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Fault-tolerant placement of phasor measurement units based on control reconfigurability
Control Engineering Practice 21 (2013) 1–11
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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac
Fault-tolerant placement of phasor measurement units based on control reconfigurability Jianzhuang Huang n, N. Eva Wu Department of Electrical and Computer Engineering, Binghamton University, Binghamton, NY 13902-6000, USA
a r t i c l e i n f o
abstract
Article history: Received 2 June 2011 Accepted 1 September 2012 Available online 28 September 2012
This paper formulates and solves the problem of placing additional phasor measurement units (PMUs) into a power grid, with focus on enhancing the tolerance to loss of a specified number of PMUs anywhere in the grid under a placement constraint based on a previously defined concept called control reconfigurability. Control reconfigurability imposes a requirement on the minimum Hankel singular value for all anticipated fault scenarios. This requirement causes the placement of PMUs to benefit the use of analytic redundancy in the controlled dynamic grid with respect to anticipated PMU outages, thus enhances the grid resilience. The same principle is applicable to placing additional control devices into the grid. A simple power network model is used to explain the fault-tolerant PMU placement method, and the particle swarm optimization algorithm is employed to minimize the number of additional PMUs placed without violating the required control reconfigurability. The results of placing PMUs for the IEEE 118-bus system are presented. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Fault tolerance Power systems Sensor placement Network reliability Phasor measurement units Control reconfigurability
1. Introduction In 2009, the U.S. Department of Energy announced the Smart Grid Investment Grant Program, which invests billions of dollars on the US power system, or the grid (U.S. Department of Energy [USDoE], 2009a). Part of the program is to install phasor measurement units (PMUs) into the grid to enhance the ability for its monitoring and control (USDoE, 2009b). PMUs are synchronized measurement devices designed to measure voltages and currents at designated buses, and the measurements are synchronized through the Global Positioning System (GPS) (Phadke & Thorp, 2008, chap. 5). Unlike observability of a dynamic system, static observability concerns solvability of the instantaneous state of a system without invoking the dynamics governing the evolution of the state. Thus PMU placement under a static observability criterion is unable to make use of the analytic redundancy inherent in the dynamic grid. Nevertheless, the topic of placement of PMUs has been fairly extensively investigated. The aim was mostly to minimize the total installation cost, which is strongly correlated to the total number of PMUs, while maintaining a full static observability. The existing optimization algorithms include integer linear programming (Gou, 2008), topology-based formulated algorithms (Mohammadi-Ivatloo,
n
Corresponding author. Tel.: þ1 607 777 4375; fax: þ1 607 777 4464. E-mail addresses: [email protected] (J. Huang), [email protected] (N. Eva Wu). 0967-0661/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conengprac.2012.09.001
2009), and particle swarm optimization (PSO) algorithms (Hajian, Ranjbar, Amraee, & Shirani, 2007). Loss of measurement devices, however, has not been considered in the aforementioned optimal placement problems. The expected expansion of the role of PMUs requires resilient structure and operation of the network of PMUs. In addition to hardware failures, some common failures encountered in PMUs include invalid data caused by improper installation, calibration and maintenance; inaccurate data due to time alignment failures with the GPS; and data delivery failures due to network congestion. To prevent continuous and reliable measurement data from being interrupted by loss of PMUs or their data, sufficient redundancy must exist to maintain the ability to fully monitor the grid. The simplest of sensor network redundancy configuration is to have a second sensor placed right next to each original sensor, which not only doubles the number of sensors being used, but also suffers from vulnerability of greater loss due to physical attack. Rakpenthai, Premrudeepreechacharn, Uatrongjit, and Watson (2007) proposed a PMU placement method that ensures full static observability in case of single PMU loss or single branch outage. The method uses minimum condition number of a static measurement model as a criterion to find the essential measurement set and select the optimal redundant measurements. It then minimizes the number of PMU placement sites. The method, however, does not deal with the case of multiple PMU losses. Emami and Abur (2010) and Xu and Abur (2005) discussed a placement scheme that finds two mutually exclusive PMU sets, both of which have the ability to fully monitor the system. A primary set is first chosen to provide the
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desired observability using the previously mentioned algorithms such as integer linear programming, or PSO. A backup set is then chosen by excluding the primary set from the selection criteria. Since each set of PMUs is optimally chosen, the backup set inevitably contains at least as many PMUs as the primary set, which increases the total number of PMUs by at least twice. This scheme can tolerate a single sensor loss in an arbitrary location or multiple sensor losses in the same set. To tolerate multiple sensor losses in arbitrary locations, multiple backup sets are required. Since all sets of sensors are mutually exclusive from each other, the sensors are quickly exhausted. The above sensor placement problems assume steady-state operation of the power grid, though monitoring, diagnosis, and feedback control call for use of dynamic models in order to deal with the transient behaviors of systems during and post faults. This paper develops a method to place a minimum number of additional sensors to facilitate the existing set of sensors in mitigating the effects of sensor losses. The method is based on dynamic grid models as it adopts a previously defined concept of control reconfigurability (Wu, Zhou, & Salomon, 2000) for placing the additional sensors. In 2004, Staroswiecki et al. discussed a general fault tolerant sensor network design approach based on three defined criteria: redundancy degrees, sensor network reliability, and mean time to non-observability. Their approach, however, is difficult to scale up for a large system such as a power grid. Measurement of observability in this paper uses reconfigurability, as the algebraic observability used in the Staroswiecki, Hoblos, and Aitouche (2004) paper cannot quantify the ability to observe states and lacks robustness. Reconfigurability specializes to observability when the minimum Hankel singular value is replaced by the minimum singular value of the observability gramian (Moore, 1981) of the essential system states. This paper does not consider life time distributions. The problem is formulated to search for a minimum number of sensors that guarantees a minimum required level of reconfigurability (or observability as measured by the singular values of observability gramian when controllability is not of a concern) when a specified number of sensors fail either permanently or intermittently. Reconfigurability in this paper refers to the potentiality to reconfigure a control law that relies on the adequate availability of sensors and actuators to maintain the system performance, rather than the need to reconfigure the sensor network upon sensor losses. Thus the tolerance to sensor data loss resulted from satisfying a reconfigurability requirement directly leads a more resilient grid operation. Frequency control and voltage control are the two basic types of control involved in power systems (Kundur, 1994, chap. 11). A change in real power demand causes a change in frequency. Since the frequency should remain nearly constant in a power system, control effort is required on the input mechanical power to the generators in order to match the real electrical power demand so as to maintain a constant frequency. Voltage control is achieved by supplying adequate level of reactive power, and is implemented through voltage regulators at generators and reactive power compensators throughout the grid. Because of the immediate impact of the PMU network in the grid, the paper focuses on the sensor placement problem, though reconfigurability can be equally effectively applied to solving an actuator placement problem. For this reason, the control variables used in the power system example presented in this paper are simplified to be the internal voltages of the generators. The selection of control variables has an effect on the value of reconfigurability, but not the problem nature for sensor placement. The particle swarm optimization (PSO) algorithm is employed in this paper to minimize the number of additional sensors placed to maintain a required reconfigurability under a specified
maximum number of sensor losses. PSO was first introduced by Kennedy and Eberhart in 1995, intended for simulating social behavior. It was inspired by the movement of bird flocks, fish schools and bee swarms. It was then simplified and used for optimization. Two years later, Kennedy and Eberhart (1997) introduced a binary version of particle swarm optimization (BPSO), which converts the position vector into binary forms. Since then, several improved versions of BPSO were published (Afshinmanesh, Marandi, & Rahimi-Kian, 2006; Gao, Hu, He, & Liu, 2008), and the version discussed by Hajian et al. (2007) changes all the variables to discrete binary variables that can only take values of 0 or 1. The sensor or actuator placement problem formulated in this paper uses binary variables to represent whether sensors or actuators ought to be placed at certain locations, and therefore is solved by BPSO. The remainder of the paper is organized as follows. Section 2 reintroduces the concept of control reconfigurability. Section 3 discusses the formulation of PMU placement problem that tolerates a specified number of PMU losses in a power system. The PMU placement using the binary particle swarm algorithm is discussed in Section 4, which minimizes the number of the additionally placed PMUs. Section 5.1 uses a simple power system model to demonstrate the placement problem formulation and solution. Simulation results of placing PMUs on the IEEE 118-bus system are presented in Section 5.2. Section 6 concludes the paper.
2. Control reconfigurability The concept of control reconfigurability (Wu et al., 2000) is now reintroduced, following the original presentation closely. Control reconfigurability measures the level of redundancy of a given linear system using the minimum Hankel singular value (smin ). Since Hankel singular values are equal to the square roots of eigenvalues of the product of controllability gramian Wc and observability gramian Wo, reduction of controllability and/or observability caused by faults in the system reduces smin . When smin is reduced to zero, the system has deficient gramian ranks and loses its full controllability and/or observability. A specified level of control reconfigurability is met when the lowest smin is above a positive threshold for the class of all fault scenarios considered. The input to output mapping of a system should remain sufficiently controllable and observable despite the loss of some measurement and control devices. This is necessary in order for the system to be effectively controlled in the face of such sensor/ actuator faults. Assume that the small signal input–output characteristics of the system can be reasonably well simulated by a model. The transparency from input to output implies that a requirement must be imposed on the combined level of controllability and observability of the model of the system. Let y denote a vector in the fault parameter space, which is defined as the Euclidean space of all parameters that change their values as the result of some fault occurrence. Let O denote the set over which y resides when faults occur. Without loss of generality, let y ¼0 be the fault-free parameter vector. Suppose a linear model of a system in the form x_ ¼ AðyÞx þ BðyÞu,
y ¼ CðyÞx
ð1Þ
has been validated, where x, u, y are the state, input, and output vectors, and A, B, C are state-space matrices defined over y. It can be seen that all Hankel singular values, and therefore the smallest Hankel singular value smin , are functions of vector y.
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Control reconfigurability rS over a subset S of O is given by
rS min smin ðyÞ: yASDO
ð2Þ
rS indicates how well the model remains controllable and observable from its input to the output for all fault scenarios represented by points in S. For a particular application, a minimal acceptable value rmin can be specified, from which a corresponding largest set S containing the origin of the fault parameter space can be determined. In the case of sensor/actuator placement for improving the system redundancy configuration, the set within O for which the requirement is met is given by SR y A O9smin ðyÞ Z rmin g: ð3Þ Therefore, if S1R ¼ fy A O9smin ðyÞ Z r1,min g, 2 SR ¼ fy A O9smin ðyÞ Z r2,min g, it follows from (2) that S1R DS2R ) r1,min Z r2,min :
and ð4Þ
In other words, a less specific set of possible faults produce a smaller control reconfigurability. In particular, rf0g ¼ smin ð0Þ Z rS , i.e., without adding new sensors and/or actuators, control reconfigurability is always bounded above by the minimum Hankel singular value of the fault-free system. Assuming that smin ð0Þ has a sufficient combined controllability and observability, the adequacy of redundancy can be provided by adding sensors and/or actuators to maintain rS Z rmin ¼ smin ð0Þ. A problem solely focused on observability without considering controllability can be solved by redefining rS in (2) using the minimum singular value of the observability gramian Wo of (1) instead of minimum Hankel singular value. On the other hand, if controllability is the only consideration, rS can be redefined using the minimum singular value of the controllability gramian Wc of (1). In order to establish a criterion that most benefits feedback control through PMU measurements as feedback variables, a combined controllability and observability measure is used, allowing the relaxation or tightening of observability requirements based on the strength of controllability along each of the directions defined by singular vectors (Moore, 1981) in the statespace. A pure observability requirement would result in a placement ignoring the corresponding controllability. Since this paper does not consider actuator placement, controllability is fixed, and adding sensors would not change the controllability of the system.
