45 46 49 50 51 53 54 56 59 62 65 66 70 71 75 76 77 78 80 83 85 86 87 89 90 92 94 96 100 102 105 106 109 110 111 112 115 117
Nub. = 22 1 2 4 9 12 15 18 19 25 28 29 30 32 33 38 41 47 50 51 53 54 56 Nub. = 62 1 3 7 8 10 11 12 15 17 19 21 22 24 25 27 28 29 32 34 35 40 41 44 45 46 49 50 51 52 54 56 59 62 66 68 72 73 74 75 76 77 78 80 83 85 86 87 89 90 92 94 96 100 101 105 107 109 110 111 112 115 117
Nub. = 17 1 3 5 7 10 11 12 13 15 16 19 20 23 24 26 27 30 Nub. = 26 1 2 4 6 9 12 14 19 20 24 25 27 29 30 32 33 36 38 41 44 46 50 51 53 54 56 Nub. = 65 1 3 5 7 8 10 11 12 15 17 19 21 22 24 25 27 28 29 32 34 35 37 40 41 44 45 46 49 50 51 52 54 56 59 62 66 68 72 73 74 75 76 77 78 80 83 85 86 87 89 90 92 94 96 100 101 105 107 109 110 111 112 115 116 117
IEEE 118 bus
Table 9 shows that the number of PMUs under failure of a PMU or a line in the IEEE 30, 57 and 118 bus systems is less than the resulted reported in [17–19], except the number of PMUs for the IEEE 57 bus system in [18], which is 22. However, under this number of PMUs, the system is not ensured complete observability. For example, an outage of line 4–5 causes bus 5 to be unobservable. 8. Conclusion In large power systems, the buses that have injection measurements may be connected to each other. It is necessary to consider suitable unequal constraints for these buses; otherwise, the optimal number of PMUs may be increased. In addition, if two buses are connected to a bus that has injection measurements and also to each other, the unequal constraints should be changed to make the system completely observable. In this paper, a method is proposed for solving the aforementioned problems. A case study on IEEE 30, 57 and 118 bus systems resulted in a complete observable power system and an optimal number of PMUs that was equal or less than all published papers, whereas the measurement redundancies were higher. Consideration of conventional measurements and outages of a PMU or a line is also discussed. References [1] G. Phadke, Synchronized phasor measurements in power systems, IEEE Comput. Appl. Power 6 (April (2)) (1993) 10–15. [2] A.G. Phadke, J.S. Thorp, Synchronized Phasor Measurements and Their Applications, Springer Science Business Media, LLC, 2008. [3] Monticelli, State Estimation in Electric Power Systems. A Generalized Approach, vol. 88, Kluwer, Norwell, MA, 1999, pp. 262–282. [4] Abur, A.G. Exposito, Power System State Estimation: Theory and Implementation, Marcel Dekker, New York, 2004. [5] A.B. Antonio, J.R.A. Torreao, M.B. Do Coutto Filho, Meter placement for power system state estimation using simulated annealing, Proc. IEEE Porto Power Technol. Conf. 3 (September) (2001) 10–13. [6] H. Mori, Y. Sone, Tabu search based meter placement for topological observability in power system state estimation, Proc. IEEE Transm. Distrib. Conf. 1 (April) (1999) 172–177. [7] R.F. Nuqui, A.G. Phadke, Phasor measurement unit placement techniques for complete and incomplete observability, IEEE Trans. Power Deliv. 20 (October (4)) (2005) 2381–2388. [8] G.B. Denegri, et al., A security oriented approach to PMU positioning for advanced monitoring of a transmission grid, Proc. Power Syst. Technol., Power Conf. 2 (2002) 798–803.
Nub. = 29 – Nub. = 64 –
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