3. PMU placement problem formulation PMU placement problems for power systems so far have been formulated using static measurement models for the purpose of estimating synchrophasors (Phadke & Thorp, 2008, chap. 5). For PMUs to monitor the change in a dynamic grid, the criterion for placement must be suitable for a dynamic system. Control reconfigurability is chosen as the criterion for formulating the PMU placement problem. Suppose the following linear differential inclusion (Boyd, Ghaoui, Feron, & Balakrishnan, 1994) x_ ¼ Ax þ Bu, y ¼ Cx, A B n A G Rðdx þ dy Þ ðdx þ du Þ C 0
ð5Þ
describes the dynamic behavior of a power system, where x, u, y are the state, input, and output vectors with dimensions dx, dy and du, A, B, and C are known constant matrices, and R is the set of real numbers. Suppose a set of M PMUs is already in place in the system for estimating the critical set of states so that baseline monitoring
3
and control for the system under normal condition is satisfied. N new PMUs are to be placed in K new possible locations to enhance tolerance to loss of n PMUs, where KZNZ1 is assumed without loss of generality. To lessen the impact of physical attack at the PMU locations, each candidate location is allowed to have only a single PMU placed, including the original M PMU locations, and all candidate locations are sufficiently far away from each other. To represent N PMUs being placed at K new locations, the C matrix is redefined as " CðzÞ ¼
Co C K ðzÞ
# ,
ð6Þ
where Co is the original C matrix for the M-PMU network, and 2
3 z1 C M þ 1 6 7 ^ 7 6 6 7 7 C K ðzÞ ¼ 6 6 z k C M þ k 7, 6 7 ^ 4 5 zK C M þ K
ð7Þ
with zkA{0,1}. zk ¼1 indicates a new PMU is being placed at location k, whereas zk ¼0 indicates no PMU placement at the location. The vector z represents a particular PMU placement with N PMUs being placed at N out of the K candidate locations. Since N new PMUs are placed, K X
zk ¼ N,
zk A f0,1g:
ð8Þ
k¼1
Assuming linearity for both the system and the measurement models, rmin , which is set to be the minimum Hankel singular value of the original M-PMU system, can be calculated using A, B and Co (Wu et al., 2000). Without loss of generality, rmin is regarded as an acceptable threshold of reconfigurability in the absence of PMU faults. Now consider the case of loss of n PMUs, where 1rn rM, for which set Sn is defined as 8 > > > > > > > > > > > > > > > > > > > <
2
ð1y1 ÞC 1 6 ^ 6 6 6 yi ÞC i ð1 6 6 ^ 6 6 6 ð1yM ÞC M 6 Sn ¼ y9CðyÞ ¼ 6 ð1 yM þ 1 Þz1 C M þ 1 > 6 > > 6 > > 6 > ^ > 6 > > 6 > > 6 y ð1 M þ j Þzj C M þ j > > 6 > > 6 > ^ > 4 > > : Þz C ð1y MþK
0
yi , yM þ j A f0,1g, @
M X
i¼1
yi þ
K
3 7 7 7 7 7 7 7 7 7 7 7, 7 7 7 7 7 7 7 7 5
MþK
K X
1
) A zj yM þ j r n :
ð9Þ
j¼1
where yi ¼ 1 indicates a loss of PMU data at location i. For the K new locations, yM þ j is undefined whenever zj ¼0. Therefore, y is an (MþN)-vector containing all defined variable yi. With the N new PMUs placed in the system, control reconfigurability is given by rSn miny A Sn smin ðyÞ, where smin ðyÞ is the minimum Hankel singular value calculated using A, B, and C(y). The case n¼1 represents a loss of data at a single PMU, and defines a cost effective and practical fault-tolerant placement problem. In this case, the condition on the fault parameters in (9)
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can be simplified to 8 2 3 ð1y1 ÞC 1 > > > 6 7 > > ^ > 6 7 > > 6 7 > < 6 ð1yi ÞC i 7 6 7 S1 ¼ y9CðyÞ ¼ 6 7, ^ > 6 7 > > 6 7 > > 6 ð1yM ÞC M 7 > > 4 5 > > : C K ðzÞ
Minimize
yi A f0,1g,
M X
yi r1g:
ð10Þ
i¼1
Thus the dimension of the fault-parameter set is reduced to the number of original PMUs due to the fact that losing one newly placed PMU does not cause control reconfigurability rS1 to fall below rmin , as rmin is specified as the minimum Hankel singular value of the original M-PMU system. The optimization for the new PMU placement that tolerates the loss of n PMUs can be formulated as Minimize
N
Subject to rSn Z rmin , ð8Þ
rSn
Subject to ð8Þ, and N ¼ N0 ,
ð11Þ
The constraint rSn Z rmin enforces a sufficient redundancy level through PMU placement in order to tolerate n PMUs losses. A variation of Problem (11) is to maximize rSn for a fixed N, i.e.,
ð12Þ
where N0 is a fixed positive integer less than or equal to K. Problems (11) and (12) in principle can be solved using branch and bound methods (Boyd & Mattingley, 2006), for they can be viewed as mixed-Boolean convex problems. Maximizing the minimum Hankel singular values using semidefinite program (Body & Vandenberghe, 2004), however, is nontrivial. One issue encountered is inadequate tools capable of handling large scale matrix variables. As a result, an alternative search algorithm is applied to solve (11) and (12). K For a exhaustive linear search algorithm, it takes steps to N find the solution of z representing a placement that maximize rSn and meets the constraints in (12). To solve (11), all values of N that are less than K have to be considered. When K and N are large, finding the solution can be time consuming. In order to improve the speed of finding a solution, a randomized search algorithm called particle swarm optimization (PSO) is implemented. A brief discussion of the PSO algorithm is given in next section.
Fig. 1. Flow chart of the binary particle swarm optimization algorithm.
J. Huang, N. Eva Wu / Control Engineering Practice 21 (2013) 1–11
5
4. Particle swarm optimization
5. Application to PMU placement
The principle of PSO (Kennedy & Eberhart, 1995) is that a set of particles is employed to find the optimal solution of the assigned problem. Each particle adjusts its own position and velocity based on the memory of its own best known position pbest and knowledge of the best position gbest of all particles in the swarm. At iteration k, position vector x and velocity vector v of particle i are adjusted by
5.1. A simple power network
xki þ 1 ¼ xki þ vki þ 1 ,
ð13Þ
vki þ 1
ð14Þ
¼ wi vki þ c1 n r 1 n ðpbesti xki Þ þ c2 n r 2 n ðgbestxki Þ, n
A simple power network as shown in Fig. 2 is used to explain the concept discussed in Section 3. The system has two generators, four transformers, four loads and eight buses. Using resistance R, inductance L of each equipment shown in Table 1 and Kirchhoff’s Voltage Law (KVL), the dynamic behavior of the system can be described by ½Lx_ ¼ ½Rx þ ½Vu,
where wi is the weight function for velocity of particle i, c1 and c2 are the acceleration coefficients, r1 and r2 are the random numbers generated between 0 and 1 with a uniform distribution. Once the position and velocity of a particle is updated, the objective function of the optimization problem is recalculated based on the new position xki þ 1. pbesti is then replaced by xki þ 1 if xki þ 1 can better optimize the objective function than pbesti. Similarly, gbest is updated to xki þ 1 if xki þ 1 can better optimize the objective function than gbest. Certain constraints can be placed on vki þ 1. For example, if vki þ 1 excesses a predefined limit vmax, vmax replaces vki þ 1 as the velocity vector for the next iteration. The first binary version of particle swarm optimization (BPSO) (Kennedy & Eberhart, 1997) is similar to the conventional PSO described above with modification only to the position Eq. (13) is replaced by ( 1 0 if r 3 Z 1 þ expðv kþ1 Þ kþ1 ij xij ¼ ð15Þ 1 otherwise
y ¼ Cx, where 2 3 iTL1 6i 7 6 TL2 7 6 7 6 iTL3 7 6 7 6 7 x ¼ 6 iTL4 7, 6 7 6 iTL5 7 6 7 6i 7 4 LD1 5 iLD2
ð18Þ 2
" u¼
V1 V2
# ,
3 iG1 6i 7 6 G2 7 6 7 6 iLD1 7 6 7 6 7 y ¼ 6 iLD2 7 : 6 7 6 iLD3 7 6 7 6i 7 4 TL1 5 iTL2
ð19Þ
To simplify the calculations, all quantities used for this example are in Per Unit (pu) (Glover, Sarma & Overbye, 2008, chap. 3), including those shown in Table 1. The states of (18) are currents at transmission lines TL1, TL2, TL3, TL4, TL5 and loads LD1 and LD2.
where j is the element number of the position vector xki þ 1 and velocity vector vki þ 1, and r3 is a randomly generated number ranged between 0 and 1. This paper uses the version of BPSO introduced by Hajian et al. (2007) with some modifications as described later on to improve the performance of optimizing (11) and meet the constraints of (12). The position Eq. (13) and the velocity Eq. (14) is replaced by xki þ 1 ¼ xki vki þ 1 ,
ð16Þ
vki þ 1 ¼ vki ðr 1 ðpbesti xi k ÞÞ ðr 2 ðgbest xki ÞÞ,
ð17Þ
where and represent two different logic operations. Experiments indicate that choosing NOR for and XOR for is less likely to cause position vector x to repeat itself and therefore gives better results. r1 and r2 in (17) are randomly generated binary vectors. All variables in (16) and (17) are binary vectors. In order to improve the convergence rate of Problem (11), a simple modification is added to the algorithm. Let G equal to the sum of gbest, indicating G new PMUs are being placed in the current best solution. Since the primary objective function of (11) is to minimize the number of PMUs being placed, if the sum of xki þ 1 is greater than G, the number of 1s in xki þ 1 will be randomly reduced until it become less than or equal to G. For Problem (12), since N is fixed, G must be equal to N. If the sum of xki þ 1 is not equal to G, the number of 1s in xki þ 1 is randomly reduced or increased until it is equal to G. A flow chart of BPSO algorithm for solving (11) described above is shown in Fig. 1. The end condition shown in the flow chart can be set to be the maximum number of iterations without finding a new gbest. When the search ends, the current value of gbest would be the current best solution for the placement z as discussed in the previous section.
Fig. 2. A simple power network with two generators, four transformers, four loads, and eight buses.
Table 1 Per unit impedances of equipment of the simple power network.
TL1a TL2 TL3 TL4 TL5 TF1a TF2 TF3 TF4 G1a G2 LD1a LD2 LD3 LD4 a
Z (pu)
R (pu)
L (pu)
0.0012 þj0.078 0.0022 þj0.033 0.0011 þj0.081 0.0019 þj0.125 0.0015 þj0.115 0.0025 þj0.040 0.0015 þj0.039 0.0024 þj0.068 0.0013 þj0.049 0.016 þj0.012 0.013 þj0.011 9.5 þj5.6 7.6 þj5.3 6.7 þj5.2 5.3 þj4.7
0.0012 0.0022 0.0011 0.0019 0.0015 0.0025 0.0015 0.0024 0.0013 0.016 0.013 9.5 7.6 6.7 5.3
2.069 10 4 8.754 10 5 2.149 10 4 3.316 10 4 3.050 10 4 1.061 10 4 1.035 10 4 1.804 10 4 1.300 10 4 3.183 10 5 2.918 10 5 1.485 10 2 1.406 10 2 1.379 10 2 1.247 10 2
TL: Transmission line, TF: Transformer, G: Generator, LD: Load.
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J. Huang, N. Eva Wu / Control Engineering Practice 21 (2013) 1–11
The inputs of (18) are internal voltages V1 and V2 of generators G1 and G2. Output y measures currents at generators G1, G2, loads LD1, LD2, LD3 and transmission lines TL1 and TL2. As a result, matrix C is equal to 2
C1
C 1 ¼ C G1 ¼ 1 1 0 1 0 1 0 , C 2 ¼ C G2 ¼ 1 0 1 0 1 0 1 , C 3 ¼ C LD1 ¼ 0 0 0 0 0 1 0 , C 4 ¼ C LD2 ¼ 0 0 0 0 0 0 1 , C 5 ¼ C LD3 ¼ 0 1 0 0 1 0 0 , C 6 ¼ C TL1 ¼ 1 0 0 0 0 0 0 , C 7 ¼ C TL2 ¼ 0 1 0 0 0 0 0 :
3
6C 7 6 27 6 7 6 C3 7 6 7 6 7 C ¼ 6 C 4 7, 6 7 6 C5 7 6 7 6C 7 4 65 C7
ð20Þ
To represent N new PMUs being placed in the 6 candidate locations, (7) and (8) become 2 3 z1 C 8 6 7 6 z2 C 9 7 6 7 6 z3 C 10 7 6 7 C 6 ðzÞ ¼ 6 ð23Þ 7, 6 z4 C 11 7 6 7 6 z5 C 12 7 4 5 z6 C 13
6 X
zk ¼ N,
zk A f0,1g,
ð24Þ
k¼1
Using matrices [L], [R] and [V], whose numerical values are shown in Appendix A, matrices A and B can be calculated as 2
49:2 6 23:1 6 6 6 16:4 6 6 A ¼ ½L1 n½R ¼ 6 16:5 6 6 22:8 6 6 4 0:9 0:7
27:7
4:1
39:5
69:7
6:0
360:6 9:4
3:3 255:4
19:9 233:4
328:8 25:3
3:3 2:2
11:2
159:6
181:6
27:1
2:3
115:1
1:8
21:1
146:5
3:1
0:4 0:3
0:3 0:4
0:6 0:4
0:5 0:4
639:3 0:02
3:8
3
2:0 7 7 7 1:5 7 7 1:4 7 7, 7 2:0 7 7 7 0:02 5 540:4
ð21Þ 2
1378:6 6 757:9 6 6 6 500:2 6 6 B ¼ ½L1 n½V ¼ 6 536:1 6 6 717:6 6 6 61:5 4 5:4
1383:4
3
705:7 7 7 7 549:1 7 7 507:4 7 7: 7 735:8 7 7 7 5:6 5 65:5
ð22Þ
Eq. (19) implies that 7 PMUs are in place for monitoring the system so that baseline control for the system under normal condition is satisfied. N new PMUs are to be placed at locations chosen from 6 candidate locations to measure currents at transformer TF1, TF2, load LD4 and transmission lines TL3, TL4 and TL5. It is assumed that each branch has only one location for only one single PMU placement, including the original 7 locations. One of the goals for this example is to tolerate one single PMU loss in the system.
where C 8 ¼ C TF1 ¼ 1 1 0 1 0 0 0 , C 9 ¼ C TF2 ¼ 1 0 1 0 1 0 0 , C 10 ¼ C LD4 ¼ 0 0 1 1 0 0 0 , C 11 ¼ C TL3 ¼ 0 0 1 0 0 0 0 , C 12 ¼ C TL4 ¼ 0 0 0 1 0 0 0 , C 13 ¼ C TL5 ¼ 0 0 0 0 1 0 0 :
ð25Þ
N and zk are the optimization variables in (11) to be solved by the BPSO algorithm. rmin is set to be the minimum Hankel singular value of (18), which is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rmin ¼ minð eigðW c W o Þ Þ: ð26Þ Wc and Wo are the controllability gramian and observability gramian of (18), for which AW c þ W c A0 ¼ 2BB0 ,
ð27Þ
A0 W o þ W o A ¼ 2C 0 C,
ð28Þ
where A and B are given in (21) and (22), and C is given in (20). The set that represents all possible data loss at a single PMU at the original 7 PMUs is given in (19) with a particular placement z
x 10−3
ρS1
1.1 1 0.9 0.8
0
20
40
60
80
100
120
140
0
20
40
60 80 Iteration (k)
100
120
140
# of PMUs
5 4 3 2 1
Fig. 3. (a) Control reconfigurability rS1 calculated using gbest at each iteration and (b) number of additional PMUs being placed to tolerate loss of data at any single PMU.
J. Huang, N. Eva Wu / Control Engineering Practice 21 (2013) 1–11
can be described as 8 2 3 ð1y1 ÞC 1 > > > 6 7 > > ^ > 6 7 > > 6 7 > < 6 ð1yi ÞC i 7 6 7 S1 ¼ y9CðyÞ ¼ 6 7, ^ > 6 7 > > 6 7 > > 6 7 > y ÞC ð1 7 7 > 4 5 > > : C 6 ðzÞ
yi A f0,1g,
7 X
yi r1g,
ð29Þ
i¼1
Control reconfigurability rS1 is defined as miny A S1 smin ðyÞ,where minimum Hankel singular value smin ðyÞ is calculated using A, B and C(y). The results of solving (11) for this example using BPSO are shown in Fig. 3. Fig. 3 (a) shows control reconfigurability rS1 at each iteration calculated using the current best solution of gbest. The dashed line in Fig. 3 (a) represents rmin . In the first several iterations, rS1 is below the level of rmin indicating that the constraint rS1 Z rmin is not met. In this state, the BPSO algorithm tries to find a better position that will raise rS1 above rmin . Once this is achieved, the algorithm focuses on minimizing the number of PMUs while still maintaining rS1 above rmin . Fig. 3 (b) shows the sum of gbest which is the number of PMUs being placed for the current best solution. The process is terminated after about 100 iterations without finding a position vector better than the current gbest. The final solution shows that only 2 additional PMUs are required to tolerate the loss of data at any single PMU in the system. The final value of z is equal to [0 0 1 0 1 0], which
7
indicates that two PMUs should be placed into the system to measure currents at load LD4 and transmission line TL4. When N¼ 3 is fixed as a constraint, or 3 PMUs are to be placed in the six candidate locations to tolerate loss of data at any one of the original 7 PMUs, (12) is solved to maximize rS1 . Fig. 4 shows the results of solving (12) with N¼3. The final value of z is [1 0 1 0 1 0], which indicates that the three PMUs should be placed to measure currents at transformer TF1, transmission line TL4 and load LD4. To tolerate data losses at any two PMUs in the system of (18), N new PMUs are to be placed in the 6 candidate locations. The corresponding fault parameter set is given by 3 ð1y1 ÞC 1 7 6 ^ 7 6 7 6 7 6 y ÞC ð1 i i 7 6 7 6 ^ 7 6 7 6 7 6 ð1y7 ÞC 7 7 6 S2 ¼ y9CðyÞ ¼ 6 7, ð1 y Þz C > 7 6 8 1 8 > > 7 6 > > 7 6 > ^ > 7 6 > > 7 6 > > 6 ð1y7 þ j Þzj C 7 þ j 7 > > 7 6 > > 7 6 > ^ > 5 4 > > : ð1y Þz C 2
8 > > > > > > > > > > > > > > > > > > > <
13
6
0
yi , y7 þ j A f0,1g, @
7 X
i¼1
ρS1
0.9
40
60 Iteration (k)
80
100
120
Fig.4. Control reconfigurability rS1 with N (number of additional PMUs) fixed to 3.
ρS2
1
# of PMUs
x 10−3
0.5
0
0
20
40
60
80
100
120
0
20
40
60 Iteration (k)
80
100
120
6 4 2 0
j¼1
The result of solving (11) for tolerating data loss in any two PMUs is shown in Fig. 5. Due to physical limitation, even if PMUs are installed in all 6 candidate locations, rS2 can never reach rmin . In this case, maximizing rS2 becomes a more practicable objective. Fig. 6 shows the result of solving (12) for rS2 with N ¼3.
1
20
1 zj y7 þ j A r 2g,
ð30Þ
1.1
0
6 X
13
x 10−3
0.8
yi þ
Fig. 5. (a) Control reconfigurability rS2 calculated using gbest at each iteration and (b) number of PMUs being placed.
8
J. Huang, N. Eva Wu / Control Engineering Practice 21 (2013) 1–11
1
x 10−3
0.8
ρS2
0.6 0.4 0.2 0
0
10
20
30
40
50 60 Iteration (k)
70
80
90
100
110
Fig. 6. Control reconfigurability rS2 with N fixed to 3.
1.5
x 10−3
ρS2
1 0.5
# of PMUs
0
0
20
40
60
0
20
40
60
80
100
120
140
160
80 100 Iteration (k)
120
140
160
10 8 6
Fig. 7. (a) Control reconfigurability rS2 when each branch current is allowed to be measured at two placement locations, (b) Number of PMUs being placed.
The result of z is [1 1 0 0 0 1], which indicates that the three PMUs are placed to measure the currents at transformers TF1, TF2, and transmission line TL5 to achieve a maximum control reconfigurability in the face of loss of data at two PMUs or less, although rS2 cannot reach fault-free control reconfigurability rmin . If each branch is allowed to have up to two locations for new PMUs to measure currents, and all locations are far away from each other, there are totally 19 candidate locations to install new PMUs. There are 7 locations from the original 7 branches with PMUs already placed, and there are 12 locations from the original 6 candidate branches. The result of solving (11) for this case to tolerate data loss in any two PMU is shown in Fig. 7. The final z indicates that 6 new PMUs should be placed and 3 out of the 6 new PMUs should be placed in the original 7 branches. 5.2. IEEE 118-bus test system The IEEE 118-bus test system obtained from the power systems test case archive in the website of Washington University is now used for application of this method. This system contains 34 generators and a total of 186 branches among 118 buses. 27 single PMUs are placed to measure currents at 27 different branches for monitoring the behavior of the system. New PMUs are to be added to the positions selected out of the remaining 159 branches to
tolerate data loss at any single PMU and any two PMUs in the system, respectively. This PMU placement problem can be solved by repeating the steps described in Section 5.1, and the results of solving (11) for rS1 and rS2 are shown in Fig. 8 and Fig. 9. Similar to Figs. 3, 8(a) and 9(a) show the control reconfigurabilities of the current best solutions. Figs. 8(b) and 9(b) show the number of PMUs being placed for the current best solutions. The final results found by BPSO show that only 17 new PMUs are required to tolerate data loss at a single PMU for this system, which is a little more than half of the number of the original 27 PMUs. For tolerating data loss at any two PMUs, 74 additional PMUs are required, assuming only one location is allowed for measuring current at each branch. Fig. 10 shows the result of solving (12) for rS1 with N fixed to be 20. One drawback common to all sensor placement methods is the time to calculate a solution, which is the time to calculate control reconfigurability in this case, for a very large system. With a modern computer featured a 1.6 GHz 32-bit duo-core Intel Atom CPU, calculating the minimum Hankel singular value for the IEEE 118-bus system using Matlab takes about one second. For each iteration, calculating the control reconfigurability rS1 takes about half minute since it involves 27 steps of calculating the minimum Hankel singular values. It takes several hours to obtain Fig. 8. For calculating control reconfigurability rS2 , if a placement z
J. Huang, N. Eva Wu / Control Engineering Practice 21 (2013) 1–11
4
9
x 10−6
ρS1
3 2 1
0
200
400
0
200
400
600
800
1000
1200
800
1000
1200
# of PMUs
100
50
0
600 Iteration (k)
Fig. 8. (a) Control reconfigurability rS1 of the IEEE 118-bus test system calculated using gbest at each iteration; (b) minimum number of additional PMUs needed to tolerate data loss at any single PMU without violating control reconfigurability requirement.
1.7
x 10−6
ρS2
1.4 1.1 0.8 0.5
0
20
40
0
20
40
60
80
100
120
60 80 Iteration (k)
100
120
# of PMUs
95 90 85 80 75 70
Fig. 9. (a) Control reconfigurability rS2 of the IEEE 118-bus test system calculated using gbest at each iteration; (b) minimum number of additional PMUs needed to tolerate data loss at any two PMUs without violating the control reconfigurability requirement.
contains 60 additional PMUs, it takes more than half an hour to obtain the result. To obtain Fig. 9, it requires several days of computation. Of course, the results can be improved with a more powerful computing machine, such as a computer featured a 3 GHz 64-bit quad-core CPU.
6. Conclusions A new formulation of fault-tolerant PMU placement problem using a previously defined control reconfigurability concept is presented. Control reconfigurability is used as a constraint for placing PMUs in the system that can tolerate data loss at a given number of PMUs. The fault-tolerant placement method in this paper differs from the other placement methods in that the method directly results in a
more resilient grid assuming that the PMU measurements are used as feedback variable for control of a system. The minimization of the number of new PMUs is solved using the binary particle swarm optimization algorithm. Simulation results obtained for a simple power network and the IEEE 118-bus test system are presented to demonstrate the feasibility of the proposed solution. Planned future work includes the generalization to allow loss of multiple types of sensors and actuators for large scale nonlinear systems. Though the current placement criterion uses a combined controllability and observability measure, sensor placement dominates the interest, and actuator placement has not been considered for a realistic control access. Thus, another focus of future work is to incorporate the voltage control problem with reactive power compensation into the control reconfigurability-based placement problem.
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x 10−7
14 12
ρS1
10 8 6 4 2
0
100
200
300
400
500 600 Iteration (k)
700
800
900
1000
Fig. 10. Control reconfigurability rS1 of the IEEE 118-bus test system with N (number of additional PMUs to be placed) fixed to 20.
Acknowledgement The authors gratefully acknowledge the guidance and insights provided by Dr. Joe H. Chow of RPI, which greatly helped take their first step toward studying the issues related to power systems.
½R1,1 ¼ 2ðRG1 þ RTF1 þ RTL1 þ RTF2 þ RG2 Þ ¼ 20:0342, ½R2,2 ¼ 2ðRG1 þ RTF1 þ RTL2 þ RTF3 þ RLD3 Þ ¼ 26:723, ½R3,3 ¼ 2ðRG2 þ RTF2 þ RTL3 þ RTF4 þ RLD4 Þ ¼ 25:317, ½R4,4 ¼ 2ðRG1 þ RTF1 þ RTL4 þ RTF4 þ RLD4 Þ ¼ 25:322, ½R5,5 ¼ 2ðRG2 þ RTF2 þ RTL5 þ RTF3 þ RLD3 Þ ¼ 26:718, ½R6,6 ¼ 2ðRG1 þ RLD1 Þ ¼ 29:516,
Appendix A. Numerical values of matrices [V], [L] and [R] of the Simple Power Network [V] is a 7 2 matrix representing connections to the generators, which 2 3 1 1 61 0 7 6 7 6 7 60 1 7 6 7 6 7 ½V ¼ 6 1 0 7 6 7 60 1 7 6 7 6 7 41 0 5 0 1 Both matrices [L] and [R] are 7 7 symmetric matrices that reflect the topology of the power network. Their values can be calculated using quantities in Table 1 as follows: ½L1,1 ¼ ðLG1 þ LTF1 þ LTL1 þ LTF2 þ LG2 Þ ¼ 4:775 104 , ½L2,2 ¼ ðLG1 þ LTF1 þ LTL2 þ LTF3 þ LLD3 Þ ¼ 1:420 102 , ½L3,3 ¼ ðLG2 þ LTF2 þ LTL3 þ LTF4 þ LLD4 Þ ¼ 1:295 102 , ½L4,4 ¼ ðLG1 þ LTF1 þ LTL4 þ LTF4 þ LLD4 Þ ¼ 1:307 102 , ½L5,5 ¼ ðLG2 þ LTF2 þ LTL5 þ LTF3 þ LLD3 Þ ¼ 1:441 102 , ½L6,6 ¼ ðLG1 þ LLD1 Þ ¼ 1:489 102 , ½L7,7 ¼ ðLG2 þ LLD2 Þ ¼ 1:409 102 , ½L1,2 ¼ ½L1,4 ¼ ½L2,4 ¼ ðLG1 þLTF1 Þ ¼ 1:379 104 , ½L1,3 ¼ ½L1,5 ¼ 2½L3,5 ¼ 2ðLTF2 þ LG2 Þ ¼ 21:326 104 , ½L1,6 ¼ ½L2,6 ¼ ½L4,6 ¼ ðLG1 Þ ¼ 3:183 105 , ½L1,7 ¼ 2½L3,7 ¼ 2½L5,7 ¼ 2ðLG2 Þ ¼ 22:918 105 , ½L2,5 ¼ ðLTF3 þ LLD3 Þ ¼ 1:379 102 , ½L3,4 ¼ ðLTF4 þ LLD4 Þ ¼ 1:260 102 , ½L2,3 ¼ ½L2,7 ¼ ½L3,6 ¼ ½L4,5 ¼ ½L4,7 ¼ ½L5,6 ¼ ½L6,7 ¼ 0,
½R7,7 ¼ 2ðRG2 þ RLD2 Þ ¼ 27:613, ½R1,2 ¼ ½R1,4 ¼ ½R2,4 ¼ 2ðRG1 þ RTF1 Þ ¼ 20:0185, ½R1,3 ¼ ½R1,5 ¼ 2½R3,5 ¼ ðRTF2 þ RG2 Þ ¼ 0:0145, ½R1,6 ¼ ½R2,6 ¼ ½R4,6 ¼ 2ðRG1 Þ ¼ 20:016, ½R1,7 ¼ 2½R3,7 ¼ 2½R5,7 ¼ ðRG2 Þ ¼ 0:013, ½R2,5 ¼ 2ðRTF3 þ RLD3 Þ ¼ 26:702, ½R3,4 ¼ 2ðRTF4 þ RLD4 Þ ¼ 25:301, ½R2,3 ¼ ½R2,7 ¼ ½R3,6 ¼ ½R4,5 ¼ ½R4,7 ¼ ½R5,6 ¼ ½R6,7 ¼ 0:
References Afshinmanesh, F., Marandi, A., & Rahimi-Kian, A. (2006). A novel binary particle swarm optimization method using artificial immune system. In Proceedings of the IEEE 2006 EUROCON Conference (Vol. 5, pp. 4104–4108). Boyd, S., Ghaoui, L. E., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory. Studies in applied mathematics, Vol. 15. Society for Industrial and Applied Mathematics (SIAM). Boyd, S., & Vandenberghe, L. (2004). Convex optimization. Cambridge University Press. Boyd, S., & Mattingley, J. (2006). Branch and bound methods. Notes for EE364b. Stanford University. /http://www.stanford.edu/class/ee364b/notes/bb_notes.pdfS. Emami, R., & Abur, A. (2010). Robust measurement design by placing synchronized phasor measurements on network branches. IEEE Transactions on Power Systems, 25(1), 38–43. Gao, Y., Hu, Z., He, X., & Liu, D. (2008). Optimal placement of PMUs in power systems based on improved PSO algorithm. In Proceedings of the 3rd IEEE conference on industrial electronics and applications (pp. 2464–2469). Glover, J. D., Sarma, M. S., & Overbye, T. J. (2008). Power system analysis and design (4th ed., pp. 108–115). USA: Thomson Learning. Gou, B. (2008). Optimal placement of PMUs by integer linear program. IEEE Transactions on Power Systems, 23(3). Hajian, M., Ranjbar, A. M., Amraee, T., & Shirani, A. R. (2007). Optimal placement of phasor measurement units: Particle swarm optimization approach. In Proceedings of the 14th international conference on intelligent system applications to power system, ISAP (pp. 152–157). Kennedy, J., & Eberhart, R. (1995). Particle swarm optimization. In Proceedings of the IEEE international conference on neural networks (Vol. IV, pp. 1942–1948). Kennedy, J., & Eberhart, R. (1997). A discrete binary version of particle swarm algorithm. IEEE Computational Cybernetics and Simulation Conference, 5, 4104–4109. Kundur, P. (1994). Power system stability and control (pp. 581–698). USA: McGrawHill.
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Mohammadi-Ivatloo, B. (2009). Optimal placement of PMUs for power system observability using topology based formulated algorithms. Journal of Applied Sciences, 9, 2463–2468. Moore, B. C. (1981). Principal component analysis in linear systems: Controllability, observability, and model reduction. IEEE Transactions on Automatic Control, 26, 17–32. Phadke, A. G., & Thorp, J. S. (2008). Synchronized phasor measurements and their applications (pp. 93–105). New York, NY: Springer Science þBusiness Media, LLC. Rakpenthai, C., Premrudeepreechacharn, S., Uatrongjit, S., & Watson, N. (2007). An optimal PMU placement method against measurement loss and branch outage. IEEE Transactions on Power Delivery, 22(1), 101–107. Staroswiecki, M., Hoblos, G., & Aitouche, A. (2004). Sensor network design for fault tolerant estimation. International Journal of Adaptive Control and Signal Processing, 18, 55–72.
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U.S. Department of Energy (2009a). Obama administration announces availability of $3.9 Billion to invest in smart grid technologies and electric transmission infrastructure. Press Release, April–June. Retrieved October, 2010, from /http://www.energy.gov/news2009/7503.htmS. U.S. Department of Energy (2009b). Locke, Chu announce significant steps in smart grid development. Press Release, April–June. Retrieved October, 2010, from /http://www.energy.gov/news/7408.htmS. Wu, N. E., Zhou, K., & Salomon, G. (2000). Control reconfigurability of linear timeinvariant systems. Automatica, 36, 1767–1771. Xu, B., & Abur, A. (2005). Optimal placement of phasor measurement units for state estimation. PSERC Publication 05-58. Power System Engineering Research Center.
Contents lists available at SciVerse ScienceDirect
Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac
Fault-tolerant placement of phasor measurement units based on control reconfigurability Jianzhuang Huang n, N. Eva Wu Department of Electrical and Computer Engineering, Binghamton University, Binghamton, NY 13902-6000, USA
a r t i c l e i n f o
abstract
Article history: Received 2 June 2011 Accepted 1 September 2012 Available online 28 September 2012
This paper formulates and solves the problem of placing additional phasor measurement units (PMUs) into a power grid, with focus on enhancing the tolerance to loss of a specified number of PMUs anywhere in the grid under a placement constraint based on a previously defined concept called control reconfigurability. Control reconfigurability imposes a requirement on the minimum Hankel singular value for all anticipated fault scenarios. This requirement causes the placement of PMUs to benefit the use of analytic redundancy in the controlled dynamic grid with respect to anticipated PMU outages, thus enhances the grid resilience. The same principle is applicable to placing additional control devices into the grid. A simple power network model is used to explain the fault-tolerant PMU placement method, and the particle swarm optimization algorithm is employed to minimize the number of additional PMUs placed without violating the required control reconfigurability. The results of placing PMUs for the IEEE 118-bus system are presented. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Fault tolerance Power systems Sensor placement Network reliability Phasor measurement units Control reconfigurability
1. Introduction In 2009, the U.S. Department of Energy announced the Smart Grid Investment Grant Program, which invests billions of dollars on the US power system, or the grid (U.S. Department of Energy [USDoE], 2009a). Part of the program is to install phasor measurement units (PMUs) into the grid to enhance the ability for its monitoring and control (USDoE, 2009b). PMUs are synchronized measurement devices designed to measure voltages and currents at designated buses, and the measurements are synchronized through the Global Positioning System (GPS) (Phadke & Thorp, 2008, chap. 5). Unlike observability of a dynamic system, static observability concerns solvability of the instantaneous state of a system without invoking the dynamics governing the evolution of the state. Thus PMU placement under a static observability criterion is unable to make use of the analytic redundancy inherent in the dynamic grid. Nevertheless, the topic of placement of PMUs has been fairly extensively investigated. The aim was mostly to minimize the total installation cost, which is strongly correlated to the total number of PMUs, while maintaining a full static observability. The existing optimization algorithms include integer linear programming (Gou, 2008), topology-based formulated algorithms (Mohammadi-Ivatloo,
n
Corresponding author. Tel.: þ1 607 777 4375; fax: þ1 607 777 4464. E-mail addresses: [email protected] (J. Huang), [email protected] (N. Eva Wu). 0967-0661/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conengprac.2012.09.001
2009), and particle swarm optimization (PSO) algorithms (Hajian, Ranjbar, Amraee, & Shirani, 2007). Loss of measurement devices, however, has not been considered in the aforementioned optimal placement problems. The expected expansion of the role of PMUs requires resilient structure and operation of the network of PMUs. In addition to hardware failures, some common failures encountered in PMUs include invalid data caused by improper installation, calibration and maintenance; inaccurate data due to time alignment failures with the GPS; and data delivery failures due to network congestion. To prevent continuous and reliable measurement data from being interrupted by loss of PMUs or their data, sufficient redundancy must exist to maintain the ability to fully monitor the grid. The simplest of sensor network redundancy configuration is to have a second sensor placed right next to each original sensor, which not only doubles the number of sensors being used, but also suffers from vulnerability of greater loss due to physical attack. Rakpenthai, Premrudeepreechacharn, Uatrongjit, and Watson (2007) proposed a PMU placement method that ensures full static observability in case of single PMU loss or single branch outage. The method uses minimum condition number of a static measurement model as a criterion to find the essential measurement set and select the optimal redundant measurements. It then minimizes the number of PMU placement sites. The method, however, does not deal with the case of multiple PMU losses. Emami and Abur (2010) and Xu and Abur (2005) discussed a placement scheme that finds two mutually exclusive PMU sets, both of which have the ability to fully monitor the system. A primary set is first chosen to provide the
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J. Huang, N. Eva Wu / Control Engineering Practice 21 (2013) 1–11
desired observability using the previously mentioned algorithms such as integer linear programming, or PSO. A backup set is then chosen by excluding the primary set from the selection criteria. Since each set of PMUs is optimally chosen, the backup set inevitably contains at least as many PMUs as the primary set, which increases the total number of PMUs by at least twice. This scheme can tolerate a single sensor loss in an arbitrary location or multiple sensor losses in the same set. To tolerate multiple sensor losses in arbitrary locations, multiple backup sets are required. Since all sets of sensors are mutually exclusive from each other, the sensors are quickly exhausted. The above sensor placement problems assume steady-state operation of the power grid, though monitoring, diagnosis, and feedback control call for use of dynamic models in order to deal with the transient behaviors of systems during and post faults. This paper develops a method to place a minimum number of additional sensors to facilitate the existing set of sensors in mitigating the effects of sensor losses. The method is based on dynamic grid models as it adopts a previously defined concept of control reconfigurability (Wu, Zhou, & Salomon, 2000) for placing the additional sensors. In 2004, Staroswiecki et al. discussed a general fault tolerant sensor network design approach based on three defined criteria: redundancy degrees, sensor network reliability, and mean time to non-observability. Their approach, however, is difficult to scale up for a large system such as a power grid. Measurement of observability in this paper uses reconfigurability, as the algebraic observability used in the Staroswiecki, Hoblos, and Aitouche (2004) paper cannot quantify the ability to observe states and lacks robustness. Reconfigurability specializes to observability when the minimum Hankel singular value is replaced by the minimum singular value of the observability gramian (Moore, 1981) of the essential system states. This paper does not consider life time distributions. The problem is formulated to search for a minimum number of sensors that guarantees a minimum required level of reconfigurability (or observability as measured by the singular values of observability gramian when controllability is not of a concern) when a specified number of sensors fail either permanently or intermittently. Reconfigurability in this paper refers to the potentiality to reconfigure a control law that relies on the adequate availability of sensors and actuators to maintain the system performance, rather than the need to reconfigure the sensor network upon sensor losses. Thus the tolerance to sensor data loss resulted from satisfying a reconfigurability requirement directly leads a more resilient grid operation. Frequency control and voltage control are the two basic types of control involved in power systems (Kundur, 1994, chap. 11). A change in real power demand causes a change in frequency. Since the frequency should remain nearly constant in a power system, control effort is required on the input mechanical power to the generators in order to match the real electrical power demand so as to maintain a constant frequency. Voltage control is achieved by supplying adequate level of reactive power, and is implemented through voltage regulators at generators and reactive power compensators throughout the grid. Because of the immediate impact of the PMU network in the grid, the paper focuses on the sensor placement problem, though reconfigurability can be equally effectively applied to solving an actuator placement problem. For this reason, the control variables used in the power system example presented in this paper are simplified to be the internal voltages of the generators. The selection of control variables has an effect on the value of reconfigurability, but not the problem nature for sensor placement. The particle swarm optimization (PSO) algorithm is employed in this paper to minimize the number of additional sensors placed to maintain a required reconfigurability under a specified
maximum number of sensor losses. PSO was first introduced by Kennedy and Eberhart in 1995, intended for simulating social behavior. It was inspired by the movement of bird flocks, fish schools and bee swarms. It was then simplified and used for optimization. Two years later, Kennedy and Eberhart (1997) introduced a binary version of particle swarm optimization (BPSO), which converts the position vector into binary forms. Since then, several improved versions of BPSO were published (Afshinmanesh, Marandi, & Rahimi-Kian, 2006; Gao, Hu, He, & Liu, 2008), and the version discussed by Hajian et al. (2007) changes all the variables to discrete binary variables that can only take values of 0 or 1. The sensor or actuator placement problem formulated in this paper uses binary variables to represent whether sensors or actuators ought to be placed at certain locations, and therefore is solved by BPSO. The remainder of the paper is organized as follows. Section 2 reintroduces the concept of control reconfigurability. Section 3 discusses the formulation of PMU placement problem that tolerates a specified number of PMU losses in a power system. The PMU placement using the binary particle swarm algorithm is discussed in Section 4, which minimizes the number of the additionally placed PMUs. Section 5.1 uses a simple power system model to demonstrate the placement problem formulation and solution. Simulation results of placing PMUs on the IEEE 118-bus system are presented in Section 5.2. Section 6 concludes the paper.
2. Control reconfigurability The concept of control reconfigurability (Wu et al., 2000) is now reintroduced, following the original presentation closely. Control reconfigurability measures the level of redundancy of a given linear system using the minimum Hankel singular value (smin ). Since Hankel singular values are equal to the square roots of eigenvalues of the product of controllability gramian Wc and observability gramian Wo, reduction of controllability and/or observability caused by faults in the system reduces smin . When smin is reduced to zero, the system has deficient gramian ranks and loses its full controllability and/or observability. A specified level of control reconfigurability is met when the lowest smin is above a positive threshold for the class of all fault scenarios considered. The input to output mapping of a system should remain sufficiently controllable and observable despite the loss of some measurement and control devices. This is necessary in order for the system to be effectively controlled in the face of such sensor/ actuator faults. Assume that the small signal input–output characteristics of the system can be reasonably well simulated by a model. The transparency from input to output implies that a requirement must be imposed on the combined level of controllability and observability of the model of the system. Let y denote a vector in the fault parameter space, which is defined as the Euclidean space of all parameters that change their values as the result of some fault occurrence. Let O denote the set over which y resides when faults occur. Without loss of generality, let y ¼0 be the fault-free parameter vector. Suppose a linear model of a system in the form x_ ¼ AðyÞx þ BðyÞu,
y ¼ CðyÞx
ð1Þ
has been validated, where x, u, y are the state, input, and output vectors, and A, B, C are state-space matrices defined over y. It can be seen that all Hankel singular values, and therefore the smallest Hankel singular value smin , are functions of vector y.
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Control reconfigurability rS over a subset S of O is given by
rS min smin ðyÞ: yASDO
ð2Þ
rS indicates how well the model remains controllable and observable from its input to the output for all fault scenarios represented by points in S. For a particular application, a minimal acceptable value rmin can be specified, from which a corresponding largest set S containing the origin of the fault parameter space can be determined. In the case of sensor/actuator placement for improving the system redundancy configuration, the set within O for which the requirement is met is given by SR y A O9smin ðyÞ Z rmin g: ð3Þ Therefore, if S1R ¼ fy A O9smin ðyÞ Z r1,min g, 2 SR ¼ fy A O9smin ðyÞ Z r2,min g, it follows from (2) that S1R DS2R ) r1,min Z r2,min :
and ð4Þ
In other words, a less specific set of possible faults produce a smaller control reconfigurability. In particular, rf0g ¼ smin ð0Þ Z rS , i.e., without adding new sensors and/or actuators, control reconfigurability is always bounded above by the minimum Hankel singular value of the fault-free system. Assuming that smin ð0Þ has a sufficient combined controllability and observability, the adequacy of redundancy can be provided by adding sensors and/or actuators to maintain rS Z rmin ¼ smin ð0Þ. A problem solely focused on observability without considering controllability can be solved by redefining rS in (2) using the minimum singular value of the observability gramian Wo of (1) instead of minimum Hankel singular value. On the other hand, if controllability is the only consideration, rS can be redefined using the minimum singular value of the controllability gramian Wc of (1). In order to establish a criterion that most benefits feedback control through PMU measurements as feedback variables, a combined controllability and observability measure is used, allowing the relaxation or tightening of observability requirements based on the strength of controllability along each of the directions defined by singular vectors (Moore, 1981) in the statespace. A pure observability requirement would result in a placement ignoring the corresponding controllability. Since this paper does not consider actuator placement, controllability is fixed, and adding sensors would not change the controllability of the system.
3. PMU placement problem formulation PMU placement problems for power systems so far have been formulated using static measurement models for the purpose of estimating synchrophasors (Phadke & Thorp, 2008, chap. 5). For PMUs to monitor the change in a dynamic grid, the criterion for placement must be suitable for a dynamic system. Control reconfigurability is chosen as the criterion for formulating the PMU placement problem. Suppose the following linear differential inclusion (Boyd, Ghaoui, Feron, & Balakrishnan, 1994) x_ ¼ Ax þ Bu, y ¼ Cx, A B n A G Rðdx þ dy Þ ðdx þ du Þ C 0
ð5Þ
describes the dynamic behavior of a power system, where x, u, y are the state, input, and output vectors with dimensions dx, dy and du, A, B, and C are known constant matrices, and R is the set of real numbers. Suppose a set of M PMUs is already in place in the system for estimating the critical set of states so that baseline monitoring
3
and control for the system under normal condition is satisfied. N new PMUs are to be placed in K new possible locations to enhance tolerance to loss of n PMUs, where KZNZ1 is assumed without loss of generality. To lessen the impact of physical attack at the PMU locations, each candidate location is allowed to have only a single PMU placed, including the original M PMU locations, and all candidate locations are sufficiently far away from each other. To represent N PMUs being placed at K new locations, the C matrix is redefined as " CðzÞ ¼
Co C K ðzÞ
# ,
ð6Þ
where Co is the original C matrix for the M-PMU network, and 2
3 z1 C M þ 1 6 7 ^ 7 6 6 7 7 C K ðzÞ ¼ 6 6 z k C M þ k 7, 6 7 ^ 4 5 zK C M þ K
ð7Þ
with zkA{0,1}. zk ¼1 indicates a new PMU is being placed at location k, whereas zk ¼0 indicates no PMU placement at the location. The vector z represents a particular PMU placement with N PMUs being placed at N out of the K candidate locations. Since N new PMUs are placed, K X
zk ¼ N,
zk A f0,1g:
ð8Þ
k¼1
Assuming linearity for both the system and the measurement models, rmin , which is set to be the minimum Hankel singular value of the original M-PMU system, can be calculated using A, B and Co (Wu et al., 2000). Without loss of generality, rmin is regarded as an acceptable threshold of reconfigurability in the absence of PMU faults. Now consider the case of loss of n PMUs, where 1rn rM, for which set Sn is defined as 8 > > > > > > > > > > > > > > > > > > > <
2
ð1y1 ÞC 1 6 ^ 6 6 6 yi ÞC i ð1 6 6 ^ 6 6 6 ð1yM ÞC M 6 Sn ¼ y9CðyÞ ¼ 6 ð1 yM þ 1 Þz1 C M þ 1 > 6 > > 6 > > 6 > ^ > 6 > > 6 > > 6 y ð1 M þ j Þzj C M þ j > > 6 > > 6 > ^ > 4 > > : Þz C ð1y MþK
0
yi , yM þ j A f0,1g, @
M X
i¼1
yi þ
K
3 7 7 7 7 7 7 7 7 7 7 7, 7 7 7 7 7 7 7 7 5
MþK
K X
1
) A zj yM þ j r n :
ð9Þ
j¼1
where yi ¼ 1 indicates a loss of PMU data at location i. For the K new locations, yM þ j is undefined whenever zj ¼0. Therefore, y is an (MþN)-vector containing all defined variable yi. With the N new PMUs placed in the system, control reconfigurability is given by rSn miny A Sn smin ðyÞ, where smin ðyÞ is the minimum Hankel singular value calculated using A, B, and C(y). The case n¼1 represents a loss of data at a single PMU, and defines a cost effective and practical fault-tolerant placement problem. In this case, the condition on the fault parameters in (9)
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J. Huang, N. Eva Wu / Control Engineering Practice 21 (2013) 1–11
can be simplified to 8 2 3 ð1y1 ÞC 1 > > > 6 7 > > ^ > 6 7 > > 6 7 > < 6 ð1yi ÞC i 7 6 7 S1 ¼ y9CðyÞ ¼ 6 7, ^ > 6 7 > > 6 7 > > 6 ð1yM ÞC M 7 > > 4 5 > > : C K ðzÞ
Minimize
yi A f0,1g,
M X
yi r1g:
ð10Þ
i¼1
Thus the dimension of the fault-parameter set is reduced to the number of original PMUs due to the fact that losing one newly placed PMU does not cause control reconfigurability rS1 to fall below rmin , as rmin is specified as the minimum Hankel singular value of the original M-PMU system. The optimization for the new PMU placement that tolerates the loss of n PMUs can be formulated as Minimize
N
Subject to rSn Z rmin , ð8Þ
rSn
Subject to ð8Þ, and N ¼ N0 ,
ð11Þ
The constraint rSn Z rmin enforces a sufficient redundancy level through PMU placement in order to tolerate n PMUs losses. A variation of Problem (11) is to maximize rSn for a fixed N, i.e.,
ð12Þ
where N0 is a fixed positive integer less than or equal to K. Problems (11) and (12) in principle can be solved using branch and bound methods (Boyd & Mattingley, 2006), for they can be viewed as mixed-Boolean convex problems. Maximizing the minimum Hankel singular values using semidefinite program (Body & Vandenberghe, 2004), however, is nontrivial. One issue encountered is inadequate tools capable of handling large scale matrix variables. As a result, an alternative search algorithm is applied to solve (11) and (12). K For a exhaustive linear search algorithm, it takes steps to N find the solution of z representing a placement that maximize rSn and meets the constraints in (12). To solve (11), all values of N that are less than K have to be considered. When K and N are large, finding the solution can be time consuming. In order to improve the speed of finding a solution, a randomized search algorithm called particle swarm optimization (PSO) is implemented. A brief discussion of the PSO algorithm is given in next section.
Fig. 1. Flow chart of the binary particle swarm optimization algorithm.
J. Huang, N. Eva Wu / Control Engineering Practice 21 (2013) 1–11
5
4. Particle swarm optimization
5. Application to PMU placement
The principle of PSO (Kennedy & Eberhart, 1995) is that a set of particles is employed to find the optimal solution of the assigned problem. Each particle adjusts its own position and velocity based on the memory of its own best known position pbest and knowledge of the best position gbest of all particles in the swarm. At iteration k, position vector x and velocity vector v of particle i are adjusted by
5.1. A simple power network
xki þ 1 ¼ xki þ vki þ 1 ,
ð13Þ
vki þ 1
ð14Þ
¼ wi vki þ c1 n r 1 n ðpbesti xki Þ þ c2 n r 2 n ðgbestxki Þ, n
A simple power network as shown in Fig. 2 is used to explain the concept discussed in Section 3. The system has two generators, four transformers, four loads and eight buses. Using resistance R, inductance L of each equipment shown in Table 1 and Kirchhoff’s Voltage Law (KVL), the dynamic behavior of the system can be described by ½Lx_ ¼ ½Rx þ ½Vu,
where wi is the weight function for velocity of particle i, c1 and c2 are the acceleration coefficients, r1 and r2 are the random numbers generated between 0 and 1 with a uniform distribution. Once the position and velocity of a particle is updated, the objective function of the optimization problem is recalculated based on the new position xki þ 1. pbesti is then replaced by xki þ 1 if xki þ 1 can better optimize the objective function than pbesti. Similarly, gbest is updated to xki þ 1 if xki þ 1 can better optimize the objective function than gbest. Certain constraints can be placed on vki þ 1. For example, if vki þ 1 excesses a predefined limit vmax, vmax replaces vki þ 1 as the velocity vector for the next iteration. The first binary version of particle swarm optimization (BPSO) (Kennedy & Eberhart, 1997) is similar to the conventional PSO described above with modification only to the position Eq. (13) is replaced by ( 1 0 if r 3 Z 1 þ expðv kþ1 Þ kþ1 ij xij ¼ ð15Þ 1 otherwise
y ¼ Cx, where 2 3 iTL1 6i 7 6 TL2 7 6 7 6 iTL3 7 6 7 6 7 x ¼ 6 iTL4 7, 6 7 6 iTL5 7 6 7 6i 7 4 LD1 5 iLD2
ð18Þ 2
" u¼
V1 V2
# ,
3 iG1 6i 7 6 G2 7 6 7 6 iLD1 7 6 7 6 7 y ¼ 6 iLD2 7 : 6 7 6 iLD3 7 6 7 6i 7 4 TL1 5 iTL2
ð19Þ
To simplify the calculations, all quantities used for this example are in Per Unit (pu) (Glover, Sarma & Overbye, 2008, chap. 3), including those shown in Table 1. The states of (18) are currents at transmission lines TL1, TL2, TL3, TL4, TL5 and loads LD1 and LD2.
where j is the element number of the position vector xki þ 1 and velocity vector vki þ 1, and r3 is a randomly generated number ranged between 0 and 1. This paper uses the version of BPSO introduced by Hajian et al. (2007) with some modifications as described later on to improve the performance of optimizing (11) and meet the constraints of (12). The position Eq. (13) and the velocity Eq. (14) is replaced by xki þ 1 ¼ xki vki þ 1 ,
ð16Þ
vki þ 1 ¼ vki ðr 1 ðpbesti xi k ÞÞ ðr 2 ðgbest xki ÞÞ,
ð17Þ
where and represent two different logic operations. Experiments indicate that choosing NOR for and XOR for is less likely to cause position vector x to repeat itself and therefore gives better results. r1 and r2 in (17) are randomly generated binary vectors. All variables in (16) and (17) are binary vectors. In order to improve the convergence rate of Problem (11), a simple modification is added to the algorithm. Let G equal to the sum of gbest, indicating G new PMUs are being placed in the current best solution. Since the primary objective function of (11) is to minimize the number of PMUs being placed, if the sum of xki þ 1 is greater than G, the number of 1s in xki þ 1 will be randomly reduced until it become less than or equal to G. For Problem (12), since N is fixed, G must be equal to N. If the sum of xki þ 1 is not equal to G, the number of 1s in xki þ 1 is randomly reduced or increased until it is equal to G. A flow chart of BPSO algorithm for solving (11) described above is shown in Fig. 1. The end condition shown in the flow chart can be set to be the maximum number of iterations without finding a new gbest. When the search ends, the current value of gbest would be the current best solution for the placement z as discussed in the previous section.
Fig. 2. A simple power network with two generators, four transformers, four loads, and eight buses.
Table 1 Per unit impedances of equipment of the simple power network.
TL1a TL2 TL3 TL4 TL5 TF1a TF2 TF3 TF4 G1a G2 LD1a LD2 LD3 LD4 a
Z (pu)
R (pu)
L (pu)
0.0012 þj0.078 0.0022 þj0.033 0.0011 þj0.081 0.0019 þj0.125 0.0015 þj0.115 0.0025 þj0.040 0.0015 þj0.039 0.0024 þj0.068 0.0013 þj0.049 0.016 þj0.012 0.013 þj0.011 9.5 þj5.6 7.6 þj5.3 6.7 þj5.2 5.3 þj4.7
0.0012 0.0022 0.0011 0.0019 0.0015 0.0025 0.0015 0.0024 0.0013 0.016 0.013 9.5 7.6 6.7 5.3
2.069 10 4 8.754 10 5 2.149 10 4 3.316 10 4 3.050 10 4 1.061 10 4 1.035 10 4 1.804 10 4 1.300 10 4 3.183 10 5 2.918 10 5 1.485 10 2 1.406 10 2 1.379 10 2 1.247 10 2
TL: Transmission line, TF: Transformer, G: Generator, LD: Load.
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J. Huang, N. Eva Wu / Control Engineering Practice 21 (2013) 1–11
The inputs of (18) are internal voltages V1 and V2 of generators G1 and G2. Output y measures currents at generators G1, G2, loads LD1, LD2, LD3 and transmission lines TL1 and TL2. As a result, matrix C is equal to 2
C1
C 1 ¼ C G1 ¼ 1 1 0 1 0 1 0 , C 2 ¼ C G2 ¼ 1 0 1 0 1 0 1 , C 3 ¼ C LD1 ¼ 0 0 0 0 0 1 0 , C 4 ¼ C LD2 ¼ 0 0 0 0 0 0 1 , C 5 ¼ C LD3 ¼ 0 1 0 0 1 0 0 , C 6 ¼ C TL1 ¼ 1 0 0 0 0 0 0 , C 7 ¼ C TL2 ¼ 0 1 0 0 0 0 0 :
3
6C 7 6 27 6 7 6 C3 7 6 7 6 7 C ¼ 6 C 4 7, 6 7 6 C5 7 6 7 6C 7 4 65 C7
ð20Þ
To represent N new PMUs being placed in the 6 candidate locations, (7) and (8) become 2 3 z1 C 8 6 7 6 z2 C 9 7 6 7 6 z3 C 10 7 6 7 C 6 ðzÞ ¼ 6 ð23Þ 7, 6 z4 C 11 7 6 7 6 z5 C 12 7 4 5 z6 C 13
6 X
zk ¼ N,
zk A f0,1g,
ð24Þ
k¼1
Using matrices [L], [R] and [V], whose numerical values are shown in Appendix A, matrices A and B can be calculated as 2
49:2 6 23:1 6 6 6 16:4 6 6 A ¼ ½L1 n½R ¼ 6 16:5 6 6 22:8 6 6 4 0:9 0:7
27:7
4:1
39:5
69:7
6:0
360:6 9:4
3:3 255:4
19:9 233:4
328:8 25:3
3:3 2:2
11:2
159:6
181:6
27:1
2:3
115:1
1:8
21:1
146:5
3:1
0:4 0:3
0:3 0:4
0:6 0:4
0:5 0:4
639:3 0:02
3:8
3
2:0 7 7 7 1:5 7 7 1:4 7 7, 7 2:0 7 7 7 0:02 5 540:4
ð21Þ 2
1378:6 6 757:9 6 6 6 500:2 6 6 B ¼ ½L1 n½V ¼ 6 536:1 6 6 717:6 6 6 61:5 4 5:4
1383:4
3
705:7 7 7 7 549:1 7 7 507:4 7 7: 7 735:8 7 7 7 5:6 5 65:5
ð22Þ
Eq. (19) implies that 7 PMUs are in place for monitoring the system so that baseline control for the system under normal condition is satisfied. N new PMUs are to be placed at locations chosen from 6 candidate locations to measure currents at transformer TF1, TF2, load LD4 and transmission lines TL3, TL4 and TL5. It is assumed that each branch has only one location for only one single PMU placement, including the original 7 locations. One of the goals for this example is to tolerate one single PMU loss in the system.
where C 8 ¼ C TF1 ¼ 1 1 0 1 0 0 0 , C 9 ¼ C TF2 ¼ 1 0 1 0 1 0 0 , C 10 ¼ C LD4 ¼ 0 0 1 1 0 0 0 , C 11 ¼ C TL3 ¼ 0 0 1 0 0 0 0 , C 12 ¼ C TL4 ¼ 0 0 0 1 0 0 0 , C 13 ¼ C TL5 ¼ 0 0 0 0 1 0 0 :
ð25Þ
N and zk are the optimization variables in (11) to be solved by the BPSO algorithm. rmin is set to be the minimum Hankel singular value of (18), which is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rmin ¼ minð eigðW c W o Þ Þ: ð26Þ Wc and Wo are the controllability gramian and observability gramian of (18), for which AW c þ W c A0 ¼ 2BB0 ,
ð27Þ
A0 W o þ W o A ¼ 2C 0 C,
ð28Þ
where A and B are given in (21) and (22), and C is given in (20). The set that represents all possible data loss at a single PMU at the original 7 PMUs is given in (19) with a particular placement z
x 10−3
ρS1
1.1 1 0.9 0.8
0
20
40
60
80
100
120
140
0
20
40
60 80 Iteration (k)
100
120
140
# of PMUs
5 4 3 2 1
Fig. 3. (a) Control reconfigurability rS1 calculated using gbest at each iteration and (b) number of additional PMUs being placed to tolerate loss of data at any single PMU.
J. Huang, N. Eva Wu / Control Engineering Practice 21 (2013) 1–11
can be described as 8 2 3 ð1y1 ÞC 1 > > > 6 7 > > ^ > 6 7 > > 6 7 > < 6 ð1yi ÞC i 7 6 7 S1 ¼ y9CðyÞ ¼ 6 7, ^ > 6 7 > > 6 7 > > 6 7 > y ÞC ð1 7 7 > 4 5 > > : C 6 ðzÞ
yi A f0,1g,
7 X
yi r1g,
ð29Þ
i¼1
Control reconfigurability rS1 is defined as miny A S1 smin ðyÞ,where minimum Hankel singular value smin ðyÞ is calculated using A, B and C(y). The results of solving (11) for this example using BPSO are shown in Fig. 3. Fig. 3 (a) shows control reconfigurability rS1 at each iteration calculated using the current best solution of gbest. The dashed line in Fig. 3 (a) represents rmin . In the first several iterations, rS1 is below the level of rmin indicating that the constraint rS1 Z rmin is not met. In this state, the BPSO algorithm tries to find a better position that will raise rS1 above rmin . Once this is achieved, the algorithm focuses on minimizing the number of PMUs while still maintaining rS1 above rmin . Fig. 3 (b) shows the sum of gbest which is the number of PMUs being placed for the current best solution. The process is terminated after about 100 iterations without finding a position vector better than the current gbest. The final solution shows that only 2 additional PMUs are required to tolerate the loss of data at any single PMU in the system. The final value of z is equal to [0 0 1 0 1 0], which
7
indicates that two PMUs should be placed into the system to measure currents at load LD4 and transmission line TL4. When N¼ 3 is fixed as a constraint, or 3 PMUs are to be placed in the six candidate locations to tolerate loss of data at any one of the original 7 PMUs, (12) is solved to maximize rS1 . Fig. 4 shows the results of solving (12) with N¼3. The final value of z is [1 0 1 0 1 0], which indicates that the three PMUs should be placed to measure currents at transformer TF1, transmission line TL4 and load LD4. To tolerate data losses at any two PMUs in the system of (18), N new PMUs are to be placed in the 6 candidate locations. The corresponding fault parameter set is given by 3 ð1y1 ÞC 1 7 6 ^ 7 6 7 6 7 6 y ÞC ð1 i i 7 6 7 6 ^ 7 6 7 6 7 6 ð1y7 ÞC 7 7 6 S2 ¼ y9CðyÞ ¼ 6 7, ð1 y Þz C > 7 6 8 1 8 > > 7 6 > > 7 6 > ^ > 7 6 > > 7 6 > > 6 ð1y7 þ j Þzj C 7 þ j 7 > > 7 6 > > 7 6 > ^ > 5 4 > > : ð1y Þz C 2
8 > > > > > > > > > > > > > > > > > > > <
13
6
0
yi , y7 þ j A f0,1g, @
7 X
i¼1
ρS1
0.9
40
60 Iteration (k)
80
100
120
Fig.4. Control reconfigurability rS1 with N (number of additional PMUs) fixed to 3.
ρS2
1
# of PMUs
x 10−3
0.5
0
0
20
40
60
80
100
120
0
20
40
60 Iteration (k)
80
100
120
6 4 2 0
j¼1
The result of solving (11) for tolerating data loss in any two PMUs is shown in Fig. 5. Due to physical limitation, even if PMUs are installed in all 6 candidate locations, rS2 can never reach rmin . In this case, maximizing rS2 becomes a more practicable objective. Fig. 6 shows the result of solving (12) for rS2 with N ¼3.
1
20
1 zj y7 þ j A r 2g,
ð30Þ
1.1
0
6 X
13
x 10−3
0.8
yi þ
Fig. 5. (a) Control reconfigurability rS2 calculated using gbest at each iteration and (b) number of PMUs being placed.
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J. Huang, N. Eva Wu / Control Engineering Practice 21 (2013) 1–11
1
x 10−3
0.8
ρS2
0.6 0.4 0.2 0
0
10
20
30
40
50 60 Iteration (k)
70
80
90
100
110
Fig. 6. Control reconfigurability rS2 with N fixed to 3.
1.5
x 10−3
ρS2
1 0.5
# of PMUs
0
0
20
40
60
0
20
40
60
80
100
120
140
160
80 100 Iteration (k)
120
140
160
10 8 6
Fig. 7. (a) Control reconfigurability rS2 when each branch current is allowed to be measured at two placement locations, (b) Number of PMUs being placed.
The result of z is [1 1 0 0 0 1], which indicates that the three PMUs are placed to measure the currents at transformers TF1, TF2, and transmission line TL5 to achieve a maximum control reconfigurability in the face of loss of data at two PMUs or less, although rS2 cannot reach fault-free control reconfigurability rmin . If each branch is allowed to have up to two locations for new PMUs to measure currents, and all locations are far away from each other, there are totally 19 candidate locations to install new PMUs. There are 7 locations from the original 7 branches with PMUs already placed, and there are 12 locations from the original 6 candidate branches. The result of solving (11) for this case to tolerate data loss in any two PMU is shown in Fig. 7. The final z indicates that 6 new PMUs should be placed and 3 out of the 6 new PMUs should be placed in the original 7 branches. 5.2. IEEE 118-bus test system The IEEE 118-bus test system obtained from the power systems test case archive in the website of Washington University is now used for application of this method. This system contains 34 generators and a total of 186 branches among 118 buses. 27 single PMUs are placed to measure currents at 27 different branches for monitoring the behavior of the system. New PMUs are to be added to the positions selected out of the remaining 159 branches to
tolerate data loss at any single PMU and any two PMUs in the system, respectively. This PMU placement problem can be solved by repeating the steps described in Section 5.1, and the results of solving (11) for rS1 and rS2 are shown in Fig. 8 and Fig. 9. Similar to Figs. 3, 8(a) and 9(a) show the control reconfigurabilities of the current best solutions. Figs. 8(b) and 9(b) show the number of PMUs being placed for the current best solutions. The final results found by BPSO show that only 17 new PMUs are required to tolerate data loss at a single PMU for this system, which is a little more than half of the number of the original 27 PMUs. For tolerating data loss at any two PMUs, 74 additional PMUs are required, assuming only one location is allowed for measuring current at each branch. Fig. 10 shows the result of solving (12) for rS1 with N fixed to be 20. One drawback common to all sensor placement methods is the time to calculate a solution, which is the time to calculate control reconfigurability in this case, for a very large system. With a modern computer featured a 1.6 GHz 32-bit duo-core Intel Atom CPU, calculating the minimum Hankel singular value for the IEEE 118-bus system using Matlab takes about one second. For each iteration, calculating the control reconfigurability rS1 takes about half minute since it involves 27 steps of calculating the minimum Hankel singular values. It takes several hours to obtain Fig. 8. For calculating control reconfigurability rS2 , if a placement z
J. Huang, N. Eva Wu / Control Engineering Practice 21 (2013) 1–11
4
9
x 10−6
ρS1
3 2 1
0
200
400
0
200
400
600
800
1000
1200
800
1000
1200
# of PMUs
100
50
0
600 Iteration (k)
Fig. 8. (a) Control reconfigurability rS1 of the IEEE 118-bus test system calculated using gbest at each iteration; (b) minimum number of additional PMUs needed to tolerate data loss at any single PMU without violating control reconfigurability requirement.
1.7
x 10−6
ρS2
1.4 1.1 0.8 0.5
0
20
40
0
20
40
60
80
100
120
60 80 Iteration (k)
100
120
# of PMUs
95 90 85 80 75 70
Fig. 9. (a) Control reconfigurability rS2 of the IEEE 118-bus test system calculated using gbest at each iteration; (b) minimum number of additional PMUs needed to tolerate data loss at any two PMUs without violating the control reconfigurability requirement.
contains 60 additional PMUs, it takes more than half an hour to obtain the result. To obtain Fig. 9, it requires several days of computation. Of course, the results can be improved with a more powerful computing machine, such as a computer featured a 3 GHz 64-bit quad-core CPU.
6. Conclusions A new formulation of fault-tolerant PMU placement problem using a previously defined control reconfigurability concept is presented. Control reconfigurability is used as a constraint for placing PMUs in the system that can tolerate data loss at a given number of PMUs. The fault-tolerant placement method in this paper differs from the other placement methods in that the method directly results in a
more resilient grid assuming that the PMU measurements are used as feedback variable for control of a system. The minimization of the number of new PMUs is solved using the binary particle swarm optimization algorithm. Simulation results obtained for a simple power network and the IEEE 118-bus test system are presented to demonstrate the feasibility of the proposed solution. Planned future work includes the generalization to allow loss of multiple types of sensors and actuators for large scale nonlinear systems. Though the current placement criterion uses a combined controllability and observability measure, sensor placement dominates the interest, and actuator placement has not been considered for a realistic control access. Thus, another focus of future work is to incorporate the voltage control problem with reactive power compensation into the control reconfigurability-based placement problem.
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J. Huang, N. Eva Wu / Control Engineering Practice 21 (2013) 1–11
x 10−7
14 12
ρS1
10 8 6 4 2
0
100
200
300
400
500 600 Iteration (k)
700
800
900
1000
Fig. 10. Control reconfigurability rS1 of the IEEE 118-bus test system with N (number of additional PMUs to be placed) fixed to 20.
Acknowledgement The authors gratefully acknowledge the guidance and insights provided by Dr. Joe H. Chow of RPI, which greatly helped take their first step toward studying the issues related to power systems.
½R1,1 ¼ 2ðRG1 þ RTF1 þ RTL1 þ RTF2 þ RG2 Þ ¼ 20:0342, ½R2,2 ¼ 2ðRG1 þ RTF1 þ RTL2 þ RTF3 þ RLD3 Þ ¼ 26:723, ½R3,3 ¼ 2ðRG2 þ RTF2 þ RTL3 þ RTF4 þ RLD4 Þ ¼ 25:317, ½R4,4 ¼ 2ðRG1 þ RTF1 þ RTL4 þ RTF4 þ RLD4 Þ ¼ 25:322, ½R5,5 ¼ 2ðRG2 þ RTF2 þ RTL5 þ RTF3 þ RLD3 Þ ¼ 26:718, ½R6,6 ¼ 2ðRG1 þ RLD1 Þ ¼ 29:516,
Appendix A. Numerical values of matrices [V], [L] and [R] of the Simple Power Network [V] is a 7 2 matrix representing connections to the generators, which 2 3 1 1 61 0 7 6 7 6 7 60 1 7 6 7 6 7 ½V ¼ 6 1 0 7 6 7 60 1 7 6 7 6 7 41 0 5 0 1 Both matrices [L] and [R] are 7 7 symmetric matrices that reflect the topology of the power network. Their values can be calculated using quantities in Table 1 as follows: ½L1,1 ¼ ðLG1 þ LTF1 þ LTL1 þ LTF2 þ LG2 Þ ¼ 4:775 104 , ½L2,2 ¼ ðLG1 þ LTF1 þ LTL2 þ LTF3 þ LLD3 Þ ¼ 1:420 102 , ½L3,3 ¼ ðLG2 þ LTF2 þ LTL3 þ LTF4 þ LLD4 Þ ¼ 1:295 102 , ½L4,4 ¼ ðLG1 þ LTF1 þ LTL4 þ LTF4 þ LLD4 Þ ¼ 1:307 102 , ½L5,5 ¼ ðLG2 þ LTF2 þ LTL5 þ LTF3 þ LLD3 Þ ¼ 1:441 102 , ½L6,6 ¼ ðLG1 þ LLD1 Þ ¼ 1:489 102 , ½L7,7 ¼ ðLG2 þ LLD2 Þ ¼ 1:409 102 , ½L1,2 ¼ ½L1,4 ¼ ½L2,4 ¼ ðLG1 þLTF1 Þ ¼ 1:379 104 , ½L1,3 ¼ ½L1,5 ¼ 2½L3,5 ¼ 2ðLTF2 þ LG2 Þ ¼ 21:326 104 , ½L1,6 ¼ ½L2,6 ¼ ½L4,6 ¼ ðLG1 Þ ¼ 3:183 105 , ½L1,7 ¼ 2½L3,7 ¼ 2½L5,7 ¼ 2ðLG2 Þ ¼ 22:918 105 , ½L2,5 ¼ ðLTF3 þ LLD3 Þ ¼ 1:379 102 , ½L3,4 ¼ ðLTF4 þ LLD4 Þ ¼ 1:260 102 , ½L2,3 ¼ ½L2,7 ¼ ½L3,6 ¼ ½L4,5 ¼ ½L4,7 ¼ ½L5,6 ¼ ½L6,7 ¼ 0,
½R7,7 ¼ 2ðRG2 þ RLD2 Þ ¼ 27:613, ½R1,2 ¼ ½R1,4 ¼ ½R2,4 ¼ 2ðRG1 þ RTF1 Þ ¼ 20:0185, ½R1,3 ¼ ½R1,5 ¼ 2½R3,5 ¼ ðRTF2 þ RG2 Þ ¼ 0:0145, ½R1,6 ¼ ½R2,6 ¼ ½R4,6 ¼ 2ðRG1 Þ ¼ 20:016, ½R1,7 ¼ 2½R3,7 ¼ 2½R5,7 ¼ ðRG2 Þ ¼ 0:013, ½R2,5 ¼ 2ðRTF3 þ RLD3 Þ ¼ 26:702, ½R3,4 ¼ 2ðRTF4 þ RLD4 Þ ¼ 25:301, ½R2,3 ¼ ½R2,7 ¼ ½R3,6 ¼ ½R4,5 ¼ ½R4,7 ¼ ½R5,6 ¼ ½R6,7 ¼ 0:
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